How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^4 \cdot 5^6}$?


If $y$ is a real number and $\lceil y \rceil = 15,$ how many possible values are there for $\lceil y^2 \rceil$?


Three consecutive prime numbers have a sum that is divisible by 5. What is the smallest possible value of this sum?


Both $x$ and $y$ are positive integers and $y > 1$. When $x^y$ is the greatest possible value less than 500, what is the sum of $x$ and $y$?


Let $a$, $n$, and $l$ be real numbers, and suppose that the roots of the equation \[x^4 - 10x^3 + ax^2 - nx + l = 0\] are four distinct positive integers. Compute $a + n + l.$


Bag X has 4 white marbles and 5 black marbles. Bag Y has 7 yellow marbles and 3 blue marbles. Bag Z has 3 yellow marbles and 6 blue marbles. A marble is drawn at random from Bag X. If it is white, a marble is drawn at random from Bag Y, otherwise, if it is black, a marble is drawn at random from Bag Z. What is the probability that the second marble drawn is yellow?


Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?


Sheila has been invited to a picnic tomorrow. The picnic will happen regardless of the weather. If it rains, there is a 25% probability that Sheila will decide to go, but if it is sunny, there is an 80% probability that she will decide to go. Additionally, there is a 10% chance of a transport strike happening tomorrow, which would prevent Sheila from attending the picnic. The forecast predicts a 50% chance of rain. What is the probability that Sheila will attend the picnic? Express your answer as a percent.


A torus (donut shape) with an inner radius of $3$ and an outer radius of $5$ rests on a flat table. Determine the radius of the largest spherical ball that can be placed on top of the center of the torus so that the ball touches the horizontal plane. Assume the torus is formed by revolving a circle in the $xz$-plane centered at $(4,0,1)$ with radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis and rests either on the table or the torus itself.


Find the smallest positive five-digit multiple of $18$.


What three-digit number with units digit 5 and hundreds digit 6 is divisible by 11?


For real numbers $a,$ $b,$ and $c,$ the matrix
\[
\begin{pmatrix}
a^2 & b^2 & c^2 \\
b^2 & c^2 & a^2 \\
c^2 & a^2 & b^2
\end{pmatrix}
\]
is not invertible. List all possible values of
\[
\frac{a^2}{b^2 + c^2} + \frac{b^2}{a^2 + c^2} + \frac{c^2}{a^2 + b^2}.
\]


The function \(g(x)\) satisfies
\[xg(y) = 2yg(x)\] for all real numbers \(x\) and \(y.\) If \(g(10) = 15,\) find \(g(2).\)


Of the numbers 1, 2, 3, ..., 20, which number has the greatest number of divisors, and what is the least common multiple (LCM) of the numbers with the most divisors?


What is the base ten equivalent of $76543_7$?


How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


What is the greatest integer less than 200 for which the greatest common divisor of that integer and 18 is 6?


A box contains nine cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and three of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is red? Express your answer as a common fraction.


What is the greatest four-digit number which is a multiple of 17?


How many ways are there to choose 4 cards from a standard deck of 52 cards, if all four cards must be of different suits?


Find $x$ such that $\lfloor x \rfloor + x = \dfrac{17}{4}$. Express $x$ as a common fraction.


How many different 8-digit positive integers exist? (Note that we don't allow "8-digit" integers that start with 0, such as 01234567; this is actually a 7-digit integer.)


What is the arithmetic mean of all positive three-digit multiples of 8?


What is the smallest prime whose digits sum to $23$?


Find the smallest set of four consecutive prime numbers greater than 13 such that their sum is divisible by four. What is this sum?


If $y < 0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil \cdot \lfloor{y}\rfloor = 132$. Express your answer using interval notation.

Solution: \boxed{(-12, -11)}


What is the greatest integer less than 200 for which the greatest common factor of that integer and 24 is 4?


What is the greatest common divisor of $12345$ and $6789$?


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{500}$?


If ten apples cost the same as five bananas, and three bananas cost the same as four cucumbers, how many cucumbers can Tyler buy for the price of 20 apples?


Suppose that all four of the numbers \[2 - \sqrt{5}, \; -2-\sqrt{5}, \; 3+\sqrt{11}, \; 3-\sqrt{11}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


If $\cos x,$ $\sin x,$ $\cot x$ form a geometric sequence, in this order, then find $\tan^6 x - \tan^2 x.$


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, and the area of face $BCD$ is $100$, with $BC=12$. Find the volume of tetrahedron $ABCD$.


If two distinct numbers are selected at random from the first nine prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.


What is the product of all real numbers that are tripled when added to their reciprocals?


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2012} \times 7$?


The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 8x + 6.$ Find the leading coefficient of $g(x).$


Find the sum of the first eight terms in the geometric sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$.


Emily reorganized her playing cards with a new pattern: the numbers repeat in the sequence $A, 2, 3, \ldots, K$, but the suits change every four cards: Hearts, Diamonds, Clubs, and Spades in that order. She wonders what the suit and number of the 101st card would be in this new arrangement.

Solution: \boxed{10 \text{ of Hearts}}


If two distinct numbers are selected at random from the first ten prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.


If \( A\,864\,B07 \) is divisible by 9, where \( A \) and \( B \) each represent a single digit, what is the sum of all possible values of the sum \( A+B \)?


Pirate Pete shares his treasure with Pirate Paul in an interesting way. Pete first says, ``One for me, one for you,'' giving himself one coin and starting Paul's pile with one coin. Then Pete says, ``Two for me, and two for you,'' giving himself two more coins but making Paul's pile two coins in total. This pattern continues until Pete gives himself $y$ more coins but makes Paul's pile $y$ coins in total. At this point all of the coins have been distributed, and Pirate Pete has exactly five times as many coins as Pirate Paul. How many gold coins do they have in total?


Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy?


How many integers fall between $\sqrt{10}$ and $\sqrt{50}$ on a number line?


If $x$ is real, calculate the maximum integer value of
\[
\frac{4x^2 + 8x + 19}{4x^2 + 8x + 5}.
\]


The coefficients of the polynomial
\[ a_{12} x^{12} + a_{11} x^{11} + \dots + a_2 x^2 + a_1 x + a_0 = 0 \]
are all integers, and its roots $s_1, s_2, \dots, s_{12}$ are all integers. Furthermore, the roots of the polynomial
\[ a_0 x^{12} + a_1 x^{11} + a_2 x^{10} + \dots + a_{11} x + a_{12} = 0 \]
are also $s_1, s_2, \dots, s_{12}.$ Find the number of possible multisets $S = \{s_1, s_2, \dots, s_{12}\}.$


For what base is the representation of $625_{10}$ a four-digit number whose final digit is even?


What integer $n$ satisfies $0 \leq n < 23$ and $$-135 \equiv n \pmod{23}~?$$


Define a set of integers as "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 9\}$, including the empty set, are spacy?


For \( \mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix} \) and \( \mathbf{w} = \begin{pmatrix} 5 \\ 10 \end{pmatrix} \),
\[
\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 10 \\ 20 \end{pmatrix}.
\]
Find \( y \).


Let $P, Q$ be functions defined as $P(x) = 3\sqrt{x}$ and $Q(x) = x^2$. Calculate the value of $P(Q(P(Q(P(Q(5))))))$.


When you simplify $\sqrt[3]{40a^6b^3c^{14}}$, what is the sum of the exponents of the variables that are outside the radical?


If twelve apples cost the same as six bananas, and three bananas cost the same as five cucumbers, how many oranges can Tyler buy for the price of 24 apples if two cucumbers cost the same as one orange?


Calculate the sum of the coefficients in the polynomial $(3x^8 - 2x^7 + 4x^6 - x^4 + 6x^2 - 7) - 5(x^5 - 2x^3 + 2x - 8) + 6(x^6 + x^4 - 3)$.


I am playing a new walking game where on move 1, I do nothing, but on move \( n \) where \( 2 \le n \le 30 \), I take two steps forward if \( n \) is prime and one step backward if the number is composite. After all 30 moves, I stop and walk back to my original starting point. How many steps long is my walk back?


The base of a triangle is of length $b$, and the altitude is of length $h$. Two rectangles are inscribed in the triangle. The first rectangle has a height of $x$, and the second rectangle, placed directly above the first, has a height of $2x$. Both rectangles share the same base, which lies along the base of the triangle. Find the area of the second rectangle.


$ABCD$ is a rectangle whose area is 20 square units. $AB = 10$ units and $BC = 2$ units. Within the rectangle, points $E$ and $F$ are on $DC$, such that $DE = 2$ units and $FC = 4$ units, respectively. Calculate the area of trapezoid $EFBA$.


In an equilateral triangle $ABC$, points $D$, $E$, and $F$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. Calculate the ratio of the shaded area (triangle $FGH$) to the non-shaded area in triangle $ABC$.

[asy]
draw((0,0)--(1,0)--(0.5,sqrt(3)/2)--cycle,linewidth(1));
filldraw((0,0)--(0.5,0)--(0.5,sqrt(3)/4)--(0,sqrt(3)/2)--cycle,gray, linewidth(1));
filldraw((0.5,0)--(0.5,sqrt(3)/8)--(0.25,sqrt(3)/2)--(0,sqrt(3)/2)--cycle,white,linewidth(1));

label("A", (0.5,sqrt(3)/2), N);
label("B", (0,0), SW);
label("C", (1,0), SE);
label("D", (0.75,sqrt(3)/4), NE);
label("E", (0.5,0), S);
label("F", (0.25,sqrt(3)/4), NW);
label("G", (0.5,sqrt(3)/4), N);
label("H", (0.5,sqrt(3)/8), E);
[/asy]


Find the remainder when $5^{5^{5^5}}$ is divided by 500.


Tom, Sarah, and Mia go on a bicycle ride. Tom cycles at a rate of 6 miles per hour. If Sarah cycles $\frac{3}{4}$ as fast as Tom, and Mia cycles $\frac{4}{3}$ as fast as Sarah, how fast does Mia cycle?

Solution: \boxed{6 \text{ miles per hour}}


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 832^\circ.$


Find the integer $m$, $0 \le m \le 180$, such that $\cos m^\circ = \cos 1234^\circ$.


If the consecutive integers from $60$ to $1$ are written as $$6059585756...,$$ what would be the $80^{\text{th}}$ digit to be written?


