Express the number 2023 as a sum of distinct powers of 2 and find the least possible sum of the exponents of these powers.


In how many ways can 100 be written as the sum of two positive perfect squares and one positive perfect cube, if the order does not matter and all numbers involved must be different?


Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD$. If $\angle BAC = 40^\circ$, then what is the measure of $\angle BDC$?


A certain integer has $5$ digits when written in base $16$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?


The graph of the parabola $x = 3y^2 - 9y + 4$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


In a circle $Q$, angle $XBY$ measures 60 degrees. Calculate the measure of minor arc $XB$ in degrees.
[asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
draw(unitcircle);
draw(dir(120)--(-1,0)--(1,0));
dot("$B$",(-1,0),W); dot("$X$",dir(120),NNE); dot("$Y$",(1,0),E); dot("$Q$",(0,0),S);
[/asy]


Maria is trying to unlock a four-digit code on her phone. She remembers that she used digits from 1 to 6 (possibly repeated), where every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Maria need to try?


Let $\mathcal P$ be a parabola defined by $y = 4x^2$. Let $V_1$ and $F_1$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal P$ such that $\angle AV_1 B = 90^\circ$. Let $\mathcal Q$ be the locus of the midpoint of $\overline{AB}$. Determine the vertex $V_2$ and focus $F_2$ of $\mathcal Q$, and compute the ratio $\frac{F_1F_2}{V_1V_2}$.


Consider the function $f$ defined by $f(x) = \frac{ax+b}{cx+d}$, where $a, b, c, d$ are nonzero real numbers. The function has properties $f(10) = 10$, $f(50) = 50$ and $f(f(x)) = x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $f$.


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


At Oakwood Junior High, the math team is to be selected from a club consisting of 5 girls and 5 boys. How many different teams consisting of 3 girls and 2 boys could be formed, given that at least one boy or one girl must be on the team?


Suppose that $\gamma$ is directly proportional to $\delta$. When $\gamma = 5$ then $\delta = -10$. Find $\gamma$ when $\delta = 25$.


In the coordinate plane, the curve $xy = 1$ intersects a circle at four points, three of which are $\left(3, \frac{1}{3}\right),$ $\left(-4, -\frac{1}{4}\right),$ and $\left(\frac{1}{5}, 5\right).$ Find the fourth point of intersection.

Solution: \boxed{\left(-\frac{5}{12}, -\frac{12}{5}\right)}


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


In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.

Note: A point is a lattice point if all its coordinates are integers.


Jane and her brother each spin a spinner once. The spinner now has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 3, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


What is the greatest divisor of 480 that is smaller than 60 and also a factor of 90?


A circle is circumscribed around quadrilateral $ABCD$ with the following known angles: $\angle ACB = 50^\circ$ and $\angle CAD = 40^\circ$. Calculate the sum of the angles $\angle CAB + \angle ACD$.


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


In a large bag of bows, $\frac{3}{20}$ are red, $\frac{3}{10}$ are blue, $\frac{1}{5}$ are green, $\frac{1}{20}$ are purple, and the remaining 24 are white. How many of the bows are green?


Consider triangle \( ABC \) with vertices \( A(0, 10) \), \( B(3, 0) \), and \( C(10, 0) \). A line through \( B \) cuts the area of \( \triangle ABC \) in half; find the sum of the slope and \( y \)-intercept of this line.


Below is a graph of $y = a \csc(bx)$, where $b=2$ and the function has a critical point at $y = 4$. Determine the value of $a$ based on given properties.


A $200 \times 300 \times 350$ rectangular solid is made by gluing together $1 \times 1 \times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1 \times 1 \times 1$ cubes?


The sum of the first \( n \) terms in the infinite geometric sequence \( \left\{1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots \right\} \) is \( \frac{85}{64} \). Find \( n \).


How many numbers are in the first $21$ rows of Pascal's Triangle (from the $0$th row to the $20$th row)?


How many prime numbers between 50 and 100 have a prime remainder when divided by 7?


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,7)$. Find $m+b$.


Points $E$, $F$, $G$, and $H$ lie on a line in that order. If $EF = 3$ units, $FG = 7$ units, and $EH = 20$ units, what is the ratio of $EG$ to $FH$?


Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 3 \\ -2 \\ 1 \\ -4 \end{pmatrix} + t \begin{pmatrix} 8 \\ 4 \\ -2 \\ 6 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} 5 \\ 3 \\ 7 \\ 2 \end{pmatrix}.\]


Let \( F_n \) be the Fibonacci sequence, where \( F_0 = 0 \), \( F_1 = 1 \), and \( F_{n+2} = F_{n+1} + F_n \) for \( n \geq 0 \). Compute
\[
\sum_{n=0}^{\infty} \frac{F_n}{5^n}.
\]

Solution: \boxed{S = \frac{5}{19}}


What is the greatest number of Mondays that can occur in the first $51$ days of a year?


Suppose that $x^5 - 3x + 2 = 0$ and $x \neq 1$. Find the value of
\[x^4 + x^3 + x^2 + x + 1.\]


Two concentric circles are centered at point Q. The sides of a 60-degree angle at Q form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 48-degree angle at Q. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.


At Premier Pups Training Academy, dogs can learn three tricks: fetching, jumping, and barking. The following data is known:
\begin{tabular}{l@{\qquad}l}
70 dogs can fetch & 25 dogs can fetch and jump \\
40 dogs can jump & 15 dogs can jump and bark \\
45 dogs can bark & 20 dogs can fetch and bark \\
12 dogs can do all three & 15 dogs can do none
\end{tabular}
How many dogs are in the academy?


Suppose $q(x)$ is a monic cubic polynomial with real coefficients such that $q(2-3i)=0$ and $q(1)=26$.

Determine $q(x)$ (in expanded form).

Solution: \boxed{q(x) = x^3 - 2.4x^2 + 6.6x + 20.8}


For a given positive integer $n > 2^5$, what is the greatest common divisor of $n^3 + 5^3$ and $n + 5$?


Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. Calculate the number of ways this arrangement can be done.


How many integers are solutions to the equation $$(x-1)^{(16-x^2)}=1?$$


A rectangle and an isosceles triangle share the same base. The height of the rectangle, $r$, equals the height of the triangle, $h$. Both geometric shapes together cover the same area as a square with side length $s$. If the base, equal for both the rectangle and the triangle, is known to be $s$, what is the height $h$ of the triangle in terms of $s$?


Determine all $x$ where the inequality \[\frac{x+1}{x-2} + \frac{x-3}{3x} \ge 4.\]

Solution: \boxed{\left[-\frac{1}{4}, 0\right) \cup (2, 3]}


The letters of the alphabet are assigned values based on the pattern $2, 1, 2, 0, -1, -2, -1, 0$, repeated throughout the alphabet. Calculate the sum of the numeric values of the letters in the word "dynamics".


Three of the four vertices of a rectangle are \((2, 9)\), \((13, 9)\), and \((13, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 2)^2 + (y + 4)^2 = 16\)?


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?


Let $m=\underbrace{55555555}_{\text{8 digits}}$ and $n=\underbrace{111111111}_{\text{9 digits}}$.

What is $\gcd(m,n)$?


Suppose that $b$ is a positive integer greater than or equal to $3.$ When $256$ is converted to base $b$, the resulting representation must have $3$ digits. What is the number of possible values for $b$?


How many right angles are in a right triangle?


What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $71$?


What is the smallest five-digit palindrome divisible by both 3 and 5?


In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled triangles with $\angle AEB=\angle BEC = \angle CED = 45^\circ$ and $AE=32$. Find the length of $CE.$


Let $f$ be a function defined on the positive integers, such that
\[ f(xy) = f(x) + f(y) \]
for all positive integers $x$ and $y.$ Given $f(10) = 16$, $f(40) = 26$, and $f(8) = 12$, find $f(1000)$.


Find the range of the function $f(x) = \arcsin x + \arccos x + \arccot x$ over the interval $-1 \leq x \leq 1$ where all functions are calculated in radians.

Solution: \boxed{\left[\frac{3\pi}{4}, \frac{5\pi}{4}\right]}


Jessica has two colors available to color a 3 by 3 grid. In this grid, any two adjacent cells (sharing a side) must not be the same color. How many different ways can Jessica color the grid?


What is the least integer value of $x$ such that $\lvert3x - 4\rvert \leq 18$?


Determine the common ratio of the infinite geometric series: $$\frac{7}{8} - \frac{35}{72} + \frac{175}{432} - \dots$$


Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or silver) and the eighth-graders will pick the color of the jersey (black, white, or gold), but the two groups of players will not confer together. If, for both garments, each possible color is equally likely to be chosen, what is the probability that the shorts will be a different color than the jersey? Express your answer as a common fraction.


Let $T_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{T_{4n}}{T_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term.


How many positive multiples of 3 that are less than 150 have a units digit of 3?


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


Find the greatest common divisor of $8!$ and $9!$.


Find the smallest positive integer $b$ for which the quadratic expression $x^2 + bx + 1764$ factors into a product of two binomials, each having integer coefficients.


Paola places 720 marbles into $m$ total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of $m$ can this be done?


For how many integers \( n \) between 1 and 150 is the greatest common divisor of 21 and \( n \) equal to 3?


Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit numbers, $WX$ and $YZ$, where both numbers must be two-digit primes between 40 and 60. The digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. Calculate how many such house numbers are possible?


How many positive factors of 180 that are perfect cubes?


Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$What is the value of $b_2 - b_1$? Express your answer as a common fraction.


My friend has 12 unique postcards, including 4 that feature famous landmarks. If she displays these postcards in a line, what is the probability that all four landmark postcards are displayed consecutively?


Given the polynomial \( q(x) = x^5 - 4x^2 + 3 \), find the coefficient of the \( x^4 \) term in the polynomial \( (q(x))^2 \).


In triangle $ABC$, $AB = 10$, $BC = 15$, and $\angle B = 90^\circ$. Points $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively; $\overline{CD}$ and $\overline{BE}$ intersect at point $X$. Point $F$ is the foot of the perpendicular from $A$ to $\overline{BC}$. Compute the ratio of the area of triangle $AFX$ to the area of triangle $BXC$.


Find the smallest constant $m$ so that for any positive real numbers $a, b, c, d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} \geq m.\]


Susan is taking a 30-question multiple choice test. Each question has four answer choices. Susan guesses random answers for the last five questions. What is the probability that she will get at least two of these final five questions right?


The positive four-digit integers that use each of the digits $1,$ $2,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $10^{\text{th}}$ integer in the list?


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 6.$ Find $|\alpha|$.


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}}


How many positive factors of 144 are also multiples of 18 and even numbers?


In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is 10, and the sum of any three consecutive terms is $40$. What is the value of $A + H$?


