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

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


The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$


A convex pentagon has interior angles with measures \( 2x+2 \), \( 3x-3 \), \( 4x+1 \), \( 5x \), and \( 6x-5 \) degrees. Find the measure of the largest angle.


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

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


How many 12-step paths are there from $E$ to $G$ which pass through $F$ and then $H$? Assume the points are placed as follows on a grid: $E$ at the top-left corner, $F$ 3 steps right and 2 steps down from $E$, $H$ 2 steps right and 2 steps down from $F$, and $G$ 3 steps right from $H$ at the bottom-right corner.


If $BBB_5$ can be expressed as $44_c$, where $B$ is a digit in base 5 and $c$ is a base greater than 6, what is the smallest possible sum $B+c$?


Let $M$ be the largest integer whose square has exactly $4$ digits when written in base 7.

What is $M$, expressed in base 7?


A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems.


Christina draws a pair of concentric circles. She draws chords $\overline{DE}$, $\overline{EF}, \ldots$ of the larger circle, each chord being tangent to the smaller circle. If $m\angle DEF = 85^\circ$, how many segments will she draw before returning to her starting point at $D$?


A soccer team has 23 players in total. 12 players start the game, while the remaining 11 are available as substitutes. The coach is allowed to make up to 5 substitutions during the game. Any one of the players in the game can be replaced by one of the substitutes, but substituting players cannot return to the game after being removed. The order of substitutions and players involved is considered in calculating the number of possible substitutions. Let $m$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Calculate the remainder when $m$ is divided by $100$.


Georgia issues license plates where the first five slots are for letters and the last two slots are for digits. Texas, on the other hand, issues license plates with the first four slots for letters followed by three slots for digits. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, how many more license plates can Georgia issue than Texas?


In a circle with center $O$ and a radius of 10 units, chord $EF$ has a length of 12 units and is parallel to diameter $GH$. Point $J$, outside the circle, is on line $GH$, such that $HJ = 20$ units and points $J, H, O, G$ are collinear. Determine the area of triangle $JEF$.

Solution: \boxed{48 \text{ square units}}


Imagine a scenario where 25 people attend a meeting. Each person shakes hands with at most 20 other people. Calculate the maximum number of handshakes if each handshake occurs only once between any two people.


An $\textit{annulus}$ is formed between two concentric circles with radii $R$ and $r$ where $R>r$. Let $\overline{OP}$ be a radius of the larger circle, let $\overline{PQ}$ be tangent to the smaller circle at point $Q$, and let $\overline{OR}$ be the radius of the larger circle that contains $Q$. Let $x = PQ$, $y = QR$, and $z = PR$. Calculate the area of the annulus. Express your answer in terms of $\pi$ and at most one of the variables $x, R, r, y, z$.

Solution: \boxed{\pi x^2}


What is the greatest common factor of all three-digit palindromes that are multiples of 3? (Note: A palindrome is a number that reads the same forwards as backwards.)


The positive integer divisors of 245, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7?


A very large number $y$ is given by $y = 2^33^24^35^36^67^58^69^6$. What is the smallest positive integer that, when multiplied with $y$, produces a product that is a perfect square?


Thirty tiles are numbered 1 through 30 and are placed into box $A$. Thirty other tiles numbered 15 through 44 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than or equal to 20 and the tile from box $B$ is either odd or greater than 35? Express your answer as a common fraction.


If the probability that a baby born in a certain hospital will laugh in the next day is 1/3, what is the probability that at least 3 babies out of a group of 7 babies will laugh tomorrow?


Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\]Among all such vectors $\mathbf{p},$ there exists constants $s$ and $v$ such that $\mathbf{p}$ is at a fixed distance from $s \mathbf{a} + v \mathbf{b}.$ Find the ordered pair $(s,v).$

Solution: \boxed{\left(\frac{9}{8}, -\frac{1}{8}\right)}


If the consecutive integers from $75$ to $1$ were written out as a continuous string of digits $$7574737271...,$$ what would be the $110^{\text{th}}$ digit?


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


In triangle $ABC$, angle $A$ measures $40^\circ$ and angle $B$ is twice angle $C$. What is the measure of angle $C$?


Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b) = 255$ and $\gcd(a,c) = 855$. Determine the smallest possible value of $\gcd(b,c)$.


How many divisors of 84 are also divisors of 90? (Take into account both positive and negative divisors.)


A circular clock face is marked with twelve equally spaced dots representing the hours. The minute hand points directly at 12 o'clock. What is the measure in degrees of the smaller angle formed between the minute hand and the hour hand when the time is 5:30?


Determine the value of $u + v$ if the sequence $3,~9,~15,~...,~u,~v,~39$ is an arithmetic sequence.


Given the five digits 1, 3, 5, 7, and 9, how many different positive two-digit integers can be formed using these digits if:
1. A digit may not be repeated in an integer.
2. The digit 5 cannot be used as the second digit.


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


Express 3125 as a sum of distinct powers of 2. What is the least possible sum of the exponents of these powers?


Find the next year after 2020 where the sum of the digits equals 4.


Compute the largest integer $k$ such that $2006^k$ divides $2006!$.


Find the smallest positive angle $x$ that satisfies this equation:
\[\sin 4x \sin 6x = \cos 4x \cos 6x\]
in degrees.


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


What is the greatest divisor of 456 that is smaller than 60 and also a factor of 108?


I have a drawer with 6 shirts, 7 pairs of shorts, and 8 pairs of socks in it. If I reach in and randomly remove four articles of clothing, what is the probability that I get two shirts, one pair of shorts, and one pair of socks?


Let \(C\) be the circle with equation \(x^2 - 8x - 4 = -y^2 + 2y\). Find the sum of the coordinates of the center \((a, b)\) and the radius \(r\) of \(C\).

Solution: \boxed{5 + \sqrt{21}}


What is the remainder when 928,927 is divided by 6?


June and Julia live 1.5 miles apart. It takes June 6 minutes to ride her bike directly to Julia's house. From Julia's house, Bernard's house is another 4 miles away. At the same rate, how many minutes would it take June to ride from her own house to Bernard's house after stopping at Julia's?

Solution: \boxed{22 \text{ minutes}}


Find the product and the sum of the solutions of the equation $-42 = -x^2 - 4x.$


Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $6$ more, she will have a multiple of $9$. If she loses $7$, she will have a multiple of $4$. Find the smallest possible value of $n$.


Simplify
\[\sin ^2 x + \sin^2 (x + y) - 2 \sin x \sin y \sin (x + y).\]

Solution: \boxed{1 - \cos^2 y = \sin^2 y}


A jar has $15$ red candies and $5$ blue candies. Terry picks two candies at random, then Mary picks one of the remaining candies at random. Calculate the probability that all picked candies are of the same color.


Find the smallest positive integer $b$ for which $x^2+bx+1800$ factors into a product of two binomials, each having integer coefficients.


What is the shortest distance from the origin to the circle defined by \(x^2 + 6x + y^2 - 8y + 18 = 0\)?

Solution: \boxed{5 - \sqrt{7}}


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


Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If the sum of the numbers at the vertices equals $1176$, determine the sum of the numbers written on the faces.


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that all its digits are different?


Compute $\left(\sqrt[3]{7}\right)^2 \cdot \sqrt[5]{7}$ and express it as $7$ raised to what power?


Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 4x^2 - 3x)(3x^2 - 8x - 5)$.


What is the base ten equivalent of $63524_7$?


The line $y = -\frac{5}{3}x + 15$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$ If $g(81) = 81,$ compute $[g(9)]^4.$


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that
\[\mathbf{a} + \mathbf{b} + 2 \mathbf{c} = \mathbf{0}.\] Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.


Express $\frac{47}{2^3 \cdot 5^7}$ as a terminating decimal.


Express as a common fraction: $$\cfrac{ \frac{3}{7}+\frac{4}{5} }{ \frac{5}{12}+\frac{2}{9}}.$$


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


Six points are placed inside or on a circle with a diameter of 1 unit. Determine \( b \), the smallest possible number such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than \( b \).


$\triangle PQR$ is similar to $\triangle STU$. $\overline{PQ}$ measures 10 cm, $\overline{QR}$ measures 15 cm, and $\overline{ST}$ is 6 cm. What is the length of $\overline{TU}$? Express your answer as a decimal to the nearest tenth.

Solution: \boxed{9.0 \text{ cm}}


What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$?


How many integers between 200 and 500 have both 22 and 16 as factors?


Determine the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2+\frac{1}{3+\frac{1}{x}}}.$$


Quantities \( a \) and \( b \) vary inversely. When \( a \) is \( 800 \), \( b \) is \( 0.5 \). What is the value of \( b \) when \( a \) is \( 3200 \)?


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.


Let $a, b, c$ be real numbers such that $a + b + c = 2$, and $a \ge -1$, $b \ge -2$, $c \ge -3$. Find the maximum value of
\[
\sqrt{4a + 2} + \sqrt{4b + 8} + \sqrt{4c + 14}.
\]


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


When the binary number $101100110011_2$ is divided by 8, what is the remainder (give your answer in base 10)?


Determine the real number $k'$ such that the following equation has no solutions in $t$ and $s$:
\[\begin{pmatrix} 5 \\ 7 \end{pmatrix} + 2t \begin{pmatrix} 3 \\ -5 \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} + s \begin{pmatrix} -2 \\ k' \end{pmatrix}\]


My school's math club has 10 boys and 12 girls. I need to select a team of 8 people to send to the state math competition. The team must include at least one boy and one girl. How many ways can I select the team?


Find the complex number $z$ such that
\[ |z - 2| = |z + 4i| = |z + 2i|. \]

Solution: \boxed{3 - 3i}


Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 75^\circ.$


Given twelve points on the circumference of a circle, how many distinct convex polygons with three or more sides can be formed by choosing some or all of these twelve points as vertices?


In the diagram, square $ABCD$ has sides of length $5,$ and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=y.$ Determine the measure of angle $BPC.$


Consider the parabolas defined by $y = 2(x - 2)^2$ and $x - 1 = 2(y + 1)^2$. Find the sum of all $x$ and $y$ coordinates of their intersection points.