Find $x^2+y^2$ if $x$ and $y$ are positive integers such that
\[
\begin{aligned}
xy + x + y &= 35 \\
x^2y + xy^2 &= 306.
\end{aligned}
\]


Suppose all four of the numbers: \[2 - \sqrt{5}, \; -2 - \sqrt{5}, \; -1 + \sqrt{3}, \; -1 - \sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


Express the quotient $1324_5 \div 12_5$ in base 5.


Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, such that the first outcome is the thousand digit, the second outcome is the hundreds digit, the third outcome is the tens digit, and the fourth outcome is the units digit, what is the probability that you will end up with a four-digit number that is divisible by 5? Use the spinner with digits 0, 1, and 5.


Suppose $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse. Find $g^{-1}(f(12))$.


Find the value of the series $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$


Three dice are stacked vertically. Eight of the eighteen faces are visible due to the orientation and stacking of the dice. The visible faces show the numbers 2, 2, 3, 4, 5, 5, 6, and 6. What is the total number of dots not visible from this view?


Evaluate the maximum integer value of the function \[\frac{4x^2 + 8x + 21}{4x^2 + 8x + 5}.\]


A square pyramid has a base edge of 48 inches and an altitude of 18 inches. A smaller square pyramid whose altitude is one-third of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the original pyramid?


Let $\mathbf{u}$ and $\mathbf{v}$ be two three-dimensional unit vectors such that the angle between them is $60^\circ$. Find the area of the parallelogram whose diagonals correspond to $\mathbf{u} + 3\mathbf{v}$ and $3\mathbf{u} + \mathbf{v}$.


Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 35$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $400$ square units, what is the area of triangle $EYH$?

Solution: \boxed{84 \text{ square units}}


Let \(a,\) \(b,\) \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 1.\) Find the maximum value of
\[6a + 3b + 10c.\]


Compute
$$\sum_{k=1}^{1500} k(\lceil \log_{3}{k}\rceil- \lfloor\log_{3}{k} \rfloor).$$


Consider the equation \[\sqrt{(x-3)^2+(y-1)^2} - \sqrt{(x+3)^2+(y-1)^2} = 4\]. The graph of this equation consists of one branch of a hyperbola. Compute the positive value for the slope of an asymptote of this hyperbola.


Triangle $XYZ$ has a perimeter of 2020 units. The sides have lengths that are all integer values with $AB \leq BC < AC$. What is the smallest possible value of $AC - BC$?


Find the smallest solution to the equation \[\lfloor x \rfloor = 2 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$.


Determine the value of \(b\) if the remainder is constant when \(12x^3 - 9x^2 + bx + 8\) is divided by \(3x^2 - 4x + 2\).


How many three-digit whole numbers contain at least one digit 6 or at least one digit 8?


Given that $\sin A + \sin B = \frac{3}{2}$ and $\cos A + \cos B = 1$, find the value of $\cos(A - B)$.


Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(a + b + c = 9.\) Find the minimum value of
\[
\frac{a^2 + b^2}{a + b} + \frac{a^2 + c^2}{a + c} + \frac{b^2 + c^2}{b + c}.
\]


The solution of the equation \(9^{x+5} = 16^x\) can be expressed in the form \(x = \log_b 9^5\). What is \(b\)?


What is the greatest number of points of intersection that can occur when $3$ different circles and $1$ straight line are drawn on the same piece of paper?


In the given diagram, there are two polygons: a regular octagon and a regular pentagon, sharing a common vertex. What is the sum, in degrees, of the measures of angles $PQR$ and $PQS$ located at the shared vertex?


Alex, Sam, and Jamie are cycling together. Alex cycles at a rate of 6 miles per hour. If Sam cycles $\frac{3}{4}$ as fast as Alex, and Jamie cycles $\frac{4}{3}$ as fast as Sam, how fast does Jamie cycle?


A new rule proposes that years ending in double zeros will only be considered leap years if they leave a remainder of either 200 or 600 when divided by 900. How many leap years ending in double zeros will there be between 2000 and 5000 under this new rule?


How many integers $n$ satisfy $(n-3)(n+5)<0$?


Let $w$ be a complex number such that $|w| = 3.$  Find the largest possible distance between $(6 + 5i)w^2$ and $w^4$ when plotted in the complex plane.

Solution: \boxed{9 \sqrt{61} + 81}


Find the point where the line passing through $(2, 3, 2)$ and $(6, -1, 7)$ intersects the $xz$-plane.

Solution: \boxed{\left(5, 0, \frac{23}{4}\right)}


$\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 7 cm, and the length of $\overline{QR}$ is 10 cm. If the length of $\overline{ST}$ is 4.9 cm, find the length of $\overline{TU}$. Express your answer as a decimal to the nearest tenth.


Calculate $\frac{4 \cdot 6! + 20\cdot 5!}{7!}$


Calculate the product $\left(\frac{2}{5}\right)\left(\frac{5}{8}\right)\left(\frac{8}{11}\right)\cdots\left(\frac{2004}{2007}\right)$. Express your answer as a common fraction.


The image of the point with coordinates $(-2,0)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(6,4)$. Find $m+b$.


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2008} \times 7$?


Let $A$ and $B$ be the endpoints of a semicircular arc of radius $3$. The arc is divided into five congruent arcs by four equally spaced points $D_1$, $D_2$, $D_3$, $D_4$. All chords of the form $\overline {AD_i}$ or $\overline {BD_i}$ are drawn. Find the product of the lengths of these eight chords.


Let $\omega$ be a nonreal root of $z^4 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega + b| = 1.$


Find the smallest positive integer that is both an integer power of 13 and is not a palindrome.


The quadratic equation $ax^2 + 15x + c = 0$ has exactly one solution. If $a + c = 24$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{24 - \sqrt{351}}{2}, \frac{24 + \sqrt{351}}{2}\right)}


Suppose that $\sec x+\tan x=3$ and that $\csc x+\cot x=\frac{m}{n},$ where $\frac{m}{n}$ is in lowest terms. Find $m+n.$


Find the integer $n$, $0 \le n \le 180$, such that $\cos n^\circ = \cos 331^\circ$.


Suppose $f(x) = ax - b$ and $g(x) = -4x + 6$. If $h(x) = f(g(x))$, and it is given that $h^{-1}(x) = x + 9$, find the value of $2a+b$.


Find the $3 \times 3$ matrix $\mathbf{P}$ such that for a $3 \times 3$ matrix $\mathbf{Q},$ $\mathbf{P} \mathbf{Q}$ is the result of rotating each row one position upwards (the first row moves to the place of the second row, the second row to the third, and the third row to the first) and tripling the original first row of $\mathbf{Q}$. In other words,
\[
\mathbf{P} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} d & e & f \\ g & h & i \\ 3a & 3b & 3c \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & 0 & 0 \end{pmatrix}}


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, what is the least possible value of $xy$?


What is the coefficient of $x^4$ when $$2x^5 - 4x^4 + 3x^3 - 5x^2 + x - 1$$ is multiplied by $$3x^4 - 2x^3 + x^2 + 5x + 6$$ and the like terms are combined?


Eight distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the second smallest is $4$?


Determine how many integer values of $b$ make it so that the quadratic equation \(x^2 + bx + 7b = 0\) always has integer roots.


Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 8$ and the diameter of the inscribed circle of triangle $OCF$ is 4, compute the product $(AB)(CD).$


Each of the numbers $b_1, b_2, \dots, b_{97}$ is either $1$ or $-1$. Find the smallest possible positive value of
\[
\sum_{1 \le i < j \le 97} b_i b_j.
\]


Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 4016, \\
pr &=& qs &=& 2000.
\end{array}
\]
If \( T = p+q+r+s\), compute the value of \(\lfloor T \rfloor\).


Circle $\omega$ has radius 3 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=10$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=9$.

Solution: \boxed{2\sqrt{91} + 9}


Find the value of $x$ if a cube's volume is $8x$ cubic units and its surface area is $\frac{x}{2}$ square units.


In $\triangle ABC$, we have $AC=BC=10$ and $AB=4$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=12$. What is $BD$?

Solution: \boxed{4\sqrt{3} - 2}


A circular piece of cardboard has a radius of 18 cm, from which a sector is removed as shown in the glossary. This larger shaded sector is then used to form a cone by bringing edge BC to edge BA. The resulting cone has a radius of 15 centimeters and a volume of $1350\pi$ cubic centimeters. Determine the degrees in the measure of angle ABC, representing the sector that Jeff did not use.


The fourteenth and twentieth terms of a geometric sequence are 10 and 80, respectively. Determine the value of the twenty-sixth term of this progression.


The first $35$ numbers of an arrangement are shown as follows. What would be the value of the $75^{\mathrm{th}}$ number if the arrangement were continued?

$\bullet$ Row 1: $3,$ $3$

$\bullet$ Row 2: $6,$ $6,$ $6,$ $6$

$\bullet$ Row 3: $9,$ $9,$ $9,$ $9,$ $9,$ $9$

$\bullet$ Row 4: $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12$


A rectangular box has a volume of $5000$ cubic inches and a surface area of $1800$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{7048 \text{ cubic inches}}


Two circles of radius 15 cm overlap such that each circle passes through the center of the other. Calculate the length of the common chord (dotted segment) of the two circles.


There are four points that are $7$ units from the line $y=20$ and $15$ units from the point $(10,20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$


Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\]

Solution: \boxed{2 + \sqrt{3}}


An ellipse has its foci at $(1, 5)$ and $(1, 1)$. Given that it passes through the point $(7, 3)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$


How many of the 729 smallest positive integers written in base 9 use 6 or 7 (or both) as a digit?


The seven digits in Joe's phone number and the four digits in his house number have the same sum. The four digits in his house number are distinct, and his phone number is 364-1528. What is the largest possible value of Joe's house number?


Let \(a\), \(b\), and \(c\) be real numbers such that \(a+b+c = t\). Additionally, let \(k\) be a positive real constant. Find, in terms of \(t\) and \(k\), the minimum value of \(ka^2 + b^2 + kc^2\).


Consider the set of points that are inside or within one unit of a rectangular parallelepiped with dimensions 2 units by 5 units by 6 units. Find the volume of this set in the standard form $\frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n$ and $p$ relatively prime. Determine $m+n+p$.


Let $\Delta ABC$ be an isosceles triangle with $AB = AC$. How many squares in the same plane as $\Delta ABC$ share two vertices with the triangle?


How many different positive values of \( x \) will make this statement true: there are exactly \( 3 \) positive two-digit multiples of \( x \)?


Find the smallest positive integer $a$ such that $x^4 + a^2 + 16$ is not prime for any integer $x$.


