I have fifteen books, of which I want to bring two to read on vacation. However, out of these, there are three specific books that cannot be paired together (let's say they are volumes of the same series). How many different pairs can I choose?


Given that $a$ is a multiple of $532$, find the greatest common divisor of $5a^3 + 2a^2 + 6a + 76$ and $a$.


Determine the matrix $\mathbf{M}$ that adds 5 to each element of the second column of any given 2x2 matrix $\mathbf{A}$. In other words, find $\mathbf{M}$ such that:
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b+5 \\ c & d+5 \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


Two circles of radius 12 cm overlap such that the distance between their centers is 16 cm. How long, in cm, is the common chord of the two circles?

Solution: \boxed{8\sqrt{5} \text{ cm}}


Murtha has decided to expand her pebble collection. She already has 3 pebbles to start with. On the first day, she collects two pebbles and on the second day, she collects three pebbles. On each subsequent day, she collects one more pebble than the previous day. How many pebbles will she have collected at the end of the fifteenth day?


How many perfect squares are two-digit and divisible by $5$?


Let $f(x) = x^4 + x^3 + x^2 + x + 1.$ Compute the remainder when $f(x^{10})$ is divided by $f(x)$.


How many rectangles can be formed where each vertex is a point on a 4x4 grid of equally spaced points?


How many even natural-number factors does $n = 2^2 \cdot 3^1 \cdot 7^2 \cdot 5^1$ have?


Two standard 8-sided dice are rolled. What is the probability that the sum rolled is a perfect square?


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


Find the matrix $\mathbf{N}$ that swaps the rows of a $2 \times 2$ matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & d \\ a & b \end{pmatrix}.\]
If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

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


An ellipse has its foci at $(-4, 1)$ and $(-4, 5).$ Given that it passes through the point $(1, 3),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$


A square pyramid has a base edge of 36 inches and an altitude of 2 feet. A smaller square pyramid, whose altitude is one-third of the original altitude, is cut away from the apex of the original pyramid. What is the volume of the remaining frustum as a fraction of the volume of the original pyramid?


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


Find the minimum value of $(\sin x + \cos x)^4.$


A rhombus has sides of length 60 units each and a shorter diagonal of length 56 units. What is the length, in units, of the longer diagonal?

Solution: \boxed{32\sqrt{11} \text{ units}}


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \begin{pmatrix} 2 & -5 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} -20 & 5 \\ 16 & -4 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ -4 & 1 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} \frac{16}{7} & -\frac{36}{7} \\ -\frac{12}{7} & \frac{27}{7} \end{pmatrix}}


If $a$ is a constant such that $9x^2 + 27x + a$ is the square of a binomial, then what is $a$?


Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n$ for all positive integers $n,$ find the first term.


How many positive integer multiples of \(3003\) can be expressed in the form \(10^j - 10^i\), where \(i\) and \(j\) are integers and \(0 \leq i < j \leq 50\)?


Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$.


Two different numbers are randomly selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. The probability that their sum is 14 would be greater if the number $n$ had first been removed from the set $S$. What is the value of $n$?


Consider the quadratic equation $x^2 - 6x + 11 = 23$. Find the value of $3a + 2b$ where $a$ and $b$ are the roots of the equation, with $a \geq b$.

Solution: \boxed{15 + \sqrt{21}}


Sandy wants to cover the exterior wall, which measures 8 feet wide by 10 feet high, and two identical roof faces, each 8 feet wide that slant upward at 45 degrees over an additional 7 feet, with special weather-resistant siding. Each sheet of siding measures 8 feet by 12 feet and costs \$32.80. Assuming Sandy can cut sections to fit as needed, calculate the total cost Sandy would need to spend to buy enough siding.


Suppose $z$ is a complex number such that $z^2 = -48+64i$. Find $|z|$.


Two circles with radii 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the shaded region between these circles. Express your answer in terms of \(\pi\).


In a chess tournament, the top 3 players enter a playoff. First #3 plays against #2. The loser receives the 3rd prize and the winner plays against #1. The loser of this final match receives the 2nd prize and the winner receives the 1st prize. In how many orders can players #1 through #3 receive the prizes?


Given acute angles $\alpha$ and $\beta$ such that $\tan \alpha = \frac{1}{3}$ and $\cos \beta = \frac{3}{5}$, find $\alpha + 2\beta$, measured in radians.

Solution: \boxed{\alpha + 2\beta = \pi - \arctan\left(\frac{13}{9}\right)}


A hyperbola is centered at (0,0) and opens either horizontally or vertically. It passes through the points $(4, -3),$ $(0, -2),$ and $(2, t).$ Find $t^2.$


Calculate the units digit of the quotient $\frac{4^{2065} + 6^{2065}}{7}$, given that $4^{2065} + 6^{2065}$ is a multiple of $7$.


For what values of $b$ is 3 not in the range of the function $g(x)=x^2 - bx + c$? Express your answer in interval notation.

Solution: \boxed{(-\sqrt{4c-12}, \sqrt{4c-12}) \text{ for } c \geq 3}


Consider the paper triangle whose vertices are $(0,0), (40,0),$ and $(20,30)$. The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle with the height from the apex to the base of the pyramid being $15$ units. Calculate the volume of this pyramid.


The 5-digit number $63\,47\square$ is a multiple of 6. Which digit is represented by $\square$?


A penny, nickel, dime, quarter, half-dollar, and dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.


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


How many four-digit numbers whose digits add up to $10$ are divisible by $11$?


Determine if the points $P = (1, -2, 1)$, $Q = (4, -7, 4)$, $R = (2, -1, -1)$, and $S = (5, -6, 2)$ form a parallelogram in space. If so, calculate the area of this parallelogram.


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

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

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


In the diagram, segment $AB$ is parallel to segment $YZ$. If $AZ = 48$ units, $BQ = 15$ units, and $QY = 30$ units, what is the length of segment $QZ$?


Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1$, $\|\mathbf{b}\| = 7$, $\|\mathbf{c}\| = 4$, and
\[
\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}.
\]
If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, then find $\sin \theta$.


Find the sum of all two-digit primes between 20 and 99 that remain prime when their digits are reversed, and the reversed number ends in 1 or 7.


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

Solution: \boxed{2 \csc x}


Eight 8-sided dice are rolled. What is the probability that exactly four of the dice show a 2? Express your answer as a decimal rounded to the nearest thousandth.


Two diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that they intersect inside the nonagon?


Consider the given functions:
$$\begin{array}{ccc}
f(x) & = & 7x^2 - \sin(x) + 6\\
g(x) & = & x^2 - k
\end{array}$$
If $f(3) - g(3) = 5$, what is the value of $k$?

Solution: \boxed{k = -55 + \sin(3)}


Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron, with each edge measuring 1 meter. A bug begins at vertex $A$ and moves according to the rule: at each vertex, it chooses randomly one of the three connecting edges to travel to the next vertex. Each edge has an equal probability of being chosen. Let $p = \frac{n}{19683}$ represent the probability that the bug is back at vertex $A$ after crawling exactly 9 meters. Determine the value of $n$.


Let $a$, $b$, and $c$ be the roots of the equation
$$
9x^3 - 27x + 54 = 0.
$$
Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$


Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[ f(x - y) = \sqrt{f(xy) + 2} \]
for all $x > y > 0.$ Determine $f(2023).$


A flower is located at $(10,12)$ in a coordinate plane. A bee starts from the point $(0,0)$ and follows the path defined by the line $y = 3x$. At what point $(c,d)$ along this line does the bee start moving away from the flower, and what is $c + d$?


Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=5$, and $|r-s|=7$. What is the sum of all possible values of $|p-s|$?


Solve the inequality
\[\frac{x - 1}{(x - 3)^2} < 0.\]

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


Find the smallest positive integer \(n > 3\) for which there exist positive real numbers \(a\) and \(b\) such that:
\[(a + bi)^n = (a - bi)^n,\]
and compute \(\frac{b}{a}\).


Find the equation of the directrix of the parabola $y = 4x^2 - 3.$

Solution: \boxed{y = -\frac{49}{16}}


The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y,$ and $f(6) = 8.$  Find $f(8).$


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


A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?


Find the four positive integers that are divisors of each number in the list $$48, 60, -30, 180, 90.$$ Sum these four integers.


Solution: \boxed{2016 - \frac{1}{2015}}


If $5x \equiv 15 \pmod{25}$ and $4x \equiv 20 \pmod{25}$, then what is the remainder when $x^2$ is divided by $25$?


For how many positive integral values of \( a \) is it true that \( x = 2 \) is the only positive integer solution of the system of inequalities $$
\begin{cases}
2x > 3x - 3\\
4x - a > -8
\end{cases}
$$


What is the degree of the polynomial when the expression $(3x^3 + 2x^2 - 4)(4x^{11} - 8x^9 + 3x^2 + 12) - (2x^2 + 4)^6$ is expanded?


Find the integer $n$, $0 \le n \le 10$, such that \[n \equiv 123456 \pmod{11}.\]


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


Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)$ function is defined as follows:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]


The sum of two positive integers $a$ and $b$ is 221. What is the largest possible value of $\gcd(a,b)$?