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


The average age of seven children is 8 years old. Each child is a different age and there is a difference of two years in the ages of any two consecutive children. In years, how old is the oldest child?


What is the smallest four-digit integer whose digits are distinct and form a geometric sequence, and the number itself is even?


Three standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction.


Determine the sum of the proper divisors of $540$.


Evaluate $\lfloor -5 -.5 \rfloor \cdot \lceil 5 +.5 \rceil \cdot \lfloor -4 -.5 \rfloor \cdot \lceil 4 +.5 \rceil \cdot \lfloor -3 -.5 \rfloor \cdot \lceil 3 +.5 \rceil \cdot \lfloor -2 -.5 \rfloor \cdot \lceil 2 +.5 \rceil \cdot \lfloor -1 -.5 \rfloor \cdot \lceil 1 +.5 \rceil \cdot \lfloor -.5 \rfloor \cdot \lceil .5 \rceil$.


What is the remainder when $8x^4 - 20x^3 + 30x^2 - 40x + 17$ is divided by $2x - 4$?


You are given a sequence of 60 terms; each term has the form \( Q+n \) where \( Q \) stands for the product \( 2 \times 3 \times 5 \times \ldots \times 67 \) of all prime numbers less than or equal to 67, and \( n \) takes, successively, the values 2, 3, 4, \ldots, 61. Determine the number of primes appearing in this sequence.


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 7 and 14 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. Determine the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$, expressing the answer as a common fraction.


Two dice with faces numbered 1 through 6 are placed side by side with three of their twelve total faces visible. The visible faces show numbers 2, 3, and 5. What is the total number of dots NOT visible in this arrangement?


Solution: \boxed{\frac{-10x + 11}{12}}


Determine the number of pairs $(a, b)$ of integers such that $6, a, b, ab+3$ is an arithmetic progression.


Let $x_1 < x_2 < x_3$ be the three real roots of the equation $\sqrt{2020} x^3 - 4040x^2 + 4 = 0$. Find $x_2(x_1 + x_3)$.


The first three stages of a pattern are shown below, where each line segment represents a straw. If the pattern continues such that at each successive stage, four straws are added to the previous arrangement, how many straws are necessary to create the arrangement for the 100th stage?


Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$?


If Ben flips 10 coins, what is the probability that he gets exactly 6 heads, given that he gets at least 4 heads?


Find the largest real number \( \lambda \) such that
\[ a^2 + b^2 + c^2 + d^2 \ge ab^2 + \lambda b^2c + c^2d \]
for all nonnegative real numbers \( a, b, c, d \).


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


What is the greatest number of Tuesdays that can occur in the first $45$ days of a year?


Given that
\[
\frac{\cos^6 \theta}{\cos^2 \phi} + \frac{\sin^6 \theta}{\sin^2 \phi} = 1,
\]
find the sum of all possible values of
\[
\frac{\sin^6 \phi}{\sin^2 \theta} + \frac{\cos^6 \phi}{\cos^2 \theta}.
\]


Evaluate the expression $b^3 \cdot b^6 \div b^2$ if $b = 4$.


The matrix
\[\begin{pmatrix} u & v \\ -\frac{3}{5} & \frac{4}{5} \end{pmatrix}\] corresponds to a reflection.  Enter the ordered pair \((u,v)\).

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


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


$ABCD$ is a trapezoid with the measure of base $\overline{AB}$ three times the measure of the base $\overline{CD}$. Point $E$ is the point of intersection of the diagonals. The measure of diagonal $\overline{AC}$ is 15. Find the length of segment $\overline{EC}$.


Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 16 and at least two of the numbers rolled are the same? Express your answer as a common fraction.


When the expression $(3x^4+3x^3+x-14)(3x^{10}-9x^7+9x^4+30)-(x^2+5)^7$ is expanded, what is the degree of the resulting polynomial?


The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{85}$. Find the greatest possible value of $k$.


Let's modify the problem slightly to make it more challenging. Paco uses a spinner to select a number from 1 through 6, each with equal probability. Manu uses a different spinner to select a number from 1 through 12, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than or equal to 36? Express your answer as a common fraction.


Suppose $a$ and $b$ are complex numbers such that $|a+b|=2$ and $|a^2+b^2|=8$. Find the smallest possible value of $|a^3+b^3|$.


Solution: \boxed{9b^4 + 4b^2 + 3b^2c - c^2}


Paco uses a spinner to select a number from 1 through 5, each with equal probability. Manu uses a different spinner to select a number from 1 through 15, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than 50? Express your answer as a common fraction.


A rectangular box has interior dimensions of 8-inches by 7-inches by 14-inches. The box is filled with as many solid 3-inch cubes as possible, with all the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes?


In the diagram, two polygons are regular. The first polygon $ABC$ is a pentagon and the second polygon $ABD$ is a triangle. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?


A circle with radius 6 cm is tangent to two longer sides and one shorter side of a rectangle. The area of the rectangle is triple the area of the circle. What is the length of the shorter side of the rectangle, in centimeters?

Solution: \boxed{12 \text{ cm}}


What is the area enclosed by the graph of $|x| + |3y| = 12$?


Find two fifths of 7.5, expressed as a simplified fraction.


A right octagonal pyramid has two cross sections obtained by slicing the pyramid with planes parallel to the octagonal base. The area of the smaller cross section is $256\sqrt{2}$ square feet and the area of the larger cross section is $576\sqrt{2}$ square feet. The distance between the two planes is $12$ feet. Determine the distance from the apex of the pyramid to the plane of the larger cross section.


Suppose I have 8 shirts and 7 ties. However, one of the shirts, a striped one, should not be worn with two of these ties, a red tie and a blue tie. How many different shirt-and-tie outfits can I make, considering this restriction?


What is the ratio of the area of a square inscribed in an ellipse with semi-major axis \(a\) and semi-minor axis \(b\) to the area of a square inscribed in a circle with radius \(b\)? Express your answer as a common fraction.


What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c+\dfrac{1}{c}\right)^8$?


Each day, Jenny ate $30\%$ of the jellybeans that were in her jar at the beginning of the day. If 24 jellybeans remained at the end of the third day, how many jellybeans were in the jar originally?


What is the probability of rolling seven standard, six-sided dice and getting exactly six distinct numbers? Express your answer as a common fraction.


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{36}$?


If the six digits 1, 2, 3, 5, 7, and 9 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 15? Express your answer as a common fraction.


Calculate \((7\frac{1}{2} - 5\frac{3}{4}) \times \left(3\frac{1}{6} + 2\frac{1}{8}\right)\) and express your answer as a mixed number.


How many numbers are greater than 250 and less than 550 that are divisible by 3?


Let $A_0=(0,0)$. Distinct points $A_1, A_2, \dots$ lie on the $x$-axis, and distinct points $B_1, B_2,\dots$ lie on the graph of $y=x^2$. For every positive integer $n$, $A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n \geq 200$?


If $x^5 - x^4 + x^3 - px^2 + qx + 4$ is divisible by $(x - 2)(x + 1),$ find the ordered pair $(p,q).$


Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = TB = 15$ and $TC = \sqrt{225 - 2TC^2}$, then what is the volume of pyramid $TABC$?

Solution: \boxed{187.5 \sqrt{3}}


Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 12x - 8y - 48$. What is $x+y$?


If $x - y - z = 8$, $x + y + z = 20$, and $x - y + 2z = 16$, what is the value of $z$?


Solution: \boxed{96 \times 12^8}


Evaluate $|\omega^2 + 4\omega + 40|$ if $\omega = 7 + 3i$.


In triangle ABC, angle B is 45 degrees, angle C is 45 degrees, and side BC is 10 units long. Find the length of side AB.


Simplify the expression $2x - 3(2-x) + 4(3+x) - 5(2+3x)$.

Solution: \boxed{-6x - 4}


Simplify $(2(3p+4)-5p\cdot2)^2 + (6 - 2\div2)(9p-12)$ to a much simpler expression of the form $cp^2 + dp - e$, where $c$, $d$, and $e$ are integers.

Solution: \boxed{16p^2 - 19p + 4}


In a diagram, there are three smaller circles positioned inside a larger circle. Each smaller circle touches the other two smaller circles exactly at the center of the larger circle. The radius of the larger circle is 9. Calculate the area of the shaded region outside the smaller circles but inside the larger circle.

[asy]
size(100);
import graph;
filldraw(Circle((0,0),3),mediumgray);
filldraw(Circle((-1.5,0),1.5),white);
filldraw(Circle((1.5,0),1.5),white);
filldraw(Circle((0,1.5*sqrt(3)),1.5),white);
[/asy]


Compute the product
\[
\prod_{n = 1}^{15} \frac{n^2 + 5n + 6}{n+2}.
\]


Suppose $\{b_n\}$ is an arithmetic sequence with
$$
b_1 + b_2 + \cdots + b_{150} = 150 \quad \text{and} \quad
b_{151} + b_{152} + \cdots + b_{300} = 300.
$$
What is the value of $b_2 - b_1$? Express your answer as a common fraction.


In the word MATHCOUNTS, each letter is represented by a magnet on a board. Three vowels and two consonants fall off and are placed into a bag. If the Ts are indistinguishable, how many distinct collections of letters are possible?


The vertical drops of five roller coasters at Fractal Funland are shown in the table.
\[
\begin{tabular}{|l|c|}
\hline
The Spiral Descent & 150 feet \\ \hline
The Accelerator & 125 feet \\ \hline
The Looping Star & 145 feet \\ \hline
The Sky Scream & 280 feet \\ \hline
The Quantum Leap & 190 feet \\ \hline
\end{tabular}
\]
What is the positive difference between the mean and the median of these vertical drops?


Calculate the value of $\frac{6x + 10y}{60xy}$, where $x = \frac{2}{3}$ and $y = \frac{8}{10}$.


The infinite sequence $T=\{t_1,t_2,t_3,\ldots\}$ is defined by $t_1=11$ and $t_n=11^{t_{n-1}}$ for each integer $n>1$. What is the remainder when $t_{100}$ is divided by $7$?


Find positive integers $a$ and $b$ where $b > 1$ such that $a^b$ is the greatest possible value less than 500. What is the sum of $a$ and $b$?


Suppose \( a \), \( b \), and \( c \) are such that \( a \) and \( b \) are inversely proportional, and likewise \( b \) and \( c \) are inversely proportional. If \( a = 40 \) when \( b = 5 \), and \( c = 10 \) when \( b = 7 \), find the value of \( c \) when \( a = 16 \).


Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,50 \pi).$


How many digits are there in the base-5 representation of 1297?


Let $g$ be a function satisfying $g(xy) = g(x)/y$ for all positive real numbers $x$ and $y$. If $g(100) = 2$, what is the value of $g(80)$?