Solve the inequality \[\frac{x^2}{(x+2)^2} \ge 0.\] Provide your answer in interval notation.

Solution: \boxed{(-\infty, -2) \cup (-2, \infty)}


In pentagon $PQRST$, two interior angles $P$ and $Q$ are $70^{\circ}$ and $100^{\circ}$. The angles $R$ and $S$ are equal, and the fifth angle $T$ is $25^{\circ}$ less than thrice $R$. Determine the measure of the largest angle in the pentagon.


Rationalize the denominator of $\frac{5}{4\sqrt{7} + 3\sqrt{13}}$ and express your answer in the form $\frac{A\sqrt{B} + C\sqrt{D}}{E}$ where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$?


Let $\mathcal{T}$ be the set $\{1,2,3,\ldots,12\}$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Find the remainder obtained when $m$ is divided by $1000$.


A rectangular box has a volume of $5400$ cubic inches and a surface area of $1920$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{7568 \text{ cubic inches}}


The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the longest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $15$ inches? Express your answer as a common fraction.


If $\frac{120}{2^4 \cdot 5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


Tom's graduating class has grown to 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 20 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?


In rectangle $ABCD$, the diagonals intersect at point $O$. If $AB = 15$ and $BC= 20$, find the value of $\cos \angle BAO$.


In a modified version of the tournament, there are 6 bowlers. First #6 bowls #5. The loser receives 6th prize, and the winner bowls #4. The loser of this game receives 5th prize, and the winner bowls #3. The loser of this game receives 4th prize, and the winner bowls #2. The loser of this game receives 3rd prize, and the winner bowls #1. The final game determines the 1st and 2nd prizes. How many different orders can bowlers #1 through #6 receive the prizes?


Ten points are marked on the circumference of a circle. However, no two adjacent points can be connected by a chord. How many different chords can be drawn under this restriction?


One of the roots of $z^2 = -72 + 21i$ is $4 + 7i.$ What is the other root?

Solution: \boxed{-4 - 7i}


What is the remainder when \(97^{50}\) is divided by 100?


A circle passes through the point $(0,0)$ and is tangent to the parabola $y = x^2$ at $(1,1)$. Find the center of the circle.

Solution: \boxed{(-1, 2)}


Simplify $\left(\dfrac{1+i}{2}\right)^8 + \left(\dfrac{1-i}{2}\right)^8.$


What is the remainder when \(98^{50}\) is divided by 50?


In right triangle $XYZ$ with $\angle YXZ = 90^\circ$, we have $XY = 40$ and $YZ = 41$. Point $W$ is on side $XZ$ such that $XW = 9$ and $WZ = XZ - 9$. Find $\tan Y$.


Line segment $\overline{CD}$ is extended past $D$ to $Q$ such that $CQ:QD = 7:2.$ Then
\[\overrightarrow{Q} = x \overrightarrow{C} + y \overrightarrow{D}\] for some constants $x$ and $y.$ Enter the ordered pair $(x,y).$

Solution: \boxed{\left(-\frac{2}{5}, \frac{7}{5}\right)}


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


What is the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{HG} \perp \overline{FG}$, $EF=7$ cm, $HG=3$ cm, and $FG=15$ cm?

Solution: \boxed{25 + \sqrt{241} \text{ cm}}


If two distinct members of the set $\{ 4, 6, 18, 28, 30, 36, 50 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 90? Express your answer as a common fraction.


Calculate $\sqrt{\sqrt[4]{0.000625}}$. Express your answer as a decimal to the nearest tenth.


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


I bike at a constant pace, and it takes me 30 minutes to bike to the park from my home. If the park is 5 miles away, and the library is 3 miles away from my home, how many minutes will it take me to bike from my home to the library?

Solution: \boxed{18\text{ min}}


Solution: \boxed{\left(\frac{9}{16}\right)^{\frac{1}{7}}}


How many integers are between $(10.1)^3$ and $(10.4)^3$?


If
\[\tan x = \frac{a^2 + b^2}{2ab},\]where $a > b > 0$ and $x$ is such that $45^\circ < x < 90^\circ$, then find $\cos x$ in terms of $a$ and $b$.

Solution: \boxed{\frac{2ab}{\sqrt{a^4 + 6a^2b^2 + b^4}}}


In a certain city, the new rate of taxation is $(x+5)\%$ for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take-home pay, where take-home pay is defined as the income minus the tax on that income?


Define a function $h(x)$, for positive integer values of $x$, by:
\[ h(x) = \left\{
\begin{aligned}
\log_2 x & \quad \text{if } \log_2 x \text{ is an integer} \\
1 + h(x + 1) & \quad \text{otherwise}.
\end{aligned}
\right. \]
Compute $h(95)$.


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


The square of $a$ and the square root of $b$ vary inversely. If $a=3$ when $b=36$, then find $b$ when $ab=108$.


Find the greatest common divisor of $12547$ and $23791$.


From a set of integers $\{1, 2, 3, \ldots, 12\}$, eight distinct integers are chosen at random. What is the probability that, among those selected, the third smallest number is $4$?


For what base is the representation of $625_{10}$ a five-digit number whose final digit is odd?


Given that $\mathbf{a}$ and $\mathbf{b}$ are nonzero vectors of equal magnitude, and $\|\mathbf{a} + \mathbf{b}\| = 2 \|\mathbf{a}\|,$ find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.


The entry fee for a concert is \$30 per adult and \$15 per child. On a particular day, the concert collected \$2250 in entry fees, with at least one adult and one child attending. Determine the ratio of adults to children on that day, such that the ratio is closest to 1.


Let $
f(n) =
\begin{cases}
n^3+1 & \text{if }n\text{ is odd} \\
\dfrac{n}{2} & \text{if }n\text{ is even}
\end{cases}.
$

For how many integers $n$ from 1 to 100, inclusive, does $f ( f (\dotsb f (n) \dotsb )) = 1$ for some number of applications of $f$?


I have a picture with dimensions $x$ and $y$ (in inches), where $x$ and $y$ are integers greater than zero. I want to place this picture in a frame such that the total outer dimensions are $(3x + 4)$ and $(y + 3)$. If the area of the frame (excluding the picture) is $54$ square inches, what is the area of the picture?


Two different numbers are randomly selected from the set $T = \{2, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$. What is the probability that their sum equals 18, and which number, if removed from $T$, would increase this probability the most?


Find $\frac{8}{19} - \frac{5}{57}$. Reduce your answer to its simplest form.


Let $\mathbf{A} = \begin{pmatrix} a & -2 \\ 1 & d \end{pmatrix}$ for some real numbers $a$ and $d.$ If $\mathbf{A} + 2\mathbf{A}^{-1} = \mathbf{0}$, then find $\det \mathbf{A}.$


If $x$ and $y$ are positive integers such that $xy - 2x + 5y = 111$, what is the minimal possible value of $|x - y|$?


Simplify the following expression: \[3x+4x^2+2-(9-3x-4x^2)+\sin(x).\]

Solution: \boxed{8x^2 + 6x - 7 + \sin(x)}


Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 6.\) Find the minimum value of
\[\frac{9}{a} + \frac{4}{b} + \frac{25}{c}.\]


If $a \equiv 54 \pmod{53}$ and $b \equiv 98 \pmod{53}$, then for what integer $n$ in the set $\{150, 151, 152, \ldots, 200\}$ is it true that $$a - b \equiv n \pmod{53}~?$$


Three identical cylindrical jars, each with a diameter of 12 cm, are stacked in a triangular formation within a box. Calculate the total height, \( h \), of this arrangement.

[asy]
draw(circle((10,10),12),black+linewidth(1));
draw(circle((34,10),12),black+linewidth(1));
draw(circle((22,32.784),12),black+linewidth(1));
draw((60,0)--(60,45),black+linewidth(1));
draw((59,0)--(61,0),black+linewidth(1));
draw((59,45)--(61,45),black+linewidth(1));
label("$h$",(60,0)--(60,45),E);
[/asy]

Solution: \boxed{12 + 6\sqrt{3} \text{ cm}}


Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_2 x & \quad \text{ if } \log_2 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(150).$


The surface area of a sphere is $400\pi \text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of $\pi$.


On modulo $20$ graph paper, two functions are graphed: $$y \equiv 7x + 3 \pmod{20}$$ and $$y \equiv 13x + 17 \pmod{20}.$$
What is the sum of the $x$-coordinates of the points where these graphs intersect?


Determine the common ratio of the infinite geometric series: $$\frac{4}{3} + \frac{16}{9} + \frac{64}{27} + \dots$$


Given the system of equations

\begin{align*}
4x - 2y &= c, \\
6y - 12x &= d,
\end{align*}
where \( d \neq 0 \), determine the value of \(\frac{c}{d}\).


Find the integer $n$, $-90 < n < 90$, such that $\tan n^\circ = \tan 1230^\circ$.


The surface area of a sphere is \(400\pi \text{ cm}^2\). Calculate the volume of the sphere in cubic centimeters and express your answer in terms of \(\pi\).

Solution: \boxed{\frac{4000}{3}\pi \text{ cm}^3}


The sides of a triangle are $12, 16,$ and $k$, where $k$ is a positive integer. Determine how many values of $k$ exist such that the triangle is obtuse.


Define a set of integers "lightly spacy" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are lightly spacy?


A dodecahedron consists of two identical pentagonal-based pyramids glued together along their pentagonal bases to form a polyhedron with twelve faces. An ant starts at the top vertex of the dodecahedron and walks to one of the five adjacent vertices, which it randomly selects and calls vertex A. From vertex A, the ant then walks to one of the five adjacent vertices it randomly selects and calls vertex B. What is the probability that vertex B will be the bottom vertex of the dodecahedron? Express your answer as a common fraction.


What is the sum of all integer values of $n$ for which $\binom{25}{n} + \binom{25}{13} = \binom{26}{13}$?


The matrix for projecting onto a certain line $\ell$, which passes through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5}
\begin{pmatrix}
\frac{4}{17} & -\frac{2}{17} & -\frac{8}{17} \\
-\frac{2}{17} & \frac{1}{17} & \frac{4}{17} \\
-\frac{8}{17} & \frac{4}{17} & \frac{15}{17}
\end{pmatrix} \renewcommand{\arraystretch}{1}.
\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

Solution: \boxed{\begin{pmatrix} 2 \\ -1 \\ -4 \end{pmatrix}}


If $81^4 = 27^x$, what is the value of $3^{-x}$? Express your answer as a common fraction.