Calculate the distance between the points \( (-3, 1, -2) \) and \( (4, -3, 2) \) in three-dimensional space.


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


What is the smallest positive integer $x$ such that $450x$ is a multiple of $800$?


How many positive integers less than $500$ are multiples of $3$ but not multiples of $9$?


Construct a quadratic function that has zeroes at $x=-1$ and $x=5$, and that takes the value $8$ when $x=2$.

Solution: \boxed{-\frac{8}{9}x^2 + \frac{32}{9}x + \frac{40}{9}}


Calculate how many factors of 9600 are perfect squares.


Find the number of positive integers \( n \leq 2000 \) such that \( 21n \) is a perfect square.


Three standard dice are tossed, and the numbers \(a\), \(b\), \(c\) appear. Determine the probability that the product \(abc = 144\).


Calculate the product of $325_8 \cdot 7_8$ and express your answer in base 8.


Two different numbers are randomly selected from the set $T = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. The probability that their sum is 14 would be greater if the number $m$ had first been removed from set $T$. What is the value of $m$?


Let $\omega$ be a complex number such that $\omega^8 = 1$ and $\omega \ne 1.$ Compute
\[\omega^{15} + \omega^{18} + \omega^{21} + \dots + \omega^{60}.\]


Simplify $\frac{12xy^3}{9x^2y^2}$ with $x=3$ and $y=4$.


Thirteen distinct points are identified on the circumference of a circle with radius \( r \). How many different triangles can be formed if each vertex must be one of these 13 points and none of the sides of the triangles can be equal to the radius \( r \)?


Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are \(324\sqrt{3}\) square feet and \(729\sqrt{3}\) square feet. The two planes are \(12\) feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 30$, what is $b_3b_6$?


What is the probability that Logan gets fewer heads than tails if he flips 8 coins?


When the base-10 integers 150 and 950 are expressed in base 2, how many more digits does 950 have than 150 (after being converted)?


The ratio of the areas of a square and a circle is $\frac{250}{196}$. After rationalizing the denominator, the ratio of the side length of the square to the radius of the circle can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$?


To prepare for her marathon, Lucy ran a total of $3526_8$ miles. How many miles (in base ten) did Lucy run?


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?

Solution: \boxed{10\sqrt{2} \text{ cm}}


When three positive integers are divided by $30$, the remainders are $15$, $7$, and $18$, respectively.

When the sum of the three integers is divided by $30$, what is the remainder?


Find the direction vector of line $\ell$ which passes through the origin, given that the matrix for projecting onto line $\ell$ is \[\begin{pmatrix} \frac{3}{13} & \frac{2}{13} & \frac{6}{13} \\ \frac{2}{13} & \frac{12}{13} & -\frac{5}{13} \\ \frac{6}{13} & -\frac{5}{13} & \frac{1}{13} \end{pmatrix}.\] Write your answer as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a$, $b$, and $c$ are integers and $\gcd(|a|,|b|,|c|) = 1$.

Solution: \boxed{\begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}}


Points \((2, -6)\) and \((12, 8)\) are opposite vertices of a rectangle. If one of the other vertices is at \((12, -6)\), find the coordinates where the diagonals of the rectangle intersect.

Solution: \boxed{(7, 1)}


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 21 + |x_1 + x_2 + \dots + x_n|.\]What is the smallest possible value of $n$?


A safari park contains five pairs of different animals, each pair consisting of one male and one female. A caretaker wishes to feed the animals in a specific pattern: he must alternate between feeding a male and a female animal. If he begins by feeding the male lion, how many ways can he complete the feeding sequence for all the animals?


Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 4$ and $PT = 2(AB - PA),$ then what is $PB$?

Solution: \boxed{PB = \frac{17 + \sqrt{33}}{2}}


Determine the volume of a cylindrical swimming pool that is 20 feet in diameter. The pool has a sloping bottom starting from a depth of 6 feet at one end and decreasing linearly to a depth of 3 feet at the other end. Express your answer in terms of $\pi$.


A rectangle is called "super-cool" if the number of square units in its area equals three times the number of units in its perimeter, and its side lengths are integers. What is the sum of all different possible areas of super-cool rectangles?


The shaded region consists of 25 congruent squares. If $PQ = 10$ cm, what is the area of the entire shaded region?
[asy]
for(int i = 0; i < 5; ++i)
{
    for(int j = 0; j < 5; ++j)
    {
        filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));
    }
}
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0)--cycle,linewidth(2));
label("P",(0,5),N);
label("Q",(5,0),E);
[/asy]

Solution: \boxed{50 \text{ square cm}}


There are 50 people in a club. 25 of them have cool dads, 28 of them have cool moms, 10 of them have cool siblings, 15 of them have cool dads and cool moms, 5 have cool dads and cool siblings, 7 have cool moms and cool siblings, and 3 have cool dads, cool moms, and cool siblings. How many people have relatives who are all uncool?


Simplify and rationalize the denominator of the expression $\frac{\sqrt{3}}{\sqrt{4}} \cdot \frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{8}}{\sqrt{9}}$.


If \(0.9 = \frac{y}{275 + x}\), find the smallest value of \(x\), if \(x\) and \(y\) are positive integers.


Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\theta$ counter-clockwise. Let $\mathbf{T}$ represent a translation matrix with translation vector $t>0$. Suppose
\[\mathbf{R} \cdot (\mathbf{D} + \mathbf{T}) = \begin{pmatrix} 10 & -5 \\ 5 & 10 \end{pmatrix},\]
where $\mathbf{T} = \begin{pmatrix} t & 0 \\ 0 & t \end{pmatrix}.$ Then find $\tan \theta.$


In $\triangle PQR,$ where $PQ=PR=17$ and $QR=15.$ Points $G,H,$ and $I$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PGHI$?


How many positive integers, not exceeding 150, are multiples of either 2 or 3 but not 5?


The dilation, centered at $1 + 2i$, with a scale factor of 2, takes the complex number $0$ to which complex number?

Solution: \boxed{-1 - 2i}


Find the sum of the first five prime numbers that have a units digit of 3.


Consider a triangle with sides of length \( 13 \), \( 14 \), and \( 15 \). Determine the length of the altitude to the longest side.


Semicircles of diameter 4 inches are lined up as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Express your answer in terms of $\pi$.


A 60-foot by 8-foot rectangular garden is enclosed by a fence. The garden needs redesigning into a circle while using the same amount of fencing. How many square feet larger or smaller than the old garden is the new garden?


Let $a$, $b$, $c$, and $d$ be positive real numbers such that $a \leq b \leq c \leq d$. What is the smallest possible value of $(a+b+c+d)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{a+d}+\frac{1}{b+c}+\frac{1}{b+d}+\frac{1}{c+d}\right)$?


Find the positive root of the equation:
\[x^3 - 4x^2 + x - 2\sqrt{2} = 0.\]

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


Solve the congruence $10x + 3 \equiv 7 \pmod{18}$ for $x$, and express the solution as $x \equiv a \pmod{m}$, where $a$ and $m$ are positive integers with $a < m$. Find the sum $a+m$.

Solution: \boxed{a+m = 4+9 = 13}


How many positive integers less than $300$ can be written as the sum of two positive perfect cubes?


A triangular region is enclosed by the lines with equations \( y = x + 4 \), \( y = -3x + 9 \), and \( y = 2 \). What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth.


Let $n$ be a three-digit positive integer and is the product of the three distinct prime factors $a$, $b$ and $10a+b$, where $a$ and $b$ are each less than 10. What is the smallest possible value of $n$?


What is the least positive integer $n$ such that $(n-1)^2$ is divisible by some but not all integer values of $k$ when $2 \leq k \leq n$?


A quadrilateral has one angle measuring 120 degrees. How many obtuse angles can this quadrilateral have at maximum?


A rectangle, with integer length and width, has a perimeter of 150 units. What is the number of square units in the least possible area?


A stock investment increased by 30% in one year. Starting at this increased value, what percent decrease is needed in the following year to return the stock price to its original value?


Given an isosceles triangle with sides 12 inches and 24 inches, find the perimeter of a similar triangle if its shortest side is 30 inches.

Solution: \boxed{150 \text{ inches}}


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.


When the decimal point of a certain positive decimal number is moved three places to the right, the new number is three times the reciprocal of the original number. What is the original number?


Simplify the expression $(5 - 4x) - (2 + 7x - y)$.

Solution: \boxed{3 - 11x + y}


Javier plans to visit Disneyland during spring break and aims to explore six attractions before lunch. However, he wants to visit the roller coaster and the water ride one after another in any order. In how many different orders can he visit all six attractions, given this condition?


Let \(Q\) be the plane passing through the origin with normal vector \(\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}\). Find the matrix \(\mathbf{Q}\) such that for any vector \(\mathbf{v}\), \(\mathbf{Q} \mathbf{v}\) is the projection of \(\mathbf{v}\) onto plane \(Q\).