For what values of $x$ is $x^2 - 36x + 323 \leq 5$? Express your answer in interval notation.

Solution: \boxed{[18 - \sqrt{6}, 18 + \sqrt{6}]}


Four marbles are randomly selected, without replacement, from a bag containing three red, two blue, and two green marbles. What is the probability that one marble of each color is selected along with an extra red marble? Express your answer as a common fraction.


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 10$, then what is the volume of pyramid $SPQR$?

Solution: \boxed{240 \text{ cubic units}}


Let \(a\) and \(b\) be positive real numbers such that \(4a + 5b < 100\). Find the maximum value of
\[
ab (100 - 4a - 5b).
\]


Calculate $\tan 30^\circ + 3 \sin 30^\circ.$

Solution: \boxed{\frac{2\sqrt{3} + 9}{6}}


Name the sum of all the greatest whole numbers less than $500$ that have an odd number of positive factors.


What is the largest $5$-digit integer congruent to $18 \pmod{25}$?


If $8$ chunks can be traded for $5$ zunks, and $3$ zunks will buy $7$ bananas, how many chunks are needed to purchase two dozen bananas?


A palindrome is a number that reads the same forward and backward. How many four-digit perfect squares with exactly '11' as the middle two digits are palindromes?


Let $a, b, c, d$ be nonzero real numbers such that $a + b + c = d$. Simplify
\[
\frac{1}{b^2 + c^2 - a^2} + \frac{1}{a^2 + c^2 - b^2} + \frac{1}{a^2 + b^2 - c^2}.
\]


What is the value of $n$ such that $10^n = 10^{-6}\times \sqrt{\frac{10^{75}}{0.01}}$?


Let $x_1, x_2, \ldots, x_n$ be real numbers such that $|x_i| < 2$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 30 + |x_1 + x_2 + \dots + x_n|.\] Determine the smallest possible value of $n$.


In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 800$ and $PS = 1600$. Let $\angle P = 45^\circ$, $\angle S = 45^\circ$, and $T$ and $U$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $TU$.


Find $g(x)$, with terms in order of decreasing degree, if \[4x^4 + 2x^2 - x + 7 + g(x) = x^3 - 4x^2 + 6.\]

Solution: \boxed{g(x) = -4x^4 + x^3 - 6x^2 + x - 1}


Two circles of radius 15 cm overlap such that each circle passes through the center of the other. Calculate the length of the common chord in simplest radical form.

Solution: \boxed{15\sqrt{3} \text{ cm}}


Solve the following equation:
\[
\frac{1}{x + 11} + \frac{1}{x + 8} = \frac{1}{x + 12} + \frac{1}{x + 7}.
\]

Solution: \boxed{x = -\frac{19}{2}}


What is the least positive integer value of $x$ such that $(3x)^2 + 2 \cdot 41 \cdot 3x + 41^2$ is a multiple of 53?


Let \(a, b, c\) be positive real numbers such that \(a + b + c = 6.\) Find the minimum value of
\[\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.\]


A certain natural number $m$ satisfies $m^2$ giving a remainder of 1 when divided by 7, and $m^3$ gives a remainder of 6 when divided by 7. What is the remainder when $m$ is divided by 7?


Take variable \( b \), triple it, and add six. Then subtract \( 6b \) from this new expression, and divide the resulting difference by three. What is the final expression in simplest form?

Solution: \boxed{-b + 2}


Three identical spherical balls, each with a diameter of 8 cm, are stacked in a pyramid-shaped crate, such that each ball touches the other two. Determine the total height, \( h \), from the bottom of the bottom ball to the top of the top ball.

Solution: \boxed{h = 8 + 4\sqrt{3} \text{ cm}}


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}.$ 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} 1 \\ 4 \\ 1 \end{pmatrix}}


Twelve points are placed on a circle. How many distinct convex polygons of four or more sides can be drawn using some (or all) of these points as vertices?


Convert the binary number $1101010101_2$ to base 4.


How many two-digit numbers are there in which the tens digit is greater than the units digit by at least 2?


Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 3 \mathbf{q}$ and $3 \mathbf{p} + \mathbf{q}.$


A sphere is inscribed in a cube with edge length 12 inches. Then a smaller cube is inscribed in the sphere such that its space diagonal is half the diameter of the sphere. Calculate the volume of this smaller inscribed cube. Express your answer in simplest radical form.


A small safari park has six pairs of different animals, one male and one female for each pair: lions, tigers, bears, wolves, elephants, and rhinos. The zookeeper must feed the animals in a specific pattern: each time he feeds an animal, the next one must be of a different gender. Considering the zookeeper starts by feeding the female lion, how many ways can he feed all the animals?


Calculate the number of digits in the value of $2^{15} \times 5^{10} \times 3^5$.


For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?


What is the remainder when $5x^9 - 3x^7 + 4x^6 - 8x^4 + 3x^3 - 6x + 5$ is divided by $3x - 6$?


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


Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn$ for all positive integers $m$ and $n$. Find $b_{15}$.


A list of five positive integers features these properties:
- The only integer in the list that occurs more than once is $7$.
- Its median is $10$.
- Its average (mean) is $12$.

What is the largest possible integer that could appear in the list?


Emily needs to calculate the sum of 68, 74, and 59, subtract 20 from the result, and then round it to the nearest ten. She considers rounding each number before performing the operations. What should be the correct result following standard mathematical procedures?


What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6?


When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers all of which are nonprime?


In Pascal's Triangle, each number is the sum of the two numbers directly above it to the left and right. What is the sum of all numbers in Row 12 of Pascal's Triangle? Also, what is the value of the middle number in this row?


An 8-sided die has numbers from 1 to 8 on its faces. How many ways can three faces of this die be painted red such that no pair of the red-painted faces adds up to 9?


Let $C$ be a point not on line $AE$. Points $D$ and $F$ lie on line $AE$ such that $CD \perp AE$ and $CF \perp AE$. Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE$. Given $AB = 3$, $CD = 10$, and $AE = 6$, find the lengths of $CE$ and $CF$.


Find the positive value of $x$ which satisfies
\[\log_5 (x + 3) + \log_{5^2} (x^2 + 5) + \log_{5^{-1}} (x + 3) = 3.\]


A rectangular prism has dimensions 4 units, 3 units, and 5 units. It has twelve edges and eight vertices. A segment, such as $x$, which joins two vertices not joined by an edge is considered a diagonal. How many diagonals does this rectangular prism have, including face diagonals and space diagonals?


A positive two-digit number is odd and is a multiple of 13. The sum of its digits is a perfect square. What is this two-digit number?


In a local badminton club, the number of matches played by each player over a month is recorded. Calculate the average number of matches played, rounding off to the nearest whole number. Below is the schematic representation:

- 1 match: 4 players
- 2 matches: 3 players
- 4 matches: 2 players
- 6 matches: 2 players
- 8 matches: 4 players


How many positive odd integers greater than 1 and less than $200$ are square-free?


Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $R$ is inside $\triangle PQR$, $\angle PQR = 50^\circ$, and $\angle PRS = 110^\circ$. What is the degree measure of $\angle QPS$?


Brad has a cylindrical barrel with a radius of 8 inches and a height of 20 inches. He plans to fill it from a well using a bucket in the shape of a hemisphere with a radius of 8 inches. Due to a hole in the bucket, each trip results in a loss of 10% of the water. How many trips must Brad make to the well to fill the barrel completely?


In triangle $PQR$, $\sin P = \frac{4}{5}$ and $\cos Q = \frac{12}{13}$. Find $\cos R$.


A sphere is inscribed in a cube. Given that one edge of the cube is 8 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of \( \pi \).


The volume of the parallelepiped determined by the three-dimensional vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6.  Find the volume of the parallelepiped determined by the vectors $2\mathbf{a} - \mathbf{b},$ $\mathbf{b} + 4\mathbf{c},$ and $\mathbf{c} + 5\mathbf{a}.$


A spiral staircase turns $180^\circ$ as it rises 12 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.


If the least common multiple of $X$ and $Y$ is $180$, and the ratio of $X$ to $Y$ is $2:5$, what is their greatest common divisor?


What is the area, in square units, of a triangle with vertices at $A(2, 3), B(8, 3), C(5, 10)$?


In a biology experiment, Thomas needs to prepare a nutrient solution by mixing 0.05 liters of nutrient concentrate with 0.03 liters of water. He aims to prepare 0.72 liters of this solution for his experiment. How many liters of water will he need to use?


Four identical circles, each of radius 1, are externally tangent to each other forming a square arrangement and internally tangent to a larger circle. What is the radius of the larger circle?

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


Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{34}\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$


Suppose the sequence \( (v_n) \) satisfies the recurrence relation
\[ v_{n+2} = 2v_{n+1} + v_n. \]
Given that \( v_2 = 6 \) and \( v_5 = 58 \), determine \( v_4 \).


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


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 $(AB)(CD).$


Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors such that $\mathbf{u} + 3\mathbf{v}$ and $3\mathbf{u} - 2\mathbf{v}$ are orthogonal. Find the angle between $\mathbf{u}$ and $\mathbf{v}$, in degrees.

Solution: \boxed{\cos^{-1}\left(\frac{3}{7}\right)}


Liam has $\frac{60}{7}$ pounds of chocolate. He wants to divide this chocolate into $5$ equal piles and then needs to triple one of these piles before giving it to his friend Noah. How many pounds of chocolate will Noah receive?


Let $x$, $y$, $z$ be real numbers, all greater than 4, such that
\[\frac{(x + 3)^2}{y + z - 3} + \frac{(y + 5)^2}{z + x - 5} + \frac{(z + 7)^2}{x + y - 7} = 45.\]
Find the ordered triple $(x, y, z)$.


A quadratic is given as $x^2 + cx + \frac{1}{4}$ where $c$ is a specific number. Olivia rewrites this quadratic using the technique of completing the square, resulting in the form $(x+m)^2 + \frac{1}{16}$. Determine $c$ assuming $c$ is negative.


Find $x$, where $x$ is a square number which is divisible by six, and between 24 and 150.


A projectile is fired such that its height (in feet) as a function of time (in seconds) is given by $-20t^2 + 120t + 36$. Determine the maximum height reached by the projectile and the time at which it occurs.

Solution: \boxed{3 \text{ seconds}}


A $6 \times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. Express your answer as a common fraction.


How many ways can we arrange 4 math books and 6 English books on a shelf if all the math books must stay together, all the English books must also stay together, and a trophy must be placed at one of the ends of the shelf? (The math books are all different and so are the English books, and the trophy is distinct from the books.)


If $x = \frac{3}{5}$ and $y = \frac{5}{3}$, find the value of $\frac{1}{3}x^8y^9$.