Calculate the area of the parallelogram formed by vectors $3\mathbf{a} + 2\mathbf{b}$ and $4\mathbf{a} - 6\mathbf{b}$ given that the area of the parallelogram formed by vectors $\mathbf{a}$ and $\mathbf{b}$ is 12.


The operation $\oplus$ is defined by
\[a \oplus b = \frac{a + b}{1 + ab}.\]Compute
\[1 \oplus (2 \oplus (3 \oplus (\dotsb \oplus (999 \oplus 1000) \dotsb))).\]


Let $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 0 \\ 2 \end{pmatrix}$. Find the vector $\mathbf{v}$ such that $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$, $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$, and $\mathbf{v} = \mathbf{a} + \mathbf{b} + t\mathbf{c}$, where $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$.

Solution: \boxed{\mathbf{v} = \begin{pmatrix} 0 \\ 2 \\ 2 \end{pmatrix}}


Define a function $B(m, n)$ identical to the Ackermann function, given by \[ B(m,n) = \left\{ \begin{aligned} &n+1& \text{ if } m = 0 \\ &B(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\ &B(m-1, B(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]
Compute $B(3, 1).$


How many positive integers less than $1000$ are either a perfect cube or a perfect square?


The admission fee at an art gallery is $30$ per adult and $15$ per child. The gallery collected a total of $2250$ from at least one adult and one child on a special event day. Determine the ratio of adults to children on that day which is closest to $2$.


Six small circles, each with a radius of 5 units, are tangent to a larger circle and also tangent to their neighboring small circles. Determine the diameter of the larger circle.


How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?


Determine the largest integer that divides the product \[
(n+2)(n+4)(n+6)(n+8)(n+10)
\] for all positive odd integers $n$.


A prism has 27 edges. How many faces does this prism have?


Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 6,$ $CD = 10,$ and $AE = 7,$ then what is the length of $CE?$
Consider also that $F$ is another point on line $AE$ that is distinct from $D$, and the distance $DF = 3$. Find the length of $CE$.


How many four-digit positive integers are there with the thousands digit $3$ and the hundreds digit even?


A triangle has sides of length 15 cm, 11 cm, and 19 cm. What is the perimeter of this triangle?


Ten people of different weights are at a gathering. Each person decides to only shake hands with people lighter than themselves. How many handshakes occur?


Convert the point \( (6, 6, -10) \) in rectangular coordinates to cylindrical coordinates. Enter your answer in the form \( (r, \theta, z) \), where \( r > 0 \) and \( 0 \le \theta < 2\pi \).

Solution: \boxed{(6\sqrt{2}, \frac{\pi}{4}, -10)}


A basketball team has 12 players. I have to choose a starting lineup consisting of 2 guards, 2 forwards, and 1 center. How many different starting lineups can I choose?


In how many ways can five people line up in a straight line if the two youngest people cannot be first in line?


The equation $x^2 - 6x + 15 = 25$ has two solutions, $c$ and $d$, with $c \geq d$. What is the value of $3c + 2d$?

Solution: \boxed{15 + \sqrt{19}}


Determine the coefficient of $x^4$ in the expression $5(x^2 - 2x^4) - 4(2x^3 - 3x^4 + x^6) + 3(4x^4 - 2x^7)$.


Determine the maximum value of the greatest common divisor (gcd) of two consecutive terms of the sequence defined by $b_n = (n+2)! - n^2$, where $n \geq 0$.


$ABCD$ is a trapezoid where base $\overline{AB}$ is three times the length of base $\overline{CD}$. Diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. The length of diagonal $\overline{AC}$ is 15 units. Determine the length of segment $\overline{EC}$.


On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 10 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be six car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 5 meters long and that the cars can travel at any speed, let $M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $M$ is divided by 10.


Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $m$ such that $Q$ is divisible by $5^m.$


I run at a constant pace, and it takes me 24 minutes to run to the park from my house. If the park is 3 miles away, and the library is 2 miles away from my house, how many minutes will it take me to run from my house to the library?

Solution: \boxed{16\text{ min}}


The surface area of a sphere with radius $r$ is $4\pi r^2$. What is the total surface area of a hemisphere with radius 10 cm, including the area of its circular base and an additional circular cutout at the top with radius 3 cm?


What is the 10th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{2}$ and $\frac{1}{4}$?


Let $x$, $y$, and $z$ be real numbers such that
\[\cos 2x + 2\cos 2y + 3\cos 2z = \sin 2x + 2\sin 2y + 3\sin 2z = 0.\] Find the sum of all possible values of $\cos 4x + \cos 4y + \cos 4z.$


Sam has 10 painting albums each containing 36 pages. Every page in the albums holds 8 paintings. Sam decides to rearrange his paintings such that each new page holds 9 paintings. After rearranging, Sam fills up 6 complete albums, each still with 36 pages per album. His seventh album contains 28 pages filled with 9 paintings per page and one last page with the remaining paintings. How many paintings are on that last page?


Juan needs to calculate the area of a circular garden to plan his planting. The actual diameter of the garden is 30 cm, however, Juan's measurement has a maximum error of up to $30\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle?


What is the $222$nd digit after the decimal point when $\frac{13}{481}$ is expressed as a decimal?


A circular piece of paper has a radius of six inches. A $300^\circ$ sector is cut from this circle. Points A and B on the edges of this sector are then glued together to form a right circular cone. What is the circumference of the base of the resulting cone? Express your answer in terms of $\pi$.


Let \( a, b, \) and \( c \) be angles such that:
\[
\sin a = \cos b, \\
\sin b = \cos c, \\
\sin c = \cos a.
\]
Find the maximum value of \( \cos a \).


A gardener plants four maple trees, three oaks, and six birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.


How many ways are there to divide a group of 7 friends among a basketball team, a soccer team, a track team, and a volleyball team, where each team can have anywhere from 0 to 7 friends?


Compute $\arccos (\sin 1.5).$ All functions are in radians.

Solution: \boxed{\frac{\pi}{2} - 1.5}


Determine $p(x)$ if the graph of $\frac{x^3 + x^2 - 4x - 4}{p(x)}$ has vertical asymptotes at $1$ and $-3$, no horizontal asymptote, and $p(2) = 8$.

Solution: \boxed{p(x) = \frac{8}{5}x^2 + \frac{16}{5}x - \frac{24}{5}}


Compute the sum and difference of the following repeating decimals: $0.\overline{6} + 0.\overline{2} - 0.\overline{9} + 0.\overline{3}$.


What is the smallest base-10 integer that can be represented as $22_a$ in one base and $33_b$ in another base, where $a$ and $b$ are any bases greater than 3?


Our club has 30 members, 16 boys and 14 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender and the president must be a senior member? Assume that there are 6 senior members, 3 boys and 3 girls, and no one can hold both offices.


Let $c$ and $d$ be real numbers. Consider the following five statements:

$\frac{1}{c} > \frac{1}{d}$
$c^2 < d^2$
$c > d$
$c > 0$
$d > 0$

What is the maximum number of these statements that can be true for any values of $c$ and $d$?


My friend June likes numbers that have an interesting property: they are divisible by 4. How many different pairings of last two digits are possible in numbers that June likes?


In triangle $XYZ$, $\angle X = 90^\circ$ and $\cos Y = \frac{3}{5}$. Find $\sin Z$.


If two numbers are randomly chosen without replacement from the set $\{4, 6, 8, 9\}$, what is the probability that their product will be a multiple of 12? Express your answer as a common fraction.


The measures of angles \(X\) and \(Y\) are both positive, integer numbers of degrees. The measure of angle \(X\) is a multiple of the measure of angle \(Y\), and angles \(X\) and \(Y\) are supplementary angles. How many measures are possible for angle \(X\)?


In the diagram below, $PQRS$ is a trapezoid with $\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \perp \overline{RS}$. If $RS = 15$, $\tan S = 2$, and $\tan Q = 3$, what is the length of $PQ$?


Find the matrix $\mathbf{M}$ that swaps both the rows and the columns of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. In other words, find $\mathbf{M}$ such that
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & c \\ b & a \end{pmatrix}.
\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}}


Andrew's grandfather is ten times as old as Andrew. If Andrew's grandfather was 60 years old when Andrew was born, how many years old is Andrew now?


A pizza with 24 slices is made with three toppings: ham, pineapple, and jalapeno. Every slice has at least one topping. Exactly 15 slices are topped with ham, 10 slices are topped with pineapple, and 14 slices are topped with jalapeno. If we know that the number of slices having all three toppings is equal to the number of slices with only jalapeno, determine how many slices have all three toppings.


Express $0.\overline{3} + 0.\overline{02} + 0.\overline{0003}$ as a common fraction.


What is the 20th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{13}$ and $\frac{1}{11}$?


If $128^7 = 16^y$, what is the value of $2^{-y}$? Express your answer as a common fraction.


What is the sum of the $x$-values that satisfy the equation $7 = \frac{x^3 - 3x^2 - 12x}{x + 3}$?


Find the greatest common divisor of $12345$ and $67890$.


In triangles $\triangle ABC\sim\triangle FGH$, the length of $BC$ is 24 cm, and side $FG$ measures 16 cm. If the length of $GH$ is 15 cm, determine how long side $AC$ is. Provide your answer as a decimal rounded to the nearest tenth.

Solution: \boxed{22.5 \text{ cm}}


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\beta,$ and the angle between $\mathbf{c}$ and $\mathbf{a} \times \mathbf{b}$ is also $\beta.$ If $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{2},$ find the smallest possible value of $\beta,$ in degrees.


Solve for $t$: $5 \cdot 3^t + \sqrt{25 \cdot 9^t} = 50$.

Solution: \boxed{t = \frac{\log 5}{\log 3}}


The expression $25x^2 - 130x - 120$ can be written as $(5x + a)(5x + b),$ where $a$ and $b$ are integers. What is $a + 3b$?


Let $a = \underbrace{6666666}_{\text{7 digits}}$ and $b = \underbrace{999999999}_{\text{9 digits}}$.

What is $\gcd(a,b)$?


Solution: \boxed{2x^3 - x^2 + 23x - 3}


What is the 417th digit to the right of the decimal point when $\frac{8}{19}$ is expressed as a decimal?


Line $l_1$ has equation $3x - 2y = 3$ and passes through $A = (0, -3)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has a positive slope, passes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $4$. What is the slope of $l_3$?


If I roll 4 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?


Consider a right triangle where one of the legs measures 9 units and the hypotenuse measures 15 units. Find $\sin C$, where $C$ is the angle opposite the leg measuring 9 units.