Solution: \boxed{1 - 9x}


What is the value of \( x \) for which \( (3012 + x)^2 = x^2 \)?


Calculate the value of
\[\text{cis } 60^\circ + \text{cis } 68^\circ + \text{cis } 76^\circ + \dots + \text{cis } 148^\circ\]
and express it in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


Let $h(x) = x^3 - 3x + 2$. Suppose $k$ is a cubic polynomial such that $k(0) = 2$, and the roots of $k$ are the squares of the roots of $h$. Find $k(4)$.

Solution: \boxed{k(4) = 0}


Find the greatest common divisor of 7980 and 6237.


The sequence $1, 4, 5, 16, 17, 20, 21, \cdots$ consists of all positive integers which are powers of 4 or sums of distinct powers of 4. Find the $50^{\text{th}}$ term of this sequence.


Consider the first 30 rows of Pascal's triangle. Rows $4$ and $8$ comprise only even numbers except for the $1$s at each end, as does row $2$. Determine how many of the first 30 rows exhibit this property of consisting only of even numbers (again, excluding row $0$ and $1$).


Let $\mathbf{v} = \begin{pmatrix} 3 \\ 2 \\ -4 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$. The columns of a matrix are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}_3$ where $\mathbf{w}_3$ is the extension of vector $\mathbf{w}$ into three dimensions as $\begin{pmatrix} 2 \\ -3 \\ 0 \end{pmatrix}$, and $\mathbf{u}$ is a unit vector. Find the largest possible determinant of the matrix.


A rectangle with dimensions 15 cm by 12 cm has four quarter circles drawn with centers at the four corners. Additionally, a circle of radius 3 cm is drawn at the center of the rectangle. Calculate the area of the shaded region outside the central circle and inside the quarter circles. Express your answer in terms of $\pi$.

Solution: \boxed{27\pi \text{ cm}^2}


Given the five digits 3, 5, 7, 8, and 9, how many different positive two-digit integers can be formed using these digits if:
1. A digit may not be repeated in an integer,
2. The integer formed must be an odd number?


Perform the calculation $15_6 - 4_6 + 20_6$. Express your answer in base 6.


In triangle $DEF$, $DE = 6$, $EF = 8$, and $FD = 10$.

Point $Q$ is randomly selected inside triangle $DEF$. What is the probability that $Q$ is closer to $F$ than it is to either $D$ or $E$?


Find the number of complex numbers \( z \) satisfying \( |z| = 1 \) and
\[
\left| \frac{z}{\overline{z}} - \frac{\overline{z}}{z} \right| = 1.
\]


In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{27}.$ Find $\tan F.$


The second hand on a clock is 8 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.

Solution: \boxed{720\pi \text{ cm}}


A square has sides of length $\sqrt{2}$. On each side of this square, a semicircular arc is constructed. Calculate the perimeter of the region bounded by these semicircular arcs.


Solution: \boxed{(-\infty, 1) \cup (2, 3) \cup (4, 5) \cup (6, \infty)}


Evaluate $180 \div \left(12 + 13 \cdot 2\right)$.


How many positive integers smaller than $500{,}000$ are powers of $2$, but are not powers of $4$?


In a school district, there are 4 math clubs with 6, 8, 9, and 10 students respectively. Each club has three leaders. If I randomly choose one club and then randomly select four members of that club to receive a mathematics handbook, what is the probability that all three leaders are among those who receive the handbook?


Let $a,$ $b,$ $c$ be distinct real numbers such that
\[
\frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} = 0.
\]
Find all possible values of
\[
\frac{a}{(b - c)^3} + \frac{b}{(c - a)^3} + \frac{c}{(a - b)^3}.
\]


We have a triangle $\triangle ABC$ with a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 12$, $BK = 6$, and $BC = 15$, what is the area of $\triangle ABC$?


In how many different ways can four students stand in a straight line if three of the students refuse to stand apart from each other?


Consider the equation
\[(x - \sqrt[3]{20})(x - \sqrt[3]{60})(x - \sqrt[3]{120}) = 1.\]
Find the value of $r^3 + s^3 + t^3$ where $r,$ $s,$ and $t$ are the distinct solutions to this equation.


I flip a biased coin (which has a probability of showing heads as 0.3) twice and roll a regular six-sided die. What is the probability of getting at least one head in the coin flips and a 6 on the die?


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


A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. The radius of the sphere can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)?


$30$ oranges are required to make $50$ liters of orange juice. How many oranges are needed to make $20$ liters of orange juice? If the required oranges decrease by $10\%$ for every $10$ liters after the first $10$ liters, how many oranges are you expected to save?


Compute
\[
\left( 1 + \cos \frac{\pi}{4} \right) \left( 1 + \cos \frac{3\pi}{4} \right) \left( 1 + \cos \frac{\pi}{2} \right) \left( 1 - \cos^2 \frac{\pi}{4} \right).
\]


A box contains nine cards. Five of the cards are black on both sides, two cards are black on one side and red on the other, and two cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?


Write the expression $\frac{6+5d}{9} + 3$ as a single fraction.

Solution: \boxed{\frac{33 + 5d}{9}}


Let \( x \), \( y \), and \( z \) be positive real numbers such that \( 5x + 8y + 3z = 90 \). Find the maximum value of \( xyz \).


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


Let \( A = (1,0) \) and \( B = (4,3) \). Let \( P \) be a point on the parabola \( y^2 = 4x \). Find the smallest possible value of \( AP + BP \).


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

Let $n$ be the number of possible values of $g(50),$ and let $s$ be the sum of the possible values of $g(50).$  Find $n \times s.$


Calculate $\tan A \tan C$ given that in triangle $ABC$, the orthocenter divides altitude $\overline{BD}$ from vertex $B$ to side $AC$ into segments with lengths $HD = 8$ and $HB = 24$.


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


Find the quotient when $x^6 + 2x^2 + 8$ is divided by $x - 1$.

Solution: \boxed{x^5 + x^4 + x^3 + x^2 + 3x + 3}


How many three-digit whole numbers have at least one 5 or at least one 8 as digits?


The length of a rectangular garden exceeds three times its width by 30 meters, and the perimeter of the garden is 800 meters. What is the area of the garden in square meters?

Solution: \boxed{28443.75 \text{ square meters}}


In triangle $ABC$, $BC = 15 \sqrt{3}$ and $\angle C = 30^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.


An infinite geometric series has a common ratio of \( \frac{1}{4} \) and a sum of \( 40 \). What is the second term of the sequence?


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3, x_4\}$, so that the graph of $y=g(x)$ consists of just four points. Suppose those four points form a quadrilateral of area $50$. The graph of $y = 3g(3x)$ also consists of just four points. What is the area of the quadrilateral formed by those four points?


What is the value of $a$ for which $\frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} + \frac{1}{\text{log}_5a} = 1$?


A ball is dropped from a height of 20 feet. After each bounce, it reaches a height that is three-quarters of the height from which it fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


A student must choose a program of five courses from a list comprising English, Algebra, Geometry, History, Art, Latin, Biology, and Chemistry. The program must include English, at least one mathematics course, and at least one science course. In how many ways can this program be chosen?


A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?


In $\triangle ABC$, both sides $AC=BC=10$ and $AB=6$. Point $D$ is on line $AB$ such that $B$ is between $A$ and $D$ and the length $CD=12$. Find the length of $BD$.

Solution: \boxed{\sqrt{53} - 3}


In how many ways can 100 be written as the sum of four positive perfect squares if no square number is repeated and the order of the squares does not matter?


Calculate the value of $m+n$ where $\sum_{k=1}^{18} \sin 10k = \tan \frac{m}{n}$, with angles measured in degrees, and $m$ and $n$ are coprime positive integers such that $\frac{m}{n} < 90$.


Acme T-Shirt Company charges a setup fee of $40 and $8 per shirt printed. Beta T-Shirt Company offers a one-time discount of $20 on the total order and charges $12 per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


The giant sculpture of a famous comedian stands at 120 feet. A scale model of the sculpture, displayed in an art gallery, measures 8 inches tall and 2 inches wide. If the actual width of the sculpture is 40 feet, calculate the following:
1. How many feet of the sculpture does one inch of the model's height represent?
2. How many feet of the sculpture does one inch of the model's width represent?


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


For how many bases between two and twelve inclusive does the representation of $625_{10}$ have a final digit of 1?


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


Let $x$, $y$, and $z$ be real numbers such that
\[\cos x + \cos y + \cos z = 1 \quad \text{and} \quad \sin x + \sin y + \sin z = 1.\]
Find the sum of all possible values of $\cos 2x + \cos 2y + \cos 2z.$


A child has a set of blocks, each distinct and categorized by 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks differ from the 'plastic medium red circle' in exactly 2 ways?


How many positive multiples of 11 that are less than 2000 end with the digit 5?


The least common multiple of two numbers is 5040, and the greatest common divisor is 24. Given that one of the numbers is 240, what is the other number?


Calculate $4535_6 + 23243_6$. Express your answer in base $6$.


Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$


Among the test scores 88, 90, 94, 95, 96, 99, four belong to Cyprian and the other two to Margaret. Cyprian's mean score is 92. What is Margaret's mean score?


Find the coordinates of the center of the hyperbola \[\frac{(3y-6)^2}{8^2} - \frac{(4x-5)^2}{3^2} = 1.\]

Solution: \boxed{\left(\frac54, 2\right)}


In a square array of 25 dots arranged in a 5x5 grid, what is the probability that five randomly chosen dots will be collinear? Express your answer as a common fraction.


How many 5-letter words with at least two vowels can be constructed from the letters $A$, $B$, $C$, $D$, $E$, and $F$? (Note that $B$, $C$, $D$, and $F$ are consonants. Any word is valid, not just English language words, and letters may be used more than once.)


Let $c$ and $d$ be the roots of the equation $x^2 - nx + 3 = 0$. Suppose that $c + \frac{1}{d}$ and $d + \frac{1}{c}$ are the roots of the equation $x^2 - rx + s = 0$. Find $s$.


