Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle. The triangle's altitude from $F$ to side $EG$ is 9 inches, and $EG = 12 \text{ inches}$. The length of segment $AB$ is one-third the length of segment $AD$. What is the area of rectangle $ABCD$?


Determine the residue of $-1277 \pmod{32}$. Your answer should be an integer in the range $0,1,2,\ldots,30,31$.


The angle XYZ is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 450 square centimeters. Additionally, the side length of square on XY is 5 cm less than side length of square on YZ. What is the number of square centimeters in the area of the largest square?

[asy]
draw((0,0)--(15,0)--(0,8)--cycle);
dot((0,0));
dot((15,0));
dot((0,8));
draw((0,0)--(0,8)--(-8,8)--(-8,0)--cycle);
draw((0,0)--(0,-15)--(15,-15)--(15,0));
draw((0,8)--(8,23)--(23,15)--(15,0)--cycle);
label("$X$",(0,8),NW);
label("$Y$",(0,0),SE);
label("$Z$",(15,0),E);
[/asy]


What is the minimum number of colors needed to shade the tessellation shown, such that no two adjacent tiles (tiles sharing a side) are the same color, if the tessellation consists of alternating squares and equilateral triangles?

[asy]
pair A=(0,0), B=(1,0), C=(0.5, sqrt(3)/2);
pair D=(1.5, sqrt(3)/2), E=(2,0), F=(2.5, sqrt(3)/2);
draw(A--B--C--cycle);
draw(B--D--C--cycle);
draw(B--E--D--cycle);
draw(E--F--D--cycle);
draw(shift((2,0))*(A--B--C--cycle));
draw(shift((2,0))*(B--D--C--cycle));
draw(shift((2,0))*(B--E--D--cycle));
draw(shift((2,0))*(E--F--D--cycle));
[/asy]


For how many integers \( n \) between 1 and 100 is the greatest common divisor (gcd) of 21 and \( n \) equal to 7?


What is the base ten equivalent of $23456_{7}$?


When a car's brakes are applied, it travels 12 feet less in each second than the previous second until it comes to a complete stop. A car goes 40 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops?


Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b) = 360$ and $\gcd(a,c) = 1170$. Determine the smallest possible value of $\gcd(b,c)$, given that $b$ is a multiple of 5 and $c$ is a multiple of 13.


Find the largest integer $n$ for which $18^n$ evenly divides $30!$.


A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Science. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen?


A number $x$ is equal to $5 \cdot 27 \cdot 64$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect fourth power?


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


If \( |x-5| = q \), where \( x < 5 \), then what is \( x+q \) in terms of \( q \)?


If 35,876,119 hot dogs are packaged in sets of 7, how many will be left over?


Let $P(x) = b_0 + b_1x + \dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \le b_i < 5$ for all $0 \le i \le n$.

Given that $P(\sqrt{5}) = 50 + 21\sqrt{5}$, compute $P(3)$.


Find the distance between the foci of the ellipse
\[\frac{x^2}{48} + \frac{y^2}{12} = 3.\]


Consider two solid spherical balls, one centered at $\left( 0, 0, 10 \right),$ with radius 3, and the other centered at $(0,0,2)$ with radius 4. How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls?


The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x$ into two quadrilaterals. What is the perimeter of one of these quadrilaterals divided by $a$? Express your answer in simplified radical form.

Solution: \boxed{4 + 2\sqrt{2}}


A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$, $S$, and the upper left corner of the square is left unshaded?

[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,4)--(2,0)--(4,2)--(0,4)--cycle,white,linewidth(1));
label("R",(2,0),S);
label("S",(4,2),E);
[/asy]


For how many different negative values of $x$ is $\sqrt{x + 200}$ a positive integer?


Thirty goblins and ten pixies are attending a Fantasy Creatures Ball. The pixies, due to a spell, can only shake hands with an even number of goblins. Goblins are friendly and shake hands with each other as well as all the pixies. How many handshakes occur during the ball?


Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{7} \begin{pmatrix} -5 & a \\ b & c \end{pmatrix}\]
and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$


Max has more than $\$2$ but less than $\$15$ worth of quarters. When he forms stacks of 4, he has 2 left. When he puts them in stacks of 7, he has 3 left. When he puts them in stacks of 9, he also has 2 left. How many quarters does Max have?


Triangle $PQR$ has $PQ=24$, $PR=26$, and $QR=30$. Points $M$ and $N$ are located on $\overline{PQ}$ and $\overline{PR}$, respectively, such that $\overline{MN}$ is parallel to $\overline{QR}$ and passes through the midpoint of the height from $P$ to $\overline{QR}$. Then $MN=\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Find $a+b$.


What is the 58th digit after the decimal point in the decimal representation of $\frac{1}{17}$?


In a new walking game variation, on move 1, I do nothing. For moves $n$ where $2 \le n \le 50$, I take one step forward if $n$ is prime and three steps backwards if $n$ is composite. After completing all 50 moves, I stop and walk back to my original starting point. How many steps long is my walk back?


A point has rectangular coordinates \((x, y, z)\) and spherical coordinates \(\left(3, \frac{3\pi}{4}, \frac{\pi}{6}\right)\). Find the spherical coordinates of the point with rectangular coordinates \((x, -y, z)\). Enter your answer in the form \((\rho, \theta, \phi)\), where \(\rho > 0\), \(0 \leq \theta < 2\pi\), and \(0 \leq \phi \leq \pi\).

Solution: \boxed{(3, \frac{5\pi}{4}, \frac{\pi}{6})}


For which integer $a$ does $x^2 - x + a$ divide $x^{15} + x + 100$?


The graphs of \( y = |x| \) and \( y = -x^2 - 5x - 4 \) are plotted. For each \( x \), a vertical line segment connecting these two graphs is drawn. Determine the smallest possible length of one of these vertical segments.


The graphs of a function \( g(x) = 2x + c \) and its inverse \( g^{-1}(x) \) intersect at the point \( (4, d) \). Given that \( c \) and \( d \) are both integers, what is the value of \( d \)?


Find the greatest common divisor of $9!$ and $(6!)^2.$


The infinite sequence $T=\{t_1, t_2, t_3, \ldots\}$ is defined by $t_1 = 3$, and $t_n = 3^{t_{n-1}}$ for each integer $n>1$. What is the remainder when $t_{50}$ is divided by $7$?


Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 5 by 6 by 7 units. Calculate the volume of this set, expressed in the form $\displaystyle \frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n$ and $p$ being relatively prime, and find the sum $m+n+p$.

Solution: \boxed{m+n+p = 1272 + 58 + 3 = 1333}


For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\]


An edge of a square can be expressed as $6x-27$ meters or as $30-2x$ meters. What is its area in square meters?

Solution: \boxed{248.0625 \text{ square meters}}


For how many positive integers \( n \geq 2 \) is \( 10101_n \) a prime number?


Let $\overline{AD},$ $\overline{BE},$ $\overline{CF}$ be the altitudes of acute triangle $ABC.$  If
\[7 \overrightarrow{AD} + 3 \overrightarrow{BE} + 5 \overrightarrow{CF} = \mathbf{0},\] then compute $\angle ACB,$ in degrees.


The sum of two positive integers $c$ and $d$ is 1023. What is the largest possible value of $\gcd(c, d)$?


There exists a scalar $j$ such that for any vectors $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ such that $\mathbf{u} - \mathbf{v} + \mathbf{w} = \mathbf{0},$ the equation
\[j (\mathbf{u} \times \mathbf{v}) + \mathbf{u} \times \mathbf{w} + \mathbf{w} \times \mathbf{v} = \mathbf{0}\]
holds. Find $j.$


Find the largest value of $n$ such that $6x^2 + nx + 144$ can be factored as the product of two linear factors with integer coefficients.


The projection of $\begin{pmatrix} 2 \\ 4 \\ z \end{pmatrix}$ onto $\begin{pmatrix} -2 \\ 6 \\ -2 \end{pmatrix}$ is
\[\frac{20}{28} \begin{pmatrix} -2 \\ 6 \\ -2 \end{pmatrix}.\]Find $z.$


What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect squares?


Let $a$ and $b$ be nonzero complex numbers such that $a^2 + ab + b^2 = 0.$  Evaluate
\[\frac{a^{15} + b^{15}}{(a + b)^{15}}.\]


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{u} = \begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} \times \mathbf{u}\] for all vectors $\mathbf{u}.$

Solution: \boxed{\begin{pmatrix} 0 & -6 & -1 \\ 6 & 0 & -3 \\ 1 & 3 & 0 \end{pmatrix}}


A line through the points $(1, -8)$ and $(k, 15)$ is parallel to the line $6x + 9y = -12$. What is the value of $k$?


Let $a, b, c$ be real numbers such that:
\begin{align*}
b+c & = 17, \\
c+a & = 18, \\
a+b & = 19.
\end{align*}
Find $\sqrt{abc(a+b+c)}$.


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these three numbers, $abc$, equals 8.


Let function $g(x)$ satisfy $g(1) = 1$ and
\[g(x + y) = 2^y g(x) + 3^x g(y)\] for all real numbers $x$ and $y$. Find the function $g(x)$.

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


What is the largest positive integer with only even digits that is less than $1,000$ and is a multiple of $9$?


For some positive integer $k$, when 120 is divided by $k^2$, the remainder is 12. What is the remainder when 160 is divided by $k$?


For how many positive integral values of $b$ is it true that $x = 3$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
3x > 4x - 4\\
4x - b > -8
\end{cases}
$$


What is the value of \( x \) if a cube's volume is \( 27x \) cubic units and its surface area is \( 3x \) square units?


How many positive factors of 1800 are perfect squares?


Let $a \diamond b = 3ab - a + b$. If $3 \diamond x = 24$, find the value of $x$.


Calculate the angle $\theta$ in degrees for the number expressed as
\[\text{cis } 55^\circ + \text{cis } 65^\circ + \text{cis } 75^\circ + \dots + \text{cis } 145^\circ\]
in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$.


Berengere and Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a pie, but neither of them has enough money alone. If the pie costs 10 euros and Emily has an American ten-dollar bill, how many euros does Berengere need to contribute to the cost of the pie if 1 euro = 1.2 USD?

Solution: \boxed{\frac{5}{3} \text{ euros}}


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 20 respectively. The height from $D$ to line $AB$ is 5. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.


If an arc of $60^{\circ}$ on circle $C$ has the same length as an arc of $40^{\circ}$ on circle $D$, what is the ratio of the area of circle $C$ to the area of circle $D$? Express your answer as a common fraction.