Solution: \boxed{\begin{pmatrix} \frac{5}{9} & \frac{2}{9} & -\frac{4}{9} \\ \frac{2}{9} & \frac{8}{9} & \frac{2}{9} \\ -\frac{4}{9} & \frac{2}{9} & \frac{5}{9} \end{pmatrix}}


A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median?


Simplify the following expression and compute the sum of the squares of the coefficients: $5(2x^3 - 3x^2 + 4) - 6(x^4 - 2x^3 + 3x - 2)$.


Five consecutive integers have a product of 15120. What is the largest of the five integers?


Emily's class consists of 420 students. At the graduation ceremony, the students will sit arranged in rows with the same number of students in each. There should be at least 12 rows and at least 18 students in each row. Determine the sum of all possible values of $x$, where $x$ is the number of students in each row.


Let $S = \{3^0, 3^1, 3^2, \ldots, 3^8\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all these differences. Find $N$.


$ABCD$ is a square with side length 4, and $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{CD}$, respectively. Find $\sin \theta$, where $\theta$ is the angle at vertex $A$ formed by lines $AM$ and $AN$.


Find the sum of the three smallest positive solutions to the equation $x - \lfloor x \rfloor = \frac{1}{\lfloor x \rfloor + 1}$. Express your answer as a mixed number.


What is the sum of the different prime factors of $1729728$?


In a cross-country meet between 2 teams, each with 6 runners, each runner finishing in the nth position contributes n points to his team's score. The team with the lower score wins. Assuming no ties among runners, determine how many different winning scores are possible.


Find the integer $n$, $-90 < n < 90$, such that $\tan n^\circ = \tan 255^\circ$.


How many three-digit numbers are left if we exclude all three-digit numbers in which there are exactly two identical digits, and these two identical digits are adjacent?


Consider a hyperbola where the center is at \((1,-4)\), one focus is located at \((1,5)\), and one vertex is at \((1,-7)\). Determine the equation of this hyperbola in the form:
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]
Calculate the sum \(h + k + a + b\).


Fully simplify the following expression: $[(3+4+5+6)\div3] + [(3\cdot6+9)\div4]$.


A large square is divided into 36 smaller squares. A shaded area is created by joining four smaller squares along one of the diagonals of the large square. What is the ratio of the area of the shaded region to the area of the large square?


Let $\mathbf{M}$ be a matrix, and $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}.\]
Compute $\mathbf{M}(\mathbf{v} - 2\mathbf{w})$.

Solution: \boxed{\begin{pmatrix} -6 \\ 14 \end{pmatrix}}


Find a unit vector that is orthogonal to both $\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{\sqrt{35}} \\ \frac{-5}{\sqrt{35}} \\ \frac{3}{\sqrt{35}} \end{pmatrix}}


Alice walked 10 meters due north from a landmark. She then turned due east and walked 30 feet. After that, she turned due south and walked 10 meters plus 40 feet. How many feet away from her original starting point is Alice?

Solution: \boxed{50 \text{ feet}}


Find the solution to $x^2|x| = 3x + 4$ which has the smallest value.


Determine the residue of $-1035 \pmod{37}$. Your answer should be an integer between $0$ and $36$.


Consider the ellipse \[4(x+2)^2 + 16y^2 = 64.\] Let $C$ be one of the endpoints of its major axis, and let $D$ be one of the endpoints of its minor axis. Find the distance $CD.$


A quadrilateral is labeled with vertices $P$, $Q$, $R$, and $S$. The lengths of sides $PQ$, $QR$, $RS$, and $SP$ are given as $9$, $11$, $15$, and $14$ respectively. Calculate how many different whole numbers could represent the length of the diagonal $PR$.

[asy]
draw((0,0)--(6,6)--(14,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(14,2),dashed);
label("9",(3,3),NW);
label("11",(10,4),NE);
label("15",(11, -2.5),SE);
label("14",(4,-3.5),SW);
[/asy]


For each pair of real numbers $a \ne b$, define the operation $\star$ as
\[
(a \star b) = \frac{a^2 + b^2}{a - b}.
\]
What is the value of $((2 \star 3) \star 4)$?


A triangle has one side of length 5 cm, another side of length 12 cm, and includes a right angle. What is the shortest possible length of the third side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth.


$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 5$ and $AE = 2.$ Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$


When a positive integer is expressed in base 9, it is $CD_9$, and when it is expressed in base 8, it is $DC_8$. What is the positive integer in decimal?


Given that $b$ is an odd multiple of $17$, find the greatest common divisor of $4b^2 + 63b + 144$ and $2b + 7$.


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

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


The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y - 2)^2 = 4$ are tangent. Compute $m$.


If $\mathbf{A}^{-1} = \begin{pmatrix} -3 & k \\ 0 & 3 \end{pmatrix},$ then find the inverse of $\mathbf{A}^2.$

Solution: \boxed{\begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}}


Find a positive integer \(N\) such that \(N\) and \(N^2\) end in the same sequence of four digits \(abcd\) in base \(10\), and \(N-2\) is divisible by 7.


Consider the simplified form of the expression \[(x+y+z)^{2008}+(x-y-z)^{2008}.\] After expanding and combining like terms, how many terms are there in the simplified expression?


Laura bought a new scooter and paid $20\%$ of the cost upfront, which was $\$240$. Additionally, she agreed to pay the remaining amount in 12 equal monthly installments. What was the price of the scooter and how much is each installment?


The sequence $3, 10, b, 24, c, 38$ is arithmetic. What is the value of $b + c$?


A box contains eight cards. Four of the cards are black on both sides, two 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.


What is the product of the numerator and the denominator when $0.\overline{037}$ is expressed as a fraction in its lowest terms?


Let $\triangle PQR$ be a right triangle with $Q$ being the right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. Given that the area of $\triangle PQR$ is $210$ and $PR = 42$, find the length of $QS$.


What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 140 and 336 units?


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $400\sqrt{2}$ square feet and $900\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


The faces of a 10-sided die are labeled with digits 1 through 10. What is the probability, expressed as a common fraction, of rolling a sum of 17 with a pair of such dice?


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


In a survey, it was found that $88.3\%$ of the participants believe that squirrels are harmful. Of those who believe squirrels are harmful, $53.8\%$ incorrectly think that squirrels often attack humans. Given that these 28 people were wrong, calculate the total number of participants surveyed.


Keisha's basketball team is updating their uniform strategy. The seventh-graders have a choice between three colors for shorts (black, gold, or silver), and the eighth-graders will pick the color of the jersey from four options (black, white, gold, or blue). The two groups will not communicate about their choices. If each possible color for both garments is equally likely to be selected, what is the probability that the shorts will be a different color than the jersey?


Determine the number of angles between 0 and $2\pi$, other than integer multiples of $\frac{\pi}{2}$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order where the common ratio is not equal to 1 or -1.


12 people attend a gathering. Each person shakes hands and high-fives every other person exactly once. How many handshakes and how many high-fives happen at this gathering?


Let $Q$ be the point on line segment $\overline{CD}$ such that $CQ:QD = 3:5.$ Then
\[
\overrightarrow{Q} = x \overrightarrow{C} + y \overrightarrow{D}
\]
for some constants $x$ and $y.$ Determine the ordered pair $(x,y)$.

Solution: \boxed{\left(\frac{5}{8}, \frac{3}{8}\right)}


Suppose two distinct integers are chosen from between 3 and 15, inclusive. What is the probability that their product is odd and their sum is even?


Given a triangle with vertices at coordinates (1,1), (4,7), and (8,1), what is the number of units in the length of the longest side of the triangle?


Two concentric circles are centered at point Q. The sides of a 60° 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° 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.


Given that $M(1,7)$ is one-third of the way from point $A(4,2)$ to point $B$, find the product of the coordinates of point $B$.


What is the smallest positive integer $n$ such that $5n$ is a perfect square and $3n$ is a perfect cube?


Interior numbers begin in the third row of Pascal's Triangle. What is the sum of the interior numbers of the ninth row?


Consider a shaded region consisting of 13 unit squares. This region is located in the first quadrant, beginning along the y-axis and stretching rightwards with the following dimensions: 1 unit wide and 5 units high from the origin and an attached section 3 units wide and 2 units high. This shape is rotated about the x-axis. Calculate the volume of the solid formed.


A girl has the following coins in her wallet: 2 pennies, 2 nickels, 1 quarter, and 2 half dollars. She randomly picks two coins, calculates the total value of these two coins, records it, and then puts them back with the other coins. This process is repeated multiple times. What is the maximum number of different sums that she can record?


If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum?


For each positive integer $n$, the mean of the first $n$ terms of a sequence is $2n$. What is the 1005th term of the sequence?


A 4-sided red die and an 8-sided green die are rolled. What is the probability that the red die shows a number divisible by 2 and the green die shows a number that is a power of 2?


Given that 12 is the arithmetic mean of the set $\{8, 15, 20, 7, y, 9\}$, what is the value of $y$?


A sphere was embedded in a block of gelatin. After removing the sphere intact, a hole 30 cm across at the top and 12 cm deep was left. What was the radius of the sphere (in centimeters)?