On a similar one-way single-lane highway, cars travel at the same speed and obey a modified safety rule: the distance from the back of the car ahead to the front of the car behind is exactly two car lengths for each 20 kilometers per hour of speed or fraction thereof. Assuming each car is 5 meters long and that cars can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye in one hour. Find the quotient when $N$ is divided by 10.


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


The graph of the quadratic \( y = ax^2 + bx + c \) has the following properties:
1. The maximum value of \( y = ax^2 + bx + c \) is 7, which occurs at \( x = 2 \).
2. The graph passes through the point \( (0, -7) \).
If the graph passes through the point \( (5, n) \), then what is the value of \( n \)?


Compute the sum:
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \le a < b < c.$)


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


In the diagram, \(O\) is the center of a circle with radii \(OA=OB=7\). A quarter circle arc from \(A\) to \(B\) is removed, creating a shaded region. What is the perimeter of the shaded region?

Solution: \boxed{14 + 10.5\pi}


The median of a set of consecutive odd integers is 179. If the greatest integer in the set is 195, what is the smallest integer in the set?


A club has 30 members and needs to elect 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Three of the members, Alice, Bob, and Charlie, will only serve as officers if at least two of them are elected as officers. In how many ways can the club elect its officers?


How many positive factors of 180 are perfect squares?


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$.


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


Ms. Thompson awards extra credit on a history quiz to her students with quiz grades that exceed the class mean. Given that 150 students take the same quiz, what is the largest number of students who can be awarded extra credit?


In a chemistry class, Mrs. Lee offers bonus points for students whose test scores exceed the class average. There are 150 students in the class. What is the maximum number of students who can receive these bonus points?


Given that $b$ is an even multiple of $7768$, find the greatest common divisor of $4b^2 + 55b + 120$ and $3b + 12$.


Given positive integers \(x\) and \(y\) such that \(\frac{1}{x} + \frac{1}{3y} = \frac{1}{9}\), what is the least possible value of \(xy\)?


What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,18]=90$?


Climbing the first flight of stairs takes Brian 25 seconds, and each following flight takes 8 seconds more than the preceding flight. Calculate the total time taken for Brian to climb the first seven flights of stairs.

Solution: \boxed{343 \text{ seconds}}


What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$?


Given a 30-60-90 triangle where the hypotenuse is $6\sqrt{2}$ units, find the area of the triangle.


Determine the maximum number of real roots for the polynomial
\[ 2x^n + x^{n-1} + \dots + x + 1 = 0, \]
where $n$ is a positive integer.


If the curve $y = \sin(x + 30^\circ)$ cuts the line whose equation is $y = \sin 30^\circ$, find the successive ratios of the segments formed. Assume $p < q$ and compute $(p, q)$ where they are relatively prime positive integers.


If we define $g(n)$ as the sum of all the positive divisors of the integer $n$, how many integers $j$ exist such that $1 \le j \le 4096$ and $g(j) = 1 + j + \sqrt{j}$?


What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -7, 10)$, and the other on the sphere of radius 40 with center $(-15, 20, -10)$?

Solution: \boxed{55 + \sqrt{1453}}


Let $x,$ $y,$ and $z$ be angles in degrees such as:
\begin{align*}
\sin x &= \cot y, \\
\sin y &= \cot z, \\
\sin z &= \cot x.
\end{align*}
Find the largest possible value of $\cos x.$

Solution: \boxed{\frac{\sqrt{5} - 1}{2}}


What is the product of the digits in the base 8 representation of $7254_{10}$?


Let $\{b_k\}$ be a sequence of integers such that $b_1 = 2$ and $b_{m+n} = b_m + b_n + mn + 1$, for all positive integers $m$ and $n$. Find $b_{12}$.


Suppose I have 8 shirts, 5 ties, 4 pairs of pants, and 2 jackets (with the option to wear no jacket). If an outfit requires a shirt, a pair of pants, and can optionally include a tie and/or a jacket, how many distinct outfits can I create?


Let \(p, q, r, s, t, u, v,\) and \(w\) be real numbers such that \(pqrs = 16\) and \(tuvw = 25.\) Find the minimum value of
\[(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2.\]


What is the remainder when $8x^4 - 18x^3 + 27x^2 - 14x - 30$ is divided by $4x-12$?


Consider the vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}.$ There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} 5 \\ 3 \\ -7 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).\] Find $r.$


Given two real numbers $p>1$ and $q>1$ so that $\frac{1}{p} + \frac{1}{q} = \frac{3}{2}$ and $pq = 6$, what is $q$?

Solution: \boxed{\frac{9 + \sqrt{57}}{2}}


Simplify
\[\frac{\tan^3 15^\circ + \cot^3 15^\circ}{\tan 15^\circ + \cot 15^\circ}.\]


In parallelogram $EFGH$, angle $E$ measures $125^\circ$. What is the number of degrees in the measure of angle $G$?


Let matrix $\mathbf{B}$ be such that
\[
\mathbf{B} \begin{pmatrix} 7 \\ 3 \end{pmatrix} = \begin{pmatrix} -7 \\ -3 \end{pmatrix}.
\]
Find $\mathbf{B}^4 \begin{pmatrix} 7 \\ 3 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 7 \\ 3 \end{pmatrix}}


At a women's doubles tennis tournament, there were four teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner and any member of her own team. What is the number of handshakes that occurred?


Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is:


Express $\frac{29}{160}$ as a terminating decimal.


What is the sum of the squares of the lengths of the $\textbf{medians}$ of a triangle whose side lengths are $9,$ $12,$ and $15$?


Let $x_1 < x_2 < x_3$ be the three real roots of the equation $\sqrt{2020} x^3 - 4041x^2 + 3 = 0$. Find $x_2(x_1+x_3)$.


There are 10 chairs in a row. Neither Mary nor James can choose the last chair. They each select their seat randomly from the remaining chairs. What is the probability that they don't sit next to each other?


Given the equation $2 \cdot f(x) + 5 \cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $6$ and the degree of $h(x)$ is $10$, determine the minimum possible degree of $g(x)$.


Suppose a real number $y$ satisfies \[\sqrt{64-y^2} - \sqrt{36-y^2} = 4.\] What is the value of \[\sqrt{64-y^2} + \sqrt{36-y^2}?\]


Define a modified Ackermann function \( A(m, n) \) with the same recursive relationships as the original problem:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned}
\right.\]
Compute \( A(3, 2) \).


Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\frac{35}{4}-x$.


Define a new operation $ \oplus $ by:
\[ a \oplus b = \frac{a + b}{1 + ab}. \]
Determine the value of:
\[ -1 \oplus (-2 \oplus (-3 \oplus (\dotsb \oplus (-999 \oplus -1000) \dotsb))). \]


How many two-fifths are there in $\frac{9}{3}$?


How many positive integers $N$ less than $2000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


In a revised shooting match, eight clay targets are organized into two columns with four targets each. A marksman must break all the targets with the following rules:
1) The marksman first selects a column from which a target is to be broken.
2) The marksman must break the lowest remaining unbroken target in the chosen column.
Determine the number of different sequences in which the eight targets can be broken.


Let $r_1$, $r_2$, and $r_3$ be the roots of the polynomial
\[ x^3 - 4x^2 + 5 = 0. \]
Find the monic polynomial, in $x$, whose roots are $3r_1$, $3r_2$, and $3r_3$.

Solution: \boxed{x^3 - 12x^2 + 135}


Jane wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property?


In a survey conducted by Jane, $78.4\%$ of the participants believed that squirrels cause property damage. Among these participants, $52.5\%$ incorrectly believed that squirrels frequently cause electrical fires. Given that these 31 participants were mistaken, find the total number of people Jane surveyed.


The numbers \(x\) and \(y\) are inversely proportional. When the sum of \(x\) and \(y\) is 60, \(x\) is three times \(y\). What is the value of \(y\) when \(x=-12\)?

Solution: \boxed{y = -56.25}


The number $1027$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$?


A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., and four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Determine the probability that two randomly selected cards from the remaining deck also form a pair, and express the result as the sum of the numerator and denominator of the simplified fraction.


Given that two altitudes of the scalene triangle DEF have lengths 6 and 18, find the maximum possible integer length of the third altitude.


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. In how many ways can this be done?


Evaluate the number of subsets of the set $\{1, 2, 3, \dots, 15\}$, including the empty subset, that are "spacy." A set is called "spacy" if it contains no more than one out of any three consecutive integers.


If $x$, $y$, and $z$ are positive integers such that $\gcd(x,y) = 270$ and $\gcd(x,z) = 105$, what is the smallest possible value of $\gcd(y,z)$?


Solution: \boxed{3^{456} - 81}


Below is the graph of $y = a \tan (b(x - c))$ for some positive constants $a$, $b$, and a phase shift $c$. The graph has a period of $\frac{\pi}{4}$, passes through the point $\left(\frac{\pi}{3}, -4\right)$, and has a vertical asymptote at $x = \frac{\pi}{4}$. Find the value of $4ab$.


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

size(6cm);

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

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

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

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

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

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

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

[/asy]


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


A company has daily operating costs of $\$600$, pays each worker $\$20$ per hour, and each worker makes 6 widgets per hour. The widgets are sold at $\$3.50$ each. What is the minimum number of workers the company must hire to make a profit during a 10-hour workday?


Let quadrilateral $ABCD$ be a rhombus with area $50$ square units. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$ respectively. Diagonal $AC$ is drawn. Calculate the area of triangle $AEC$.

Solution: \boxed{12.5 \text{ square units}}


Given that \(\sec x - \tan x = \frac{5}{3},\) find all possible values of \(\sin x.\)


What is the domain of the function $g(x)=\frac{x+6}{\sqrt{x^2-4x-5} \cdot (x-3)}$?

Solution: \boxed{(-\infty, -1) \cup (5, \infty)}


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le16$, then what is the largest possible value of $a+b+c$?


Compute the smallest base-10 positive integer greater than 10 that is a palindrome when written in both base 2 and base 4.


Emily organized her playing cards with two suits (Hearts and Spades) in the following repeated cycle: $$A♥, A♠, 2♥, 2♠, 3♥, 3♠, \ldots, K♥, K♠, A♥, A♠, \ldots.$$ What is the 42nd card in this sequence?