A point \((x,y)\) is randomly and uniformly chosen inside the square with vertices \((0,0)\), \((0,3)\), \((3,3)\), and \((3,0)\). What is the probability that \(x+y < 4\)?


How many positive divisors does \(8!\) have?


Find the values of $x$ and $y$ such that $\sqrt{4 - 5x + y} = 9$.

Solution: \boxed{y = 77 + 5x}


What is the positive difference between $\frac{8^2 + 8^2}{8}$ and $\frac{8^2 \div 8^2}{8}$?


Let \(a,\) \(b,\) \(c\) be distinct real numbers such that
\[\frac{a}{1 + b} = \frac{b}{1 + c} = \frac{c}{1 + a} = k.\] Find the product of all possible values of \(k.\)


Given that $-14$ is a solution to $x^2 + bx + 49 = 0$, what is the value of $b$?


What is the matrix \(\mathbf{N}\) that performs the transformation which sends square \(PQRS\) to square \(P'Q'R'S'\), where \(PQRS\) is rotated by 60 degrees counterclockwise and scaled by 2?

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


Determine the value of $T$, when $T = 2 - 4 + 6 - 8 + \ldots + 4018 - 4020$, modulo 4020.


Let $m$ and $n$ be consecutive odd numbers, with $n < m$. Determine the largest integer which divides all possible numbers of the form $m^2 - n^2$.


In a triangle $XYZ$, $\angle XYZ = \angle YXZ$. If $XZ=8$ and $YZ=11$, what is the perimeter of $\triangle XYZ$?


A circle passes through the point $(1,2),$ and is tangent to the parabola $y = x^2$ at $(3,9).$ Find the center of the circle.

Solution: \boxed{(-28.8, 14.3)}


Consider that a day has 75,600 seconds instead of 86,400. In how many ways can a day be divided into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers, and each period is evenly divisible into the day?


What is the period of $y = \sin 5x + \cos 3x$?


Find the sum of all positive integers that are divisors of each number in the list $$24, 48, -18, 108, 72.$$


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 4,$ and
\[\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.\] Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$


If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face-up integers is 17?


A quadratic expression is given as $x^2 + cx + \frac{1}{4}$, where $c$ is a specific negative number. Andrew rewrites this quadratic using the technique of completing the square, and he finds that it can be expressed in the form $(x+n)^2 + \frac{1}{16}$. Determine the value of $c$.


Sixty cards are placed into a box, each bearing a number from 1 to 12, with each number entered on five cards. Five cards are drawn from the box at random and without replacement. Let $p$ be the probability that all five cards bear the same number. Let $r$ be the probability that three of the cards bear a number $a$ and the other two bear a number $b$ that is not equal to $a$. What is the value of $r/p$?


John owns 16 pairs of shoes: 8 identical black pairs, 5 identical brown pairs, and 3 identical white pairs. If he picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


How many 12-digit positive integers have all digits either 1 or 2, and have at least one pair of consecutive 1's?


Find the value of $k$ such that $\frac{-15-\sqrt{165}}{8}$ is a root of $4x^2 + 15x + k$.


Calculate $7\cdot7! + 5\cdot5! + 6\cdot5!$.


Find the ratio of the area of $\triangle BDY$ to the area of $\triangle ADY$ in the diagram if $DY$ bisects $\angle ADB$. [asy]
import markers;

real t=30/(30+33);
pair A=(-16.17,0);
pair B=(9.83,0);
pair D=(0,26.35);
pair Y=t*A+(1-t)*B;

draw(D--A--B--D--Y);

label("$A$",A,SW);
label("$B$",B,E);
label("$D$",D,N);
label("$Y$",Y,NE);

label("$28$",.5*(B+A),S);
label("$30$",.5*(B+D),NE);
label("$33$",.5*(A+D),NW);
[/asy]


Evaluate \( \left| \frac{3}{4} - 3i \right| \).


If \( a, b, c \) are non-negative integers less than 7 such that
\begin{align*}
a + 3b + 2c &\equiv 3 \pmod 7, \\
2a + b + 3c &\equiv 2 \pmod 7, \\
3a + 2b + c &\equiv 1 \pmod 7,
\end{align*}
then determine the remainder when \( abc \) is divided by \( 7 \).


In the complex plane, plot the line segment with endpoints $-5 + 6i$ and $7 - 4i$. Find the complex number that is one-third of the way from $-5 + 6i$ to $7 - 4i$.

Solution: \boxed{-1 + \frac{8}{3}i}


If $x + \frac{1}{x} = \sqrt{2}$, then find $x^8$.


When $\sqrt[4]{2^6 \cdot 5^5}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?


Let $p=11$ and consider the set $\{1, 2, 3, \ldots, 10\}$. What is the sum of all integers in this set that are primitive roots $\pmod{11}$?


Two circles, each of radius $4$, are drawn with centers at $(20, 100)$ and $(25, 90)$. A line passing through $(20, 90)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line?


Utilizing the twelve-letter alphabet of the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, S, T, U, V), imagine the scenario for generating six-letter license plates. How many valid six-letter license plates can be formed that start with either A or E, end with R, cannot contain the letter V, and have no repeating letters?


In a right-angled triangle, one of the legs measures 24 units. If the sum of the squares of the three side lengths is 2500, what is the length of the hypotenuse?


An ordinary $8$-sided die (octahedral die) has a number on each face from $1$ to $8$. How many ways can I paint two faces of this die red, so that the numbers on the red faces don't add up to $9$?


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


In triangle $PQR$, $PQ = 24$ and $QR = 18$. Find the largest possible value of $\tan P$.


12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: He chooses a ball, adds 5 to the number of the ball, records the new number, and places the ball back in the bin. How many different lists are possible, considering the numbers continue from 1 after reaching 12 (i.e., ball 12 + 5 -> becomes 5)?


What is the smallest positive integer that satisfies the congruence $3x \equiv 14 \pmod{17}$?


Determine the domain of the function $$f(x) = \frac{1}{x+9} + \frac{1}{x^2+9} + \frac{1}{x^3+9} + \frac{1}{e^x + 9}.$$

Solution: \boxed{(-\infty, -9) \cup (-9, -\sqrt[3]{9}) \cup (-\sqrt[3]{9}, \infty)}


How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments on a 3x2 grid?

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


Sandy and Sam each selected a positive integer less than 250. Sandy's number is a multiple of 15, and Sam's number is a multiple of 20. What is the probability that they selected the same number? Express your answer as a common fraction.


If 1023 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?


How many different four-digit numbers can be formed by arranging the digits in 3003?


For how many digits $C$ does the positive three-digit number $2C4$ become a multiple of 5?


A map of the town where Annie, Barbara, and David live can be represented by the Cartesian plane. Annie is located at $(6,-20)$, Barbara is at $(1, 14)$, and David is at $(0, -6)$. They agree to meet at the closest point that is equidistant from all their locations before walking upwards together to get to Charlie's location at $\left(\frac{7}{2}, 2\right)$. How many units upward do Annie, Barbara, and David walk together to get to Charlie?


Simplify $\sqrt{32y} \cdot \sqrt{75z} \cdot \sqrt{14y}$. Express your answer in simplest radical form in terms of $y$ and $z$.


Find the largest prime divisor of 13! + 14! * 2.


Cindy wishes to arrange her coins into \( X \) piles, each containing the same number of coins, \( Y \). Each pile will have more than one coin, and no pile will have all the coins. If there are 15 possible values for \( Y \) given all of the restrictions, what is the smallest number of coins she could have?


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


Two right triangles share a side such that the common side AB has a length of 8 units, and both triangles ABC and ABD have respective heights from A of 8 units each. Calculate the area of triangle ABE where E is the midpoint of side CD and CD is parallel to AB. Assume that side AC = side BC.


A $\textit{palindrome}$ is a number which reads the same forward as backward. What is the least natural number that can be added to 56,789 to create a palindrome?


What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$?


The pages of a book are numbered consecutively from 1 to $n$. It was found that when all the page numbers were summed, one of the pages was accidentally counted twice, resulting in an incorrect sum of $2550$. Determine the page number that was counted twice.


Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$?


In triangle $PQR$, where $PQ = 6$, $QR = 10$, and $RP = 8$, find the incenter $I$ of the triangle using vector notation:
\[\overrightarrow{I} = x \overrightarrow{P} + y \overrightarrow{Q} + z \overrightarrow{R},\]
where $x, y, z$ are constants satisfying $x + y + z = 1$. Compute the ordered triple $(x, y, z)$.

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


According to a modified version of Moore's law, the number of transistors that can be placed inexpensively on a silicon chip triples every two years. In 1990, a typical CPU contained about $500,\!000$ transistors. According to this law, how many transistors did a typical CPU contain in the year 2010?

Solution: \boxed{29,\!524,\!500,\!000 \text{ transistors}}


A function $f$ is defined for all real numbers and satisfies $f(3+x)=f(3-x)$ and $f(8+x)=f(8-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1500\leq x \le 1500$?


Suppose $x$ and $y$ are positive real numbers such that $x^2 - xy + y^2 = 8$. Find the maximum possible value of $x^2 + xy + y^2$ which can be expressed in simplest form as $\frac{a + b \sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are positive integers. Determine $a + b + c + d$.


Determine the tens digit of $23^{2023}$.


What is the least integer value of $x$ such that  $\lvert3x+ 5\rvert\le 21$?


Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. After rationalizing, write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers. Calculate the value of $ABC$.


Let $A,$ $B,$ $C$ be the angles of a non-right triangle. Compute
\[\begin{vmatrix} \cot A & 1  & 1 \\ 1 & \cot B & 1 \\ 1 & 1 & \cot C \end{vmatrix}.\]


If $\mathbf{A}^{-1} = \begin{pmatrix} -5 & 3 \\ 1 & 3 \end{pmatrix},$ then find the inverse of $\mathbf{A}^3.$

Solution: \boxed{\begin{pmatrix} -146 & 66 \\ 22 & 30 \end{pmatrix}}


Hannah is interested in numbers that are divisible by 4 but dislikes any number ending in 0. How many different units digits are possible in numbers that Hannah likes?


Let \[H(x) = (3x^4 - 42x^3 + gx^2 + hx + i)(4x^4 - 96x^3 + jx^2 + kx + l),\]where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $H(x)$ is $\{2, 3, 4, 4, 5, 6, 6, 7\}.$ Find $H(7).$


Expand the following product with the inclusion of subtraction: $\frac{3}{4}\left(\frac{4}{x} - 5x^2 + \frac{20}{x^3}\right)$.