Simplify the following expression to its simplest form: $[(3+5+7+9)\div3] - [(4 \cdot 6 + 13)\div5]$.


If the least common multiple of two 5-digit integers has 9 digits, then their greatest common divisor has at most how many digits?


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


Determine the minimum possible value of the sum
\[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{7a},\]
where $a,$ $b,$ and $c$ are positive real numbers.


What is the least common multiple of the numbers 1540 and 2310?


Each row of a seating arrangement seats either 7 or 9 people. A total of 112 people needs to be seated with every seat occupied. How many rows must seat exactly 9 people?


How many positive integers less than $151$ are multiples of either $4$ or $6$, but not both at once?


Determine the total number of distinct, natural-number factors for the number $4^5 \cdot 5^2 \cdot 6^3 \cdot 7!$.


Suppose the function $g$ has all real numbers in its domain and range and is invertible. Some values of $g$ are given by the following table:
$$
\begin{array}{c || c | c | c | c | c}
x & -1 & 0 & 1 & 2 & 3 \\
\hline
g(x) & 0 & 1 & 3 & 4 & 6
\end{array}
$$
What is the value of $g(g(1)) + g(g^{-1}(2)) + g^{-1}(g^{-1}(3))?$ If there is not enough information to answer this question, enter "NEI".


If the price of a stamp is 35 cents, and there is a 5% discount on the total price for buying more than 100 stamps, what is the maximum number of stamps that can be purchased with $\$32$?


Five balls are numbered 1 through 5 and placed in a bowl. Alice will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Alice will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and greater than 12? Express your answer as a common fraction.


In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{4}{5}$ and $\sin C = \frac{2}{5}$, what is $DC$?

[asy]
pair A,B,C,D;
A = (0,0);
B = (10*sqrt(5),24);
D = (10*sqrt(5),0);
C = (10*sqrt(5) + 40*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]


How many of the 729 smallest positive integers written in base 9 use 5 or 6 (or both) as a digit?


The sum of the areas of two squares is 85, and the difference in their areas is 45. Find the difference between their perimeters.

Solution: \boxed{4(\sqrt{65} - 2\sqrt{5})}


A coin is flipped ten times. If the first flip results in heads, at least three out of the next four flips must also result in heads. How many distinct sequences are possible?


What is the probability that Andrew gets fewer heads than tails if he flips 8 coins?


Given the integer $n = 2^{12} \cdot 3^{15} \cdot 7^9$, determine how many natural number factors of $n$ are multiples of $252$.


Suppose I have 8 shirts, 5 pairs of pants, and 6 ties. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many different outfits can I create?


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. The area of $\triangle DEF$ is 18. Determine the area of $\triangle ABC$.


How many times does the digit 9 appear in the list of all integers from 1 to 700?


Let $g(n)$ denote the sum of all the positive divisors of the integer $n$. How many integers $j$ exist such that $1 \le j \le 1000$ and $g(j) = 1 + \sqrt{j} + j$?


In this square array of 9 dots arranged in a 3x3 grid, four dots are to be chosen at random. What is the probability that the four dots will be collinear? Express your answer as a common fraction.

[asy]
size(45);
for(int i = 0; i<3; ++i)
for(int j = 0; j<3; ++j)
dot((i,j),linewidth(7));
[/asy]


Jana currently takes 30 minutes to walk 2 miles. If she wants to cover 3 miles in the same amount of time, what should her new walking speed be in miles per hour?

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


Below is a portion of the graph of a function, $y=v(x)$:

[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-3.25,ymax=3.25;

pen cqcqcq=rgb(0.75,0.75,0.75);

/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return -x+2*sin(x*pi/2);}
draw(graph(f2,-3.25,3.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]

Using the provided graph, determine the exact value of $v(-3)+v(-1.5)+v(1.5)+v(3)$.


Find the number of complex solutions to the equation:
\[\frac{z^4 - 1}{z^3 - 2z^2 + z - 2} = 0.\]


Two numbers $180$ and $n$ share exactly three positive divisors. What is the greatest of these three common divisors?


Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\).

Solution: \boxed{10 - 2\sqrt{5}}


Two cylindrical cans must have the same volume. The height of one can is five times the height of the other. If the radius of the narrower can is 10 units, what is the radius of the wider can? Express your answer in simplest radical form.


Express $7.\overline{123}$ as a common fraction in lowest terms.


Express $7.\overline{3}$ as a common fraction.


Suppose a parabola has a vertex at $\left(\frac{-1}{3},-\frac{4}{3}\right)$ and is described by the equation $y = ax^2 + bx + c$, where \(a > 0\) and \(a + b + c\) is an integer. Find the smallest possible value of \(a\).


If a stock investment increased by 30% in one year, what percentage decrease is required the following year for the stock to return to its original value?


The distribution of populations in a series of towns is shown in a pie chart. What percent of the towns have fewer than 50,000 residents?

[asy]
draw(Circle((0,0),25),linewidth(1));
draw((-24,-7)--(0,0)--(24,-7),linewidth(1));
draw((0,0)--(7,-24),linewidth(1));
label("65\%",(0,12));
label("20\%",(-10,-10));
label("15\%",(10,-10));
label("\textbf{Distribution of Town Populations}",(0,25),N);
label("\tiny{20,000 to 49,999}",(30,15),N);
label("\tiny{less than 20,000}",(-30,-15),S);
label("\tiny{50,000 or more}",(30,-15),S);
label("\tiny{residents}",(30,13),N);
label("\tiny{residents}",(-30,-17),S);
label("\tiny{residents}",(30,-17),S);
[/asy]


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


Determine the value of $b$ if the inequality $-x^2 + bx + 1 < 0$ holds only when $x \in (-\infty, 2) \cup (6, \infty)$.


Sandy's son has a fort in the yard which includes one shaded exterior wall and two inclined roof panels, all requiring a special coating to protect them from weather. The coating material is only sold in 10-foot by 15-foot sections that cost $35 each. If the shaded wall is 9 feet by 7 feet and each panel of the roof is 9 feet by 6 feet, how much in total will Sandy need to pay for the required coating sections?


For constants $a$ and $b,$ consider the function $f(x)$ given by
\[
f(x) = \left\{
\begin{array}{cl}
2ax + b & \text{if } x \leq 0, \\
7 - 2x & \text{if } x > 0.
\end{array}
\right.
\]
The function $f$ has the property that $f(f(x)) = x$ for all $x$ in its domain. Determine $a + b.$


Right triangle $T$ is inscribed in circle $A$, which has radius $13$. Circle $B$ with radius $4$ is internally tangent to circle $A$ at one vertex of $T$. Circle $C$, with radius $3$, is internally tangent to circle $A$ at another vertex of $T$. The circles $B$ and $C$ are externally tangent to another circle $E$, which has radius $r$. If the angle between the radii of circle $A$ touching vertices related to circles $B$ and $C$ is $90^\circ$, find $r$.

Solution: \boxed{\frac{\sqrt{181} - 7}{2}}


In diagram, square $ABCD$ has sides of length $6,$ and $\triangle ABE$ is isosceles with $AB=BE$ and $\angle ABE=45^\circ$. Line segments $BE$ and $AC$ intersect at $P$. Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x$. Determine the measure of angle $BPC$.


Let $x$ and $y$ be nonzero real numbers. Determine the minimum value $m$ and the maximum value $M$ of
\[\left(\frac{|x + y|}{|x| + |y|}\right)^2.\]
Find $M - m$.


What is the least positive integer that can be added to 365 to yield a multiple of 5?


You have a rectangular prism box with length $x+3$ units, width $x-1$ units, and height $x^3 - 20$ units. Determine for how many positive integer values of $x$ is the volume of the box less than 1000 units?


Let $M=123456789101112\dots4950$ be the $95$-digit number formed by writing the integers from 1 to 50 in order, one after the other. What is the remainder when $M$ is divided by 45?


In a rectangular coordinate system, calculate the distance from the origin to the point (-12, 16). Additionally, find the distance from this point to another point on the grid, (3, -8).


What is the sum of all positive integers \( n \) that satisfy
\[
\mathop{\text{lcm}}[n,120] = \gcd(n, 120) + 300~?
\]


Let $f(x) = x^2 + x + 2010$. What is the greatest common divisor of $f(101)$ and $f(102)$?


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

Solution: \boxed{\left(6, \frac{5\pi}{3}, 2\right)}


Calculate the value of $\frac{\frac{1}{5} - \frac{1}{3}}{ \frac{3}{7} \div \frac{2}{9}}$. Express your answer as a simplified fraction.


What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A(-3, -2)$ and $B(2, -6)$?


Define $\mathbf{B} = \begin{pmatrix} 0 & 2 \\ 5 & 0 \end{pmatrix}.$ Find the vector $\mathbf{w}$ such that
\[(\mathbf{B}^8 + \mathbf{B}^6 + \mathbf{B}^4 + \mathbf{B}^2 + \mathbf{I}) \mathbf{w} = \begin{pmatrix} 0 \\ 31 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} 0 \\ \frac{31}{11111} \end{pmatrix}}


Triangle $PQR$ has side lengths $PQ = 10$, $QR = 20$, and $PR = 16$. A line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. What is the perimeter of $\triangle PXY$?