Suppose the numbers \[4 - \sqrt{5}, \; -4 - \sqrt{5}, \; 2 + 2\sqrt{3}, \; 2 - 2\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. Determine the smallest possible degree of the polynomial.


On a balance scale, $4$ red balls balance $8$ green balls, $3$ orange balls balance $6$ green balls, and $8$ green balls balance $6$ blue balls. How many green balls are needed to balance $3$ red, $2$ orange, and $4$ blue balls?


Determine the product $mnp$ for integers $m$, $n$, and $p$ such that the equation $b^8xz - b^7z - b^6x = b^5(c^5 - 1)$ is equivalent to the equation $(b^mx-b^n)(b^pz-b^3)=b^5c^5$.


A sequence begins with the number 1,024,000. Each term of the sequence is obtained by dividing the previous term by 4. What is the last integer in this sequence?


In a physics problem solving session, Julia encounters a quadratic \(x^2 - tx + q\) with roots \(u_1\) and \(u_2\). She discovers that for some unknown reason, \(u_1+u_2 = u_1^2+u_2^2 = u_1^4+u_2^4\). Intrigued, she decides to calculate the maximum possible value of \(\dfrac{1}{u_1^{2009}} + \dfrac{1}{u_2^{2009}}\).


Let $r(x)$ have domain $\{0, 1, 2, 3\}$ and range $\{1, 3, 5, 7\}$. Define $s(x) = x + 2$, with a domain of $\{2, 3, 4, 5\}$. Determine the sum of all possible values for $s(r(x))$.


A box contains 3 red balls, 3 white balls, and 3 black balls. They are drawn one at a time. What is the probability that all of the draws alternate in colors without repeating any color consecutively?


A company has 48,297,858 pencils and wants to distribute them equally in boxes containing 6 pencils each. How many pencils will be left over?


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 $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.


In a larger regional bowling competition, the top 6 bowlers enter the playoff matches. Initially, bowler #6 competes against bowler #5. The winner of this match plays against bowler #4, and this process continues with each subsequent winner facing the next higher-ranked bowler. The loser of each game receives the next lowest prize, starting with the 6th prize for the loser of the first game. This sequence continues until the winner of the final match receives the 1st prize, and the loser gets the 2nd prize. In how many different sequences can the prizes be distributed among bowlers #1 through #6?


In triangle $ABC$, $\angle A = 45^\circ$, $\angle B = 90^\circ$, and $AC = 6$. Find $BC$.


What is the smallest positive integer $n$ for which $n^2$ is divisible by 24 and $n^3$ is divisible by 450?


How many prime numbers between 50 and 100 have a prime remainder when divided by 9?


Determine the real number \( k \) such that the equation
\[
\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix} + s \begin{pmatrix} 2 \\ k \end{pmatrix}
\]
does not have any solutions in \( t \) and \( s \).


Pedro sells five enchiladas and two tacos for $\$$4.30 and he sells four enchiladas and three tacos for $\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of four enchiladas and five tacos? Express your answer as a decimal to the nearest hundredth.


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \frac{1}{3} x.$ On the second day, he increased the object's new length from the previous day by $\frac{1}{4}$; on the third day by $\frac{1}{5}$, and so on, with each day increasing the object's length by the next increment in the series $\frac{1}{n+3}$ for the $n^{\text{th}}$ day. If by the $n^{\text{th}}$ day Barry wants the object's length to be exactly 50 times its original length, what is the value of $n$?


The equations $x^3 + Cx^2 + 20 = 0$ and $x^3 + Dx + 100 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$


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


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


$ABCDEFGH$ is a cube where each side has length $a$. Find $\sin \angle GAC$.


Find the smallest positive four-digit number that is divisible by each of the four smallest odd prime numbers.


Find the number of distinct numbers in the list
\[
\left\lfloor \frac{1^2}{2000} \right\rfloor, \ \left\lfloor \frac{2^2}{2000} \right\rfloor, \ \left\lfloor \frac{3^2}{2000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{2000} \right\rfloor.
\]


Let \( p, q, r, s, t, u, v, \) and \( w \) be distinct elements from the set \[
\{-11, -8, -6, -1, 1, 5, 7, 12\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


The sum of the $x$-coordinates of the vertices of a quadrilateral in the Cartesian plane equals $20$. Find the sum of the $x$-coordinates of the midpoints of the sides of the quadrilateral.


A standard deck of 52 cards contains cards of ranks (Ace through King) and suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, $\clubsuit$), with each suit perfectly represented by all ranks. In a randomly shuffled deck, what is the probability that the top four cards are either all $\spadesuit$ or all $\clubsuit$?


Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^6 x - \cos^6 x = \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x}.\]


An equilateral triangle has an area of $100\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is increased by 3 cm, by how many square centimeters is the area increased?

Solution: \boxed{32.25\sqrt{3} \text{ cm}^2}


The number $1057$ can be written as $23q + r$ where $q$ and r are positive integers. What is the greatest possible value of $q - r$?


In a pentagon $FGHIJ$, there is a vertical line of symmetry. The coordinates of vertices $F$, $G$, $H$, and $J$ are given as $F(0,0)$, $G(0,6)$, $H(3,t)$, and $J(6,0)$ respectively. What is the $y$-coordinate, $t$, of vertex $H$ so that the area of pentagon $FGHIJ$ is 60 square units?


An assortment of 300 pencils is sold through a catalog for $\$28.50$. Shipping is an additional $\$8.25$. If a 10% bulk discount is applied to the total cost before shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole number.


Consider the parabola defined by the equation \(y=(x-3)^2+4\). The graph of this parabola is:
1. Rotated 180 degrees about its vertex.
2. Shifted 4 units to the right.
3. Shifted 3 units down.
Find the sum of the x-coordinates where the parabola intersects the x-axis.


What is the smallest prime whose digits sum to $22$?


Write the expression $\frac{6+5d}{9} + 3$ as a single fraction.


Madison has 15 boxes. Nine of the boxes contain pencils, five contain pens, and three boxes contain both pens and pencils. Additionally, four boxes contain markers, with two of these also having pencils but none having pens. No boxes contain all three items. How many boxes contain neither pens, pencils, nor markers?

Solution: \boxed{2 \text{ boxes}}


What is the value of the following expression: $2 - 5 + 8 - 11 + 14 - \cdots - 47 + 50 - 53 + 56$ ?


Triangle $PQR$ is similar to triangle $XYZ$. If $PQ=8$, $QR=16$, and $ZY=32$ units, what is the length of segment $XY$?


Twenty-five tiles are numbered 1 through 25 and are placed into box $A$. Thirty other tiles numbered 15 through 44 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 18 and the tile from box $B$ is either prime or greater than 40? Express your answer as a common fraction.


In a triangle, two of the side lengths are 10 and 15, and the angle between them is $60^\circ$. Find the length of the third side.


Find the matrix $\mathbf{N}$ that doubles the first row and triples the second row of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. If no such matrix $\mathbf{N}$ exists, provide the zero matrix.

Solution: \boxed{\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}}


An astronomy club has 25 members and it needs to select 3 officers: president, secretary, and treasurer.  Each person can hold at most one office.  Two of the members, Alice and Bob, will only serve together or not at all.  In how many ways can the club choose its officers?


What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$?


A region $S$ in the complex plane is defined by
\[
S = \{x + iy : - 1 \leq x \leq 1, - 1 \leq y \leq 1\}.
\]
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac12 + \frac12i\right)z$ is also in $S$?


Two circles each with a radius of 12 cm overlap such that the distance between their centers is 16 cm. Determine the length of the common chord of the two circles.


A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 7. A treasure chest contains $6532_7$ dollars worth of jewelry, $3201_7$ dollars worth of gold coins, and $526_7$ dollars worth of silverware. What is the total dollar amount that this pirate has accumulated? Express your answer in base 10.


What is the base-10 integer 784 when expressed in base 7?


Given $z = 2 + 5i$, calculate $(z^2) \cdot \overline{z}$, where $\overline{z}$ is the complex conjugate of $z$. (Note that $i^2 = -1.$)

Solution: \boxed{58 + 145i}


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


How many three-digit whole numbers have no 5's, no 6's, no 7's, and no 9's as digits?


Evaluate the ratio $\frac{10^{3000} + 10^{3004}}{10^{3001} + 10^{3003}}$ and determine which whole number it is closest to.


How many four-character words can be formed by arranging two consonants and two vowels, in the order consonant, vowel, vowel, consonant, and excluding 'Y' as a vowel?


Consider the function $g(x) = x^2 + 3\sqrt{x}$. Evaluate $3g(3) - g(9)$.

Solution: \boxed{-63 + 9\sqrt{3}}


Let $O$ be the origin. Determine if there exists a scalar $k$ such that for any points $A,$ $B,$ $C,$ $D,$ and $E$, the equation
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} + 2 \overrightarrow{OE} = \mathbf{0}\]
ensures that the five points $A,$ $B,$ $C,$ $D,$ and $E$ are coplanar. Find $k.$


The terms $120, b, \frac{1}{5}$ are the first, second, and third terms, respectively, of an arithmetic sequence. If $b$ is positive, what is the value of $b$?


When two fair 8-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and the individual numbers $a$ and $b$ are all divisible by 4?


Let $a, b,$ and $c$ be real numbers such that \[a + b + c = 6\] and \[a^2 + b^2 + c^2 = 18.\] What is the difference between the maximum and minimum possible values of $c$?


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


Two of the roots of
\[ ax^3 + (a + 2b) x^2 + (b - 3a) x + (10 - a) = 0 \]
are -2 and 3. Find the third root.


Jason is trying to remember the seven-digit combination to his safe. He knows that he only used digits 0 through 4 (possibly repeated), and that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try?


Calculate $\sqrt{98x} \cdot \sqrt{18x} \cdot \sqrt{50x}$ and express your answer in simplest radical form in terms of $x$.

Solution: \boxed{210x \sqrt{2x}}


Evaluate $\left|\frac{3}{4} + 3i\right|$.


Determine the area of the circle described by the graph of the equation:
\[ r = 3 \cos \theta - 4 \sin \theta. \]


If \(x^2 + 3x + 7 = y - 5\) and \(x = -4\), what is the value of \(y\)?