Solution: \boxed{\frac{3}{x} - \frac{15x^2}{4} + \frac{15}{x^3}}


A six-digit integer will be chosen at random from all possible positive six-digit integers. What is the probability that the number's units digit will be an even number? Express your answer as a common fraction.


Determine how many $x$-intercepts the graph of the cubic parabola $x = -3y^3 + 2y^2 - y + 2$ has.


Consider a grid where each block is 1 unit by 1 unit. You need to travel from point $C$ to point $D$ on this 7x7 grid. Point $C$ is at the bottom-left corner (0,0) and point $D$ is at (6,6). You are only allowed to move right or up and you must avoid passing through the midpoint block at (3,3). How many different paths are there from $C$ to $D$ under these conditions?


Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score?


The greatest common divisor of 36 and some number between 70 and 85 is 9. What is the number?


For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?


Compute the determinant of the sum of two matrices:
\[
\begin{vmatrix}
5 & -2 \\
3 & 4
\end{vmatrix}
+
\begin{vmatrix}
1 & 3 \\
-1 & 2
\end{vmatrix}.
\]


Three different integers are randomly chosen from the set $$\{ -5, -8, 7, 4, -2, 3 \}$$. What is the probability that their product is negative?


What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.


Determine the distance between the planes \(3x + 6y - 6z + 3 = 0\) and \(6x + 12y - 12z + 15 = 0.\)


Given that $2 - 3\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx - 37 = 0\] where \(a\) and \(b\) are rational numbers, compute \(a\).


Let \(A=(0,4)\) and \(B=(0,14)\). Points \(A'\) and \(B'\) are on the line \(y=x\), and the line segments \(\overline{AA'}\) and \(\overline{BB'}\) intersect at \(C=(3,6)\). Determine the length of \(\overline{A'B'}\).

Solution: \boxed{A'B' = \frac{90\sqrt{2}}{11}}


Let $C$ be the circle with equation $x^2 - 8y - 7 = -y^2 - 6x$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$?

Solution: \boxed{1 + 4\sqrt{2}}


Melinda will roll two standard six-sided dice and make a two-digit number with the two numbers she rolls. If she rolls a 5 and a 2, she can either form 52 or 25. What is the probability that she will be able to make an integer between 21 and 30, inclusive? Express your answer as a common fraction.


When two fair 10-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits), $a$, and $b$ are all divisible by 5?


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


Let $M$ denote the number of $8$-digit positive integers where the digits are in increasing order using only the digits from $1$ to $7$ (repetition allowed). Determine the remainder obtained when $M$ is divided by $1000$.


Solve the inequality
\[\frac{x^2 - 49}{x + 7} < 0.\]

Solution: \boxed{(-\infty, -7) \cup (-7, 7)}


What is the modulo $5$ remainder of the sum $(1 \times 3 + 2 \times 3 + 3 \times 3 + \ldots + 150 \times 3)$?


Let $a, b, c$ be non-zero distinct even integers, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1$. Find the smallest possible value of
\[|a + b \omega + c \omega^2|.\]


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


A line segment begins at $(3, -2)$. It is 15 units long and ends at the point $(1, y)$ where $y < 0$. What is the value of $y$?

Solution: \boxed{-2 - \sqrt{221}}


What is the midpoint of the segment with endpoints (-5, 8) and (11, -3)?

Solution: \boxed{(3, 2.5)}


What is the sum of the squares of the lengths of the $\textbf{medians}$ of a triangle whose sides are $13$, $13$, and $10$ units, and additionally, find the area of this triangle?


Consider the graph of the polar equation
\[ r = 4\cos\theta - 3\sin\theta. \]
Determine the area of the circle represented by this equation.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{z}$ where $z$ is an integer such that $0<z\le15$, then what is the largest possible value of $a+b+c$?


Say that an integer $B$ is delicious if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest delicious integer?


A class of 40 students receives a total of 200 pieces of candy. If the rule is that each student must take at least two pieces of candy, what is the greatest number of pieces one student could have taken?


For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?


What is the area, in square inches, of a right triangle with a 28-inch leg and a 30-inch hypotenuse?


If a complex number $z$ satisfies
\[
  z + z^{-1} = 2 \cos \left(\frac{\pi}{4}\right),
\]
what is the value of
\[
  z^6 + z^{-6} + (z^2 + z^{-2})^2 ?
\]


If the eight solutions of $x^8 = -256$ are written in the form $a+bi$, where $a$ and $b$ are real, find the product of those solutions with $a>0.$


What is the greatest common divisor of $765432$ and $654321$?


Let $a$ and $b$ be real numbers. Consider the following six statements:

1. $\frac{1}{a} > \frac{1}{b}$
2. $|a^2| < |b^2|$
3. $a > b$
4. $a > 0$
5. $b > 0$
6. $|a| > |b|$

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


What is the remainder when $1287 \cdot 1499$ is divided by $300$?


What is the greatest common divisor of 80, 180, and 450?


The integer $n$ is between $50$ and $150$ and is a multiple of $7$. When $n$ is divided by $9$, the remainder is $3$. Similarly, when $n$ is divided by $4$, the remainder is $3$. What is the value of $n$?


Suppose functions $g$ and $f$ have the properties that $g(x)=4f^{-1}(x) - 2$ and $f(x)=\frac{18}{x+2}$. For what value of $x$ does $g(x)=14$?


Determine how many positive divisors are shared by the numbers 9240 and 8820.


If $A\,568\,B72$ is divisible by $9$, where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B$?


If an arc of $60^{\circ}$ on circle $C$ has the same length as an arc of $40^{\circ}$ on circle $D$, what is the ratio of the area of circle $C$ to the area of circle $D$? Express your answer as a common fraction.


Suppose that a parabola has vertex $\left(\frac{1}{2},-\frac{5}{4}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a.$


If $3x - 2y = 7$ and $x + 3y = 8$, what is the value of $x$?


Let \( T \) be the set of 12-tuples \((b_0, b_1, \dots, b_{11})\), where each entry is either 0 or 1, so \( T \) contains \( 2^{12} \) 12-tuples. For each 12-tuple \( t = (b_0, b_1, \dots, b_{11}) \) in \( T \), let \( q_t(x) \) be the polynomial of degree at most 11 such that
\[ q_t(n) = b_n \] for \( 0 \le n \le 11 \).
Find:
\[ \sum_{t \in T} q_t(12) \].


The integer $x$ has 18 positive factors. The numbers 18 and 24 are factors of $x$. What is $x$?


Solution: \boxed{6x^{24} - 24x^9 - 48x^3 + 6}


Each row of a classroom seats either 6 or 7 students. There are fifty-three students in total who need to be seated with every seat occupied. How many rows seat exactly 7 students?


Compute the determinant of the matrix:
\[
\begin{vmatrix}
1 & \cos (a - b) & \sin a \\
\cos(a - b) & 1 & \sin b \\
\sin a & \sin b & 1
\end{vmatrix}.
\]


What is the value of $\dfrac{5 \times 7}{10}?$


Calculate the value of the alternating series sum:
\[\binom{100}{0} - \binom{100}{2} + \binom{100}{4} - \dots - \binom{100}{100}.\]


How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are "spacy", where a subset is labeled "spacy" if it contains no more than one out of any three consecutive integers?


The product $N$ of three positive integers is $8$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.

Solution: \boxed{272 + 160 + 128 = 560}


Compute \(\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}^{10}.\)

Solution: \boxed{\begin{pmatrix} 512 & 512 \\ 512 & 512 \end{pmatrix}}


Let $b_1 = 2, b_2 = 3, b_3 = 5$. For $n > 3$, let $b_n$ be the number of real numbers $x$ such that
\[ x^4 - 3b_{n-1}x^2 + b_{n-2}b_{n-3} = 0. \]
Compute the sum $b_1 + b_2 + b_3 + \dots + b_{20}$.


Let $N = 123456789101112\dots505152$ be the number obtained by writing out the integers from 1 to 52 consecutively. Compute the remainder when $N$ is divided by 45.


Two dice, one red and one blue, are rolled. The red die has 8 sides, and the blue die has 10 sides. What is the probability that the red die shows an odd number and the blue die shows a number that is either a perfect square or a prime number?


Find the number of real solutions of the equation
\[\frac{6x}{x^2 + x + 1} + \frac{7x}{x^2 - 7x + 1} = -1.\]


I have three apples, seven oranges, and four bananas. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples, oranges, and bananas are identical within their respective types. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)


The legs of a right triangle have lengths $\log_3 64$ and $\log_9 16$. If the length of the hypotenuse is $h$, compute $3^h$.


Let $x$ and $y$ be real numbers such that
\[
  -1 < \frac{y - x}{x + y} < 2.
\]
If $\frac{y}{x}$ is an integer, what is its value?


Let $M = 123456789101112\dots4849$ be the concatenation of numbers from 1 to 49. What is the remainder when $M$ is divided by $50$?


Find the number of solutions to
\[
\sin x = \left( \frac{1}{3} \right)^x
\]
on the interval $(0, 150 \pi)$.


Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + 2x + 3$. Find the sum of all possible values of $g(49)$.


What is the sum of the first fifteen positive multiples of $11$, excluding any multiples of $5$?


Alice throws a fair eight-sided die each morning. If Alice throws a perfect square number, she chooses porridge for breakfast. For any prime numbers, she chooses toast. If she throws an 8, she will throw the die again. How many times can Alice be expected to throw her die in a leap year?


At a sandwich shop, there are five types of bread, seven kinds of meat, and six kinds of cheese. A sandwich is made up of one type of bread, one type of meat, and one type of cheese. Turkey, salami, Swiss cheese, and rye bread are available at the shop. If a customer does not order a sandwich with a turkey/Swiss cheese combination nor a sandwich with rye bread/salami, how many different sandwiches can the customer order?


How many zeroes does $15!$ end with when $15!$ is written in base 12?


Let $G$ be the centroid of quadrilateral $ABCD$. If $GA^2 + GB^2 + GC^2 + GD^2 = 116$, find the sum $AB^2 + AC^2 + AD^2 + BC^2 + BD^2 + CD^2$.


Determine the coefficient of $x^3$ when the expression $2(x^2 - 2x^3 + x) + 4(x + 3x^3 - 2x^2 + 2x^5 + 2x^3) - 6(2 + x - 5x^3 - x^2)$ is simplified.


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?