If $n$ is divided by 4, the remainder is 3. What is the remainder when $8n$ is divided by 7?


What is the area enclosed by the region defined by the equation $x^2 + y^2 + 8x + 10y + 9 = 0$?


Let \(\mathbf{v} = \begin{pmatrix} 3 \\ 2 \\ -4 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} -1 \\ 5 \\ 2 \end{pmatrix}\). The columns of a matrix are \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\), where \(\mathbf{u}\) is a unit vector in the direction of \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\). Find the largest possible determinant of the matrix.


Consider the function $g(x)$ satisfying
\[ g(xy) = 2g(x)g(y) \]
for all real numbers $x$ and $y$ and $g(0) = 2.$ Find $g(10)$.


Consider a function $g(x) = 4x + c$ and its inverse $g^{-1}(x)$. They intersect at the point $(-4, d)$, where $c$ and $d$ are integers. Determine the value of $d$.

Solution: \boxed{d = -4}


Of the six points (4, 15), (8, 25), (10, 30), (14, 40), (18, 45), and (22, 55), what is the sum of the \( x \)-coordinates of the points that lie in the region above the line \( y = 3x + 5 \) in the coordinate plane?


A fair 8-sided die is rolled. What is the probability that the number rolled is a divisor of 12?


What is the area, in square units, of the region bounded by the lines $y = 3x + 6$, $y = -2x + 12$, and the $y$-axis?


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

size(6cm);

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

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

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

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

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

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

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

[/asy]


Let $x_1,$ $x_2,$ $x_3,$ $x_4$ be the roots of the polynomial $f(x) = x^4 - x^3 + x^2 + 1$. Define $g(x) = x^2 - 3$. Find the product:
\[ g(x_1) g(x_2) g(x_3) g(x_4). \]


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


Simplify $\frac{3}{504} - \frac{17}{72}$ and then multiply the result by $\frac{5}{7}$.


The isosceles trapezoid has sides labeled as follows: AB = 24 units, BC = 13 units, CD = 12 units, DA = 13 units. Calculate the length of diagonal AC. [asy]
unitsize(1.5mm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=3;

pair A=(0,0), B=(24,0), C=(18,12), D=(6,12);
pair[] dots={A,B,C,D};

draw(A--B--C--D--cycle);
dot(dots);
label("A",A,SW);
label("B",B,SE);
label("C",C,NE);
label("D",D,NW);
label("12",midpoint(C--D),N);
label("13",midpoint(D--A),NW);
label("24",midpoint(A--B),S);
label("13",midpoint(B--C),NE);
[/asy]


A right pyramid has a square base with each side measuring 12 cm. The peak of the pyramid is 15 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges and express your answer to the nearest whole number.


What is the smallest even abundant number greater than 12?


A shaded region consists of rectangles organized as follows and rests along the $x$-axis and the $y$-axis. The region is bounded by $y=4$ from $x=0$ to $x=1$, and by $y=3$ from $x=1$ to $x=5$. This region is rotated about the $x$-axis to create a solid. Calculate the volume of the solid formed in cubic units. Express your answer in terms of $\pi$.


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


The nonzero roots of the equation $x^2 + 10x + m = 0$ are in the ratio $3:2$. What is the value of $m$?


If the roots of the quadratic equation $2x^2 + 8x + d = 0$ are $x = \frac{-8 \pm \sqrt{12}}{4}$, then what is the value of $d$?


Find the maximum value of
\[\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \cos \theta & 1 \\ 1 + \sin \theta & 1 & 1 \end{vmatrix},\]as \(\theta\) ranges over all angles.


How many square units are in the area of the triangle whose vertices are the $x$ and $y$ intercepts of the cubic curve $y = x^3 - x$?


Calculate the residue of $222 \cdot 15 - 35 \cdot 9 + 2^3 \pmod{18}$.


A circular cake with a diameter of $16\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $m^2$?

Solution: \boxed{m^2 = 128}


Given that the line $y = \frac{2x - 4}{3}$ is parameterized in the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{v} + t \mathbf{d},\]
so that for $x \ge 2,$ the distance between $\begin{pmatrix} x \\ y \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$ is $t,$ find $\mathbf{d}.$

Solution: \boxed{\begin{pmatrix} \frac{3}{\sqrt{13}} \\ \frac{2}{\sqrt{13}} \end{pmatrix}}


In Mrs. Johnson's class, there are 120 students who completed a test. To boost their confidence, she decides to give extra credit to any student whose score exceeds the median score. What is the maximum number of students who can receive extra credit?


What is the largest five-digit number that is divisible by 5?


What is the number of degrees in angle $y$? Consider a triangle ABC where $\angle ABC = 2y$, $\angle BCA = y + 10^\circ$, and $\angle BAC = 4y$. Verify if this triangle can exist and if so, find the value of $y$.


In right triangle $ABC$ with $\angle A = 90^\circ$, the sides $AB = 8$ and $BC = 17$. Find $\sin C$.


Alex has 12 different kinds of lunch meat and 11 different kinds of cheese. If he wants to make a sandwich with one kind of meat and three kinds of cheese, and he has 5 different types of bread to choose from, how many different sandwiches could he make?


Two sides of an isosceles triangle are 12 inches and 18 inches. If the shortest side of a similar triangle is 30 inches, what is the perimeter of this larger triangle?

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


Tracy had a bag of candies. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy's sister took between two and six candies, leaving Tracy with ten candies. How many candies did Tracy have at the start?


The least common multiple of two numbers is 2640, and the greatest common divisor is 24. Given that one of the numbers is 240, what is the other number?


Define the function $g$ that maps nonnegative integers to real numbers, with $g(0) = 0$ and $g(1) = 2$. The function satisfies the relation
\[ g(m + n) + g(m - n) = \frac{g(2m) + g(2n)}{2} \]
for all nonnegative integers $m \geq n$. Determine the sum of all possible values for $g(7)$.


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

a) When divided by 6, the remainder is 5.

b) When divided by 8, the remainder is 7.

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


If $x\%$ of three-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth.


What is the sum of all integer values of $n$ such that $\frac{36}{2n-1}$ is an integer?


Solution: \boxed{\frac{1 + \sqrt{73}}{2}}


In a right triangle $PQR$ with $\angle P = 90^\circ$, the side lengths are $PQ = 8$ and $QR = 10$. Calculate $\cos R$.


Let the function $p(x) = |x|-3$ and the function $q(x) = -x^2$. On a set of axes, calculate the sum of the values obtained by evaluating $q(p(x))$ for $x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$.


A math conference is organizing a series with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, requiring Dr. Smith to be scheduled after Dr. Jones. Additionally, Dr. Green's lecture depends on Dr. Brown's lecture, requiring Dr. Green to be scheduled after Dr. Brown. In how many orders can the seven lecturers be scheduled?


How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $I$? Each letter can be used more than once, and $B$, $C$, $D$, $F$, $G$ are consonants.


How many solutions does the equation $\tan x = \sin (\cos x)$ have on the interval $0 \le x \le \cos^{-1} 0.1$?


Compute the product of the roots of the equation \[x^3 - 9x^2 + 27x - 8 = 0.\]


Factor the quadratic expression $2y^2 + 5y - 12$ into the form $(2y + a)(y + b)$, where $a$ and $b$ are integers. What is $a - b$?


Let
\[ f(x) = \frac{x^2 + ax}{x^2 + x + 1}. \]
Find the constant \( a \) such that \( f(f(x)) = x \) for all \( x \) where the function is defined.


What is the value of $(10! \cdot 4! \cdot 3!) / (9! \cdot 7!)$?


Add $35_6 + 14_6$. Express your answer in base $6$.


Suppose \(z\) and \(w\) are complex numbers such that
\[|z| = |w| = 2 \text{ and } z \overline{w} + \overline{z} w = 1.\] Find the largest possible value of the real part of \(z + w\).


The polynomial $x^{12} - 1$ is factored as
\[
x^{12} - 1 = q_1(x) q_2(x) \dotsm q_m(x),
\]
where each factor $q_i(x)$ is a non-constant polynomial with real coefficients. Find the largest possible value of $m$.


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 5$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right |$?


Find the positive value of $k$ such that the equation $4x^3 + 9x^2 + kx + 4 = 0$ has exactly one real solution in $x$.


Triangle \(ABC\) has its vertices \(A\), \(B\), and \(C\) on the sides of a rectangle 6 units by 8 units as displayed. What is the area of triangle \(ABC\) in square units?
- \(A\) is located at \( (0,3) \)
- \(B\) is located at \( (6,0) \)
- \(C\) is located at \( (3,8) \)


Determine the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 10 units and lateral edges measuring 7 units.


The area of a larger triangle is 1600 square feet. This triangle is composed of two identical smaller triangles each with a base of 40 feet. Find the altitude of the larger triangle given this configuration.