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 18 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $G$ is the midpoint of $CD$. Determine the area of triangle $EFG$.


A Senate subcommittee has 4 Democrats and 6 Republicans. In how many ways can they sit around a circular table such that no two Democrats sit next to each other?


How much greater, in square inches, is the area of a circle of radius 25 inches than a circle of diameter 15 inches? Express your answer in terms of $\pi$.


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


A Senate committee consists of 4 Democrats, 4 Republicans, and 3 Independents. In how many ways can they sit around a circular table without restrictions? Note that two seatings are considered the same if one is a rotation of the other, and that all politicians are distinguishable.


A sign painter paints numbers for each of 150 houses, numbered consecutively from 1 to 150. How many times does the digit 9 appear in total on all the house numbers?


In a chess club at Maplewood Middle School, there are 12 members, including Sarah and Molly. Each week, the members split into two groups of six to study different strategies. Over the course of a year, each possible group of six members meets once. How many times did Sarah study in the same group as Molly?


Points $A$ and $B$ are chosen on the graph of $y = -x^2$ such that triangle $ABO$ is equilateral where $O$ is the origin. Find the coordinates of points $A$ and $B$ and the length of one side of triangle $ABO$.


Determine the remainder when $f(x) = x^4 - 6x^3 + 11x^2 + 8x - 20$ is divided by $x + 2$.


A right triangle with integer leg lengths is called "super cool" if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of super cool right triangles?


In an isosceles right triangle, one of the angles opposite the equal sides measures $45^{\circ}$. What is the measure of the third angle?


Find the domain of the function
\[ f(x) = \sqrt{2 - \sqrt{3 - \sqrt{4 - x}}}. \]

Solution: \boxed{[-5, 4]}


Let $\alpha$ and $\gamma$ be complex numbers such that $|\gamma| = 2$ and $\gamma \neq 2\alpha$. Find the maximum value of
\[
\left| \frac{\gamma - \alpha}{2 - \overline{\alpha} \gamma} \right|.
\]


There are $30n$ members in the Stonewall marching band, and when they line up in rows of 34, there are 2 members left over. If $n$ is an integer, and there are fewer than 1500 band members, what is the maximum number of people that could be in the Stonewall marching band?


Compute without using a calculator: $\dfrac{10!}{7!3!}$.


How many three-digit numbers are composed of three distinct non-consecutive digits, such that one digit is the average of the other two?


What is the sum of all the odd divisors of $360$?


Given $m$ and $n$ are positive integers such that $\gcd(m,n) = 18$, what is the smallest possible value of $\gcd(8m, 12n)$?


What is the maximum number of consecutive positive integers starting from 3 that can be added together before the sum exceeds 500?


Five people can mow a lawn in 8 hours. How many more people will be needed to mow another slightly larger lawn, which is 1.5 times the size of the first, in 6 hours, assuming each person mows at the same rate and it takes 1 hour for setup before mowing can begin?


Winnie now has 32 cherry lollipops, 150 wintergreen lollipops, 7 grape lollipops, and 280 shrimp cocktail lollipops. She wants to distribute these lollipops equally among her 14 friends, with each friend receiving the same number of lollipops. How many lollipops will Winnie have left after distributing as many as she can to her friends?


In a right triangle $ABC$ with angle $A$ equal to 90 degrees, the length of segment $AC$ is 5 cm, and the hypotenuse $BC$ is 13 cm. What is the length of side $AB$ and the area of triangle $ABC$?

Solution: \boxed{AB = 12 \text{ cm}, \text{ Area } = 30 \text{ cm}^2}


Find all solutions to the inequality \[\sqrt[3]{x}+ \frac{4}{\sqrt[3]{x}+ 4} \le 0,\] giving your answer in interval notation.

Solution: \boxed{(-\infty, -64) \cup \{-8\}}


Chandra has five bowls, each one a different color (red, blue, yellow, green, orange). She also has five glasses, but only three unique colors among them (red, blue, yellow), with one extra red and one extra blue glass. If she chooses a bowl and a glass from the cupboard, how many pairings are possible?


Out of 1500 people surveyed, $25\%$ do not like television, and out of those who do not like television, $15\%$ also do not like books. How many people surveyed do not like both television and books?


In the modified function machine, the input is 12. What is the output?

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{Input}$ = 12",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 3",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 25",(0,0));
draw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,"Multiply by 2",(width1/2,height/2));
draw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,"Subtract 5",(width1/2,height/2));
draw(f,(0,0)--(widthE,0)--(widthE,0)+lengthE*dir(angleE)--lengthE*dir(angleE)--cycle);
label(f,"$\mbox{Output}$ = ?",lengthE/2*dir(angleE) + (widthE/2,0));
add(shift(width1/2*left)*a); draw((0,0)--(0,-2),EndArrow(4));
add(shift(5*down + width2/2*left)*b);
add(shift((7+heightC/2)*down)*c); draw((0,-5)--(0,-7),EndArrow(4));
pair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);
draw("$\leq 25$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 25?$",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));
rightpt = rightpt + 2.5E+2S;
leftpt = leftpt + 2.5W+2S;
add(shift(leftpt+height*down+.3*width3*left)*d);
add(shift(rightpt+height*down+.7*width4*left)*e);
rightpt = rightpt+.75height*down+.7*width4*left;
leftpt = leftpt+.75height*down+.7*width3*right;
draw(leftpt--rightpt);
pair midpt = (leftpt+rightpt)/2;
draw(midpt--(midpt+2down),EndArrow(4));
add(shift(midpt+.65widthE*left+(2+lengthE*Sin(angleE))*down)*f);[/asy]


The sum of the lengths of all the edges of a regular tetrahedron is 72 cm. Find the number of cubic centimeters in the volume of the tetrahedron.


How many integers fall between $\sqrt{15}$ and $\sqrt{150}$ on a number line?


Three positive integers have an arithmetic mean of 30 and median of 25. If the median is 7 less than the largest number, what is the smallest number?


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


What is the greatest divisor of 546 that is smaller than 30 and also a factor of 126?


Find the quotient of the division $(4x^4 - 3x^3 + 6x^2 - 9x + 3)/(x + 2)$.

Solution: \boxed{4x^3 - 11x^2 + 28x - 65}


Given the ellipse described by
\[\frac{x^2}{36} + \frac{y^2}{16} = 1,\]
a chord $\overline{AB}$ passes through a focus $F$ of the ellipse. If the distance $AF = 2$, find the distance $BF$.


Suppose $x+2$ and $y-2$ are multiples of $8$.

What is the smallest positive integer, $n,$ for which $x^2-xy+y^2+n$ is a multiple of $8$?


If $7s + 6t = 156$ and $s$ is three more than half of $t$, what is $s$?


The Gnollish language now consists of 4 words: `splargh`, `glumph`, `amr`, and `drung`. In a sentence, `splargh` cannot come directly before `glumph`, and `amr` cannot come directly after `drung`. How many valid 3-word sentences are there in Gnollish?


What is the area of the parallelogram where the base is 20 ft, the height is 4 ft, and one of the angles is \(60^\circ\)?


A right triangle has a side length of 24 inches and a hypotenuse of 40 inches. A second triangle is similar to the first and has a hypotenuse of 120 inches. What is the length of the shortest side of the second triangle?

Solution: \boxed{72 \text{ inches}}


In the diagram, a square has a perimeter of 40, and adjacent to it is a triangle with a height of 40. Both the square and the triangle share one side length in common. Determine the length of the base of the triangle, denoted as \( x \), given that the square and the triangle have the same area. [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(5,0)--(5,5)--cycle);
draw((4.8,0)--(4.8,0.2)--(5, 0.2));
label("$x$", (4,0), S);
label("40", (5, 2.5), E);
[/asy]


Tom's graduating class now has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, what is the sum of all possible values of $x$, where $x$ represents the number of students in each row?


Find the product of all positive integer values of $c$ such that $9x^2 + 18x + c = 0$ has two real roots.


Find the greatest possible value of $x$ such that the least common multiple of $x$, $15$, and $21$ is $105$.


Find the remainder when the polynomial $x^{500}$ is divided by the polynomial $(x^2 + 1)(x^2 - 1).$


If $47 = x^6 + \frac{1}{x^6}$, then what is the value of $x^3 + \frac{1}{x^3}$?


Suppose a function $f(x)$ is defined on the domain $[-10, 2]$. If we define a new function $h(x)$ by $$h(x) = f(-3x),$$ what is the domain of $h(x)$? Express your answer in interval notation.

Solution: \boxed{\left[-\frac{2}{3}, \frac{10}{3}\right]}


If $m$ and $n$ are positive integers such that $\gcd(m,n) = 10$, then what is the smallest possible value of $\gcd(12m,18n)$?


Given vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) such that \(\|\mathbf{a}\| = \|\mathbf{b}\| = 1\), \(\|\mathbf{a} + \mathbf{b}\| = 2\), and
\[
\mathbf{c} - \mathbf{a} - 2\mathbf{b} = 4 (\mathbf{a} \times \mathbf{b}),
\]
find \(\mathbf{b} \cdot \mathbf{c}\).


Increasing the radius of a cylinder by $4$ units results in an increase in volume by $z$ cubic units. Increasing the height of the cylinder by $10$ units also results in an increase in volume by the same $z$ cubic units. If the original radius is $3$ units, what is the original height of the cylinder?


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?


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


In the trapezoid described, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $4:1$. If $AB + CD = 150$ cm, determine the length of segment $\overline{AB}$.

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


Solution: \boxed{-8388608 + 5^{27}}


Solve the inequality
\[\frac{x^2 - 16}{x - 4} \leq 0.\]

Solution: \boxed{(-\infty, -4]}


In a drawer, there are 20 distinguishable socks: 5 white, 5 brown, 5 blue, and 5 green. In how many ways can I choose a pair of socks such that the socks are of the same color and at least one of the socks in the pair is blue?


The volume of a different box is 360 cubic units, where \(a, b,\) and \(c\) are integers with conditions \(1 < a < b < c\). Determine the smallest possible value of \(b\).