Trapezoid $ABCD$ with bases $AB$ and $CD$ where $AB = 2CD$ is given, and the area of the trapezoid is $72$ square units. Points $M$ and $N$ are the midpoints of legs $AD$ and $BC$ respectively. Determine the area of triangle $AMN.$

Solution: \boxed{18 \text{ square units}}


Let \(p,\) \(q,\) and \(r\) be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if either \(x < -6\) or \(|x-30| \le 2.\) Given that \(p < q,\) find the value of \(p + 2q + 3r.\)


The values of \(a\), \(b\), \(c\), and \(d\) are 2, 3, 5, and 8, but not necessarily in that order. What is the smallest possible value of the sum of four products \(ab\), \(bc\), \(cd\), and \(da\)?


The first $20$ numbers of an arrangement are shown below. What would be the value of the $60^\text{th}$ number if the arrangement were continued?

$\bullet$ Row 1: $2,$ $2$
$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$
$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$
$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$


Let $T$ denote the sum of all four-digit positive integers where each digit is distinct and ranges from 1 to 9. Compute the remainder when $T$ is divided by $1000$.


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point (8, -15)? Additionally, calculate the distance from the origin to the midpoint of the segment connecting the origin and the point (8, -15).


Determine how many different positive integers leave a remainder of 7 when divided into 59.


Let $x,$ $y,$ $z$ be real numbers such that $x + 2y + 3z = 5,$ and $x \ge -1,$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of
\[\sqrt{x + 1} + \sqrt{2y + 4} + \sqrt{3z + 9}.\]


If $x$, $y$, and $z$ are positive integers satisfying $xy+z = yz+x = zx+y = 47$, what is the value of $x+y+z$?


The sum of two positive integers $a$ and $b$ is 1008. What is the largest possible value of $\gcd(a, b)$?


The points $A = (2, -5, 3),$ $B = (4, -9, 6),$ $C = (3, -4, 1),$ and $D = (5, -8, 4)$ in space form a flat quadrilateral. Find the area of this quadrilateral.


Find the largest constant \( m \) such that for any positive real numbers \( x, y, z, \) and \( w \)
\[
\sqrt{\frac{x}{y+z+x}} + \sqrt{\frac{y}{x+z+w}} + \sqrt{\frac{z}{x+y+w}} + \sqrt{\frac{w}{x+y+z}} > m.
\]


Let $x = {1 + \frac{\sqrt{3}}{1 + \frac{\sqrt{3}}{1 + ...}}}$. Compute $\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\frac{A+\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, determine $|A| + |B| + |C|$.


In a certain warehouse, there are $1200$ boxes, each containing $35$ books. Before repacking, Melvin discovers that $100$ books are damaged and discards them. He is then instructed to repack the remaining books so that there are $45$ books in each new box. After packing as many such boxes as possible, how many books does Melvin have left over?


At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize and the winner bowls #2. The loser of this game receives $3^{rd}$ prize, and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes?


Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{15625}}}}$$


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 8 and altitude 10, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


Calculate the sum of the first six terms of the geometric series $1 + \left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4 + \left(\frac{1}{4}\right)^5$. Express your answer as a common fraction.


A plane reflects over line $\ell,$ which passes through the origin. The reflection matrix is
\[\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix},$ where $a,$ and $b$ are integers, $a > 0,$ and $\gcd(|a|,|b|) = 1.$

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


The perimeter of a rectangle is 30 inches. One side of the rectangle is fixed at 7 inches. What is the number of square inches in the maximum possible area for this rectangle?


How many positive factors of 108 are perfect cubes?


Find all $x$ such that $\lfloor \lfloor 3x \rfloor - 1/3 \rfloor = \lfloor x + 3 \rfloor.$

Solution: \boxed{\left[\frac{5}{3}, \frac{7}{3}\right)}


Find both the arithmetic mean and the geometric mean of the prime numbers in this list: 10, 13, 19, 28, 31.


The pages of a book are numbered from 1 through $n$. By mistake, one of the page numbers was added twice, resulting in an incorrect sum of $2530$. Determine the page number that was added twice.


Let $p$, $q$, and $r$ be real numbers, and let $A$, $B$, $C$ be points such that the midpoint of $\overline{BC}$ is $(p,2,0)$, the midpoint of $\overline{AC}$ is $(0,q,2)$, and the midpoint of $\overline{AB}$ is $(0,0,r)$. Find
\[
\frac{AB^2 + AC^2 + BC^2}{p^2 + q^2 + r^2}.
\]


The average age of Amy's, Ben's, and Chris's ages is 12. Five years ago, Chris was twice the age Amy is now. In 2 years, Ben's age will be equal to half of Amy's age at that time. How many years old is Chris now?


Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 25.\]


Find the coefficient of the \(x^2\) term in the expansion of the product \((3x^2 + 4x + 5)(6x^2 + 7x + 8)\).


Simplify and rationalize the denominator: $$\frac{1}{1+ \frac{1}{\sqrt{5}+2}}.$$


The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd?


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point \((12, 5)\)?


Determine the sum of all positive integers $n$ that satisfy the equation $$\mathop{\text{lcm}}[n,120] = \gcd(n,120) + 300.$$


Let $\mathbf{B}$ be a matrix such that
\[\mathbf{B} \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 12 \\ -4 \end{pmatrix}.\]Find $\mathbf{B}^4 \begin{pmatrix} 3 \\ -1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 768 \\ -256 \end{pmatrix}}


Convex hexagon $PQRSTU$ has exactly two distinct side lengths. Side $PQ$ measures 7 units, and side $QR$ measures 8 units. The perimeter of hexagon $PQRSTU$ is 44 units. How many sides of hexagon $PQRSTU$ have measure 8 units?


A dog has discovered $521_{9}$ different ways to bury its bones for each of its nine adventures. How many ways are there in base 10?


Let \( p, q, r \) be three consecutive terms in an arithmetic sequence with positive values such that \( pqr = 216 \). Find the smallest possible value of \( q \).


Calculate the value of $n$ if
\[
(1 + \tan 3^\circ)(1 + \tan 6^\circ)(1 + \tan 9^\circ) \dotsm (1 + \tan 45^\circ) = 2^n.
\]


Determine the number of solutions to
\[3\cos^3 x - 7\cos^2 x + 3\cos x = 0\]
in the range \(0 \le x \le 2\pi\).


Consider a modified Lucas sequence where the first term is 2, the second term is 5, and each subsequent term is the sum of the two preceding terms. What is the remainder when the $75^{\text{th}}$ term of this modified sequence is divided by 7?


Call a $4$-digit number geometric if it has $4$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.


Suppose that \( p \) is a prime number and \( 2017_p + 405_p + 114_p + 206_p + 7_p = 253_p + 372_p + 452_p \). Determine how many possible values of \( p \) are there?


What is the value of $x$ if a cube's volume is $8x^2$ cubic units and its surface area is $4x$ square units?


Determine the inverse of the matrix
\[\begin{pmatrix} 7 & 5 \\ 3 & 2 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} -2 & 5 \\ 3 & -7 \end{pmatrix}}


The graph of function $y=g(x)$ is known to satisfy $g(x) > 0.3$ for all $x > 2$. The function $g(x)$ has the form $\frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers. Find $A+B+C$. Additional graphical information suggests that the vertical asymptotes are at $x = 1$ and $x = -3$.


Given that $x^2$ and $y$ vary inversely and $y = 20$ when $x = 3$, find the value of $x$ when $y$ is 5000.


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?


What is the domain of the function $f(x)=\log_2(\log_7(\log_5x))$?

Solution: \boxed{(5, \infty)}


Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them. If $a^2+b^2=2023c^2$, find the value of
\[\frac{\cot \gamma}{\cot \alpha+\cot \beta}.\]


Suppose the roots of $x^3 + 5x^2 + 6x - 7 = 0$ are $p$, $q$, and $r$. If the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find the value of $w$.


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 4 \sqrt{2}.$  Find $|\alpha|.$


Determine the number of roots of unity that are roots of $z^3 + pz + q = 0$, where $p$ and $q$ are real numbers.


A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one 7?


Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^4 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite".


If 600 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Below is the graph of \(y = a \sin (bx + c)\) for some positive constants \(a\), \(b\), and \(c\). The graph reaches its maximum at \(x = -\frac{\pi}{4}\) and has an amplitude of 3. Find the smallest possible value of \(c\).


Five of the seven numbers 1971, 2008, 2101, 2150, 2220, 2300, and 2350 have a mean (average) of 2164. What is the mean (average) of the other two numbers?


If 3913 were to be expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1 + 2 + 3 + \cdots + n$. Determine the greatest possible value of the greatest common divisor of $6T_n$ and $n+2$.


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?


John drove continuously from 8:15 a.m. until 2:05 p.m. of the same day and covered a distance of 210 miles. What was his average speed in miles per hour?


Let $ABCD$ be a tetrahedron with vertex $D$ such that edges $AD$, $BD$, and $CD$ are mutually perpendicular. Let the areas of triangles $ABD$, $BCD$, and $ACD$ be denoted by $p$, $q$, and $r$, respectively. Calculate the area of triangle $ABC$ in terms of $p$, $q$, and $r$.

Solution: \boxed{\sqrt{p^2 + q^2 + r^2}}


I have four distinct mystery novels, four distinct fantasy novels, four distinct biographies, and four distinct science fiction novels. I'm planning a retreat and want to take two books of different genres. How many possible pairs can I choose?


Let $x, y,$ and $z$ be positive real numbers such that $x + y + z = 5.$ Find the minimum value of:
\[\frac{9}{x} + \frac{4}{y} + \frac{25}{z}.\]


What integer $n$ satisfies $0 \leq n < 23$ and $$38635 \equiv n \pmod{23}~?$$


Alice took a mathematics test consisting of 90 questions: 15 arithmetic, 35 algebra, and 40 geometry problems. She answered $80\%$ of the arithmetic, $50\%$ of the algebra, and $70\%$ of the geometry problems correctly. However, she failed to pass the test as she got less than $65\%$ of the total problems right. How many more questions would she have needed to answer correctly to earn a $65\%$ passing grade?


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}.$ Find the vector $\mathbf{v}$ such that $\mathbf{v} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a})$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$.

Solution: \boxed{\begin{pmatrix} 0 \\ 5 \\ 5 \end{pmatrix}}


For all values of $x$ for which it is defined, let
\[ g(x) = \cot \frac{x}{3} - \cot \frac{x}{2}. \]
The function $g(x)$ can be rewritten in the form
\[ g(x) = \frac{\sin lx}{\sin \frac{x}{3} \sin \frac{x}{2}}. \]
Find the value of $l$.


Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x-1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?


Find the equation of the directrix of the parabola \( y = -3x^2 + 6x - 7 \).