The sum of two positive integers $a$ and $b$ is 1980. What is the largest possible value of $\gcd(a,b)$?


For $k > 0$, let $J_k = 20\ldots 064$, where there are $k$ zeros between the $2$ and the $6$. Let $M(k)$ be the number of factors of $2$ in the prime factorization of $J_k$. What is the maximum value of $M(k)$?


How many possible distinct arrangements are there of the letters in the word "BALL"?


The sum of 36 consecutive integers is $6^4$. What is their median?


An equilateral triangle and a rectangle have the same perimeter of 24 inches. The rectangle's length is twice its width. What is the ratio of the side length of the triangle to the width of the rectangle? Express your answer as a common fraction.


How many one-thirds are there in $\frac{9}{5}$?


A cylinder has a height of $15$ units and a radius of $5$ units. Calculate the total surface area of the cylinder, including the two ends, and determine the volume of the cylinder.


When an integer is divided by 21, the remainder is 13. Find the sum of the remainders when the same integer is divided by 3 and by 7.


Two distinct positive integers from 1 to 70 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?


A function $f$ from the integers to the integers is defined as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $k$ is odd and $f(f(f(k))) = 31.$  Find $k.$


A fair 12-sided die is rolled once. If I roll a number \(n\), then I win \(10 - n\) dollars if \(n\) is less than or equal to 5. If \(n\) is greater than 5, I win \(n - 5\) dollars. What is the expected value of my win, in dollars?


In the diagram, if $\angle PQR = 60^\circ$, what is the measure of $\angle PMN$? [asy]

size(6cm);

pair p = (0, 0); pair m = dir(180 - 30); pair n = dir(180 + 30); pair r = 1.3 * dir(30); pair q = 2 * 1.3 * Cos(60) * dir(-30);

label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);

draw(m--q--r--n--cycle);

add(pathticks(m--p, s=4));

add(pathticks(n--p, s=4));

add(pathticks(r--p, 2, spacing=0.9, s=4));

add(pathticks(r--q, 2, spacing=0.9, s=4));

[/asy]


What is the value of the least positive base ten number which requires eight digits for its binary representation?


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 44. If one integer is 60, what is the smallest possible value of the other integer?


A hexagon $ABCDEF$ has two interior angles, $A$ and $B$, which are $90^\circ$ and $120^\circ$ respectively. Angles $C$ and $D$ are equal, and angle $E$ exceeds twice angle $C$ by $20^\circ$. Finally, angle $F$ is $30^\circ$ less than angle $A$. Calculate the measure of angle $D$.


Let $x$ be a real number. Consider the following six statements:
1. $0 < x^3 < 1$
2. $x^3 > 1$
3. $-1 < x < 0$
4. $0 < x < 1$
5. $0 < x - x^3 < 1$
6. $x^3 - x > 1$

What is the maximum number of these statements that can be true for any value of $x$?


If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?


The Euler family has two boys aged $8,$ twin girls aged $10,$ a girl aged $15,$ and a boy aged $12.$ What is the mean (average) of the ages of the children?


What is the smallest possible area, in square units, of a right triangle with two sides measuring $6$ units and $8$ units?


In triangle $XYZ$, the sides measure $XY = 5$, $YZ = 7$, and $XZ = 8$. Two ants start simultaneously from vertex $X$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They stop when they first meet at point $P$. What is the length of $YP$?

Solution: \boxed{YP = 5}


Let $p,$ $q,$ and $r$ be real numbers, and let $X,$ $Y,$ $Z$ be points such that the midpoint of $\overline{YZ}$ is $(p,0,0),$ the midpoint of $\overline{XZ}$ is $(0,q,0),$ and the midpoint of $\overline{XY}$ is $(0,0,r).$  Find
\[\frac{XY^2 + XZ^2 + YZ^2}{p^2 + q^2 + r^2}.\]


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex $A$, follows this rule: at each vertex, it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = \frac{n}{2187}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 8 meters. Find the value of $n$.


Simplify $\frac{\sqrt{800}}{\sqrt{200}} \times \left( \frac{\sqrt{180}}{\sqrt{72}} - \frac{\sqrt{224}}{\sqrt{56}} \right)$.

Solution: \boxed{\sqrt{10} - 4}


What is the greatest three-digit number that is divisible by 4, divisible by 8, and ends in 4?


Complex numbers $a$, $b$, $c$ form an equilateral triangle with side length 24 in the complex plane. If $|a + b + c| = 48$, find $|ab + ac + bc|$.


Sara baked 60 pies. Of these, one-third contained berries, half contained cream, three-fifths contained nuts, and one-fifth contained coconut. What is the largest possible number of pies that had none of these ingredients?


How many distinct sequences of five letters can be made from the letters in MATHEMATICS if each sequence must begin with E, end with S, and no letter can appear in a sequence more than once?


If $x = 201$ and $x^3y - 2x^2y + xy = 804000$, what is the value of $y$?


When \( n \) is divided by 4, the remainder is 3. What is the remainder when \( 7n \) is divided by 5?


You have a certain number of gold coins that you intend to distribute equally among your 12 best friends. However, when you attempt to divide your gold coins into 12 equal piles, you realize that if you give away all your gold coins, 3 people will receive an extra gold coin. You have less than 120 gold coins. What is the largest number of gold coins you could have to make this happen?


What is the value of $x$ for which $(10 - x)^2 = 2x^2$?

Solution: \boxed{-10 - 10\sqrt{2}}


In a school class of 45 students, 15 prefer chocolate pie, 10 prefer apple, and 9 prefer blueberry. Half of the remaining students prefer cherry pie and the other half prefer lemon. How many degrees should be designated for lemon pie on a pie chart?


For the parabolas $y = (x + 2)^2$ and $x + 5 = (y - 4)^2$, find the sum of the x-coordinates and y-coordinates of all points where they intersect:
\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\]


In triangle $PQR$, $p = 8$, $q = 6$, $r = 10$. Let $J$ be the incenter. Represent $\overrightarrow{J}$ as a linear combination of $\overrightarrow{P}$, $\overrightarrow{Q}$, $\overrightarrow{R}$ where $x + y + z = 1$. Find the ordered triple $(x, y, z)$.

Solution: \boxed{\left(\frac{1}{3}, \frac{1}{4}, \frac{5}{12}\right)}


In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm, and all are initially shaded. Six unshaded circles of different sizes are then placed on the grid as illustrated. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$?

[asy]
for(int i = 0; i < 6; ++i)
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    for(int j = 0; j < 6; ++j)
    {
        filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle,gray,linewidth(2));
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}
filldraw(circle((1.5,1.5),0.5),white,linewidth(2));
filldraw(circle((4.5,1.5),0.5),white,linewidth(2));
filldraw(circle((1.5,4.5),0.5),white,linewidth(2));
filldraw(circle((4.5,4.5),0.5),white,linewidth(2));
filldraw(circle((3,3),1),white,linewidth(2));
filldraw(circle((3,3),0.5),white,linewidth(2));
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In rectangle $ABCD$, let $AB = 120$. Point $E$ is the midpoint of $\overline{AD}$. It is given that lines $AC$ and $BE$ are perpendicular to each other. Find the greatest integer less than the length of $AD$.


Consider the hyperbolas \[\frac{x^2}{9} - \frac{y^2}{16} = 1\] and \[\frac{x^2}{25} - \frac{y^2}{M} = 1\] which share the same asymptotes. Find the value of \(M\).


On a "prime date," both the month and the day are primes as before. Determine how many prime dates occurred in February 2024, considering it is a leap year.


Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits, and ensure the sum of the digits is divisible by 6.


Bonnie constructs the frame of a cube using 12 pieces of wire, each four inches long. Roark uses 1-inch-long pieces of wire to create several unconnected unit cube frames. The total volume of Roark's cubes equals the volume of Bonnie's cube. Determine the ratio of the total length of Bonnie's wire to the total length of Roark's wire.


Objects $A$ and $B$ move simultaneously in the coordinate plane. Object $A$ starts at $(0,0)$, and each of its steps is either right or up, both equally likely. Object $B$ starts at $(6,8)$, and each of its steps is either left or down, both equally likely. Calculate the probability that the objects meet after each has moved exactly four steps.


Let \((x,y)\) be an ordered pair of real numbers that satisfies the equation \(x^2+y^2=20x+64y\). What is the minimum value of \(y\)?

Solution: \boxed{32 - 2\sqrt{281}}


What are the last two digits in the sum of the factorials of the first 15 positive integers?


How many positive divisors does 8! have?


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 10, and the third card is a Queen? Assuming the dealing is done without replacement.


Find the volume of the region in space defined by
\[|x + y + 2z| + |x + y - 2z| \le 12\]
and $x, y, z \ge 0$.


For \(\mathbf{v} = \begin{pmatrix} x \\ 4 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\),
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ 1.2 \end{pmatrix}.\] Find \(x\).


Let
\[ g(x) =
\begin{cases}
-x^3 & \text{if } x \geq 0,\\
x+10 & \text{if } x < 0.
\end{cases}
\]
Compute $g(g(g(g(g(2)))))$.


A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. Determine how many relatively prime dates are in February of a non-leap year?


Every vector on the line $y = \frac{3}{4} x + 3$ is projected onto a certain vector $\mathbf{w'},$ and the result is always the specific vector $\mathbf{p'}$. Find this vector $\mathbf{p'}$.

Solution: \boxed{\begin{pmatrix} -\frac{36}{25} \\ \frac{48}{25} \end{pmatrix}}


How many ways are there to arrange the letters of the word $\text{BA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{X}_1\text{X}_2$, in which the three A's and the two N's are considered different, now adding two X's as well, each considered different?


Find the $x$-intercept and $y$-intercept points of the line $4x + 7y = 28$. Provide your answers as ordered pairs.

Solution: \boxed{(7, 0)}


Let $g(n)$ be the sum of all the divisors of a positive integer $n$. If $g(g(n)) = n+3$, define $n$ as hyperdeficient. Determine how many hyperdeficient positive integers there are.


What is the shortest distance from the origin to the circle defined by $x^2 - 6x + y^2 - 8y + 15 = 0$?

Solution: \boxed{5 - \sqrt{10}}


A cylindrical drum has a height of 2 yards and a diameter of 4 feet. Calculate the volume of the drum in cubic feet.


Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 500, what is the sum of $a$ and $b$?


A right circular cylinder with radius 3 is inscribed in a sphere with radius 7 so that its bases are parallel to the base of the sphere. What is the height of this cylinder?


What is the greatest common divisor of all three-digit palindromes?