Solution: \boxed{40 \text{ feet}}


Define the operation $*$ by
\[a * b = \frac{a - b}{1 - ab}.\]
Compute
\[2 * (3 * (4 * 5)).\]


$ABC$ is a triangle with vertices $A=(0,0), B=(30,18)$, and $C$ having integer coordinates. Determine the minimum area of $\triangle ABC$.


Find $4 \cdot 7^{-1} + 12 \cdot 13^{-1} - 6 \cdot 17^{-1} \pmod{60}$.

Express your answer as an integer from $0$ to $59$, inclusive.


A university has $3000$ students, each studying either History or Psychology, and some study both. The number of students studying History is between $70\%$ and $75\%$ of the university population, and the number studying Psychology is between $40\%$ and $50\%$. Let $m'$ be the smallest number of students who could study both subjects, and let $M'$ be the largest number of students who could study both subjects. Find $M'-m'$.


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


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$


Let $t$ represent all real values. Consider the curve formed by the set of points $(x, y)$ defined by the equations:
\[
x = t^3 - 3t + 1,
\]
\[
y = t^4 - 4t^2 + 4.
\]
Determine the ordered pair $(x,y)$ where the curve crosses itself.


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


Find both the largest and smallest values of \( x \) that satisfy the equation \( |x-3|=10 \).


A spiral staircase turns $315^\circ$ as it rises 12 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.


The sequence 2,000,000; 400,000; 80,000 and so on, is made by repeatedly dividing by 5. What is the last integer in this sequence?


The equation $x^2 + ax = -18$ has only integer solutions for $x$. If $a$ is a positive integer, what is the greatest possible value of $a$?


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


Fifteen points are placed on a circle. Determine how many distinct convex polygons with at least three sides can be formed using some or all of the fifteen points as vertices.


Five partners in a business decide to split the profits of their company in the ratio 2:3:4:4:6. If the profit one year is $\$38,\!000$, what is the largest number of dollars received by any of the partners?


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


The sum of 81 consecutive integers is $3^9$. What is their median?


Lucy places a total of 30 purple Easter eggs in some pink baskets and a total of 45 yellow Easter eggs in several brown baskets. Each basket contains the same number of eggs, and there are at least 5 eggs in each basket. How many eggs did Lucy put in each basket?


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{3}{2^7 \cdot 5^{10}}$?


When a polynomial is divided by \(3x^2 - 7x + 5\), what are the possible degrees of the remainder? Enter all possible values, separated by commas.

Solution: \boxed{0, 1}


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 150^\circ & -\sin 150^\circ \\ \sin 150^\circ & \cos 150^\circ \end{pmatrix}^n = \mathbf{I}.\]


How many obtuse interior angles can be in an isosceles triangle?


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


Anna will randomly select three letters from the word PLANET, three letters from the word CIRCUS, and two letters from the word GAMES. What is the probability that she will have all the letters from the word PLUM? Express your answer as a common fraction.


What is the largest three-digit multiple of 8 whose digits' sum is 24?


Let \( x \) and \( y \) be real numbers such that \( \frac{\sin x}{\sin y} = 4 \) and \( \frac{\cos x}{\cos y} = \frac{1}{2} \). Find the value of
\[ \frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}. \]


In the diagram, $G$ and $H$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Determine the sum of the $x$ and $y$ coordinates of $I$, the point of intersection of $\overline{AG}$ and $\overline{BH}$. Assume $A(0,8)$, $B(0,0)$, and $C(10,0)$.


Three of the four vertices of a rectangle are $(2, 7)$, $(13, 7)$, and $(13, -6)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 2)^2 + (y + 6)^2 = 16$? Express your answer in terms of $\pi$.


A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community each year. Determine the total number of members in the community five years later.


Determine the value of the following expression: $\frac{1}{2} - \frac{1}{6} + \frac{1}{18} - \frac{1}{54} + \frac{1}{162} - \frac{1}{486}$. Express your answer as a common fraction.


For any positive integer $n$, the factorial of $n$ ($n!$) is the product of all positive integers from $1$ to $n$. Find the greatest common divisor of $10!$ and $12!$.


In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 41^\circ$ and $\angle ACB = 27^\circ$, find the measure of $\angle BHC$, in degrees.


Cara is sitting at a round table with her eight friends. How many different pairs of friends could Cara be potentially sitting between?


Find the modular inverse of $32$, modulo $33$.

Express your answer as an integer from $0$ to $32$, inclusive.


In Professor Sterling's class, there are 150 students who took a recent test. Assume one student scored significantly lower than the others, allowing for the possibility that a large majority's scores exceeded the class mean. What is the maximum number of students who can receive extra credit, given these conditions?


What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{22}{70}$?


Let $p(x), q(x), r(x), s(x)$ be invertible functions. Define the function $f$ by
\[ f = s \circ q \circ p \circ r. \]
Determine the correct expression for $f^{-1}$.

Solution: \boxed{r^{-1} \circ p^{-1} \circ q^{-1} \circ s^{-1}}


There are 25 people in the drama club. 10 of them are left-handed. 18 of them like musical theater. 3 of them are right-handed and dislike musical theater. How many club members are left-handed and like musical theater? Assume people are either left-handed or right-handed, but not both.


If $q(x) = x^5 - 4x^3 + 3$, then find the coefficient of the $x^6$ term in the polynomial $(q(x))^2$.


Find the sum of the first ten terms in the geometric sequence \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots \). Express your answer as a common fraction.


Catherine has 8 blue marbles and 4 red marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly five of the marbles that she removes are blue? Express your answer as a decimal rounded to the nearest thousandth.


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


Determine the number of positive real solutions to the equation $x^{8} + 6x^7 + 14x^6 + 1429x^5 - 1279x^4 = 0$ that are less than 2.


The volume of a sphere is increased to $72\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\pi$.

Solution: \boxed{36\pi \cdot 2^{2/3}}


Let $x$ and $y$ be positive real numbers such that $2x + 5y < 105.$ Find the maximum value of
\[xy (105 - 2x - 5y).\]


Let $a, b, c$ be the roots of the polynomial $x^3 - 6x^2 + 14x - 6 = 0$. Define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Find the value of $s^4 - 12s^2 - 24s$.


Check if $x=4$ is a root of the polynomial $x^3 - 5x^2 + 7x - 12$. If it is not, find the correct numerical coefficient of $x$ in place of $7$ so that $x=4$ would be a root of the polynomial $x^3 - 5x^2 + ax - 12$.


The Happy Valley Kennel now has 3 chickens, 3 dogs, and 4 cats, and they need to be placed in a row of 10 cages. How many ways can the 10 animals be placed in the cages, such that all of the animals of each type are in adjacent cages, with each type of animal being distinguishable?


Marcelle and Jaclyn each think of a polynomial. Each of their polynomials is monic, has degree 3, and has the same positive constant term. The product of their polynomials is \[z^6 + 2z^5 + 4z^4 + 6z^3 + 4z^2 + 2z + 9.\] What is the constant term of Jaclyn's polynomial?


In a recent examination, a teacher collected the following data from a test taken by her $120$ students. Calculate the mean score (out of maximum score $120$) based on the data provided:

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Score}&\textbf{Number of Students}\\\hline
120&12\\\hline
110&19\\\hline
100&33\\\hline
90&30\\\hline
75&15\\\hline
65&9\\\hline
50&2\\\hline
\end{tabular}


What is the domain of the function $$v(x) = \frac{1}{\sqrt{x} + \sqrt{x + 4}}~?$$ Express your answer in interval notation.

Solution: \boxed{[0, \infty)}


Find the greatest common divisor of 9486, 13524, and 36582.


A regular polygon has exterior angles each measuring 18 degrees. Calculate how many sides the polygon has and determine the measure of each interior angle.


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


I was planning to work for 25 hours a week for 15 weeks to save $4000 for a vacation. Unfortunately, due to unexpected circumstances, I couldn't work for the first three weeks. How many hours per week must I now work for the remaining weeks if I still need to save the same amount of money?


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

Solution: \boxed{19x + 1}


According to the table below, how many dollars are in the median value of the 71 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$130{,}000$\\\hline
Vice-President&15&$\$90{,}000$\\\hline
Director&10&$\$80{,}000$\\\hline
Associate Director&8&$\$50{,}000$\\\hline
Administrative Specialist&37&$\$25{,}000$\\\hline
\end{tabular}


In right triangle $ABC$, where $AB = BC$, point $D$ is the midpoint of $\overline{BC}$. Point $E$ is on the extension of $\overline{AC}$ such that $CE = 15$ units. If $D$ is also the midpoint of $\overline{AE}$, what is the length of $\overline{BD}$? Express your answer as a decimal to the nearest tenth.


A cubic polynomial \( p(x) \) satisfies
\[ p(n) = n^2 \] for \( n = 1, 2, 3, \) and \( 4 \). Find \( p(5) \).


In right triangle $XYZ$, we have $\angle Y = \angle Z$ and $XY = 8\sqrt{2}$. What is the area of $\triangle XYZ$?