Solution: \boxed{y = -\frac{47}{12}}


How many integers $n$ satisfy the condition $150 < n < 250$ and the condition $n$ has the same remainder whether it is divided by $7$ or by $9$?


A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors?


Simplify
\[\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.\]

Solution: \boxed{2 \csc x}


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


A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of members in this club.


Consider a function $f$ defined on a subset of real numbers such that for each real number $x$ in its domain, $1/x$ is also in the domain and satisfies the relation
\[
f(x) + f\left(\frac{1}{x}\right) = 3x.
\]
Determine the largest set of real numbers that can be in the domain of $f$.

(a) ${\{x\mid x\ne0\}}$

(b) ${\{x\mid x<0\}}$

(c) ${\{x\mid x>0\}}$

(d) ${\{x\mid x\ne-1\ \text{and}\ x\ne0\ \text{and}\ x\ne1\}}$

(e) ${\{-1,1\}}$

Solution: \boxed{\{-1, 1\}}


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

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


From a circular piece of paper with radius $BC$, Kim removes the unshaded sector shown. Using the larger shaded sector, she joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 10 centimeters and volume $250\pi$ cubic centimeters. What is the number of degrees in the measure of angle $ABC$ of the sector that is not used?


Calculate the remainder when $1 + 11 + 11^2 + \cdots + 11^{2500}$ is divided by $500$.


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5 and its thousands digit is odd? Express your answer as a common fraction.


In a group of 75 students in the drama club, 42 take mathematics, 35 take physics, and 25 students take both mathematics and physics. Additionally, 10 students take drama exclusively. How many drama club students take neither mathematics, physics, nor drama exclusively?


Consider resolving the rational function:
\[\frac{1}{(x-1)x(x+1)(x+2)(x+3)} = \frac{A}{x-1} + \frac{B}{x} + \frac{C}{x+1} + \frac{D}{x+2} + \frac{E}{x+3}\]
into partial fractions and find \( A + B + C + D + E \).


If 20 bahs are equal to 40 rahs, and 12 rahs are equal to 24 yahs, how many bahs are equal in value to 1200 yahs?


Let $x$, $y$, and $z$ be nonnegative real numbers such that $x + y + z = 7.$ Find the maximum value of \[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\]


Given that a sphere is perfectly inscribed within a cube whose edge length is 10 inches, calculate the volume and surface area of the sphere. Express your answers in terms of $\pi$.

Solution: \boxed{100 \pi}


Simplify the following expression:
\[ 3x + 4x^2 + 2 - (5 - 3x - 5x^2 + 2y). \]

Solution: \boxed{9x^2 + 6x - 2y - 3}


Compute $4 \begin{pmatrix} 3 \\ -6 \end{pmatrix} + 5 \begin{pmatrix} 2 \\ -9 \end{pmatrix} - \begin{pmatrix} -1 \\ 3 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 23 \\ -72 \end{pmatrix}}


What is the greatest integer less than 150 for which the greatest common factor of that integer and 24 is 3?


How many ways are there to put 6 balls into 4 boxes if the balls are not distinguishable and neither are the boxes, with at least one ball in each box?


Given that $\mathbf{v} \times \mathbf{u} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$ and $\mathbf{w} = 2\mathbf{v} + 3\mathbf{u}$, find $(2\mathbf{v} - \mathbf{u}) \times \mathbf{w}.$

Solution: \boxed{\begin{pmatrix} 24 \\ -8 \\ 16 \end{pmatrix}}


The integers $r$ and $k$ are randomly chosen such that $-5 < r < 10$ and $1 < k < 10$. Calculate the probability that the result of $r \div k$ is an integer. Express your answer as a simplified fraction.


Five standard, six-sided dice are to be rolled. If the product of their values turns out to be an odd number, what is the probability their sum is also odd? Express your answer as a common fraction.


A parabola with equation $y = x^2 + bx + c$ passes through the points $(-2, -16)$, $(2, 8)$, and $(4, 36)$. Determine the values of $b$ and $c$.

Solution: \boxed{b = 6, c = -8}


Each of $a_1,$ $a_2,$ $\dots,$ $a_{50}$ is equal to $1$ or $-1.$  Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} a_i a_j.\]


Determine the numerical value of $p$ such that
\[\frac{5}{a + b} = \frac{p}{a + c} = \frac{8}{c - b}.\]

Solution: \boxed{p = 13}


Calculate the angle $\theta$ for the complex number sum
\[ e^{11\pi i/120} + e^{31\pi i/120} + e^{51 \pi i/120} + e^{71\pi i /120} + e^{91 \pi i /120} \]
expressed in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$.


Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front (driver's seat and passenger seat) and the other two sit in the back. Either Mr. or Mrs. Lopez must sit in the driver's seat, and the other must also sit in the front passenger seat. How many seating arrangements are possible?


Consider the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$. This equation reduces to an equivalent form involving the product of cosines:
\[\cos ax \cos bx \cos cx = 0 \text{ for some positive integers } a, b, \text{ and } c.\]
Find $a + b + c$.


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 $b$?


Below is a portion of the graph of a function, $y=v(x)$, defined as $v(x) = 2\cos(x) - x$:

Consider the function within the interval $[-4, 4]$. What is the exact value of $v(-3) + v(-1) + v(1) + v(3)$?

Solution: \boxed{4\cos(3) + 4\cos(1)}


An environmental agency is expanding its pollution monitoring capacities and must hire new employees accordingly. The agency needs 95 new employees to monitor water pollution, 80 for air pollution, and an additional 50 to monitor land pollution. Of these employees, 30 will be responsible for both water and air pollution tasks, 20 for both air and land pollution, and 15 for both water and land pollution. In total, 10 employees will be tasked with monitoring all three types of pollution. Determine the minimum number of new hires needed for this expansion.


The number 96 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?


Let $\mathbf{N} = \begin{pmatrix} 3 & 0 \\ 2 & -4 \end{pmatrix}$. Find constants $c$ and $d$ such that
\[\mathbf{N}^{-1} = c \mathbf{N} + d \mathbf{I}.\]

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


Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 30$ and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = 3xyz.\] Find $\frac{x^3 + y^3 + z^3}{xyz}.$


A triangle is made of wood sticks of lengths 9, 16, and 18 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. Determine the length of the smallest piece that can be cut from each of the three sticks to achieve this.


The internal angles of a hexagon are in the ratio 2:3:3:4:5:6. What is the measure of the largest angle?


Points $A$, $B$, $C$, $D$, and $E$ are located on line segment $\overline{AB}$ such that $AB = 4AD = 8BC$, and $E$ divides segment $CD$ into two equal parts. What is the probability that a point selected at random on $\overline{AB}$ is between $C$ and $E$?


Find the integer \( n \), \( -150 < n < 150 \), such that \( \tan n^\circ = \tan 1340^\circ \).


Compute $4 \begin{pmatrix} 3 \\ -5 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -6 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 6 \\ -2 \end{pmatrix}}


Let \(m\) be the smallest integer whose cube root is of the form \(n + r\), where \(n\) is a positive integer and \(r\) is a positive real number less than \(1/500\). Find \(n\).


What is the sum of all two-digit positive integers whose squares end with the digits 25?


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 15,$ $b_{10} = 55,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n-1$ terms. Find $b_2$ and then find $b_{11}$.


Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $150$ with exactly three positive divisors. What is $m+n$?


Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,100\pi).$


One caterer charges a basic fee of \$150 plus \$17 per person. A second caterer charges a basic fee of \$250 but offers a discount: if more than 40 people are served, the charge per person is \$13; otherwise, it is \$15. What is the least number of people for which the second caterer is cheaper?


Find the modular inverse of \( 5 \), modulo \( 23 \).

Express your answer as an integer from \( 0 \) to \( 22 \), inclusive.


The function $g(x)$ satisfies $g(1) = 2$ and
\[g(x + y) = 4^y g(x) + 3^x g(y)\]
for all real numbers $x$ and $y.$ Find the function $g(x).$

Solution: \boxed{g(x) = 2(4^x - 3^x)}


Find the value of \( a \) such that the line \( y = 2x + a \) is tangent to the circle \( x^2 + y^2 = 9 \).

Solution: \boxed{a = \pm 3\sqrt{5}}


Stuart has drawn two concentric circles. He starts drawing chords $\overline{AB}$, $\overline{BC}, \ldots$ of the larger circle, each tangent to the smaller circle. If the interior angle $m\angle ABC = 60^\circ$, find how many segments Stuart will draw before returning to his starting point at $A$.


A number $x$ is equal to $5 \cdot 32 \cdot 45$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect square?


Determine how many five-letter words can be formed where the first and last letters are the same vowel (a, e, i, o, u), and the remaining three letters can be any letters from the alphabet.


Let \( P, Q, R, \) and \( S \) be positive real numbers such that
\begin{align*}
\log_{10} (PQ) + \log_{10} (PS) &= 3, \\
\log_{10} (QS) + \log_{10} (QR) &= 4, \\
\log_{10} (RP) + \log_{10} (RS) &= 5.
\end{align*}
Compute the value of the product \( PQRS \).


What is the units digit of the product of all the odd positive integers between 10 and 200?


The square root of $y$ is greater than 3 and less than 6. How many integer values of $y$ satisfy this condition?


A triangular region is enclosed by the lines with equations $y = 2x + 4$, $y = -3x + 9$, and $y = 2$. Additionally, a point $P$ lies on line $y = 2$ such that it is equidistant from the two vertical lines passing through the vertices of the triangle on this line. Find the area of the triangular region and the coordinates of point $P$. Express the area as a decimal to the nearest hundredth.


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{5}.$  Find $|\alpha|.$


Let \( g(x) = 9x + 7 \). Find the sum of all \( x \) that satisfy the equation \( g^{-1}(x) = g((3x)^{-1}) \).


Expand \( (1+0.1)^{500} \) using the binomial theorem without further simplification to express it as
\[ {500 \choose 0}(0.1)^0 + {500 \choose 1}(0.1)^1 + {500 \choose 2}(0.1)^2 + \cdots + {500 \choose 500}(0.1)^{500} = B_0 + B_1 + B_2 + \cdots + B_{500}, \]
where \( B_k = {500 \choose k}(0.1)^k \) for \( k = 0, 1, 2, \ldots, 500 \). For which \( k \) is \( B_k \) the largest?


Compute without using a calculator: $10! - 8! + 6!$.


Two cylindrical tanks have the same volume. The height of one tank is double the height of the other. If the radius of the taller tank is 10 units, what is the radius of the shorter tank? Provide your answer in simplest radical form.


How many three-digit perfect squares are divisible by $4$?