A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 65.$ Find $m.$


Solve for $z$ in the following equation: $4 + 2iz = 2 - 6iz$.


Let \( p \) and \( q \) be positive integers such that
\[
\frac{6}{11} < \frac{p}{q} < \frac{5}{9}
\]
and \( q \) is as small as possible. What is \( p+q \)?


The terms $140, a, \frac{45}{28}$ are the first, second, and third terms, respectively, of an arithmetic sequence. If $a$ is positive, what is the value of $a$?

Solution: \boxed{a = \frac{3965}{56}}


If two numbers are randomly chosen without replacement from the set $\{4, 6, 8, 9\}$, what is the probability that their product will be a multiple of 12 and at least one of the numbers is even? Express your answer as a common fraction.

Solution: \boxed{\frac{4}{6} = \frac{2}{3}}


Find the smallest value of \( t \) such that
\[
\frac{16t^2 - 36t + 15}{4t - 3} + 4t = 7t + 6.
\]

Solution: \boxed{\frac{51 - \sqrt{2073}}{8}}


What is the modular inverse of $13$, modulo $997$?

Express your answer as an integer from $0$ to $996$, inclusive.


Triangle $DEF$ has a perimeter of 2050 units. The sides have lengths that are all integer values with $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{1024}}}}$$


Point \(O\) is the center of an ellipse with major axis \(\overline{AB}\) and minor axis \(\overline{CD}\). Point \(F\) is one focus of the ellipse. If \(OF = 5\) and the diameter of the inscribed circle of triangle \(OCF\) is 3, compute the product \((AB)(CD)\).

Solution: \boxed{(AB)(CD) = 152.25}


The least common multiple of two integers is 72, and their greatest common divisor is 8. If one of the integers is four times the greatest common divisor, find the product of the two numbers.


For what digit $d$ is the six-digit number $45678d$ a multiple of 9?


A new rule is proposed where a year ending in double zeros will be considered a leap year only if it leaves a remainder of either 300 or 700 when divided by 1000. How many leap years will there be that end in double zeros between 2000 and 5000?


A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. February in a leap year has 29 days and the number for February is 2. Determine how many relatively prime dates are in February of a leap year.


How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?


Convert the point $(3,-3)$ in rectangular coordinates to polar coordinates. Enter your answer in the form \((r,\theta),\) where \( r > 0 \) and \( 0 \le \theta < 2\pi \).

Solution: \boxed{\left( 3\sqrt{2}, \frac{7\pi}{4} \right)}


Given two vectors $\mathbf{a}$ and $\mathbf{b}$ with an angle of $60^\circ$ between them, what is the angle between the vector $2\mathbf{a}$ and the vector obtained by rotating $\mathbf{b}$ by $45^\circ$ clockwise?


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 942^\circ.$


If $\sqrt{5 + x} + \sqrt{25 - x} = 8$, what is the value of $(5 + x)(25 - x)$?


There are 30 different complex numbers \( z \) such that \( z^{30} = 1 \). For how many of these is \( z^6 \) a real number?


The positive difference between the two roots of the quadratic equation $5x^2 - 8x - 7 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$.


The integers 342 and 78 are expressed in base 5 and added. What is the resulting sum expressed in base 5?


Expand and simplify the expression $2(c-2)(c^2 + c(4 - c))$. What is the sum of the coefficients of the expanded form?


Crestview's school colors are purple and gold. The students are designing a flag using three solid-colored horizontal stripes. Using one or both of the school colors, how many different flags are possible if no two adjacent stripes may have the same color?


Starting with the number 200, Dina repeatedly divides her number by 3 and then takes the greatest integer less than or equal to that number. How many times must she do this before she reaches the number 1?


In a different ellipse, the center is at \((-2, 1)\), one focus is at \((-3, 0)\), and one endpoint of a semi-major axis is at \((-2, 4)\). Find the semi-minor axis of this ellipse.


If \( y \) is real, compute the maximum integer value of
\[
\frac{4y^2 + 8y + 19}{4y^2 + 8y + 5}.
\]


In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively of a right-angled triangle $\triangle ABC$ with right angle at $B$. Point $F$ is the midpoint of $\overline{BC}$. Calculate the area of quadrilateral $BDFE$.

[asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,4);
B=(0,0);
C=(6,0);
D=(0,2);
E=(3,2);
F=(3,0);
draw(A--B--C--A);
draw(B--D--F--E--B);
label("$A(0,4)$", A, NW);
label("$B(0,0)$", B, S);
label("$C(6,0)$", C, S);
label("$D$", D, W);
label("$E$", E, NE);
label("$F$", F, S);
[/asy]


What is the smallest positive integer that can be subtracted from 425 to result in a multiple of 5?


The perimeter of a rectangle is 84 meters. The ratio of its length to its width is 5:2. What is the length in meters of a diagonal of the rectangle?


Find the distance between the points (0, 6) and (8, 0) and calculate the area of the triangle formed by these points and the origin (0,0).


Jenna is 74 inches tall. Using the conversion 1 inch = 2.58 cm, how tall is Jenna in centimeters? Express your answer as a whole number.


Gina is studying a quadratic expression, \[ x^2 - ax + b, \] with roots $u$ and $v$. She finds that \[ u + v = u^2 + v^2 = u^4 + v^4 = \ldots = u^{18} + v^{18}. \] Curiosity piqued, she wants to calculate the greatest possible value of \[ \dfrac{1}{u^{20}} + \dfrac{1}{v^{20}}. \]


Both roots of the quadratic equation \(x^2 - 65x + k = 0\) are prime numbers. How many possible values of \(k\) are there?


Find the minimum value of
\[
\frac{x^2}{(x-8)^2}
\]
for \(x > 8.\)


In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 25 units, respectively, and the altitude is 15 units. Points $T$ and $U$ are the midpoints of sides $PS$ and $QR$, respectively. What is the area of quadrilateral $TURS$ in square units?


A circle centered at $O$ is circumscribed around $\triangle PQR$ such that $\angle POQ = 120^\circ$ and $\angle QOR = 130^\circ$. Determine the measure of $\angle PQR$.


Determine the area of the region defined by the equation $x^2 + y^2 + 6x - 10y + 5 = 0$.


Let \( x \) and \( y \) be two-digit integers such that:
1. \( x \) is an integer between 10 and 99, inclusive;
2. \( y \) is the number formed by reversing the digits of \( x \);
3. Let \( z = |x - y| \).

How many distinct values of \( z \) are possible?


Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[3g(a^2 + b^2) = [g(a)]^2 + 2[g(b)]^2\] for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $g(25),$ and let $s$ be the sum of the possible values of $g(25).$  Find $n \times s.$


The expression $x^2 - 18x + 72$ can be written in the form $(x - c)(x - d)$, where $c$ and $d$ are both nonnegative integers and $c > d$. What is the value of $4d - c$?


The bacteria in a jar quadruple every 30 seconds. After five minutes, there are 4,194,304 bacteria in the jar. How many were in the jar at the beginning of the experiment?


What is the smallest positive integer \( n \) for which \( 8n-3 \) and \( 6n + 4 \) share a common factor greater than \( 1 \)?

Solution: \boxed{n = 1}


Compute the following sum:
$$\sum_{k=1}^{2000} k(\lceil \log_{\sqrt{2}}{k}\rceil - \lfloor\log_{\sqrt{2}}{k} \rfloor).$$


Find the sum of angles, in degrees, of the solutions to the equation \(z^6 = -32\), where each solution \(z_k\) can be expressed in the form:
\[
z_k = r_k (\cos \theta_k + i \sin \theta_k)
\]
for \(k=1, 2, \dots, 6\), where \(r_k > 0\) and \(0^\circ \leq \theta_k < 360^\circ\).


Triangle $PQR$ has $\angle Q = 45^{\circ}$ and $QR = 6$. Point $S$ is the midpoint of $QR$. If the altitude from point $P$ to $QR$ is 3 units, what is the largest possible value of $\tan{\angle PSQ}$?


In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, where $j$ and $k$ are positive integers, what is the sum of all possible values for $k$?


Find the number of positive integers $n \le 1500$ such that $21n$ is a perfect square.


Determine how many integers, $x$, satisfy the inequality $|7x + 2| \le 9$?


Consider a right triangle $DEF$ with right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7$. A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF$. If $Q$ is the point where the circle and side $DF$ meet, then what is $FQ$?


In $\triangle ABC$, the ratio $AC:CB$ is $4:5$. The bisector of the interior angle at $C$ intersects $AB$ at point $Q$. If $AB = 18$ units, find the ratio $AQ:QB$.


How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^3)(x-2^3)(x-3^3)\dotsm(x-50^3)} = 0
\]
have for \(x\)?


The quadratic equation $x^2+ax+b=0$ has roots that are three times those of $x^2+cx+a=0$, where none of $a,$ $b,$ and $c$ is zero. What is the value of $b/c?$


In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form.


If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$


Jessica is curious about how many different ways she can rearrange the letters in her friend's name, "Daniel." If she can write fifteen rearrangements per minute, how many hours will it take her to write all possible rearrangements?

Solution: \boxed{0.8 \text{ hours}}


What integer is closest to the value of $\sqrt[3]{7^3 + 9^3 + 10}$?


A rectangular band formation is a formation with $n$ band members in each of $s$ rows, where $n$ and $s$ are integers. A particular band has less than 150 band members. The director arranges them in a rectangular formation and finds that he has three members left over. If he increases the number of members in each row by 2 and reduces the number of rows by 3, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have?


Lucas has a quadratic of the form $x^2+cx+\frac{1}{4}$, where $c$ is a specific negative number. He uses the method of completing the square to rewrite this quadratic in the form $(x+n)^2+\frac{1}{8}$. Determine the value of $c$.


What is the first year after 2010 for which the sum of the digits equals to 15?


I own a collection of 18 books, including 4 novels that are each part of a series. To keep the variety on my vacation, I decide not to pick two books from this series. With this restriction, how many different pairs of books can I choose to take?


Max picks two different cards without replacement from an extended 54-card deck which includes two jokers (which have no suits). What is the probability that the cards are of different suits, given that neither of the two cards picked is a joker?


Let $g(x) = x^2 + 4x + d$ for all real numbers $x$, where $d$ is some real number. For what values of $d$ does $g(g(x))$ have exactly 3 distinct real roots?


What is the domain of the function $$h(y) = \frac{1}{(y-2)^2 + (y-8)}~?$$ Express your answer in interval notation.