Given the diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 10.$ [asy]
import cse5; import olympiad;
size(120);
add(grid(8,8));
draw((0,0)--(8,0),linewidth(1.2));
draw((0,0)--(0,8),linewidth(1.2));
label("$x$",(8,0),E);
label("$y$",(0,8),N);
draw((0,5)--(3,0),Arrows);
draw((0,2)--(7,0),Arrows);
label("$l$",(0,5)--(3,0),NE);
label("$m$",(0,3)--(7,0),NE);
[/asy]


Two friends plan to meet at a park for a walk. They both decide to come at random, independent times between 3 p.m. and 5 p.m. They will stay for different durations, $m_1$ and $m_2$, respectively. The probability that either one arrives while the other is already there is $50\%$. If $m_1 + m_2 = a - b\sqrt{c}$, where $a$, $b$, and $c$ are integers, $c$ is not divisible by the square of any prime, and $m_1 = m_2$, find the value of $a + b + c$.


Find the angles sum of the solutions expressed in degrees for the equation $z^5 = -32i$.


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


Terrell usually lifts three 25-pound weights 10 times. If he uses three 20-pound weights instead, how many times must Terrell lift them to lift the same total weight?


Solution: \boxed{\frac{10y^7 + 3y^3}{3}}


During a school fruit eating contest, eight students participate where each eats a different number of oranges ranging from 1 to 8. The graph shows the number of oranges eaten by each student. Alice ate the most oranges, and Bob ate the fewest. How many more oranges did Alice eat than Bob?
```plaintext
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[9];
for (int i = 0; i < 9; ++i) {
    yaxis[i] = (0, i);
    draw(yaxis[i]--yaxis[i]+(17, 0));
}
draw((0,0)--(0,8));
draw((17,8)--(17,0));

fill((1,0)--(1,3)--(2,3)--(2,0)--cycle, lightgray);
fill((3,0)--(3,5)--(4,5)--(4,0)--cycle, lightgray);
fill((5,0)--(5,1)--(6,1)--(6,0)--cycle, lightgray);
fill((7,0)--(7,6)--(8,6)--(8,0)--cycle,lightgray);
fill((9,0)--(9,4)--(10,4)--(10,0)--cycle,lightgray);
fill((11,0)--(11,2)--(12,2)--(12,0)--cycle,lightgray);
fill((13,0)--(13,7)--(14,7)--(14,0)--cycle, darkgray); // Alice has the tallest column
fill((15,0)--(15,8)--(16,8)--(16,0)--cycle, lightgray); // Bob has the shortest column
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("Students/Participants",(8.5,0),S);
label("Results of a Fruit Eating Contest",(8.5,8),N);
label(rotate(90)*"$\#$ of Oranges Eaten",(-1,4),W);
[/asy]
```


What is the greatest common divisor of $6!$ and $\frac{9!}{4!}?$ Express your answer as an integer.


Suppose I have 8 shirts, 4 ties, 3 pairs of pants, and 2 belts. An outfit requires a shirt and a pair of pants, and can optionally include a tie and/or a belt. How many different outfits can I create?


Let $r$, $s$, and $t$ be solutions of the equation $x^3 - 4x^2 + 4x = 6$.

Compute $\frac{rs}t + \frac{st}r + \frac{tr}s$.


How many of the divisors of \( 9! \) are greater than \( 8! \)?


Evaluate the expression given by: $$2 + \cfrac{3}{4 + \cfrac{5}{6 + \cfrac{7}{8}}}$$


In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 45 square units, what is the area of trapezoid $PQRS$?


In how many ways can 100 be written as the sum of three distinct positive perfect squares if the order of the three perfect squares does not matter?


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


Find the focus of the parabola given by the equation $x = -\frac{1}{8} y^2$.

Solution: \boxed{(-2, 0)}


Let \( f(x) = -4x^3 + 2x^2 - 5x - 7 \), \( g(x) = 6x^2 + 4x - 9 \), and \( h(x) = -x^3 + 3x^2 + 2x + 8 \). Express \( f(x) + g(x) + h(x) \) as a single polynomial, with the terms in order by decreasing degree.

Solution: \boxed{-5x^3 + 11x^2 + x - 8}


Determine the value of $x$ that satisfies $\sqrt[4]{x\sqrt{x^4}}=2$.


Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where  \[f(n) = \left\{
\begin{array}{cl}
12 & \text{ if }n\text{ is a multiple of 3}, \\
4 & \text{ if }n\text{ is only a multiple of 2}, \\
0 & \text{ if }n\text{ is neither a multiple of 2 nor 3}.
\end{array}
\right.\]
Allie rolls the die five times and gets a 2, 6, 3, 1, and 6. Betty rolls four times and gets a 4, 6, 3, and 5. Calculate the product of Allie's total points and Betty's total points.


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


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

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


Four students' test scores are given as 86, 88, 90, 91, 95, and 99. Two of these scores belong to Jane and the other four to Alex. Jane's mean score is 93. What is Alex's mean score?


Simplify $\sqrt[3]{40a^7b^9c^{14}}$ and find the sum of the exponents of the variables that are outside the radical.


John arranged his cards in the same suit sequence $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, 2, 3, \ldots.$$ What is the $58$th card?


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


The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$?


What is the coefficient of $x^7$ in the expansion of $(3x - 4)^{10}$?


A rectangle $EFGH$ has dimensions $12$ by $15$. An equilateral triangle is inscribed such that no point of the triangle lies outside $EFGH$. Calculate the maximum possible area of such a triangle.


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


The line $y = 5$ intersects the graph of $y = 3x^2 + 2x - 2$ at points $C$ and $D$. Find the distance between $C$ and $D$ and express it in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. Find the value of $p - q$.


Liam runs 3 miles in one-third the time it took Mia to run 5 miles. If it took Mia 45 minutes to run 5 miles, using the same rates, how many minutes would it take Liam to run 7 miles?


Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 11}.\]


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


A butterfly starts flying from point $Q_0$. It flies $2$ inches due east to point $Q_1$. For $k \ge 1$, once the butterfly reaches point $Q_k$, it turns $45^\circ$ counterclockwise and then flies $2$ inches straight to point $Q_{k+1}$. When the butterfly reaches $Q_{1024},$ how far from $Q_0$ is it, in inches?

Solution: \boxed{0 \text{ inches}}


How many even three-digit numbers are there such that the sum of the tens and units digits is 9?


What is the smallest positive value of $x$ such that $x + 1234$ results in a palindrome?


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


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


A rectangular garden has a perimeter of 60 meters and an area of 221 square meters. What are the dimensions of the garden?

Solution: \boxed{17 \text{ meters and } 13 \text{ meters}}


A right triangle has legs of lengths $12$ and $9$. It is similar to another right triangle, where the leg corresponding to the $12$ unit leg is $7.5$ units. Find the length of the leg corresponding to the $9$ unit leg of the first triangle.


Solution: \boxed{75x^3(1 - 4x^4)}


Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*}
x + yz &= 9, \\
y + xz &= 13, \\
z + xy &= 12.
\end{align*}
be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$ Find $x_1 + x_2 + \dots + x_n.$


Calculate the expression $(3^5 \cdot 4^5) \cdot 6^2$.


A regular dodecagon \(Q_1 Q_2 \dotsb Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((1,0)\) and \(Q_7\) at \((3,0)\). If \(Q_n\) is the point \((x_n,y_n),\) compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).\]


A positive two-digit number is divisible by 6 and is a multiple of 3. The product of its digits is a perfect square. What is the smallest such number?


The polynomial
\[ax^4 + bx^3 + cx^2 + dx + e = 0\]
has coefficients that are all integers, and has roots $-3,$ $4,$ $8,$ and $-1/2.$  If $e$ is a positive integer, then find its smallest possible value.


The school offers German and Chinese language classes. There are 30 students enrolled in at least one language class. 22 students are in the German class and 19 students are in the Chinese class. If a teacher randomly selects two students from the list, what is the probability that both the German and Chinese classes will be represented in the selection? Express your answer as a fraction in simplest form.


Solution: \boxed{5x^2 + 24x}


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25,$ $b_{12} = 125,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$


How many positive factors of 360 are perfect squares?


What is the area enclosed by the graph of $|2x| + |3y| = 12$?


In the diagram, square $ABCD$ has sides of length $6$, and $\triangle ABE$ is isosceles with $AB=BE$ and $\angle ABE = 45^\circ$. Line segments $BE$ and $AC$ intersect at $P$. Point $Q$ is on $BC$ such that $PQ$ is perpendicular to $BC$ and $PQ=x.$ Determine the measure of angle $BPC.$


Determine the number of positive real solutions to the equation $x^{10} + 9x^9 + 20x^8 + 2000x^7 - 1500x^6 = 0$.


6 12-sided dice are rolled. What is the probability that the number of dice showing a two-digit number is equal to twice the number of dice showing a one-digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.)


Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Calculate the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5\}$. Find the remainder when this number is divided by $100$.


Find $x$ such that $\lceil x \rceil \cdot x = 156$. Express $x$ as a decimal.


How many of the divisors of \(10!\) are larger than \(9!\)?