Let a $44$-gon named $Q_1$ be placed in the Cartesian plane where the sum of the $x$-coordinates of its vertices is $132$. Using the same steps of finding the midpoints to form $Q_2$ from $Q_1$, and $Q_3$ from $Q_2$, calculate the sum of the $x$-coordinates of the vertices of $Q_3$.


How many pairs of positive integers \((m,n)\) satisfy \( m^2 + 3n < 50?\)


Triangle $PQR$ is similar to triangle $XYZ$. If $PQ = 8$, $QR = 16$, and $YZ = 32$ units, what are the lengths of segments $XY$ and $XZ$?


What fraction of $\frac{1}{3}$ is $\frac{1}{8}$?


Let $\mathbf{M}$ be a matrix and let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 0 \\ 4 \end{pmatrix}.\]
Compute $\mathbf{M} (\mathbf{v} - 2 \mathbf{w}).$

Solution: \boxed{\begin{pmatrix} 5 \\ -7 \end{pmatrix}}


Given \( w \) and \( z \) are complex numbers such that \( |w+z| = 2 \) and \( |w^2+z^2| = 8 \), find the smallest possible value of \( |w^3+z^3| \).


Suppose that $6$ is a factor of $b$, $b$ is a divisor of $24$, and $b$ is positive. What is the number of possible values of $b$?


Calculate the remainder when the sum $3 + 9 + 15 + 21 + \cdots + 273 + 279$ is divided by $8$.


In $\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{MN}$ and $\overline{NO}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PMNO$?


In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$  If $\tan \angle APD = 2,$ find $AB.$


What is the smallest positive integer $n$ such that $3n \equiv 1367 \pmod{26}?$


In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. What is the perimeter of parallelogram $ADEF$?


Sandy's daughter has recently expanded her playhouse with an additional wall. Now, Sandy plans to cover the two 8-foot by 6-foot exterior walls and the two triangular roof faces (each having a base of 8 feet and a height of 5 feet) with a special siding. The siding is sold in 10-foot by 12-foot sections at a cost of \$30 each. How much will Sandy spend on the siding?


Convert the point $(3, -3\sqrt{3}, 2)$ 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{\left(6, \frac{5\pi}{3}, 2\right)}


Let $a, b, c, d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{1 + \omega},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1} + \frac{1}{d + 1}.
\]


The sequence 2,048,000; 1,024,000; 512,000; and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?


You have four small congruent square corrals, each with a side length of \( s \). These corrals are rearranged to form one larger square corral using the exact same amount of fencing. What is the ratio of the total area of the four small square corrals to the area of the new large square corral?


A rectangular cow pasture is enclosed on three sides by a fence, and the fourth side is part of the side of a barn that is $300$ feet long. The total amount of fencing available is $\$900$ at a cost of $\$6$ per foot. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture.


How many integers satisfy \( -6 \leq 3x + 2 \leq 9 \) excluding \( x = 0 \)?


A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 7 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\pi$. Find $W$, as a common fraction.

Solution: \boxed{\frac{1372}{3} - 32\sqrt{33}}


A function $g$ is defined by $g(z) = -i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Determine how many values of $z$ satisfy both $|z| = 3$ and $g(z) = z$.


Caroline is interested in writing all the possible rearrangements of the letters in her name "Caroline", and she writes 15 rearrangements per minute. How many hours will it take her to write all the possible rearrangements?

Solution: \boxed{44.8 \text{ hours}}


Parallelogram $EFGH$ has vertices $E(4,4)$, $F(-2,-2)$, $G(-8,-2)$, and $H(-2,4)$. If a point is selected at random from the region determined by the parallelogram and a line $y=-1$, what is the probability that the point lies between the $x$-axis and the line $y=-1$? Express your answer as a common fraction.


Let $g$ be a function from the set of positive integers to positive integers such that

(i) $g$ is increasing, i.e., $g(n + 1) > g(n)$ for all positive integers $n$,
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$, and
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(36)$.


Compute $5 \begin{pmatrix} 3 \\ -5 \end{pmatrix} - 3 \begin{pmatrix} -1 \\ 6 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 20 \\ -44 \end{pmatrix}}


A pirate is searching for buried treasure on 5 islands. On each island, there is a $\frac{1}{5}$ chance that the island has buried treasure and no traps, a $\frac{1}{5}$ chance that the island has traps but no treasure, and a $\frac{3}{5}$ chance that the island has neither traps nor treasure. What is the probability that while searching all 5 islands, the pirate will encounter exactly 2 islands with treasure, and exactly 2 islands with traps?


Find the non-zero, real value of $x$ that satisfies $(7x)^4 = (14x)^3$. Express your answer as a common fraction.


Determine the number of integer solutions for the equation \(2^{2x} - 5^{2y} = 175\).


Determine all real numbers $a$ such that the roots of the polynomial
$$2x^3 - 10x^2 + 30x + a$$ form an arithmetic progression and include complex roots.


Suppose the point $(3, 5)$ is on the graph of $y = 3f(x)$. Determine one point that must be on the graph of $y = \frac{1}{3}f^{-1}(x)$ and find the sum of the coordinates of that point.


A pirate is exploring 8 islands for buried treasure. On each island, there is a $\frac{1}{3}$ chance that the island contains treasure and no traps, a $\frac{1}{6}$ chance that the island contains traps but no treasure, and a $\frac{1}{2}$ chance that the island has neither traps nor treasure. What is the probability that while searching all 8 islands, the pirate will find exactly 4 islands with treasure, and none with traps?


If $a$, $b$, and $c$ are positive integers satisfying $ab+c = bc+a = ac+b = 47$, what is the value of $a+b+c$?


At Gourmet Burger Palace, a customer can choose from the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, sauce, and bacon. Customers can select any combination of these condiments. Additionally, they can choose between one, two, or three meat patties, and between a white bun or a wheat bun. How many different kinds of burgers can be ordered?


How many positive 3-digit numbers are multiples of 30, but not of 75?


What is the positive difference of the solutions of $\dfrac{r^2 - 5r - 12}{r + 5} = 3r + 10$?


Let the first term of a geometric sequence be $\frac{5}{3}$, and let the second term be $10$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by $2^8$?


Find the number of complex numbers $z$ such that $|z| = 2$ and
\[\left| \frac{z^2}{\overline{z}^2} + \frac{\overline{z}^2}{z^2} \right| = 2.\]


A marathon is $26$ miles and $400$ yards. One mile equals $1760$ yards.

Mark has run fifteen marathons in his life. If the total distance Mark covered in these marathons is $m$ miles and $y$ yards, where $0 \leq y < 1760$, what is the value of $y$?


What is the greatest common divisor of $987654$ and $876543$?


The solutions to $z^5 = 32i$ can be expressed in the form
\[
z_1 = r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 = r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 = r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 = r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 = r_5 (\cos \theta_5 + i \sin \theta_5),
\]
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$ Find $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5,$ in degrees.


Given that $0.75 = \frac{y}{256 + x}$, determine the smallest value of $x$, such that $x$ is a whole number and $y$ is an integer.

Solution: \boxed{x = 0}


We have a triangle $\triangle DEF$ where $DE = DF = 13$ and $EF = 14.$ Calculate the length of the median $DM$ from $D$ to $EF$.


Circle $O$ has a circumference of $18\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $AOB$ is $45^{\circ}$, what is the length, in inches, of segment $OB$?

Solution: \boxed{OB = 9 \text{ inches}}


An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$?


If $\mathbf{B} = \begin{pmatrix} x & y \\ z & w \end{pmatrix},$ then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} x & z \\ y & w \end{pmatrix}.\] Given that $\mathbf{B}^T = 2\mathbf{B}^{-1},$ find $x^2 + y^2 + z^2 + w^2.$


In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$.


Two different integers are randomly chosen from the set $$\{ -5, 0, -8, 7, 4, -2, -3, 1 \}.$$ What is the probability that their product is negative?


Let $a$ be the number of positive multiples of $12$ that are less than $60$. Let $b$ be the number of positive integers that are less than $60$, and a multiple of both $4$ and $6$. Compute $(a - b)^2$.


The bacteria in a flask double every 30 seconds. After four minutes, there are 524,288 bacteria in the flask. How many were in the flask at the beginning of the experiment?


Compute the determinant of the following matrix:
\[
\begin{vmatrix}
1 & \sin (a - b) & \sin a \\
\sin(a - b) & 1 & \sin b \\
\sin a & \sin b & 1
\end{vmatrix}.
\]


How many positive factors of 1800 are perfect squares?


Define a function \(g(x)\) such that \(g(11) = 24\), and if there exists an integer \(a\) such that \(g(a) = b\), then \(g(b)\) is defined and
\[ g(b) = 3b + 1 \] if \(b\) is odd
\[ g(b) = \frac{b}{2} \] if \(b\) is even.
What is the smallest possible number of integers in the domain of \(g\)?


Express $\frac{47}{2^3\cdot5^4}$ as a terminating decimal.


Suppose that the roots of $x^3 + 4x^2 + 5x - 14 = 0$ are $\alpha$, $\beta$, and $\gamma$. Assume that the roots of another polynomial $x^3 + px^2 + qx + r = 0$ are $\alpha+\beta$, $\beta+\gamma$, and $\gamma+\alpha$. Find the value of $r$.


In Mr. Greene's first-grade class, there are 36 students. If there are three times as many students with red hair as with green eyes, 12 students have red hair and green eyes, and 4 students have neither red hair nor green eyes, how many students have green eyes?


Suppose $x$, $y$, $z$, and $w$ are integers satisfying:
1. $x - y + z = 7$,
2. $y - z + w = 8$,
3. $z - w + x = 4$,
4. $w - x + y = 3$.
What is the value of $x + y + z + w$?


Consider two lines parameterized as follows:
\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix} \]
and
\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} + u \begin{pmatrix} 1 \\ -1 \end{pmatrix} \]
Determine the point where these two lines intersect.

Solution: \boxed{\begin{pmatrix} 0 \\ 2 \end{pmatrix}}


Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$, provided the $\clubsuit$ is not a King?


A right circular cone sits on a table, pointing upwards. The cross-section triangle, perpendicular to the base, has a vertex angle of 45 degrees. The diameter of the cone's base is $16$ inches. A cylinder is also placed co-axially inside the cone, its height equal to the height of the cone and its diameter equal to half of the cone's base diameter. A sphere is placed inside the cylinder so that it is tangent to the sides of the cylinder and to the cone's sides. What is the volume, in cubic inches, of the sphere? Express your answer in terms of $\pi$.


Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $400$. What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$?