Solution: \boxed{(-\infty, -1) \cup (-1, 4) \cup (4, \infty)}


Find the remainder when $5^{5^{5^5}}$ is divided by 500.


The isosceles trapezoid has base lengths of 24 units (bottom) and 12 units (top), and the non-parallel sides are each 12 units long. How long is the diagonal of the trapezoid?


A store offers a discount of 25% on its products. Following this, there is an additional promotion where the reduced prices are further discounted by 15%. The store advertises that the combined discount makes the products 40% cheaper than the original price. Calculate the true total discount percentage relative to the original price and determine the difference between the true discount and the store's claimed discount.


If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $216x^3+64= (ax^2 + bx + c)(dx^2 +ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?


Find the real root of the equation \[\sqrt{x} + \sqrt{x+8} = 8.\]


A rectangular tile measures 2 inches by 5 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 6 feet?


Convert the point $(\sqrt{3},-\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$

Solution: \boxed{\left(\sqrt{6}, \frac{7\pi}{4}\right)}


In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $k$?


Solve for $z$ in the following equation: $3 - 2iz = -2 + 5iz$.

Express your answer in standard form.


A square pyramid is added to one of the faces of a rectangular prism, which consists of 6 faces, 12 edges, and 8 vertices. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the resulting solid when the pyramid is added to different faces of the prism.


Find the smaller of two different 2-digit positive integers, if the average of these two integers equals the number obtained by writing one of these integers before the decimal point and multiplying the other integer by 0.01 after the decimal point.


A $5$-foot by $10$-foot rectangular piece of plywood will be cut into $5$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?


What is the greatest common factor of the numbers 4620 and 10780?

Solution: \boxed{2^2 \times 5 \times 7 \times 11 = 1540}


The solution of the equation $9^{x+5} = 10^x$ can be expressed in the form $x = \log_b 9^5$. What is $b$?


Evaluate the infinite product $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$


What is the greatest divisor of 420 that is smaller than 50 and also a factor of 90?


Let \( S \) be the set of positive real numbers. Define a function \( f: S \to \mathbb{R} \) such that
\[ f(x)f(y) = f(xy) + 1001 \left( \frac{1}{x} + \frac{1}{y} + 1000 \right) \]
for all \( x, y > 0 \).
Determine \( n \), the number of possible values of \( f(2) \), and \( s \), the sum of all possible values of \( f(2) \). Compute \( n \times s \).


The first term of a sequential series is 3, and each successive term is twice the sum of all previous terms of the sequence. What is the value of the first term which exceeds 10,000?


Determine the number of digits in the value of \(2^{15} \times 5^{10}\).


What is the value of $\frac{15!}{6!9!}$?


Find $\cos \left( \frac{\pi}{3} \right).$


In triangle $ABC$, shown below, $AC = \sqrt{20}$ and $AB = 4$. What is $\tan A$?

[asy]
pair A,B,C;
B = (0,0);
A = (0,4);
C = (2,0);
draw(A--B--C--A);
draw(rightanglemark(A,B,C,7));
label("$B$",B,SW);
label("$A$",A,NW);
label("$C$",C,SE);
label("$4$",(A+B)/2,NW);
label("$\sqrt{20}$",(A+C)/2,NE);
[/asy]


Let's assume Jackie and Phil have now four fair coins and a third coin that comes up heads with probability $\frac{3}{5}$. Both Jackie and Phil take turns to flip these five coins. Find the smallest integer value $x$ so that the probability that Jackie gets the same number of heads as Phil is greater than $\frac{1}{x}$.


A jar initially contained only red and yellow jelly beans with probabilities of $0.4$ and $0.6$, respectively. More jelly beans (orange and green) were added, and the probability of randomly selecting a red jelly bean changed to $0.25$, and an orange jelly bean is $0.35$. What is the probability of randomly selecting a green or a yellow jelly bean from this jar now?


How many positive 4-digit numbers are divisible by 9?


Find the number of odd digits in the base-7 representation of $528_{10}$.


The average of five different positive whole numbers is 5. If the difference between the largest and the smallest number is as small as possible, what is the average of the other three numbers?


In a specific sequence, the first term is $a_1 = 4014$ and the second term is $a_2 = 4015$. The sequence is defined such that for all $n \ge 1$:
\[a_n + a_{n+1} + a_{n+2} = 2n.\]
Determine the value of $a_{1000}$.


Each of $b_1,$ $b_2,$ $\dots,$ $b_{50}$ is equal to $2$ or $-2.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} b_i b_j.\]


What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{137}$?


Nine distinct points are placed on the circumference of a circle such that no three points are collinear along any line that passes through the circle's center. How many different triangles can be formed if each vertex must be one of these 9 points, and additionally, how many distinct straight lines can be drawn connecting pairs of these points?


Consider a $3 \times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring?


How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?

[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15));
dot((15,0));
label("$A$",(0,15),NW);
label("$B$",(15,0),SE);
[/asy]


What integer $m$ satisfies $0 \leq m < 31$ and $$79453 \equiv m \pmod{31}~?$$


How many digits are located to the right of the decimal point when $\frac{5^6}{10^5 \cdot 8}$ is expressed as a decimal?


What is the nearest integer to $(3+\sqrt5)^4$?


Determine the value of $\frac{a}{b}$ for the solution of the system of equations, assuming $b \neq 0$:

\begin{align*}
4x-2y &= a, \\
5y - 10x &= b.
\end{align*}


A flagpole is supported by a wire which extends from the top of the pole to a point on the ground 4 meters from its base. When Joe walks 3 meters from the base of the pole toward the point where the wire is attached to the ground, his head just touches the wire. Joe is 1.8 meters tall. How many meters tall is the flagpole?


What is the 25th number in the row of Pascal's triangle that has 28 numbers?


How many different positive even integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, \ldots, 19, 20\}?$


For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(6,6)--(13,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(13,2),dashed);
label("9",(3,3),NW);
label("11",(9.5,4),NE);
label("17",(10.5, -2.5),SE);
label("13",(4,-2.5),SW);
[/asy]


What is the largest positive integer with only odd digits that is less than $10,000$ and is a multiple of $5$?


Calculate $\frac{1}{3} \cdot \frac{4}{7} \cdot \frac{9}{11}$.


Solve the inequality
\[\left| \frac{3x + 2}{x + 2} \right| > 3.\]

Solution: \boxed{(-\infty, -2) \cup \left(-2, -\frac{4}{3}\right)}


At a workshop with 40 professionals, 18 professionals used laptops, 14 used tablets, and 16 drank coffee. 7 professionals used both laptops and tablets, 4 used both tablets and coffee, and 5 used both laptops and coffee. Additionally, 3 professionals used all three: laptops, tablets, and coffee. How many professionals did not use any laptop, tablet, or drink coffee?


Faces $PQR$ and $QRS$ of tetrahedron $PQRS$ meet at an angle of $45^\circ$. The area of face $PQR$ is $150$, the area of face $QRS$ is $50$, and $QR=10$. Find the volume of the tetrahedron.


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 3 \\ -4 \end{pmatrix}}


Fifteen boys are standing in a circle. A ball is thrown clockwise around the circle. The first boy, Ben, starts with the ball, skips the next four boys and throws to the sixth boy, who then skips the next four boys and continues the pattern. How many total throws are necessary for the ball to return to Ben?


Jessie is asked to add 82 to 56, then subtract 15 from the result, and finally round it to the nearest five. She decides to round the numbers before performing the operations to speed up the process. She rounds 82 to 85 and 56 to 55, then subtracts 15 rounded to 15, resulting in $85 + 55 - 15 = 125$. What is the correct final result?


Let $T$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.ababab\ldots = 0.\overline{ab}$, where the digits $a$ and $b$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different numerators are required?


Find the maximum value of the expression
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_6 + \cos \theta_6 \sin \theta_1,
\]
over all real numbers $\theta_1, \theta_2, \theta_3, \theta_4, \theta_5, \theta_6.$


In a local volleyball league, there are 5 teams, with each team consisting of 8 players. Each team hosts one tournament in rotation. At each tournament, the host team selects four of its members to be on the organizing committee, while each of the other four teams selects one member to join the committee. Calculate the total number of distinct tournament committees that can be formed over one complete rotation of hosts.


Twenty-five gremlins and eighteen imps are at the Annual Mischief Convention. Due to a recent squabble, five specific gremlins refuse to shake hands with each other or with any other gremlins. However, these five gremlins will shake hands with all imps. The remaining gremlins and all of the imps shake hands with each other as before: imps do not shake hands with each other but will shake hands with any gremlin. How many handshakes occur at the convention?


Suppose we flip five coins simultaneously: a penny (1 cent), a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). What is the probability that at least 40 cents worth of coins come up heads?


Of the points (4, 20), (7, 30), (15, 50), (20, 65), and (25, 80), determine the sum of the $x$-coordinates of the points that lie above the line $y = 3x + 5$ in the coordinate plane.


What is the least integer value of $x$ such that $\lvert3x + y - 4\rvert \leq 25$ for $y = 5$?


What is the smallest positive value of $x$ such that $x + 7890$ results in a palindrome?


Six numbers are 1870, 1996, 2022, 2028, 2112, and 2124. Four of these numbers have a mean (average) of 2011. Determine the mean (average) of the other two numbers.


Given the following data, calculate how much cheaper, in cents, the cheaper store's price for a gadget $Y$ is compared to the more expensive store. \begin{tabular}{|l|l|}
\hline
\textbf{Store}& \textbf{Sale Price for Gadget $Y$} \\ \hline
Mega Deals & $12\%$ off the list price $\$78.50$, then an additional $\$5$ off \\ \hline
Quick Save & $30\%$ off the list price $\$78.50$ \\ \hline
\end{tabular}


What is the sum of the squares of integers that are both greater than 5 and less than 15?


Alice can buy pins, pans, and pots for $\$3$, $\$4$, and $\$9$ each, respectively. What is the largest number of pots she can purchase if she must buy at least one of each item and will spend exactly $\$100$?


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3 \cdot 5^6}$?


You recently bought more than 150 eggs. The eggs are sorted in containers that can store exactly 15 eggs. However, upon inspecting the containers, you realize that three containers each hold only 14 eggs, while all other containers hold 15 eggs. What is the smallest number of eggs you could have right now?


In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form.


Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 8$ and the diameter of the inscribed circle of triangle $OCF$ is 4, compute the product $(AB)(CD).$


How many distinct four-digit numbers are divisible by 5 and have 75 as their last two digits?