Find all real numbers $k$ for which there exists a nonzero, 3-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 2 & 5 & 0 \\ 0 & 2 & 3 \\ 3 & 0 & 2 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\]

Solution: \boxed{2 + \sqrt[3]{45}}


Assume $x$ is a real number such that \( x + \frac{1}{x} = 5 \). Define \( S_m = x^m + \frac{1}{x^m} \). Calculate the value of $S_5$.


Consider a constant $k = \pi$. Define the function $$ f(x) = \lfloor \pi x \rfloor - \pi x. $$
What is the range of the function $f(x)$? Express your answer in interval notation.


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 7x^2 + 12x = 18$.

Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.


Solution: \boxed{\frac{\log_4 5}{2}}


How many integers between $200$ and $250$ have three different digits in increasing order? One such integer is $234$.


Let $D$ be the circle with equation $x^2 + 14y + 72 = -y^2 - 12x$. Find the center $(c,d)$ and the radius $s$ of $D$, and calculate $c+d+s$.

Solution: \boxed{-13 + \sqrt{13}}


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
\[
\frac{PQ^2 + PR^2 + QR^2}{p^2 + q^2 + r^2}.
\]


A whole number is said to be "11-heavy" if the remainder when the number is divided by 11 is greater than 7. What is the least three-digit 11-heavy whole number?


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


Given that $m$ and $n$ be any two even numbers, with $n$ less than $m$, find the largest integer which divides all possible numbers of the form $m^2 + n^2$.


A circular dartboard is divided into six regions of equal area but different scoring points. A dart has a probability of $\frac{1}{8}$ of landing in a specific region designated as "double score." What is the measure, in degrees, of the central angle for the region with "double score"?


Let \[Q(x) = (3x^3 - 27x^2 + gx + h)(4x^3 - 36x^2 + ix + j),\] where \(g, h, i, j\) are real numbers. Suppose that the set of all complex roots of \(Q(x)\) is \(\{1, 2, 6\}\). Find \(Q(7).\)


Triangle $PQR$ is an isosceles right triangle with the measure of angle $P$ equal to 90 degrees. The length of segment $PR$ is 8 cm. What is the area of triangle $PQR$, in square centimeters?


Let $A, B, C$ be distinct non-zero digits in base $7$ such that $\overline{ABC}_7 + \overline{BCA}_7 + \overline{CAB}_7 = \overline{AAA0}_7$. Find the value of $B+C$ in base $7$.


Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\]


Let $\triangle ABC$ have vertices $A$ at the origin $(0,0)$, $B$ at $(8,0)$, and $C$ at $(1,7)$. Suppose you guess that the point $P = (3,4)$ might be a point of interest (e.g., aiming to minimize some combined distance property). Calculate the sum of distances from $P$ to $A$, $B$, and $C$ and express it as $m\sqrt{p} + n\sqrt{q}$, where $m, n, p, q$ are integers. What is $m+n+p+q$?


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

Solution: \boxed{\begin{pmatrix} -\frac{6}{5} \\ \frac{2}{5} \end{pmatrix}}


Determine the outcome when $0.05$ is divided by $0.002$.


There are \(30m\) members in the Watsonville marching band, and when they line up in rows of 31, there are 7 band members left over. If \(m\) is an integer and there are fewer than 1500 band members, what is the maximum number of people that could be in the Watsonville marching band?


Find the number of ordered pairs of positive integers $(a, b)$ such that $a+b = 500$ and neither $a$ nor $b$ is an even number.


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
\[
\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1}.
\]


Simplify $\frac{4^5 + 4^3}{4^4 - 4^2}$ and express your answer as a common fraction.


Circle $S$ has a circumference of $16\pi$ meters. Segment $AB$ represents a diameter. If the measure of angle $SAC$ is $45^\circ$, determine the length, in meters, of segment $AC$.

Solution: \boxed{AC = 8\sqrt{2}}


A circle passes through the point $(0, 9),$ and is tangent to the parabola $y = x^2$ at $(3, 9).$  Find the center of the circle.

Solution: \boxed{\left(\frac{3}{2}, \frac{37}{4}\right)}


Completely factor the expression: $$x^{10} - 1024$$

Solution: \boxed{(x-2)(x+2)(x^4 + 2x^3 + 4x^2 + 8x + 16)(x^4 - 2x^3 + 4x^2 - 8x + 16)}


Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = 2x.
\]


Three cubes with faces numbered 1 through 6 are tossed, and the numbers shown on the top faces are added. What is the probability that the sum is odd? Express your answer as a common fraction.


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


What integer \( n \) satisfies \( 0 \leq n < 151 \) and
$$150n \equiv 93 \pmod{151}~?$$


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


In the diagram, $\triangle ABC$ is an isosceles triangle where $\angle ACB = \angle ABC$. If $AB = 6$ and $BC = 8$, what is the perimeter of $\triangle ABC$?


Amanda borrowed money from her friend to purchase a bicycle. Her friend agreed to let her repay the debt by mowing the lawn under the following conditions: her first hour of mowing is worth $2, the second hour $4, the third hour $6, the fourth hour $8, the fifth hour $10, the sixth hour $12, and then the cycle repeats. If Amanda repays her debt by mowing for 45 hours, how many dollars did she borrow?


Add $26_7 + 245_7.$ Express your answer in base 7.


Square EFGH has its center at $(5, -3)$ and an area of 9 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of 3. What are the coordinates of the vertex of the image of square EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(19.5, -13.5)}


Morgan has borrowed $200$ clams from Nina at a $12\%$ simple daily interest. Meanwhile, Olivia has borrowed $300$ clams from Peter at a $4\%$ simple daily interest. In how many days will Morgan and Olivia owe the same amount, assuming that they will not make any repayments in that time period?

Solution: \boxed{\frac{25}{3} \text{ days}}


Grandma has 12 pieces of art, including 4 paintings by Monet. What is the probability that all four Monet paintings will be placed consecutively if she arranges her art pieces in a row randomly?


Given the ranges \( -3 \leq x \leq -1 \) and \( 1 \leq y \leq 3 \), find the largest possible value of \( \frac{x+y}{x-y} \).


Suppose \( f, g, h, \) and \( j \) are 4, 5, 9, and 10 respectively, assigned not in order. What is the largest possible sum of the four products \( fg, gh, hj, \) and \( fj \) given that \( f < g \)?


In a town called Gridland, the streets are 25 feet wide, and the blocks are squares with a side length of 450 feet. Sarah runs on the inner side of the block, while her friend Tim runs on the outer side of the same block. How many more feet than Sarah does Tim run for each lap around the block?


The average of 12 numbers is 90. If the numbers 80, 85, and 95 are removed from this set of numbers, what is the average of the remaining numbers?


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


Find the area of the region bounded by the graph of $r = 2\sec \theta,$ the graph of $r = 2\csc \theta,$ the $x$-axis, and the $y$-axis.


Find the value of $t$ that satisfies $\frac{1}{t+3} + \frac{3t}{t+3} - \frac{4}{t+3} = 5$.


What is the greatest three-digit number that is two more than a multiple of 6 and four more than a multiple of 7?


Let \( a \) and \( b \) be real numbers. One of the roots of the cubic polynomial \( x^3 + ax + b = 0 \) is \( 1+i\sqrt{7} \). Find \( a + b \).

Solution: \boxed{a + b = 20}


In the diagram, $O$ is the center of a circle with radii $OP=OQ=6$. Calculate the perimeter of the shaded region, given that the arc $PQ$ encompasses half of the circle.

[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(1,0),E);

fill(Arc((0,0),1,0,180)--cycle,mediumgray);
draw(Arc((0,0),1,0,180));
fill((0,0)--(-1,0)--(1,0)--cycle,white);
draw((-1,0)--(0,0)--(1,0));
draw((-.1,0)--(-.1,-.1)--(0,-.1));
[/asy]

Solution: \boxed{12 + 6\pi}


Two mathematicians were both born in the last 600 years. Each lives (or will live) to be 120 years old, then dies. Each mathematician is equally likely to be born at any point during those 600 years. What is the probability that they were contemporaries for any length of time?


What is the sum of the digits of the base $8$ representation of $888_{10}$?


Find the value of $x$ if $x$ is positive and $x \cdot \lfloor x \rfloor = 50$. Express your answer as a decimal.


Let $a_1, a_2, \ldots$ be a sequence with the following properties:

(i) $a_1 = 3$, and

(ii) $a_{2n} = n \cdot a_n$ for any positive integer $n$.

What is the value of $a_{2^{100}}$?

Solution: \boxed{3 \cdot 2^{4950}}


What is the largest $5$-digit integer congruent to $15 \pmod{24}$?


Compute $\arccos (\cos \frac{3\pi}{2}).$


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that the penny, nickel, and half-dollar all come up heads, while the dime and the quarter do not both come up heads?


A digital video library contains 15 days of videos and occupies 24,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of video in this library take up? Express your answer to the nearest whole number.


Find $\begin{pmatrix} 5 \\ -6 \end{pmatrix} + \begin{pmatrix} -2 \\ 13 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 0 \\ 13 \end{pmatrix}}


Solution: \boxed{g(f(1)) = BA - B^3 + B^2}


Two circles, one with a radius of 6 inches and the other with a radius of 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 before they meet at point P again? Additionally, calculate the total distance each bug has traveled by their next meeting.


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


Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 12?


Solution: \boxed{m = 102}


What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?


In the diagram below, we have $\cos \angle RPQ = \frac{24}{25}$. What is $\cos \angle RPT$ where $T$ is directly opposite to $Q$ with respect to point $P$ on a straight line?

[asy]
pair R,P,Q,T;
T = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aCos(24/25))*(1.5,0);
dot("$T$",T,W);
dot("$Q$",Q,E);
dot("$R$",R,N);
dot("$P$",P,S);
draw(T--Q);
draw(P--R);
[/asy]


The polynomial $ax^4 + bx^3 + 48x^2 - 24x + 4$ has a factor of $4x^2 - 3x + 1.$  Find the ordered pair $(a,b).$

Solution: \boxed{(-16, -36)}


Suppose $g(x)$ is a function defined only for $-1 \le x \le 2$, and $g(x) = cx + d$ for constants $c$ and $d$ where $c > 0$. What is the range of $g$ in terms of $c$ and $d$?