$\bullet$ Sequence $A:$ $3,$ $6,$ $12,$ $24,$ $48,$ $\ldots$

$\bullet$ Sequence $B:$ $30,$ $60,$ $90,$ $120,$ $150,$ $\ldots$


If four standard, 6-faced dice are rolled, what is the probability that the sum of the face-up integers is 22?


Let \( r_1, r_2, \) and \( r_3 \) be the roots of
\[ x^3 - 5x^2 + 12 = 0. \]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3 \).

Solution: \boxed{x^3 - 15x^2 + 324}


A square carpet of side length 12 feet contains one large shaded square and four smaller, congruent shaded squares, as illustrated. The ratios $12:\text{S}$ and $\text{S}:\text{T}$ are both equal to 4. $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares. Calculate the total shaded area.

Solution: \boxed{11.25 \text{ square feet}}


What is the least five-digit positive integer which is congruent to 7 (mod 18)?


As $t$ takes on all real values, the set of points $(x, y)$ defined by
\[
x = t^2 - 4, \quad y = t^3 - 6t + 3
\]
forms a curve. Compute the ordered pair $(x, y)$ where this curve crosses itself.

Solution: \boxed{(2, 3)}


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 $k$?


When the base-16 number $A3F52_{16}$ is written in base 2, how many base-2 digits (bits) does it have?


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


Let $a$ and $b$ be positive real numbers. Find the minimum value of
\[ a^4 + b^4 + \frac{1}{(a + b)^4}. \]


Six of the eight numbers 1871, 2011, 2059, 2084, 2113, 2167, 2198, and 2210 have a mean (average) of 2100. What is the mean (average) of the other two numbers?


Let $k, a_2, a_3$ and $k, b_2, b_3$ be nonconstant geometric sequences with different common ratios. If \[ a_3^2 - b_3^2 = 3(a_2^2 - b_2^2), \] then what is the sum of the common ratios of the two sequences?


The square root of $3x$ is greater than 4 and less than 5. How many integer values of $x$ satisfy this condition?


A right circular cone with base radius $r$ and height $h$ rolls on its side on a flat table. As it rolls without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex of the cone touches the table. The cone returns to its original position after making $20$ complete rotations. Determine the value of $h/r$ in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.


Let $x = 1 + \frac{\sqrt{3}}{1 + \frac{\sqrt{3}}{1 + \cdots}}$. Find the value of $\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\frac{A + \sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, find $|A|+|B|+|C|$.


Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Find the number of such triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find the remainder when this number is divided by $1000$.


How many positive perfect squares less than $10^8$ are multiples of 210?


Find all values of $x$ which satisfy the inequalities:
$$\frac{x}{4} \leq 3 + x \quad \text{and} \quad 3+x \leq -3(1+x).$$
Express your answer in interval notation and simplify any fractions.

Solution: \boxed{[-4, -\frac{3}{2}]}


If \(\tan \gamma = 5\) and \(\tan \beta = 3,\) find \(\tan(\gamma + \beta)\) and \(\tan(\gamma - \beta)\).


The squares of a 9x9 chessboard are labelled with numbers by the following rule: the square in row \( i \) and column \( j \) is labelled \( \frac{1}{i+9-j} \), where \( i, j \) are from 1 to 9. Nine squares are to be chosen, such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of the nine chosen squares.


Determine the value of $b$ such that the remainder is constant when $8x^3 + bx^2 - 9x + 10$ is divided by $3x^2 - 2x + 1$.


Determine $h(x)$, ordered by terms of decreasing degree, if \[4x^4 + 11x^3 + h(x) = 10x^3 - x^2 + 4x - 7.\]

Solution: \boxed{h(x) = -4x^4 - x^3 - x^2 + 4x - 7}


What is the sum of $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5$?


Let $q(x) = x^5 - 5x^2 + 4$. Find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.


Find the distance between the points $(3,5)$ and $(-2,0)$. Check if the points $(3,5)$, $(-2,0)$, and $(1,-5)$ are collinear. If not, calculate the perimeter of the triangle they form.

Solution: \boxed{5\sqrt{2} + 2\sqrt{26} + \sqrt{34}}


A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\frac{1}{4}$ and on region $B$ is $\frac{1}{3}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction.


Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*}
x + y + z &= 5, \\
x^2 + y^2 + z^2 &= 11.
\end{align*}
Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$


Find the integer \( n, 0 \le n \le 360, \) such that \( \cos n^\circ = \cos 1234^\circ. \)

Solution: \boxed{n = 154}


Assume that $h$ and $k$ are functions for which $h^{-1}(k(x)) = 3x - 4$. Find $k^{-1}(h(5))$.


Find the greatest common divisor of $765432$ and $654321$.


What is the remainder when the sum $3 + 9 + 15 + 21 + \cdots + 297 + 303$ is divided by $6$?


How many values of $x$, $-25<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1?$  (Note: $x$ is measured in radians.)


According to the table below, how many dollars are in the median value of the 74 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$140{,}000$\\\hline
Senior Vice-President&4&$\$95{,}000$\\\hline
Manager&12&$\$80{,}000$\\\hline
Senior Consultant&8&$\$55{,}000$\\\hline
Consultant&49&$\$28{,}000$\\\hline
\end{tabular}


A circular pizza with a diameter of $18\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that can be drawn in one of these pieces. What is $m^2$?


There exist constants \( b_1, b_2, b_3, b_4, b_5, b_6 \) such that
\[
\cos^6 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta
\]
for all angles \(\theta\). Find \(b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2\).


Let $a$, $b$ be nonzero real numbers. Calculate the maximum and minimum values of
\[\frac{3|a + b|}{|a| + |b|},\]
and find $M - m$ where $m$ and $M$ respectively represent the minimum and maximum values of the above expression.


A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct license plates are possible if the digits and the letter block can appear in any order?


In the diagram, triangles $PQR$ and $QRS$ are isosceles. The perimeter of $\triangle QRS$ is $24$, the perimeter of $\triangle PQR$ is $23$, and the length of $QR$ is $10$. Determine the length of $PQ$.


In triangle $PQR,$ $p = 8,$ $q = 10,$ and $r = 6.$ Let $J$ be the incenter of the triangle.

Then
\[\overrightarrow{J} = u \overrightarrow{P} + v \overrightarrow{Q} + w \overrightarrow{R},\]where $u,$ $v,$ and $w$ are constants such that $u + v + w = 1.$  Determine the ordered triple $(u,v,w).$

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


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 6 cards?


Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}^4 = \mathbf{0}$. Determine the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite".


What is the sum of the $x$- and $y$-coordinates of all points that are $6$ units from the line $y=13$ and $12$ units from the point $(7,13)$?


Triangle $PQR$ is an isosceles triangle with side lengths of 26, 26, and 30 centimeters. What is the area of triangle $PQR$, in square centimeters?

Solution: \boxed{15\sqrt{451} \text{ square centimeters}}


The point $(2,2,2)$ is rotated $90^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $90^\circ$ about the $y$-axis, reflected through the $xz$-plane, and finally reflected through the $xy$-plane. Find the coordinates of the new point.

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


Factor the following expression: $75x^2 + 50x$.

Solution: \boxed{25x(3x + 2)}


Compute without using a calculator: $10!-9!$


A right triangle has legs measuring 24 cm and 32 cm. What is the length of the hypotenuse, and what is the perimeter of the triangle in centimeters?


The side length of a regular hexagon is 10 cm. Calculate the area of a trapezoidal region created by connecting opposite vertices of the hexagon. Express your answer in simplest radical form.

[asy]
size(120);
pair A,B,C,D,E,F;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300);

fill(A--C--E--cycle,heavycyan);

draw(A--B--C--D--E--F--A);
[/asy]

Solution: \boxed{100\sqrt{3} \text{ cm}^2}


Find the maximum number of real roots to a polynomial of the form
\[x^n + x^{n - 1} + \dots + x + c = 0,\]where $n$ is a positive integer, and $c$ is a real number.


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


Let $f(x) = x^2 + 2x$. Find the number of distinct real numbers $c$ such that $f(f(f(f(c)))) = 0$.


For how many integers \( n \) between 1 and 50 is the greatest common divisor of 21 and \( n \) equal to 3?


A jovial juggler is juggling balls which, when shot in the air, describe a path on the ground represented by the equation \( x^2 + y^2 + 5 = 2x + 4y \). What is the radius of the path described by one of the balls?


Simplify
\[\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.\]

Solution: \boxed{2 \csc x}


Eight people can paint three houses in twelve hours. How many hours would it take nine people to paint four houses, assuming everyone paints at the same rate?


Suppose $33 \cdot 77 \cdot 99 \equiv m \pmod{30}$, where $0 \le m < 30$.

What is the value of $m$?


In triangle $XYZ$, altitudes $XP$, $YQ$, and $ZR$ intersect at the orthocenter $H$. If $\angle XYZ = 37^\circ$ and $\angle XZY = 53^\circ$, find the measure of $\angle YHZ$, in degrees.


Let $a$ and $b$ be positive integers such that $\gcd(a,b)$ is divisible by exactly $10$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $35$ distinct primes. If $a$ has fewer distinct prime factors than $b$, determine the maximum number of distinct prime factors $a$ can have.


Determine the number of ways to arrange the letters of the word TOOTT.


What is the smallest positive integer \(n\) such that \(7875\) is a factor of \(n!\)?


Determine the range of the function
\[ g(A) = \frac{\cos A (2 \sin^2 A + 3 \sin^4 A + 2 \cos^2 A + \cos^2 A \sin^2 A)}{\cot A (\csc A - \cos A \cot A)} \]
for \( A \neq n\pi \) where \( n \) is any integer.

Solution: \boxed{[2, 5]}


What is the greatest prime factor of 899?


The quadratic $x^2 + 900x + 1800$ can be written in the form $(x+d)^2 + e$, where $d$ and $e$ are constants. What is $\frac{e}{d}$?


"Modulo $m$ graph paper" for $m=17$ consists of a grid of $17^2$ points. Each point $(x,y)$ represents pairs of integer residues where $0\le x, y <17$. On this graph, we want to plot the congruence $$7x\equiv 3y + 2 \pmod{17}$$.
Find the value of $x_0 + y_0$, where $(x_0,0)$ and $(0,y_0)$ are, respectively, the $x$-intercept and $y$-intercept of the graph.


Find the smallest solution to the equation
\[
\frac{1}{x-3} + \frac{1}{x-5} = \frac{4}{x-4}.
\]

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


For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{2940}$ terminate?


Given that $x$ is a multiple of $54321$, what is the greatest common divisor of $f(x) = (3x+4)(8x+5)(15x+11)(x+14)$ and $x$?