Consider the function $$g(x) = \lfloor \lfloor x \rfloor - x \rfloor.$$ Determine the range of $g(x)$.

Solution: \boxed{\{-1, 0\}}


What is the last digit of the decimal expansion of $\frac{1}{2^{12}}$?


Two concentric circles share the same center O. Point Z is a point on the larger circle such that OZ is three-quarters the radius of the larger circle. Find the ratio of the area of the circle with radius OZ to the area of the circle with radius OP, where P is a point on the circumference of the larger circle.


Determine the sum of the prime factors of the expression $3^{6} - 1$.


Let $p,$ $q,$ and $r$ be real numbers, and let $P,$ $Q,$ $R$ be points such that the midpoint of $\overline{QR}$ is $(p,0,0),$ the midpoint of $\overline{PR}$ is $(0,q,0),$ and the midpoint of $\overline{PQ}$ is $(0,0,r).$  Find
\[\frac{PQ^2 + PR^2 + QR^2}{p^2 + q^2 + r^2}.\]


Wendy is trying to locate the Fermat point \( P \) of \(\triangle ABC\), where \( A \) is at the origin, \( B \) is at \( (12,0) \), and \( C \) is at \( (5,8) \). She estimates that the point is at \( P = (5,3) \), and calculates the sum of the distances from \( P \) to the vertices of \(\triangle ABC\). If she obtains \( m\sqrt{a} + n\sqrt{b} \), where \( m \), \( n \), \( a \), and \( b \) are integers, what is \( m + n \)?


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


The values of \( p, q, r, \) and \( s \) are 2, 3, 4, and 5, but not necessarily in that order. What is the largest possible value of the sum of the four products \( pq + qr + rs + sp \)?


Let the function \( g(x) \) take positive real numbers to real numbers such that
\[ xg(y) - yg(x) = g \left( \frac{x}{y} \right) + x - y \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( g(50) \).


Let \(\theta\) be an acute angle, and let
\[
\cos \frac{\theta}{2} = \sqrt{\frac{x + 1}{2x}}.
\]
Express \(\sin \theta\) in terms of \(x.\)

Solution: \boxed{\frac{\sqrt{x^2 - 1}}{x}}


What is the remainder when $1502 \cdot 2021$ is divided by $500$?


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{w}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{w}\|}.$


Square EFGH has its center at $(5, -3)$ and has an area of 16 square units. The top side of the square is horizontal. First, the square is dilated with a dilation center at $(0,0)$ and a scale factor of 3. Then, it's reflected over the x-axis. What are the coordinates of the vertex of the transformed image of square EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(21, 15)}


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$ and the second card is a king?


Define the operation $*$ such that $a * b = a + a \times \frac{1}{b} - 3$. What is the value of $4 * 8$?


In the diagram, $AB$ and $CD$ are diameters of a circle with radius 6. If $AB$ and $CD$ are perpendicular, and the shaded region consists of triangle $AOB$ and sector $COD$, where $O$ is the center of the circle and $\angle COD = 45^\circ$, what is the area of the shaded region?
[asy]
size(120);
import graph;
fill((0,0)--(1,-1)--cycle,mediumgray);
fill(Arc((0,0),sqrt(2),270,315)--cycle,mediumgray);
draw(Circle((0,0),sqrt(2)));
draw((-1,-1)--(1,1)--(1,-1)--(-1,1)--cycle);
label("$A$",(-1,1),NW); label("$C$",(1,1),NE); label("$D$",(-1,-1),SW); label("$B$",(1,-1),SE);

[/asy]

Solution: \boxed{18 + 4.5\pi}


For each value of $x$, let $f(x)$ be defined as the minimum value of three numbers $3x + 4$, $-\frac{1}{3}x + 2$, and $-\frac{1}{2}x + 8$. Find the maximum value of $f(x)$.


Let $n$ equal the number of sides in a regular polygon. For $3 \leq n < 12$, how many values of $n$ result in a regular polygon where the common degree measure of the interior angles is not an integer?


If
\[\cot x = \frac{a^2 - b^2}{2ab},\] where $a > b > 0$ and $0^\circ < x < 90^\circ,$ then find $\cos x$ in terms of $a$ and $b.$

Solution: \boxed{\cos x = \frac{a^2 - b^2}{a^2 + b^2}}


Given that $$(x+y+z)(xy+xz+yz)=49$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=17$$ for real numbers $x$, $y$, and $z$, and that $$x^3 + y^3 + z^3 = 27,$$ what is the value of $xyz$?


The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers?


Find the sum of the first seven prime numbers that have a units digit of 3 and are less than 150.


Find the value of \(b\) that satisfies the equation \(123_{b} + 234_{b} = 357_{b}\).

Solution: \boxed{b = 8}


A standard deck of 52 cards consists of 13 ranks across 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), with exactly one card per rank and suit combination. Among these, $\spadesuit$ and $\clubsuit$ are black, and $\heartsuit$ and $\diamondsuit$ are red. Assuming the deck is shuffled randomly, what is the probability that the top three cards are of the same color and of the same suit?


The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?


What is the number of square units in the area of trapezoid EFGH with vertices E(2,2), F(2,5), G(6,2), and H(6,8)?


Let the function $g$ take nonnegative integers to real numbers, defined by $g(1) = 2$, and
\[g(m + n) + g(m - n) = 2[g(m) + g(n)]\]
for all nonnegative integers $m \ge n$. Find the value of $g(10)$.


What is the value of \(a + b\) if the sequence \(3, ~8, ~13, \ldots, ~a, ~b, ~33\) is an arithmetic sequence?


A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter. What is the perimeter of this rectangle?


Suppose all six of the numbers \[3 - 2\sqrt{2}, \; -3-2\sqrt{2}, \; 1+\sqrt{7}, \; 1-\sqrt{7}, \; 4 + \sqrt{11}, \; 4 - \sqrt{11}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the positive difference between the two numbers is $3$ or greater? Express your answer as a common fraction.


How many different positive, four-digit integers can be formed using the digits 0, 0, 3, and 9?


Suppose the point $(2,3)$ is on the graph of $y=\frac{g(x)}{3}$. Identify a point that must be on the graph of $y=\frac{g^{-1}(x)}{3}$ and find the sum of that point's coordinates.


Find the degree measure of $\angle A$ in a quadrilateral $ABCD$ where $\angle A = 3\angle B = 4\angle C = 6\angle D$. Round your answer to the nearest whole number.


Let $f(x)$ be a function defined on the domain $\{x_1, x_2, x_3\}$, so that the graph of $y = f(x)$ consists of just three points forming a triangle of area 48. Consider the graph of $y = 3f(3x)$. This graph also consists of just three points. Determine the area of the triangle formed by those three points.


In trapezoid $WXYZ$, the sides $WZ$ and $XY$ are parallel, and the lengths of $WX$ and $YZ$ are equal. Given that $WZ = 10$, $XY = 18$, and the height of the trapezoid (the perpendicular distance between the lines containing $WZ$ and $XY$) is 5, what is the perimeter of $WXYZ$?

[asy]
pen p = linetype("4 4");
draw((0,0)--(5,5)--(15,5)--(20,0)--cycle);
draw((5,0)--(5,5), p);
label(scale(0.75)*"W", (0,0), W);
label(scale(0.75)*"X", (5,5), NW);
label(scale(0.75)*"Y", (15,5), NE);
label(scale(0.75)*"Z", (20, 0), E);
label(scale(0.75)*"18", (10,5), N);
label(scale(0.75)*"10", (10,0), S);
label(scale(0.75)*"5", (5, 2.5), E);
[/asy]

Solution: \boxed{28 + 2\sqrt{41}}


A number $x$ is equal to $11\cdot36\cdot54$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


Find the intersection of the lines $8x - 3y = 24$ and $10x + 2y = 14.$ Express your answer as an ordered pair $(x, y)$.

Solution: \boxed{\left(\frac{45}{23}, \frac{-64}{23}\right)}


Sides $\overline{AB}$ and $\overline{BC}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$?


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


Express $\frac{45}{64}$ as a terminating decimal.


In a shop, $36$ lemons are required to make $48$ gallons of lemonade. For lemon tea, $20$ lemons are needed to make $10$ gallons. How many lemons are necessary to make $6$ gallons of lemonade and $5$ gallons of lemon tea?


Suppose that $c$ is a positive integer greater than or equal to $2.$ When $256$ is converted to base $c$, the resulting representation has $3$ digits. What is the number of possible values for $c$?


Given that for real numbers \( a \), \( b \), and \( c \), the polynomial \[ g(x) = x^3 + ax^2 + x + 20 \] has three distinct roots, and each root of \( g(x) \) is also a root of the polynomial \[ f(x) = x^4 + x^3 + bx^2 + 50x + c. \] What is \( f(1) \)?


Find the integer $n$, $0 \le n \le 15$, such that \[ n \equiv 12345 \pmod{11}. \]


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?


The polynomial \( Q(x) = x^3 + px^2 + qx + d \) has the property that the mean of its zeros, the product of its zeros taken two at a time, and the sum of its coefficients are all equal. If the \( y \)-intercept of the graph of \( y = Q(x) \) is 5, what is \( q \)?


How many numbers in the sequence $\{7, 17, 27, 37, \ldots\}$ up to 100 can be written as the difference of two prime numbers?


In a tactical shooting competition, there are four columns of clay targets, with the first column containing two targets, the second column containing three, the third column containing two, and the fourth column containing one target. The sharpshooter must break all the targets according to the following rules:
1. The sharpshooter first chooses a column from which a target is to be broken.
2. The sharpshooter must then break the lowest remaining target in the chosen column.
Determine the number of different orders in which the sharpshooter can break all the eight targets.


In triangle $DEF,$ $\sin D = \frac{4}{5}$ and $\cos E = \frac{12}{13}.$  Find $\cos F.$


If \(x\) is real, compute the maximum integer value of
\[
\frac{4x^2 + 8x + 19}{4x^2 + 8x + 9}.
\]


A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?


The planet Zenon follows an elliptical orbit around its star with the star at one focus. At its closest approach (perigee), Zenon is 3 astronomical units (AU) from the star, and at its farthest point (apogee), it is 15 AU away. Calculate how far Zenon is from its star when it is at one vertex of the minor axis of its orbit.


Five more than four times \( A \) is subtracted from 30, resulting in three times \( B \). Also, \( B \) is twice of \( A \). What are the values of \( A \) and \( B \)?


Parallelogram \(EFGH\) with \(E(3,4)\), \(F(5,7)\), \(G(7,4)\), and \(H(5,1)\) is reflected across the \(x\)-axis to \(E'F'G'H'\) and then \(E'F'G'H'\) is reflected across the line \(y=x-1\) to \(E''F''G''H''\). This is done such that \(H'\) is the image of \(H\), and \(H''\) is the image of \(H'\). What is the ordered pair of \(H''\) in the coordinate plane?