A hexagon is created by joining, in order, the points $(0,0)$, $(2,1)$, $(3,3)$, $(2,4)$, $(0,3)$, $(1,1)$, and back to $(0,0)$. The perimeter of this hexagon can be written in the form $p+q\sqrt{2}+r\sqrt{10}$, where $p$, $q$, and $r$ are integers. Find $p+q+r$.


The diagram shows a rectangle dissected into seven non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]draw((0,0)--(52,0)--(52,37)--(0,37)--(0,0));draw((15,0)--(15,15)--(0,15)); draw((15,22)--(52,22));draw((27,22)--(27,37));draw((20,15)--(20,37)); draw((28,15)--(28,22)--(20,22)); draw((15,15)--(15,17)--(28,17));[/asy]


A pyramid with a square base has a base edge of 20 cm and a height of 40 cm. Two smaller similar pyramids are cut away from the original pyramid: one has an altitude that is one-third the original, and another that is one-fifth the original altitude, stacked atop the first smaller pyramid. What is the volume of the remaining solid as a fraction of the volume of the original pyramid?


A sphere is inscribed in a cube. The edge of the cube is 10 inches. Calculate both the volume and the surface area of the sphere. Express your answer for the volume in terms of \(\pi\).


What is the 415th digit to the right of the decimal point when $\frac{13}{19}$ is expressed as a decimal?


Natalie writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them. How many integers are in Natalie's list?


What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side are the same color, if the tessellation consists of hexagons (similar to a honeycomb)?


A right circular cylinder with an equal diameter and height is inscribed in a right circular cone. The cone has a diameter of 15 and an altitude of 15. The axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


Find the sum of all integers $k$ such that $\binom{28}{4} + \binom{28}{5} = \binom{29}{k}$.


Compute $2031 \cdot 2032 \cdot 2033 \cdot 2034$ modulo 7.


A parabola with the equation \( y = 2x^2 + bx + c \) passes through the points \( (-2, -20) \) and \( (2, 24) \). What is the value of \( c \)?


The matrix for projecting onto a certain line $\ell$ in three-dimensional space, which passes through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5}
\begin{pmatrix}
\frac{1}{7} & -\frac{2}{7} & \frac{3}{7} \\
-\frac{2}{7} & \frac{4}{7} & -\frac{6}{7} \\
\frac{3}{7} & -\frac{6}{7} & \frac{9}{7}
\end{pmatrix}
\renewcommand{\arraystretch}{1}
.\]
Find the direction vector of line $\ell$. Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

Solution: \boxed{\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}}


Find the number of positive integers $n$, $1 \le n \le 500$, for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients.


A spherical stone was sitting in a pond when the pond froze. The stone was removed, leaving a hole $30$ cm across at the top and $10$ cm deep. What was the radius of the stone (in centimeters)?


If \( a \) and \( b \) are positive 4-digit integers such that \( \gcd(a, b) = 5 \), what is the smallest possible value for \( \mathop{\text{lcm}}[a, b] \)?


Thirteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Bella, starts with the ball, skips the next four girls, and throws to the sixth girl, who then skips the next four girls and continues the pattern. How many total throws are necessary for the ball to return to Bella?


A square is completely covered by a large circle and each corner of the square touches a smaller circle of radius \( r \). The side length of the square is 6 units. What is the radius \( R \) of the large circle?


A right circular cylinder is inscribed in a right circular cone with the cone's diameter being 16 and altitude being 20. The cylinder's diameter is equal to its height, and the axes of the cylinder and cone coincide. Determine the radius of the cylinder. Express your answer as a common fraction.


When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point?


A solid box is 18 cm by 12 cm by 9 cm. A new solid is formed by removing a cube 2 cm on a side from each corner of this box. What percent of the original volume is removed?


A line contains the points $(0, 5)$, $(7, k)$, and $(25, 2)$. What is the value of $k$?


Mrs. Garcia teaches a class of 150 students who all took a recent quiz. What is the maximum number of students who can receive extra credit if extra credit is given to those who score above the class mean?


Given that $f(x) = |x|^k$ where $k > 0$, what is the range of $f(x)$ on the interval $(-\infty, -1]$?

Solution: \boxed{[1, \infty)}


Find $47^{-1} \pmod{48}$, as a residue modulo 48. (Give an answer between 0 and 47, inclusive.)


What is the greatest common multiple of 10 and 15 that is less than 120?


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


Given the product $(2x^2 - 3x + 5)(9 - 3x)$ can be expressed in the form $ax^3 + bx^2 + cx + d$, calculate $27a + 9b + 3c + d$.


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Point $G$ is where diagonal $AC$ intersects $BD$. Find the area of quadrilateral $EFCD$ in square units.


A card is chosen at random from a modified deck of 54 cards (standard 52 cards plus 2 jokers), and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace or a joker?


Rationalize the denominator of \(\frac{35}{\sqrt[3]{35}}\).


A clock face is decorated with 12 equally spaced rays emanating from the center, each pointing to the hour marks. One of these rays points towards 12 o'clock. What is the measure, in degrees, of the smaller angle formed between the ray pointing towards 3 o'clock and the ray pointing towards 7 o'clock?

Solution: \boxed{120\text{ degrees}}


Let $m$ be the smallest positive, three-digit integer congruent to 3 (mod 7). Let $n$ be the smallest positive, four-digit integer congruent to 3 (mod 7). What is $n-m$?


I have 7 books, three of which are identical copies of the same math book, and the rest are different. How many ways can I arrange them on a shelf?


The polynomial \( Q(x) = 3x^3 + dx^2 + ex + f \) has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The \( y \)-intercept of \( y = Q(x) \) is 12. What is \( e \)?


Find $100^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


Let $x$ and $y$ be positive real numbers such that $x = y$. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 2 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 2 \right).\]


$30$ lemons are required to make $40$ gallons of lemonade. If this ratio is maintained, how many lemons are needed to make $10$ gallons of lemonade? Additionally, if $2$ pounds of sugar are used for every $5$ gallons of lemonade, how much sugar is required for $10$ gallons?


Multiply $555.55$ by $\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth.


Compute the sum of all the distinct roots of the equation $(3x + 4)(x^2 - 5x + 6) + (3x + 4)(2x - 7) = 0$.


Determine the 250th term of the increasing sequence of positive integers formed by omitting not only the perfect squares but also perfect cubes.


Find the sum of all integers $n$ such that $n^2 - 17n + 72$ is a perfect square, and $\frac{18}{n}$ is an integer.


Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row.


In $\Delta ABC$, $\overline{FG} \parallel \overline{AB}, CF = 5$ cm, $FA = 15$ cm, and $CG = 9$ cm. What is the number of centimeters in the length of $\overline{CB}$?

Solution: \boxed{36 \text{ cm}}


The equation $x^2 - (2A + 1)x + 3A = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist?

Solution: \boxed{1\text{ value}}


A whole number is said to be ''7-heavy'' if the remainder when the number is divided by 7 is greater than 3. What is the least four-digit 7-heavy whole number?


Calculate the area, in square units, of triangle \(ABC\) given the coordinates \(A(-2,2)\), \(B(5,2)\), and \(C(2,-1)\).

Solution: \boxed{10.5 \text{ square units}}


For how many integer values of $n$ between 1 and 524 inclusive does the decimal representation of $\frac{n}{525}$ terminate?


A sphere is inscribed in a right cuboid where the length of each dimension of the cuboid is 8 inches, 10 inches, and 10 inches respectively. What is the volume of the inscribed sphere? Express your answer in terms of $\pi$.


How many positive divisors of $252$ are not divisible by $3$?


A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 4 meters from its base. When Leah walks 3 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.6 meters tall. How many meters tall is the pole?

Solution: \boxed{6.4\text{ meters}}


What is the value of the following expression: $\frac{1}{3} - \frac{1}{6} + \frac{1}{12} - \frac{1}{24} + \frac{1}{48} - \frac{1}{96}$? Express your answer as a common fraction.


Given that
$$(x+y+z)(xy+xz+yz) = 16$$
and that
$$x^2(y+z) + y^2(x+z) + z^2(x+y) = 4$$
for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[az^2 + 2bz + c = 0.\]
Find the largest possible value of $|z|$.

Solution: \boxed{1 + \sqrt{2}}


How many digits are there in the base-8 representation of $1728$?


Two circles are drawn in a 15-inch by 18-inch rectangle. Each circle has a diameter of 5 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{\sqrt{269} \text{ inches}}


I want to create a license plate that is 4 characters long, where the first character is a letter, the second character is either a letter or a digit, the third character must be the same as the second, and the fourth character is a digit. How many ways are there to choose such a license plate?


How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable but the boxes are not?


A sports conference has expanded to 24 teams, divided into 3 divisions of 8 teams each. Calculate the total number of games in a complete season if each team plays every other team in its division three times and each team in the other two divisions two times.


In the coordinate plane, the curve $xy = 1$ intersects a circle at four points, three of which are $\left( 3, \frac{1}{3} \right),$ $\left( -4, -\frac{1}{4} \right),$ and $\left( \frac{1}{6}, 6 \right).$ Find the fourth point of intersection.

Solution: \boxed{\left( -\frac{1}{2}, -2 \right)}


The sum of the first and third of three consecutive odd integers is 156. What is the value of the second integer?


A right circular cone is inscribed in a right rectangular prism. The base of the prism measures $3a$ by $2a$ and the height of the prism (and cone) is $h$. What is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of $\pi$.


The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ with the first two terms being 1 and each subsequent term being the sum of the two preceding ones. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 9?


In the complex plane, plot the line segment with endpoints $-7 + 5i$ and $5 - 3i$. Calculate the complex number corresponding to the midpoint of this line segment.

Solution: \boxed{-1 + i}


Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$  If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{3}{\omega},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]


Two circles are drawn in a 15-inch by 18-inch rectangle. Each circle has a diameter of 8 inches. Assuming that the circles do not extend beyond the perimeter of the rectangle, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{\sqrt{149}\text{ inches}}


- Maria goes to the market to purchase oats and rice. The amount of oats she buys, in kilograms, is at least 4 kilograms more than one-third the amount of rice, and is no more than three times the amount of rice. Determine the minimum amount of rice Maria could purchase.


Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1+\omega+2\omega^2)^3 + (1 - 2\omega + \omega^2)^3.\]


Luca decides to scale up his pizza production and needs to prepare dough using 1200 mL of flour. If he uses 60 mL of milk for every 300 mL of flour, how many mL of milk does he need to use for this quantity of flour and what would be the required amount of milk if he decides to make two batches?