Solution: \boxed{[-c + d, 2c + d]}


Let $g(n)$ be the sum of all the divisors of a positive integer $n$. Define a number $n$ as ultra-scarce if $g(g(n)) = n + 3$. Determine the number of ultra-scarce positive integers.


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

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


What is the ones digit of $8^{50}$ when expressed as an integer?


Consider the function $y=H(x)$ defined as $H(x) = (x-3)^2/2 - 2$. A portion of the graph is shown below, where the distance between grid lines is $1$ unit.

Compute $H(H(H(H(H(2)))))$.

[graph description not visually displayed]
Suppose through the graph you know that:
- $H(2) = 1/2$
- $H(1/2) = 5$
- $H(5) = 1/2$


Consider a fifth-degree polynomial:
\[ x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0 \]
where the coefficients are all integers. Define $m$ as the exact number of integer roots of this polynomial, counting multiplicity. Determine all possible values of $m$.

Solution: \boxed{0, 1, 2, 3, 5}


Uri buys three burgers and two sodas for $\$5.10$, and Gen buys two burgers and three sodas for $\$5.40$. How many cents does a soda cost if a side dish costs $\$0.30$ and Uri buys one side dish?


A segment with endpoints at $A(3, 3)$ and $B(15, 9)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair.

Solution: \boxed{(21, 12)}


Wei has updated the design of his company's logo to include more circles within the same square frame. Each circle is still tangent to two sides of the square and to its adjacent circles. If the square's side length remains 20 inches, how many square inches will be shaded now?

[asy]
size(100);
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,grey);
draw(circle((1,1),2/3)); draw(circle((2,1),2/3)); draw(circle((3,1),2/3));
draw(circle((1,2),2/3)); draw(circle((2,2),2/3)); draw(circle((3,2),2/3));
draw(circle((1,3),2/3)); draw(circle((2,3),2/3)); draw(circle((3,3),2/3));
fill(circle((1,1),2/3),white); fill(circle((2,1),2/3),white); fill(circle((3,1),2/3),white);
fill(circle((1,2),2/3),white); fill(circle((2,2),2/3),white); fill(circle((3,2),2/3),white);
fill(circle((1,3),2/3),white); fill(circle((2,3),2/3),white); fill(circle((3,3),2/3),white);
[/asy]

Solution: \boxed{400 - 100\pi}


If the consecutive integers from $60$ to $1$ were written as $$60595857...,$$ what would be the $80^{\text{th}}$ digit to be written?


A fair 12-sided die has its faces labeled from 1 to 12. Calculate the probability that the sum of the numbers resulting from rolling two such dice is 18.


Express as a common fraction: $$\cfrac{\frac{3}{7} + \frac{5}{8}}{\frac{5}{12} + \frac{2}{3}}.$$


Find the smallest positive solution to
\[\tan 4x + \tan 5x = \sec 5x\] in radians.


A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.


What is the coefficient of $x^3$ in the expansion of $$(x+3\sqrt{2})^8?$$


A ball is dropped from a height of 15 feet, and it bounces back to one-third of the height it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot?


Determine the smallest positive angle $y$ that satisfies $\sin 4y \sin 5y = \cos 4y \cos 5y,$ in degrees.


In how many ways is it possible to arrange the digits of 1225 to get a four-digit multiple of 5?


Of the numbers $\frac{9}{12}, \frac{5}{6}$, and $\frac{11}{12}$, which number is the arithmetic mean of the other two?


How many total days were there in the years 1999 through 2005?


The shaded region consists of 20 congruent squares. Suppose $RS = 10$ cm represents the length of a square's diagonal that includes exactly 4 of these shaded squares. What is the area of the entire shaded region?

Solution: \boxed{250 \text{ square cm}}


Let $f(x) = \sin x$. Calculate the area between the graph of $y = 4\sin(x-\pi)$ and the $x$-axis over the interval $[\pi, 3\pi]$.


Noelle needs to earn 30 homework points. For each of the first six points, she has to complete 1 assignment per point. For the next six points, she must complete 2 homework assignments per point, and so on. How many homework assignments does Noelle need to complete in total to earn these 30 points?


Solution: \boxed{\frac{9}{20} \text{ days}}


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


As $m$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $13m+4$ and $7m+2$?


In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\).


A ball bounces back up to $\frac{3}{4}$ of the height from which it falls. If the ball is initially dropped from a height of $160$ cm, determine after how many bounces the ball first rises to less than $20$ cm?


Calculate $\frac{2}{3} \cdot \frac{5}{7} \cdot \frac{9}{11} \cdot \frac{4}{13}$.


Karlanna places 420 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?


Find the value of $x$ such that $\sqrt{2x + 15} = 12$.


If $x^2 + 3x + 7$ is a factor of $x^4 + Rx^2 + S$, find $R + S$.


In triangle \(PQR\), \(PQ = 8\), \(QR = 15\), and \(PR = 17\). If \(S\) and \(T\) are points on \(\overline{PQ}\) and \(\overline{PR}\) respectively, such that \(PS = 3\) and \(PT = 10\), find the area of triangle \(PST\).


A line segment begins at $(2, 1)$ and is 8 units long. It ends at the point $(x, 7)$ where $x > 0$. What is the value of $x$?

Solution: \boxed{2 + 2\sqrt{7}}


What is the smallest integer value of $b$ such that $5$ is not in the range of $y=x^2+bx+10$?


A high school basketball team has 18 players, including a set of triplets: Bob, Bill, and Brenda. In how many ways can we choose 7 starters if exactly two of the triplets are in the starting lineup?


The mean of the set of numbers $\{90, 88, 85, 87, 89, y\}$ is 88. What is the median of the set of six numbers?


Find the smallest digit $d$ so that $586{,}d17$ is divisible by 9.


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


The numbers from 1 to 200, inclusive, are placed in a bag, and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1)? Express your answer as a common fraction.


In the given diagram, Alberto and Bjorn bike along straight paths with constant speeds. After 5 hours, considering that Alberto took a 30-minute break and Bjorn took a 15-minute break, how many more miles has Alberto biked than Bjorn? The graph shows Alberto's biking rate as 15 miles per hour and Bjorn's as 10.5 miles per hour.


Points \( P, Q, R, \) and \( S \) lie on a line, in that order. The distance \( PQ \) is 3 units, \( QR \) is 7 units, and \( PS \) is 22 units. What is the ratio of \( PR \) to \( QS \)?


If
\[
\frac{\cos^6 \alpha}{\cos^3 \beta} + \frac{\sin^6 \alpha}{\sin^3 \beta} = 1,
\]
then find the sum of all possible values of
\[
\frac{\sin^6 \beta}{\sin^3 \alpha} + \frac{\cos^6 \beta}{\cos^3 \alpha}.
\]

Solution: \boxed{\sin^3 \beta + \cos^3 \beta = 1}


Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between?


Given that $(1 + \sin t)(1 + \cos t) = \frac{9}{4}$ and $(1 - \sin t)(1 - \cos t) = \frac{m}{n} - \sqrt{k}$, where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$


Tom's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?


Determine the number of solutions to the equation
\[\tan (10 \pi \cos \theta) = \cot (10 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$


Consider the sequence that begins with 6400 and where each subsequent term is obtained by repeatedly dividing by 2. How many integers are in this sequence?


Alice and Bob are classical music enthusiasts. They share twelve identical CDs in their collections. Alice has twenty-three CDs in total. Eight CDs are in Bob's collection which are not in Alice's. How many CDs are in either Alice's or Bob's collection, but not both?


What is the sum of all integer solutions to $4 < (x-2)^2 < 36$?


Let $P = (5, \alpha)$ and $Q = (12, \beta)$ in polar coordinates. If $\beta - \alpha = \frac{2\pi}{3},$ find the distance $PQ.$


Find the greatest common divisor of $37500$ and $61250$.


A point $(x,y)$ is randomly picked from inside a rectangle with vertices $(0,0)$, $(6,0)$, $(6,1)$, and $(0,1)$. What is the probability that $x < 2y$?


The arithmetic mean of 15 scores is 75. When the highest and lowest scores are removed, the mean of the remaining scores rises to 78. If the highest score among the original 15 scores is 95, what is the lowest score?


Evaluate $\log_4 64\sqrt{16}$. Express your answer as an improper fraction.


Two cylindrical cans have the same volume. The height of one can is half the height of the other. If the radius of the taller can is 8 units, what is the radius of the shorter can? Provide your answer in simplest radical form.


Let
\[ f(x) = \frac{2ax}{3x + 4}. \]
Find the constant \( a \) such that \( f(f(x)) = x \) for all \( x \neq -\frac{4}{3} \).


There is an unlimited supply of congruent equilateral triangles made of colored paper, where each triangle is a solid color. A large equilateral triangle is constructed from four of these paper triangles, and two large triangles are considered distinguishable if they cannot be matched by translations, rotations, and/or reflections. Given that there are 8 different colors of triangles to choose from, how many distinguishable large equilateral triangles can be constructed?


Two scientists visit an observatory to watch the stars. They arrive independently at random times between 10 p.m. and midnight, and each stays for exactly $n$ minutes. The probability that either one arrives while the other is there is $30 \%$. Given $n = x - y\sqrt{z},$ where $x, y,$ and $z$ are positive integers, with $z$ not divisible by the square of any prime, find $x + y + z.$


For each value of \(x\), let \(f(x)\) be defined as the minimum value of the three numbers \(3x - 1\), \(-x + 4\), and \(2x + 5\). Find the maximum value of \(f(x)\).


Determine the value of $x$ such that
\[5 + \frac{5 + x}{3} + \frac{5 + 2x}{3^2} + \frac{5 + 3x}{3^3} + \dotsb = 13.\]


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


Convert the complex number $2 + 2i$ into its exponential form $re^{i \theta}$ and find $\theta$.


Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \[g(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a multiple of 3 and 4}, \\
3 & \text{ if } n \text{ is only a multiple of 3}, \\
1 & \text{ if } n \text{ is only a multiple of 4}, \\
0 & \text{ if } n \text{ is neither a multiple of 3 nor 4}.
\end{array}
\right.\]
Allie rolls the die four times and gets a 6, 3, 4, and 1. Betty rolls and gets 12, 9, 4, and 2. Compute the product of Allie's total points and Betty's total points.


What is the area enclosed by the graph of $|5x| + |3y| = 15$?


Henry rewrites the quadratic $16x^2 - 48x - 72$ in the form of $(ax + b)^2 + c,$ where $a,$ $b,$ and $c$ are all integers. What is $ab$?


How many factors of $18000 = 2^3 \cdot 3^2 \cdot 5^3$ are perfect squares?


In base $b$, there are exactly one hundred forty-four three-digit numbers whose digits are all distinct and each number includes the digit 1 at least once. What is $b$?


Determine the quadratic polynomial \( q(x) \) such that \( q(-3) = 0 \), \( q(6) = 0 \), and \( q(2) = -45 \).

Solution: \boxed{\frac{9}{4}x^2 - \frac{27}{4}x - 40.5}


Simplify the expression $3x - 5(2-x+y) + 4(1-x-2y) - 6(2+3x-y)$.

Solution: \boxed{-14x - 7y - 18}


The length of a rectangular garden is three times its width plus an additional 10 feet, and the perimeter of the garden is 400 feet. What is the area of the garden in square feet?


Each of the numbers $b_1,$ $b_2,$ $\dots,$ $b_{50}$ is $\pm 1.$  Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 50} b_i b_j.\]


The measures of angles $C$ and $D$ are both positive, integer numbers of degrees. The measure of angle $C$ is a multiple of the measure of angle $D$, and angles $C$ and $D$ are supplementary angles. How many measures are possible for angle $C$?


Let \( n \) be 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?


Suppose $x+1$ and $y-5$ are multiples of $7$.

What is the smallest positive integer, $n,$ for which $x^2+xy+y^2+3n$ is a multiple of $7$?


Find the number of real solutions of the equation
\[\frac{x}{150} = \sin x.\]


Suppose the point $(3,4)$ is on the graph of $y=\frac{f(x)}{3}$. Then there is one point which must be on the graph of $y=\frac{f^{-1}(x)}{4}$. What is the sum of that point's coordinates?


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

Solution: \boxed{4^x - 3^x}


A four-inch cube has each of its surfaces painted blue. This cube is then uniformly cut into one-inch cubes exactly along the edges marked in the provided diagram of a four-inch cube divided into one-inch cubes. How many of the resulting one-inch cubes have at least two of their faces painted blue?


Let $x, y, z$ be real numbers such that $9x^2 + 4y^2 + 25z^2 = 1$. Find the maximum value of
\[5x + 3y + 10z.\]


How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $5,$ $12,$ and $13$?


An environmental agency plans to expand its workforce to address three types of pollution: water, air, and soil. They need 90 new employees for water pollution monitoring, 80 for air pollution, and 50 for soil pollution. Some employees can handle multiple types of pollution: 30 can monitor both water and air, 20 can monitor both air and soil, and 15 can monitor both water and soil. Additionally, 10 employees are skilled enough to monitor all three types. Determine the minimum number of new employees the agency needs to hire.


The graph of $y = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are quadratic. Identify the value of $\frac{p(4)}{q(4)}$, given the following information:
- The rational function has a vertical asymptote at $x = -1$.
- The rational function has a horizontal asymptote, $y = -3$.
- There is a hole in the graph at $x = 6$.
- The graph intersects with the x-axis at $(2,0)$.


What is the least common multiple of the numbers 1428 and 924?


The equations \(x^3 + 2Ax + 20 = 0\) and \(x^3 + Bx^2 + 100 = 0\) share exactly two common roots. Find the product of these common roots expressed in the form \(a \sqrt[b]{c}\), where \(a\), \(b\), and \(c\) are positive integers, and determine the value of \(a + b + c\).


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 $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=y.$
Determine the measure of angle $BPC.$


Let $a$ and $b$ be complex numbers such that
\[\frac{a + b}{a - b} + \frac{a - b}{a + b} = 4.\] Find
\[\frac{a^4 + b^4}{a^4 - b^4} + \frac{a^4 - b^4}{a^4 + b^4}.\]


Suppose $w$ is a complex number such that $w^2 = 45-21i$. Find $|w|$.


For any vector $\mathbf{v}$ in $\mathbb{R}^3$, consider a new operation where
\[
\mathbf{i} \times (\mathbf{v} \times (\mathbf{i}+\mathbf{j})) + \mathbf{j} \times (\mathbf{v} \times (\mathbf{j}+\mathbf{k})) + \mathbf{k} \times (\mathbf{v} \times (\mathbf{k}+\mathbf{i})) = d \mathbf{v}.
\]
Determine the scalar $d$.


For some positive integer \( k \), when 90 is divided by \( k^2 \), the remainder is 10. What is the remainder when 150 is divided by \( k \)?


What is the largest positive integer \( n \) for which \( n^3 + 64 \) is divisible by \( n+10 \)?


On a complex plane map of a fictional continent, city A is located at the origin $0$, city B is at $3900i$, and city C is at $1170 + 1560i$. Calculate the distance from city C to city A on this plane.


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty cent coin. What is the probability that the penny, the dime, and the fifty cent coin all come up showing the same side?


Find all real values of $x$ that satisfy \[\frac{1}{x(x+2)}-\frac{1}{(x+2)(x+3)} < \frac{1}{4}.\] (Give your answer in interval notation.)

Solution: \boxed{(-\infty, -3) \cup (-2, 0) \cup (1, \infty)}


A bookstore has a sale on days of the month that are multiples of 3 (such as July 3, July 6, etc.). A shoe store has a sale every 7 days, starting from July 2. Determine how many times in July do the two stores have sales on the same date.


The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y^2}\]
for all positive real numbers $x$ and $y.$ If $f(40) = 30,$ find $f(60).$


At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per batch. The other details remain the same. How many full recipes should they prepare?


I have 12 distinguishable socks in my drawer: 5 black, 3 green, and 4 red. In how many ways can I choose a pair of socks, provided that I must select one sock of each color?


What is the positive difference of the solutions of $\frac{q^2 - 4q - 21}{q - 3} = 3q + 8$?


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


Container I holds 8 red balls and 4 green balls; container II holds 3 red balls and 5 green balls; container III holds 4 red balls and 6 green balls. A container is randomly selected and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction.


The integer $y$ has 8 positive factors. Both 18 and 20 are factors of $y$. What is the smallest possible value of $y$?


In triangle $DEF$, $d = 5$, $e = 8$, and $f = 6$. Let $J$ be the incenter of triangle $DEF$.

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

Solution: \boxed{\left(\frac{5}{19}, \frac{8}{19}, \frac{6}{19}\right)}


The sequence $b_1, b_2, \ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\log_4 b_1+\log_4 b_2+\cdots+\log_4 b_{10} = 2010,$ find the number of possible ordered pairs $(b,s).$


Let $R'$ be the set of all possible remainders when a number of the form $3^n$, where $n$ is a nonnegative integer, is divided by 500. Let $S'$ be the sum of the elements in $R'$. Find the remainder when $S'$ is divided by 500.


In triangle ABC, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular, and their lengths are $AM = 15$ and $BN = 20$. Find the length of side $\overline{AB}$.


Solution: \boxed{6xy + 30x + 8xz + 15y + 20z + 75}


Compute $\sqrt{48} \cdot \sqrt{27} \cdot \sqrt{8} \cdot \sqrt{3}$.


In a right triangle $\triangle XYZ$, medians are drawn from vertex $X$ to divide side $\overline{YZ}$ and from vertex $Y$ to divide side $\overline{XZ}$ into two equal parts. The lengths of these medians are $5$ units and $3\sqrt{5}$ units respectively. Determine the length of the hypotenuse $\overline{XY}$.

Solution: \boxed{\sqrt{56} = 2\sqrt{14}}


If $0.75= \frac{y}{210+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.


In convex quadrilateral $EFGH$, $\angle E = \angle G = \beta$, $EF = GH = 200$, and $EH \ne FG$. The perimeter of $EFGH$ is 720. Find $\cos \beta$.


Each page number of a 600-page book is printed one time. The first page is page 1 and the last page is page 600. When printing all of the page numbers, how many more 5's are printed than 9's?


The sum of four positive integers that form an arithmetic sequence is 50. Of all such possible sequences, what is the greatest possible third term?


Marie has $10$ identical candies and $5$ identical boxes. How many ways are there for her to distribute all of the candies among the boxes, if only the number of candies in each box matters?


Let $\theta$ be the angle between the planes $3x - 2y + z - 4 = 0$ and $9x - 6y - 4z + 2 = 0.$ Find $\cos \theta.$

Solution: \boxed{\cos \theta = \frac{35}{\sqrt{1862}}}


Let $z$ be a complex number such that
\[|z^2 + 9| = |z(z + 3i)|.\]Find the smallest possible value of $|z + i|$.


What is the smallest positive value of $x$ such that $x + 2004$ results in a palindrome?


What is the greatest common divisor of 3060 and 561?


Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2020$, and $a^2 - b^2 + c^2 - d^2 = 2020$. Find the number of possible values of $a$.


Let
\[ g(x) = (\cosh x)^3 + (\sinh x)^3. \]
Find the range of $g(x)$ for all real numbers $x$.

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


A bag contains 10 red, 12 green, 15 yellow, 18 blue, and 5 purple jelly beans. A jelly bean is selected at random. What is the probability that it is purple? Additionally, what is the probability that the jelly bean is either blue or purple?


Real numbers $a$ and $b$ are chosen such that $2 < a < b$ such that no triangle with positive area has side lengths $2, a,$ and $b$ or $\frac{1}{b}, \frac{1}{a},$ and $\frac{1}{2}$. What is the smallest possible value of $b$?

Solution: \boxed{3 + \sqrt{5}}


The cafe has enough chairs to accommodate $310_5$ people. If $3$ people are supposed to sit on one bench, how many benches does the cafe have?