Solution: \boxed{(0, 4)}


Determine the area of trapezoid EFGH with vertices E(0,0), F(0,-3), G(6,0), and H(6,8).


What is the sum of all the three-digit even positive integers?


What is the smallest integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?


A bowling ball is a solid sphere with a diameter of 24 cm. To customize the ball, three holes are drilled into it. The holes drilled for Bowler Sam are 10 cm deep with diameters of 3 cm, 3 cm, and 4 cm. Assuming these holes are right circular cylinders, calculate the remaining volume of the bowling ball in cubic centimeters, expressing your answer in terms of $\pi$.


Solve for $t$: $2 \cdot 3^t + \sqrt{9 \cdot 9^t} = 27$.

Solution: \boxed{3 - \log_3 5}


If \( n = 2^{12} \cdot 3^{15} \cdot 5^{9} \), how many of the natural-number factors of \( n \) are multiples of 180?


How many numbers are in the following list: $$-6, -2, 2, 6,\ldots, 50$$


A six-digit integer will be chosen at random from all possible positive six-digit integers. What is the probability that both of the number's last two digits will be less than 5? Express your answer as a common fraction.


Calculate the period of the function $y = \sin(10x) + \cos(5x)$.


Find the largest integer $m$ such that every digit of $m$ is either $8$ or $0$ and $m$ is a multiple of $18$. Compute $\frac{m}{18}$.


The members of a choir are arranged in rectangular formations. When arranged in 10 rows, there are 4 positions unoccupied. When they are arranged in 11 rows, there are 5 positions unoccupied. Determine how many members are in the choir if the total number is between 200 and 300.


Completely factor the following expression: \[(20x^3 - 100x^2 + 90x - 10) - (5x^3 - 10x^2 + 5).\]

Solution: \boxed{15(x^3 - 6x^2 + 6x - 1)}


Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic sequence with common difference 5. If \( \frac{S_{2n}}{S_n} \) is a constant for all positive integers \( n \), then find the first term.


A function $f$ has domain $[1,3]$ and range $[-1,2]$. Define
\[g(x) = 2 - f(x + 2).\]
Determine the domain and range of $g(x)$, and express your answer as an ordered quadruple $(a,b,c,d)$.


Let \[f(x) =
\begin{cases}
3x - 7 &\text{if } x < 0, \\
x^2 + 4x + 4 & \text{if } x \geq 0.
\end{cases}
\]
Find $f(-1)$ and $f(2)$.


A wire is cut into two pieces, one of length $x$ and the other of length $y$. The piece of length $x$ is bent to form a square, and the piece of length $y$ is bent to form a regular pentagon. The square and the pentagon have equal perimeter. What is $\frac{x}{y}$?


Find the product of the nonreal roots of the equation $x^4 - 4x^3 + 6x^2 - 4x = 2010.$

Solution: \boxed{1 + \sqrt{2011}}


Below is the graph of an ellipse situated in a coordinate plane where major and minor axes lie vertically and horizontally respectively. Endpoint of the major axis on a vertical line are at $(2, 0)$ and $(2, 8)$. Endpoint of the minor axis on a horizontal line are at $(0, 4)$ and $(4, 4)$.

Compute the coordinates of the focus of the ellipse with the greater $y$-coordinate.

Solution: \boxed{(2, 4 + 2\sqrt{3})}


An 8-sided die is rolled. If the roll is even, then you win twice the amount of the number rolled in dollars (for example, if you roll a 4, then you win $8). If you roll an 8, you additionally win a $5 bonus on top of the doubled roll value. If the roll is odd, you win nothing. Calculate the expected value of your winnings in dollars.


If $\left\lfloor \frac{n^2}{5} \right\rfloor - \left\lfloor \frac{n}{2} \right\rfloor^2 = k$, find all integer values of $n$ such that $k = 1$.


Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D,$ $H_E$ denote the centroids of triangles $BCDE,$ $ACDE,$ $ABDE,$ $ABCE,$ and $ABCD$ respectively. Find $\frac{[H_A H_B H_C H_D H_E]}{[ABCDE]}.$


Round $73.\overline{26}$ to the nearest hundredth.


For what value of $n$ is $3 \times 4 \times 5 \times n = 6!$?


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


In a basketball match, Natasha attempted only three-point shots, two-point shots, and free-throw shots. She was successful on $25\%$ of her three-point shots and $40\%$ of her two-point shots. Additionally, she had a free-throw shooting percentage of $50\%$. Natasha attempted $40$ shots in total, given that she made $10$ free-throw shot attempts. How many points did Natasha score?


Simplify the expression $\frac{5}{4x^{-4}} \cdot \frac{4x^{3}}{3} \div \frac{x}{2}$.


Solution: \boxed{x^4 - 3x^2 + 9}


How many distinct sequences of four letters can be made from the letters in FREEDOM if each letter can be used only once and each sequence must begin with F and not end with M?


A point is selected at random from the segment of the number line from 0 to 6. What is the probability that the point is closer to 5 than to 1?


Find the smallest constant $D$ such that for all real numbers $x$ and $y$,
\[ x^2 + 2y^2 + 5 \ge D(2x + 3y) + 4. \]


Mrs. Crabapple has slightly changed her rules during British Literature period visits. Now, each day she picks a student randomly to give a crabapple, but a student cannot be selected two days in a row. With 11 students in the class and the class meeting three times a week instead of four, how many different sequences of crabapple recipients are possible in a week?


The arithmetic mean of 14 scores is 85. When the highest and lowest scores are removed, the new mean becomes 88. If the highest of the 14 scores is 105, what is the lowest score?


Determine $x$ given that \[4 \log_3 x = \log_3 (4x).\]


Determine how many more digits the base-4 representation of $1573_{10}$ has than the base-7 representation of $1573_{10}$.


The two lines $y = 5x - 28$ and $3x + y = 120$ intersect. What is the value of $x$ at the point of intersection?


Let \[g(x) =
\begin{cases}
2x - 4 &\text{if } x < 0, \\
5 - 3x &\text{if } x \geq 0.
\end{cases}\]
Find $g(-2)$ and $g(3)$.


Determine the coefficient of $x^2$ when the polynomial $3x^3 - 4x^2 + 5x - 2$ is multiplied by $2x^2 - 3x + 4$.


Define a new operation $x\#y = xy - 3x + y$. What is the value of $(6\#5) - (5\#6)$?


In a triangle, two of the side lengths are 10 and 11, and the angle between them is $150^\circ.$ Find the length of the third side.

Solution: \boxed{\sqrt{221 + 110\sqrt{3}}}


An equilateral triangle is inscribed in the ellipse whose equation is \( x^2 + 9y^2 = 9 \). One vertex of the triangle is \( (0,1) \), one altitude is contained in the y-axis, and the square of the length of each side is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. Find \( p+q \).


Let $T = 1 - 2 + 3 - 4 + \cdots + 2011 - 2012$. What is the residue of $T$, modulo 2012?


Yan is at a point between his home and the park. To get to the park, he can either walk directly or walk back home and then ride his scooter to the park. He rides 6 times as fast as he walks, and both choices take the same amount of time. What is the ratio of Yan's distance from his home to his distance from the park?


A wizard is mixing a magical concoction. To prepare this, she needs to select one of four herbs and one of six crystals. However, one of the crystals is incompatible with one of the herbs, and another crystal is incompatible with two different herbs. How many valid combinations can the wizard use to prepare her concoction?


Six couples are at a gathering. Each person shakes hands exactly once with everyone else except his/her spouse and one other person who they choose not to shake hands with. How many handshakes were exchanged?


A region is formed by arranging four congruent squares in a 2x2 grid. If the total area of the region is 144 square centimeters, what is the perimeter of this region, in centimeters?

Solution: \boxed{48 \text{ centimeters}}


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


The polynomial $$Q(x)=(1+x+x^2+\ldots+x^{20})^2-x^{20}$$has 39 complex zeros of the form $z_k=s_k\left[\cos(2\pi\beta_k) + i\sin(2\pi\beta_k)\right]$, $k=1,2,3,\ldots,39$, with $0<\beta_1\le\beta_2\le\beta_3\le\dots\le\beta_{39}<1$ and $s_k>0$. Find $\beta_1+\beta_2+\beta_3+\beta_4+\beta_5.$


Find the intersection of the lines $12x - 5y = 40$ and $8x + 2y = 20.$ Express your answer as an ordered pair $(x, y)$.

Solution: \boxed{\left(\frac{45}{16}, -\frac{5}{4}\right)}


The greatest common divisor of two integers is $(x+1)$ and their least common multiple is $x(x+1)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one?


Let $\omega$ be a nonreal root of $z^3 = 1$. Let $a_1, a_2, \dots, a_n$ be real numbers such that
\[
\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 3 + 4i.
\]
Compute
\[
\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1}.
\]


If the three points $(1, b, a), (b, 2, a),$ and $(b, a, 3)$ are collinear, determine the value of $a + b$.


Compute the largest integer $k$ such that $1764^k$ divides $1764!$.


For certain ordered pairs $(a,b)\,$ of integers, the system of equations
\[\begin{aligned} ax+by&=3 \\ x^2 + y^2 &= 85  \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. In addition, ensure that $3a - 5b = 0$. How many such ordered pairs $(a,b)\,$ are there?


The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number, if the least common multiple of these two numbers is also between 200 and 300?


What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~39$ is an arithmetic sequence?


A classroom contains 60 chairs arranged in a rectangular array. Each row must contain at least two chairs, as should each column, which necessitates that all chairs be used in forming the array. Determine how many different rectangular arrays can be formed under these conditions.


Determine the sum of all integral values of $c$ such that $c \leq 18$ for which the equation $y = x^2 - 5x - c$ has exactly two rational roots.


Suppose the point $(2,6)$ is on the graph of $y = 3f(x)$. Determine the point that must be on the graph of $y = \frac{f^{-1}(x)}{3}$ and find the sum of that point's coordinates.


Let \( P \) be a point inside triangle \( ABC \) such that
\[ \overrightarrow{PA} + 3 \overrightarrow{PB} + 2 \overrightarrow{PC} = \mathbf{0}. \]
Find the ratio of the area of triangle \( ABC \) to the area of triangle \( APB \).


Joe, during breakfast, lunch, and dinner, can now choose either an apple, an orange, a banana, or a pear, each with equal probability. What is the probability that Joe will eat at least two different kinds of fruit in a day?