Four points $B,$ $A,$ $E,$ and $L$ are on a straight line. The point $G$ is off the line making $\angle BAG = 130^\circ$ and $\angle GEL = 70^\circ.$ If the reflex angle at $G$ is $x^\circ,$ then what does $x$ equal?


Consider a modified deck of 60 cards, which includes an additional suit called $\star$ with 8 ranks (1, 2, 3, 4, 5, 6, 7, 8) along with the standard four suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, $\clubsuit$) each still containing 13 ranks. The deck is randomly arranged. What is the probability that the top card is a King from any suit?


Chris will roll two standard six-sided dice and make a two-digit number with the two numbers he rolls. For example, if he rolls a 5 and a 2, he can either form 52 or 25. What is the probability that he will be able to make an integer between 50 and 60, inclusive? Express your answer as a common fraction.


Calculate the area of the trapezoid EFGH with vertices E(1, 1), F(1, -3), G(6, 1), and H(6, 7).

Solution: \boxed{25 \text{ square units}}


Marty wants to paint a box with one of five different colors: blue, green, yellow, black, or white. He can use either a brush, a roller, a sponge, or a spray. However, if he chooses white paint, he cannot use a spray due to restrictions in paint properties. How many different combinations of color and painting method can Marty choose?


Let $x$, $y$, and $k$ be real numbers with $k > 0$. Let $m$ and $M$ be the minimum value and maximum value of
\[\frac{|kx + y|}{|x| + |y|},\]
respectively. Find $M - m$.


Calculate the number of ways to arrange the letters in the word MISSISSIPPI that start with the letter P.


Chewbacca has 30 pieces of cherry gum and 40 pieces of grape gum. Each complete pack contains $x$ pieces of gum. If Chewbacca loses 2 packs of cherry gum, then the ratio of cherry gum to grape gum will be the same as if he finds 4 packs of grape gum. Determine $x$.


Jenny now decides to eat 25% of the jellybeans in her jar at the beginning of each day. At the end of the third day, 27 jellybeans remained in the jar. How many jellybeans did Jenny originally have in her jar?


Twelve points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the twelve points as vertices?


Let $\mathbf{B} = \begin{pmatrix} 4 & 1 \\ 0 & 3 \end{pmatrix}.$  Find $\mathbf{B}^{15} - 3 \mathbf{B}^{14}$.

Solution: \boxed{\begin{pmatrix} 4^{14} & 4^{14} \\ 0 & 0 \end{pmatrix}}


Determine the number of intersections for the graphs of $r = 5 \cos \theta$ and $r = 10 \sin \theta$.


How many even perfect square factors does \(2^6 \cdot 5^3 \cdot 7^8\) have?


Suppose we need to divide 12 dogs into three groups, where one group contains 4 dogs, another contains 6 dogs, and the last contains 2 dogs. How many ways can we form the groups so that Rover is in the 4-dog group and Spot is in the 6-dog group?


Given that $-1 - 3\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx + 15 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Let $x,$ $y,$ $z$ be nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$  Find all possible values of
\[
\frac{x^7 + y^7 + z^7}{xyz (xy + xz + yz)}.
\]


A rectangular park measures $30$ meters by $60$ meters. A landscaper has 2400 meters of fencing that can be used for internal fencing to create congruent, rectangular flower beds. Each flower bed must measure $5$ meters by $3$ meters, and the entire park must be partitioned. The sides of the rectangles must be parallel to the edges of the park. What is the largest number of flower beds into which the park can be partitioned using all or some of the available fencing?


Define operation $X \spadesuit Y = 4X - 3Y + 7$. What is the value of $X$ if $X \spadesuit 5 = 23$?


The eight digits in Jane's phone number and the four digits in her office number have the same sum. The digits in her office number are distinct, and her phone number is 539-8271. What is the largest possible value of Jane's office number?


Consider the line parameterized by
\[
\begin{align*}
x&= 5t + 3,\\
y&= 2t + 4.
\end{align*}
\]
Find a vector $\begin{pmatrix} a \\ b \end{pmatrix}$ pointing from point $(1,1)$ to this line that is parallel to $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{37}{4} - 1 \\ \frac{13}{2} - 1 \end{pmatrix} = \begin{pmatrix} \frac{33}{4} \\ \frac{11}{2} \end{pmatrix}}


Two boxes each contain black and white marbles, with a total of 36 marbles between them. One box contains only black marbles. One marble is taken from each box randomly. The probability that both marbles are black is $\frac{18}{25}$. Calculate the probability that both marbles are white, given in the form $m/n$ where $m$ and $n$ are coprime integers. What is $m + n$?


If a 4'' by 4'' square is added at each successive stage, what will be the area of the rectangle at Stage 8, in square inches?


Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(400) = 4$, what is the value of $f(800)$?


A collection of six positive integers has a mean of 4.5, a unique mode of 4, and a median of 5. If a 9 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.


If the least common multiple of two 5-digit integers has 8 digits, then their greatest common divisor has at most how many digits?


The same amount of steel used to create four solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball?


Medians are drawn from point $A$ and point $B$ in a right triangle to divide segments $\overline{BC}$ and $\overline{AC}$ in half, respectively. The lengths of the medians are 5 and $3\sqrt{5}$ units, respectively. How many units are in the length of segment $\overline{AB}$?


What is the smallest positive integer that satisfies the following conditions?

a) When divided by 5, the remainder is 4.

b) When divided by 7, the remainder is 6.

c) When divided by 9, the remainder is 8.

d) When divided by 11, the remainder is 10.


Let \[f(n) = \left\{
\begin{array}{cl}
n^2-2 & \text{ if }n<0, \\
2n-20 & \text{ if }n \geq 0.
\end{array}
\right.\]Determine the positive difference between the two values of \( a \) that satisfy the equation \( f(-3) + f(3) + f(a) = 0 \).


Let vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be non-parallel and nonzero such that
\[
(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = -\frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.
\]
Let \(\theta\) be the angle between \(\mathbf{b}\) and \(\mathbf{c}\). Determine \(\sin \theta\).


Let $g(m)$ be a function that, given an integer $m$, returns an integer $p$, where $p$ is the smallest possible integer such that $p!$ is divisible by $m$. Given that $m$ is a multiple of 21, what is the smallest value of $m$ such that $g(m) > 21$?

Solution: \boxed{m = 483}


Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a 50% chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n$.


In the row of Pascal's triangle that starts with 1 and then 14, what is the fifth number?


For how many positive integers $n$ does the sum $1+2+\cdots+n$ evenly divide $10n$?


The 5-digit number $43\,56\square$ is a multiple of 6. Which digit is represented by $\square$?


A right triangle has legs measuring 24 inches and 25 inches. First, calculate the length of the hypotenuse, and then find the perimeter of the triangle.

Solution: \boxed{49 + \sqrt{1201}}


If $x$ is a positive multiple of 9, and $x^2 > 200$, but $x < \sqrt{x^2 - 144} \cdot 5$, what is $x$?


There are four complex numbers \(z\) such that
\[z \overline{z}^3 + \overline{z} z^3 + 100 = 450,\]
and both the real and imaginary parts of \(z\) are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four complex numbers as vertices.


Suppose \( a \) is inversely proportional to \( b \). Let \( a_1, a_2 \) be two nonzero values of \( a \) such that \( \frac{a_1}{a_2} = \frac{3}{4} \). Let the corresponding \( b \) values be \( b_1, b_2 \). If \( b_1, b_2 \) are nonzero, find the value of \( \frac{b_1}{b_2} \).


Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle.


Let $\mathbf{a} = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$

Solution: \boxed{\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}}


How many divisors of 81 are also divisors of 90? (Recall that divisors may be positive or negative.)


The number of degrees in the measures of the interior angles of a convex hexagon are six consecutive integers. What is the number of degrees in the largest angle?


What is $3^{18} \div 27^2$? Write your answer as an integer.


For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{2100}$ terminate?


If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row of the board contains no more than one pawn?


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(x^2 + yg(z)) = xg(x) + zg(y) \] for all real numbers \( x, y, \) and \( z \).

Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \).


Find the area of the triangle bounded by the $y$-axis and the lines $y-4x=-2$ and $2y+x=12$.


On a $9\times9$ checkerboard, the ten horizontal and ten vertical lines define $r$ rectangles, of which $s$ are squares. The fraction $s/r$ can be expressed in the reduced form $m/n$. Determine $m + n$.


Alice has exactly enough paint to paint the surface of a rectangular prism whose dimensions are 4 by 2 by 1. This amount of paint is also exactly enough to cover the surface of a sphere. If the volume of this sphere is $\frac{L \sqrt{2}}{\sqrt{\pi}}$, then what is $L$?

Solution: \boxed{L = \frac{14 \sqrt{14}}{3}}


Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD$. If $\angle BAC = 30^\circ$, then what is $\angle BDC$?


A freight train travels at a speed of 40 miles per hour initially but decelerates at a constant rate and stops after travelling for 2 hours. How far does the train travel before it stops?

Solution: \boxed{s = 40}


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

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


In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?

[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P, S);
draw(X--P--Y--Z--cycle);
[/asy]


A math conference is organizing a lecture series with eight different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture so that Dr. Smith must be scheduled at some time after Dr. Jones, and in addition, Dr. Lee's lecture depends on Dr. Kim's lecture so Dr. Lee must lecture at some time after Dr. Kim, in how many orders can the eight lecturers be scheduled?


Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters?


Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, 0, $z_1,$ and $z_2$ form the vertices of an isosceles right triangle, where $z_1$ and $z_2$ are at equal distance from 0. Find $\frac{a^2}{b}$.


Determine the largest constant $m$ such that for any positive real numbers $a, b, c, d, e$,
\[\sqrt{\frac{a}{b+c+d+e}} + \sqrt{\frac{b}{a+c+d+e}} + \sqrt{\frac{c}{a+b+d+e}} + \sqrt{\frac{d}{a+b+c+e}} + \sqrt{\frac{e}{a+b+c+d}} > m.\]


The prime factorization of 3456 is $2^7 \times 3^3$. How many of its positive integer factors are perfect squares?


A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\triangle DEQ$ is greater than both $\triangle DFQ$ and $\triangle EFQ$?


Given three vertices of a parallelogram $(2, -3)$, $(8, 5)$, and $(5, 0)$ where $(2, -3)$ and $(8, 5)$ are opposite vertices, determine the coordinates of the fourth vertex and find the point where the diagonals of this parallelogram intersect.

Solution: \boxed{(5, 1)}


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 $(AB)(CD).$