Liam has recorded times, in seconds, of 104, 111, 109, 103, and 106 for running 400 meters in his previous five attempts. After his sixth attempt, he managed to bring his median time down to 107 seconds. What was his time, in seconds, for his sixth attempt?


Let $O$ be the origin. Determine a scalar $m$ such that for any points $E,$ $F,$ $G,$ and $H$ satisfying the equation
\[4 \overrightarrow{OE} - 3 \overrightarrow{OF} + 6 \overrightarrow{OG} + m \overrightarrow{OH} = \mathbf{0},\]
the points $E,$ $F,$ $G,$ and $H$ are coplanar. Find $m$.


The circular top of a small table has an area of 616 square inches. Oliver wants to cover the edge of the table with a decorative trim and decides to buy 5 inches more of the trim than the circumference of the table. Using the approximation \( \pi = \frac{22}{7} \), how many inches of trim will Oliver need to purchase?

Solution: \boxed{93 \text{ inches}}


What is the quotient when $12x^3 + 20x^2 - 7x + 4$ is divided by $3x + 4$?

Solution: \boxed{4x^2 + \frac{4}{3}x - \frac{37}{9}}


If the least common multiple of $A$ and $B$ is $180$, and the ratio of $A$ to $B$ is $2:5$, then what is their greatest common divisor?


Given that $x$ is a positive integer less than 150, how many solutions does the congruence $x + 17 \equiv 75 \pmod{46}$ have?


The graph of $y = \frac{p(x)}{q(x)}$ is given, where $p(x)$ and $q(x)$ are quadratic polynomials. The graph has a horizontal asymptote at $y = 3$ and a vertical asymptote at $x = -3$. There is also a hole in the graph at $x = 4$. Given that the graph passes through the point $(2,0)$, find the value of $\frac{p(0)}{q(0)}$.


Given that $$(x+y+z)(xy+xz+yz)=49$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=19$$ for real numbers $x$, $y$, and $z$, find the value of $xyz$.


Determine the vertex of the parabola given by the equation $y^2 + 10y + 3x + 9 = 0.$

Solution: \boxed{\left(\frac{16}{3}, -5\right)}


The council at a local community center has 12 members of which 5 are council officers. A five-person subcommittee is to be formed such that it includes at least two council officers. How many distinct subcommittees are possible?


Lee can make 24 cookies using 3 cups of flour. However, Lee wastes half a cup of flour during the process. How many cookies can he make with 5 cups of flour without wasting any?


A standard deck of 52 cards is used in a game where 2 cards are drawn simultaneously. Calculate the probability that one of the cards drawn is the Ace of $\heartsuit$.


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x - y)) = \frac{f(x) f(y) - f(x) + f(y)}{x + y + 1}\]
for all $x, y \neq -1$ and $x + y \neq -1$. Find the value of $f(1)$ if $f(0) = 0$.


Find the third-smallest three-digit palindromic prime.


A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display?

Solution: \boxed{n = 9}


Contractor Lucy agreed to complete a project in 40 days. After working for 10 days with 12 workers, they completed $\frac{2}{5}$ of the project. If each worker maintains the same pace, what is the minimum number of workers Lucy must keep on the project to ensure that it is completed on time?


Construct a quadratic function that has zeroes at $x=1$ and $x=5$, and that has a value of $8$ when $x=3$.

Solution: \boxed{-2x^2 + 12x - 10}


An isosceles, obtuse triangle has one angle with a degree measure that is 75$\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a fraction.


The algae in a local pond doubles every day, and it took 20 days for the pond to be completely covered in algae. Determine the day on which the pond was 87.5% algae-free.


All the numbers 1, 2, 3, 4, 5, 6 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?


Eight positive integers from a list of eleven positive integers are $5, 6, 9, 3, 7, 10, 8, 4$. What is the largest possible value of the median of this list of eleven positive integers?


The surface area of a sphere is \(256\pi\text{ cm}^2\). What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of \(\pi\).


What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $19,$ $7,$ and $3?$


In a class of 80 students, all students participate in a math competition. To encourage higher performance, the teacher decides to give out special awards to those whose scores are above the average score of the class. What is the maximum number of students who can receive this award?


Find the point on the line defined by
\[
\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}
\]
that is closest to the point $(3, 0, 1)$.

Solution: \boxed{\left( \frac{31}{13}, \frac{20}{13}, \frac{24}{13} \right)}


A store advertises a series of two discounts on a product. The first discount is 40% off the original price. Subsequently, a second discount of 30% off the new reduced price is applied. The store claims that the total discount offered is 70% off the original price. Calculate the actual total discount as a percentage of the original price and determine the difference between the store's claimed discount and the actual discount.


An isosceles triangle $PQR$ of area 180 square inches is divided by line segment $ST$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $PQR$ from $P$ is 30 inches, calculate the length of $\overline{ST}$.

Solution: \boxed{6\text{ inches}}


A batch of 350 books is available for purchase at $\$15.30$ per lot. An extra delivery fee of $\$9.25$ is charged. Calculate the average cost per book in cents, rounding to the nearest whole number.


How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|, \quad y = -|2x - 4|
\]


Two rays with a common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, and point $B$ on the other ray, with $AB=1$. What is the maximum possible length of $\overline{OB}$?


Find the product of the nonreal roots of the equation $x^4 - 4x^3 + 6x^2 - 4x = 1007$.

Solution: \boxed{1 + \sqrt{1008}}


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


Consider a shaded region that consists of 13 unit squares along the $x$-axis and $y$-axis. It is divided into two regions. The bottom region has height 4 units, and the top region is directly above it with a narrow strip extending to the right. The shaded region is rotated about the $x$-axis to form a solid. Determine the volume of the resulting solid in cubic units, expressed in terms of $\pi$.

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A relatively prime date is defined as a date for which the number of the month and the number of the day are relatively prime. Determine how many relatively prime dates are in February, considering February has 28 days.


The table below is partially filled in, representing the results of a survey conducted by radio station KMAT. Calculate how many of the female respondents reported listening to this station:

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males & 75 & 85 \\ \hline
females & & 135 \\ \hline
undeclared & 20 & 15 \\ \hline
total & 160 & 235 \\
\hline
\end{tabular}


A math conference is hosting a series of lectures by seven distinct lecturers. Dr. Smith's lecture depends on Dr. Jones’s lecture, and additionally, Dr. Brown's lecture depends on Dr. Green’s lecture. How many valid orders can these seven lecturers be scheduled, given these dependencies?


Let $p$ and $q$ be the roots of $x^2 - 5x + 7 = 0$. Compute
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \]


Consider a set of 150 cards numbered from 1 to 150. Each card is randomly placed in a box. A card is selected at random from this box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction.


A six-digit integer is chosen at random from all possible positive six-digit integers. What is the probability that the number's units digit is a prime number?


Yann and Camille go to a restaurant. If there are 12 items on the menu and each orders one dish, but one particular dish can only be ordered by one of them (either Yann or Camille), how many different combinations of meals can Yann and Camille order?


Suppose the function $g$ has all real numbers in its domain and range and is invertible. Some values of $g$ are given by the following table:
$$\begin{array}{c || c | c | c | c | c | c}
x & -1 & 1 & 2 & 3 & 4 & 6 \\
\hline
g(x) & 3 & 5 & 1 & 8 & 6 & 7
\end{array}$$
What is the value of $g(g(2)) + g(g^{-1}(6)) + g^{-1}(g^{-1}(7))?$ If there is not enough information to answer this question, enter "NEI".


Find a quadratic equation with rational coefficients and leading coefficient 1 that has $\sqrt{5} - 3$ as one of its roots.

Solution: \boxed{x^2 + 6x + 4}


The number $0.324375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


For a finite sequence \( B = (b_1, b_2, \dots, b_n) \), the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_n}{n},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq n \).

If the Cesaro sum of the 99-term sequence \( (b_1, \dots, b_{99}) \) is 800, what is the Cesaro sum of the 100-term sequence \( (10, b_1, \dots, b_{99}) \)?


Points $E$, $F$, $G$, and $H$ lie on a line, in that order. If $EF=3$ units, $FG=6$ units, and $EH=20$ units, what is the ratio of $EG$ to $FH$? Express your answer as a common fraction.


Pascal's triangle exhibits a unique pattern in which certain rows consist entirely of even numbers except for the $1$s at each end. Determine how many of the first 50 rows of Pascal’s triangle, excluding row $0$ and row $1$, consist solely of even numbers.


In the diagram, square $ABCD$ has sides of length $6,$ and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ such that $PQ$ is perpendicular to $BC$ and $PQ = x$. Determine the measure of angle $BPC.$


Leo scored 92, 87, and 91 on his first three physics exams. If Leo scores 95 on his fourth exam, what will be the new average, and by how much will this new score change his previous average?


Let \(r\) and \(s\) denote the two real roots of the equation \(x^2 - 2x\sqrt{5} + 2 = 0\). Determine \(r^6 + s^6\).


Let $a$, $b$, $c$, $x$, $y$, and $z$ be real numbers satisfying the equations:
\begin{align*}
  17x + by + cz &= 0 \\
  ax + 29y + cz &= 0 \\
  ax + by + 53z &= 0.
\end{align*}
Assume that $a \ne 17$ and $x \ne 0$. Determine the value of
\[
\frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53}.
\]


In Central High School, a dodgeball league includes 12 students, among them are Dexter and Sam. Students split into two teams of 6 players each during their sports period. Over the course of a term, each group of six players team up exactly once. Determine how many times Dexter played in the same team as Sam?


Let $m$ be the integer such that $0 \le m < 41$ and $5m \equiv 1 \pmod{41}$. Calculate $\left(3^m\right)^2 - 3 \pmod{41}$.


If $\log (xy^4) = 2$ and $\log (x^3y) = 2$, what is $\log (xy)$?


12 coins are simultaneously flipped. What is the probability that heads are showing on at most 3 of them?


What is the area, in square units, of a trapezoid bounded by the lines $y = x + 1$, $y = 15$, $y = 8$ and the $y$-axis?


Let \(\mathbf{p}, \mathbf{q},\) and \(\mathbf{r}\) be unit vectors such that
\[
\mathbf{p} + 2\mathbf{q} + \mathbf{r} = \mathbf{0}.
\]
Find the angle between \(\mathbf{p}\) and \(\mathbf{r}\), in degrees.


Let $\mathbf{N}$ be a matrix such that
\[
\mathbf{N} \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}.
\]
Compute $\mathbf{N} \begin{pmatrix} 4 \\ 5 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 4 \\ 5 \end{pmatrix}}


If the least common multiple of $A$ and $B$ is $180$, and the ratio of $A$ to $B$ is $2:5$, then what is their greatest common divisor?


A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit "7-heavy" whole number?


Evaluate the expression $\log_{y^3}{x}\cdot\log_{x^4}{y^3}\cdot\log_{y^5}{x^2}\cdot\log_{x^2}{y^5}\cdot\log_{y^3}{x^4}$ and express it as $b\log_y{x}$ for some constant $b$.


How many positive integers less than $200$ are multiples of either $3$ or $11$, but not both at once?


By starting with one hundred million and alternatively dividing by 2 and multiplying by 5, Rahim created a sequence of integers that starts 100000000, 50000000, 250000000, 125000000, and so on. What is the last integer in his sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as possible.


An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB$ and $\angle DBC = \angle DCB = \frac{k}{2} \cdot \angle ABC$, where $k$ is a constant coefficient, find the value of $k$ when $\angle BAC = 60^\circ$.


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the original triangle, dividing the triangle into a rectangle and two smaller right triangles. The area of one of the two smaller right triangles is $n$ times the area of the rectangle. The side lengths of the rectangle are in a ratio of 1:3. What is the ratio of the area of the other small right triangle to the area of the rectangle? Express your answer as a fraction in terms of $n$.


A list of five positive integers has all of the following properties:

$\bullet$ The only integer in the list that occurs more than once is $7,$ and it occurs three times.
$\bullet$ Its median is $10,$
$\bullet$ Its average (mean) is $11.$

What is the largest possible integer that could appear in the list?


How many distinct digits can appear as the units digit of an integral perfect-square number when considered modulo 16?


Piravena needs to make a trip from $X$ to $Y$, then from $Y$ to $Z$, and then from $Z$ to $X$. The cities form a right-angled triangle, with $Z$ a distance of 4000 km from $X$ and $Y$ a distance of 5000 km from $X$. To take a bus, it costs Piravena $\$0.20$ per kilometer. To take an airplane, it costs her a $\$150$ booking fee, plus $\$0.15$ per kilometer. Determine the distance she travels for her complete trip.

Solution: \boxed{12000 \text{ km}}


Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]


Write the following expression as a polynomial: $$(3x^2+4x+8)(x-2)-(x-2)(x^2+5x-72)+(4x-15)(x-2)(x+6).$$

Solution: \boxed{6x^3 - 4x^2 - 26x + 20}


In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to the nearest thousandth.


The graphs of $y=|x-1|$ and $y=-x^2-4x-3$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.


The values of $x$ and $y$ are always positive, and $3x^2$ and $y$ vary inversely. If $y$ is 30 when $x$ is 3, find the value of $y$ when $x$ is 6.


A damaged portion of a mathematical worksheet hides the coefficient of $x$ in a quadratic equation. The equation is known to have two distinct positive, integer solutions. Determine the sum of all possible integers that could be obscured by the damage.

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label("$=24$",(3,1.5),E);
[/asy]


In the diagram below, trapezoid $WXYZ$ has $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. Given $YZ = 15$, $\tan Z = 2$, and $\tan X = 2.5$, what is $XY$?

[asy]
pair WW, X, Y, Z;
Z = (0,0);
Y = (15,0);
WW = (15,30);
X = (27,30);
draw(WW--X--Y--Z--WW);
label("$W$", WW, N);
label("$X$", X, N);
label("$Y$", Y, S);
label("$Z$", Z, S);
[/asy]


Square $ABCD$ has an area of $256$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $50$, what is the area of triangle $ABF$?

Solution: \boxed{18 \text{ units}^2}


An isosceles triangle has side lengths 7 cm, 7 cm, and 12 cm. The longest side of a similar triangle is 30 cm. Calculate the perimeter of this larger triangle.


The product of three different positive integers is equal to $5^4$. What is the sum of these three integers?


Define a function $g(x),$ for positive integer values of $x,$ by
\[
g(x) = \left\{
\begin{aligned}
\log_3 x & \quad \text{if } \log_3 x \text{is an integer} \\
1 + g(x + 1) & \quad \text{otherwise}.
\end{aligned}
\right.
\]
Compute $g(50)$.


What is the value of the sum $\frac{3}{4}+\frac{3^2}{4^2}+\frac{3^3}{4^3}+ \ldots +\frac{3^{10}}{4^{10}}$? Express your answer as a common fraction.


If $a$, $b$, and $c$ are positive numbers with $ab = 32$, $ac = 48$, and $bc = 80$, what is $abc$?


In rectangle $ABCD$, $AB = 6$ cm and $BC = 3$ cm, $M$ is the midpoint of $\overline{CD}$. Let $O$ be the intersection of $\overline{AC}$ and $\overline{BM}$. Determine the ratio of $OC$ to $OA$.


If the two roots of the quadratic $5x^2 + 4x + k$ are $\frac{-4 \pm i \sqrt{379}}{10}$, what is $k$?


Solution: \boxed{\frac{\log_3 7}{4}}


Andrew's grandfather's age is fifteen times Andrew's age. Yet, when Andrew was born, his grandfather was already 55 years old. How many years old is Andrew now?


A positive real number $x$ is such that
\[
\sqrt{1-x^2} + \sqrt{1+x^2} = 2
\]
Find $x^4$.


There are three ${\bf positive}$ integers $c$ for which the equation $$7x^2 + 17x + c = 0$$ has rational solutions. What is the product of those three values of $c$?


Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9?


Suppose that $p$ is prime and $1009_p + 307_p + 115_p + 126_p + 7_p = 143_p + 274_p + 361_p$. How many possible values of $p$ are there?


There are 8 blue chips, 5 red chips, 4 yellow chips, and 3 green chips in a bag. One chip is drawn from the bag and then returned to the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Given six positive integers from a list of ten positive integers as $3, 4, 5, 7, 8, 9$, what is the largest possible value of the median of this list of ten positive integers?


Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be nonzero vectors, no two of which are parallel, 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 \(\mathbf{c}\). Find \(\sin \theta\).


Alex has 7 pieces of paper, each with a different math problem. In how many ways can he distribute these problems to his 9 friends, if each friend can receive at most one problem?


How many positive factors of 180 are perfect squares?


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

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


A circular garden has a diameter of 10 meters. Calculate half the area of this garden and express your answer in terms of $\pi$.


Given that the roots of the polynomial $x^2 - 2x - 1$ are also roots of the polynomial $x^7 - bx - c$, find the product $bc$.


Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[
\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 12 & 0 \\ 0 & 12 \end{pmatrix}.
\]
Find the smallest possible value of $|p| + |q| + |r| + |s|$.


Simplify the following expression:

$$7x + 9 - 2x + 3y$$

Solution: \boxed{5x + 3y + 9}


Determine the scalar $k$ such that for any vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ satisfying $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$, the equation
\[ k (\mathbf{v} \times \mathbf{u}) + \mathbf{v} \times \mathbf{w} + \mathbf{w} \times \mathbf{u} = \mathbf{0} \]
holds.


Jane starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ and wants to find the three-digit number formed by the 998th, 999th, and 1000th digits. Help her determine what those digits are.


Consider two points on the surface of a sphere, representing Earth. Point $A$ is located at $0^\circ$ latitude and $90^\circ \text{ E}$ longitude, and point $B$ is at $30^\circ \text{ N}$ latitude and $90^\circ \text{ W}$ longitude. Calculate the degree measure of $\angle ACB$ where $C$ is the center of the Earth.


Find the minimum value of $3x^2 - 12x + 908$.


How many four-digit positive integers have at least one digit that is a 5 or a 7?


If $5x + 8 \equiv 3 \pmod{19}$, then $5x + 9$ is congruent $\pmod{19}$ to what integer between $0$ and $18$, inclusive?


Simplify the following expression:
\[\frac{\sin 20^\circ + \sin 40^\circ + \sin 60^\circ + \sin 80^\circ + \sin 100^\circ + \sin 120^\circ + \sin 140^\circ + \sin 160^\circ}{\cos 10^\circ \cos 20^\circ \cos 40^\circ}.\]


Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.


In triangle $DEF$, $DE = 7$, $EF = 10$, and the length of median $DN$ is 5. Find $DF$.


When the product \((2x + 3y + 2)(x + 5y + 6)\) is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of \(y\)?


Let \( x, y, z \) be complex numbers such that
\[
xy + 5y = -25, \\
yz + 5z = -25, \\
zx + 5x = -25.
\]
Find all possible values of \( xyz \).


A right triangle has sides $a$, $b$, and hypotenuse $c$. Find the largest possible value of
\[
\frac{a+b}{\frac{ab}{c}}
\]


Determine \( q(x) \) if the graph of \( \frac{x^3-2x^2-5x+6}{q(x)} \) has vertical asymptotes at \( 3 \) and \( -3 \), no horizontal asymptote, and \( q(2) = 18 \).

Solution: \boxed{q(x) = -\frac{18}{5}x^2 + \frac{162}{5}}


The sides of a triangle have lengths \( 13, 17, \) and \( k \), where \( k \) is a positive integer. Determine the number of possible values for \( k \) for which the triangle is obtuse.


Calculate $X+Y$ (in base 10), given the following addition problem in base 7:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
 & & & 5 & X & Y_{7}\\
+& & & & 6 & 2_{7}\\
\cline{2-6}
 & & & 6 & 4 & X_{7}\\
\end{array}
\]


The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,000 miles from the center, the gravitational force is 800 Newtons. What is the gravitational force (in Newtons) that the Earth exerts on her when she's standing on a space station, 360,000 miles from the center of the earth? Express your answer as a fraction.


Consider all four-digit numbers (including leading zeros) from $0000$ to $9999$. A number is considered balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. Calculate the total number of such balanced four-digit numbers.


What is the least integer value of \(x\) such that \( \lvert 2x + 3\rvert \leq 12\)?


In trapezoid $EFGH$, sides $EF$ and $GH$ are equal. $GH$ is parallel to $EF$, $EF = 10$ units, and $GH = 22$ units. The perpendicular distance between $EF$ and $GH$ is 5 units. Calculate the perimeter of trapezoid $EFGH$.

Solution: \boxed{32 + 2\sqrt{61}}


What is the smallest three-digit multiple of 9 whose digits' sum is 27?


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


Find the sum of all real values of \( x \) that satisfy
\[
x = 2 - x + 2x^2 - 2x^3 + 2x^4 - 2x^5 + \dotsb.
\]


Suppose I have 7 shirts, 5 pairs of pants, and 4 ties. If an outfit requires a shirt and pants, and can either have one of the ties or not have a tie at all, how many different outfits can I make?


Consider the modified Fibonacci sequence defined as \( A_1 = 2, A_2 = 5 \), and each subsequent term is the sum of the two preceding terms. What is the remainder when the \( 50^{\mathrm{th}} \) term of the sequence is divided by 9?


Determine the average student headcount for the spring terms of the `02-`03, `03-`04, `04-`05, and `05-`06 academic years. Additionally, calculate the percentage change in student headcount from spring `02-`03 to spring `05-`06.

For the following years:
- '02-'03: 10,000 students in Spring,
- '03-'04: 11,000 students in Spring,
- '04-'05: 9,500 students in Spring,
- '05-'06: 10,500 students in Spring.


Determine how many possible integral values of $k$ make a triangle with sides of lengths $8, 17,$ and $k$ obtuse.


A jar of maple syrup which is 4 inches in diameter and 5 inches high sells for \$1.20. At the same rate, what would be the price for a jar that is 8 inches in diameter and 10 inches high?


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.1\overline{35}$?


How many integers $n$ satisfy the condition $200 < n < 300$ and $n$ has the same remainder whether it is divided by $7$ or by $9$, and additionally, the remainder must be less than $5$?


Expand the product $(3x + 4y)(2x - 5y + 7)$.

Solution: \boxed{6x^2 - 7xy + 21x - 20y^2 + 28y}


Simplify the expression $\frac{2-3i}{4-5i}$ where $i^2 = -1$.

Solution: \boxed{\frac{23}{41} - \frac{2}{41}i}


What is the least integer greater than $\sqrt{500}$?


Circle $\Gamma$ is the circumcircle of $\triangle ABC$ and also serves as the incircle of $\triangle XYZ$. $X$, $Y$, and $Z$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. If $\angle A = 50^\circ$, $\angle B = 70^\circ$, and $\angle C = 60^\circ$, find the measure of $\angle YZX$.


A company needs to hire new employees so that 120 of the new employees will manage software projects, 90 will handle hardware integration, and exactly 40 of these employees will handle both tasks. (These 40 are included in the 120 and 90 mentioned above.) What is the minimum number of employees that the company needs to hire?


Let $a,$ $b,$ $x,$ and $y$ be real numbers such that $ax+by=5,$ $ax^2+by^2=11,$ $ax^3+by^3=25,$ and $ax^4+by^4=58.$ Find $ax^5+by^5.$


Below is a portion of the graph of a function, \( y = F(x) \):

\[ F(x) = \sqrt{\left| x - 1 \right|} + \left( \frac{10}{\pi} \right) \arctan(\sqrt{\left| x \right|}) \]

Determine the integer value of \( F(2) \).


Mary can mow a lawn in 3 hours and Tom can mow the same lawn in 6 hours. If Tom mows for 3 hours alone and then Mary joins him for another 1 hour, what fractional part of the lawn remains to be mowed?


If \( f(x) \) and \( g(x) \) are polynomials such that \( f(x) + g(x) = 3x - x^2 \), then what is \( g(x) \) if \( f(x) = x^2 - 4x + 3 \)?

Solution: \boxed{-2x^2 + 7x - 3}


What is the smallest whole number $b$ such that 125 can be expressed in base $b$ using only three digits?


Calculate the sum of the first eight terms in the geometric sequence starting with $\frac14$ and having a common ratio of 2.


What is the greatest common factor of 36 and 54?


Determine the area enclosed by the region defined by the equation \(x^2 + y^2 - 8x + 10y + 1 = 0\).


In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 10, sides $\overline{CD}$ and $\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $120^\circ$. What is the length of diagonal $\overline{AC}$?

Solution: \boxed{AC = \sqrt{867}}


How many different counting numbers will each leave a remainder of 7 when divided into 59?


You have $5$ red shirts, $5$ green shirts, $5$ blue shirts, $6$ pairs of pants, $8$ green hats, and $8$ red hats. All items are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, without having the same color of shirts and hats?


In a new diagram based on the previous problem, point F is added between points C and D. The triangles formed are ABE, ABF, AEF, BCF, BDF, and CDF. The shading remains the same, covering CDF. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded?


Jonathan has $m$ stickers, where $m$ is a positive integer with three digits. If he buys $6$ more, he will have a multiple of $9$. If he loses $9$, he will have a multiple of $6$. What is the smallest possible value of $m$?


A deck of fifty cards consists of five $1$'s, five $2$'s,..., and five $10$'s. A matching triplet (three cards with the same number) is removed from the deck. Calculate the probability that two randomly selected cards from the remaining deck also form a pair, where the result is presented as $m/n$ with $m$ and $n$ relatively prime. Determine the value of $m+n$.


Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]


Find the matrix that corresponds to reflecting over the vector $\begin{pmatrix} 4 \\ 1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{15}{17} & \frac{8}{17} \\ \frac{8}{17} & -\frac{15}{17} \end{pmatrix}}


How many positive integers less than $150$ are multiples of either $6$ or $9$, but not both at once?


The square root of $3x$ is greater than 4 and less than 6. How many integer values of $x$ satisfy this condition?


How many distinct, positive factors does $1320$ have?


Using the same binary operation $\diamondsuit$ defined by $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$, solve the equation $504 \,\diamondsuit\, (12\,\diamondsuit\, x)=50.$


Four partners in a business decide to split the profits of their company in the ratio 3:4:4:6. If the profit one year is $\$35,000$, what is the largest number of dollars received by any of the four partners?


Given positive integers $x$ and $y$ such that $x \neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, what is the smallest possible value for $x + y$?


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


Let \(\omega\) be a nonreal root of \( z^4 = 1 \). Find the number of ordered pairs \((a, b)\) of integers such that \( |a \omega + b| = 2 \).


What is the sum of all the two-digit primes that are greater than 20 but less than 80 and are still prime when their two digits are interchanged, and one digit is exactly double the other?


A woman labels the squares of a very large chessboard $1$ through $64$. On each square $k$, she puts $2^k$ grains of rice. How many more grains of rice are placed on the $20^{th}$ square than on the first $15$ squares combined?


A list of five positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?


Cora needs to buy 18 identical binders for her class and only has $160. There is a flat $5 entrance fee for shopping at the store where she plans to buy the binders. The price of each binder is the same whole-dollar amount. Assuming an $8\%$ sales tax is added to the price of each binder, what is the greatest possible price (in dollars) of a binder that would allow Cora to buy the binders?


Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + 2b + 3c = 1.$ Find the maximum value of $a^2 + b^2 + c^2.$


How many four-digit positive integers have at least one digit that is a 5 or a 7?


The mean of the set of numbers $\{90, 88, 85, 86, 87, y\}$ is 87.5. What is the median of this set of six numbers?


A coach has a team of $30$ players and wants to form teams for a mini-tournament. Each team must have an equal number of players and consist of no more than $8$ players. What is the minimum number of teams needed?


Each of the eight letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.


Let $P$ be a point inside triangle $ABC$, such that $P$ divides triangle $ABC$ into three smaller triangles of equal areas. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is 27, find the area of triangle $G_1 G_2 G_3$.


The sum of the interior numbers in the sixth row of Pascal's Triangle is 30. What is the sum of the interior numbers of the eighth row?


Consider the function $g$ defined by $g(x) = \frac{ax+b}{cx+d}$, where $a, b, c, d$ are real numbers and $a, c \neq 0$. The function satisfies $g(23)=23$, $g(101)=101$, and $g(g(x))=x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $g$.


Find the distance between the foci of the ellipse:
\[\frac{x^2}{56} + \frac{y^2}{14} = 8.\]

Solution: \boxed{8 \sqrt{21}}


A pyramid with a volume of 72 cubic inches has a square base with side $s$ and height $h$. If the side of the square base is tripled and the height is doubled, what is the volume of the modified pyramid?


Given that $16^{-1} \equiv 31 \pmod{101}$, find $256^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


Each outcome on a new spinner has equal probability, with digit options being 1, 4, and 8. If you spin the spinner four times, forming a four-digit number where each consecutive spin outcome assigns the thousands, hundreds, tens, and units digit respectively, what is the probability that the resulting four-digit number is divisible by 4? Express your answer as a common fraction.

[asy]
draw(Circle((0,0),10));
draw((0,0)--(8.7,-5));
draw((0,0)--(-8.7,-5));
draw((0,0)--(0,10));
label("1",(7,7),SW);
label("4",(-7,7),SE);
label("8",(0,-2),S);
draw((0,0)--(3,5),Arrow);
[/asy]


Let $\alpha_1, \alpha_2,$ and $\alpha_3$ be complex numbers such that:
\[\alpha_1+\alpha_2+\alpha_3=2,\]
\[\alpha_1^2+\alpha_2^2+\alpha_3^2=6,\]
\[\alpha_1^3+\alpha_2^3+\alpha_3^3=14.\]
Compute $\alpha_1^{7} + \alpha_2^{7} + \alpha_3^{7}$.


The seventh, eighth, and ninth terms of an arithmetic sequence are 4, 10, and 16, respectively. What is the sum of the first five terms of the sequence?


Two circles, one of radius 7 inches and the other of radius 3 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at \(4\pi\) inches per minute, the other crawling along the smaller circle at \(3\pi\) inches per minute. How many minutes will it take for them to meet again at point P?


In triangle $PQR$, the angle bisectors are $PS$, $QT$, and $RU$, which converge at the incenter $I$. Given that $\angle PRQ = 42^\circ$, find the measure of $\angle QIS$, in degrees.


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


A point $(x,y)$ is randomly and uniformly chosen inside the rectangle with vertices (0,0), (0,3), (4,3), and (4,0). What is the probability that $x + 2y < 6$?


Let $a$, $b$, and $c$ be the sides of a triangle, and $\alpha$, $\beta$, $\gamma$ be the angles opposite these sides respectively. If $a^2 + b^2 = 2500c^2$, calculate the value of:
\[\frac{\cot \gamma}{\cot \alpha + \cot \beta}.\]


Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards in a row. Each leopard has a unique collar color. The two shortest leopards have inferiority complexes and demand to be placed at the ends of the row. Moreover, the two tallest leopards, which have a superiority complex, insist on being placed next to each other somewhere in the row. How many ways can Marius arrange the leopards under these conditions?


An ordinary 6-sided die has a number on each face from 1 to 6. How many ways can I paint three faces of a die red, such that none of the sums of any two of these red faces equal 7?


Calculate the magnitude of the complex number \(\left(\frac{2}{3} + \frac{5}{6}i\right)^8\).


The parabola $y = ax^2 + bx + c$ has vertex $(h, k)$ where $h$ and $k$ are positive numbers, and a $y$-intercept at $(0, -k)$. Calculate the value of $b$ if $k = 2h$.


What is the value of $\frac{1}{2} \cdot \frac{4}{1} \cdot \frac{1}{8} \cdot \frac{16}{1} \cdot \ldots \cdot \frac{1}{8192} \cdot \frac{16384}{1}$?


If eight people decide to come to a concert, where five of them are only 1/3 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 7 people stayed the entire time?