For \(-1 < r < 1\), let \( T(r) \) denote the sum of the geometric series \[ 15 + 15r + 15r^2 + 15r^3 + \cdots . \] Let \( b \) between \(-1\) and \(1\) satisfy \( T(b)T(-b) = 3240 \). Find \( T(b) + T(-b) \).


A specific organization consists of five leaders and some number of regular members. Each year, the current leaders are dismissed, and each member loses 10% of its regular members due to non-renewal. Then, remaining regular members must each recruit two additional members. Afterward, five new leaders are recruited from outside the organization. Initially, there are twenty-five people in total in the organization. Determine the total number of people in the organization after five years.


An area in the shape of a right-angled triangle is defined by points at $(0,0)$, $(0,5)$, and $(5,0)$. If a point $(x,y)$ is chosen at random within this triangle, what is the probability that $x + y \le 3$? Express your answer as a common fraction.


In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). Determine the ratio $PA:AB$.


**How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 3, 6, \ldots, 45\}$, where the numbers form an arithmetic sequence? Already given that the common difference between successive elements is 3.**


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


Solve for $x$ in the equation $9^{x+9} = 12^x$ such that the solution can be expressed as $x = \log_b 9^9$. Find the base $b$.


Three positive integers have an arithmetic mean of 30 and median of 28. If the median is 6 less than the largest number, what is the smallest number?


What is the smallest whole number $b$ such that 100 can be expressed in base $b$ using only three digits?


The degree measures of the angles in a convex 15-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.


Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form.


The large square seen in the diagram is constructed from 16 smaller unit squares. Four of these small squares are diagonally split into two equal triangles. If only the triangles are shaded, and each triangle covers half of its respective square, what fraction of the large square is shaded?

[asy]
size(100);
filldraw((0,0)--(1,0)--(0,1)--cycle, gray(0.6));  // shaded triangle in the bottom left square
filldraw((3,3)--(4,3)--(3,4)--cycle, gray(0.6));  // shaded triangle in the top right square
filldraw((3,0)--(4,0)--(4,1)--cycle, gray(0.6));  // shaded triangle in the bottom right square
filldraw((0,3)--(0,4)--(1,4)--cycle, gray(0.6));  // shaded triangle in the top left square
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);  // large square
draw((1,0)--(1,4));  // vertical lines
draw((2,0)--(2,4));
draw((3,0)--(3,4));
draw((0,1)--(4,1));  // horizontal lines
draw((0,2)--(4,2));
draw((0,3)--(4,3));
[/asy]


Find the product $ABC$ from the partial fraction decomposition of
\[\frac{x^2 + 5x - 14}{x^3 + x^2 - 14x + 24}\]
into the form
\[\frac{A}{x - 2} + \frac{B}{x + 3} + \frac{C}{x - 4}.\]


What is the largest integer that is a divisor of \[
(n)(n+2)(n+4)(n+6)(n+8)
\] for all positive odd integers $n$?


The sum of the first \(n\) terms in the infinite geometric sequence \(\left\{1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\dots \right\}\) is \(\frac{31}{16}\). Find \(n\).


If \( 12s + 7t = 154 \) and \( s \) is three less than twice \( t \), what is the value of \( t \)?


Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3.


A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 8 and an altitude of 10. The height of the cylinder is twice its radius. Find the radius of the cylinder. Express your answer as a common fraction.


I flip a fair coin twice and roll a regular six-sided die. What is the probability that at least one coin will show heads and the die will show a 3?


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


Calculate the number of square units in the area of the shaded region within a larger square with vertices at (0,0), (40,0), (40,40), and (0,40). The shaded region is formed by the polygon with vertices at (0,0), (20,0), (40,30), (40,40), (10,40), and (0,10). Calculate the area of the shaded region.


Solution: \boxed{14 + \frac{14\pi}{3}}


Determine the value of the following expression: $\frac{1}{5}-\frac{1}{25}+\frac{1}{125}-\frac{1}{625}+\frac{1}{3125}$.


In triangle $PQR$, the side lengths $p$, $q$, and $r$ satisfy $p + 3q + 3r = p^2$ and $p + 3q - 3r = 3$. Find the largest angle of the triangle, assuming $p$ is opposite $\angle P$, $q$ opposite $\angle Q$, and $r$ opposite $\angle R$.


A circle is positioned inside the parabola with equation \(y = 4x^2\) so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?


A rectangular box has a volume of $4500$ cubic inches and a surface area of $1800$ square inches. The sum of the lengths of its $12$ edges is $216$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by one inch?

Solution: \boxed{5455 \text{ cubic inches}}


A function $g$ is defined by $g(z) = (3 + i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$?


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


Add 78.2514 to 34.7859 and round to the nearest hundredth.


Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 9? Express your answer as a common fraction.


If $x$ and $y$ are positive integers less than $20$ for which $x + y + xy = 119$, what is the value of $x + y$?


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


I had planned to work for 15 hours a week for 15 weeks during fall to earn $4500 to travel to Europe. Unfortunately, I got a part-time internship for the first three weeks of fall, during which I can only work 5 hours each week. How many hours a week will I need to work for the remaining weeks if I still plan to travel?


What is the value of $\sqrt{49+81}-\sqrt{36-9}$?

Solution: \boxed{\sqrt{130} - 3\sqrt{3}}


The sequence $3, x, 13, 18, y, 28, 33$ is arithmetic. What is the value of $x + y$?


For certain real numbers $p$, $q$, and $r$, the polynomial \[ h(x) = x^3 + px^2 + 2x + 15 \] has three distinct roots, and each root of $h(x)$ is also a root of the polynomial \[ k(x) = x^4 + x^3 + qx^2 + 150x + r. \] What is $k(1)$?


Alex has 8 pieces of paper, each with a different math problem. He must give all these problems to his 10 friends. However, problems 1 and 2 must be given to the same friend. In how many ways can he distribute these problems?


Let the first term of a geometric sequence be $\frac{8}{9}$, and the second term be $40$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by two million?


At a regional competition, 30 athletes are assigned individual rooms in a hotel, with room numbers ranging from 1 to 30. If the athletes staying in rooms 15, 16, and 17 do not arrive, what is the median room number of the athletes who are present?


Let $A,$ $B,$ and $C$ be constants such that the equation \[\frac{(x+B)(Ax+16)}{(x+C)(x+5)} = 3\]has infinitely many solutions for $x.$ For these values of $A,$ $B,$ and $C,$ it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the sum of these values of $x.$


Find $65^{-1} \pmod{66}$, as a residue modulo 66. (Give an answer between 0 and 65, inclusive.)


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


John and Jack live 2 miles apart. It takes John 6 minutes to walk directly to Jack's house. At the same rate, how many minutes would it take John to walk the 4 miles from his own house to Ben's house?


The cost of two pencils and three pens is $4.10, and the cost of three pencils and one pen is $2.95. What is the cost of one pencil and four pens?


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(3000,0),(3000,3000),$ and $(0,3000)$. What is the probability that $x > 3y$? Express your answer as a common fraction.


Both roots of the quadratic equation $x^2 - 65x + k = 0$ are prime numbers. Find the number of possible values of $k$.


How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable, the boxes are not distinguishable, and no box can be empty?


A hexagon is formed by joining, in order, points $(0,0)$, $(1,1)$, $(2,1)$, $(3,0)$, $(2,-1)$, $(1,-1)$, and back to $(0,0)$. Calculate the perimeter of this hexagon and represent it in the form $$a+b\sqrt{2} + c\sqrt{3}$$ where $a$, $b$, and $c$ are integers. Find the sum $a + b + c$.


Find the integer $n$, $1 \le n \le 10$, such that \[n \equiv 123456 \pmod{11}.\]


Suppose positive numbers $a$, $b$, and $c$ satisfy:
\begin{align*}
a^2/b &= 2, \\
b^2/c &= 3, \text{ and}\\
c^2/a &= 4.
\end{align*}
Find $a$.


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


Given that $a$, $b$, $c$, and $d$ are nonzero real numbers, find all possible values of the expression
\[
\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{d}{|d|} + \frac{abcd}{|abcd|}.
\]

Solution: \boxed{5, 1, -3}


Given that $$(x+y+z)(xy+xz+yz)=40$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y) + 2xyz = 20$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$?

Solution: \boxed{xyz = 20}


If $\mathbf{B} = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$, then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} p & r \\ q & s \end{pmatrix}.\]
Given that $\mathbf{B}^T = 2\mathbf{B}^{-1}$, find $p^2 + q^2 + r^2 + s^2.$


How many non-congruent triangles with perimeter 8 have integer side lengths?


Let $x, y, z$ be real numbers such that:
\[
x + y + z = 5,
\]
\[
x^2 + y^2 + z^2 = 9.
\]
Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$


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


What is the greatest common factor and the least common multiple of 42, 126, and 105?


How many prime numbers between 50 and 90 have a prime remainder when divided by 12?


In Pascal's Triangle, each entry is the sum of the two entries directly above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $4:5:6$?


Calculate the sum of all integral values of $c$ with $c \leq 20$ for which the equation $y = x^2 - 9x - c$ produces two rational roots.


Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 16x - 12y + 100$. What is $x-y$?


Let $a, b, c, d$ be four terms in an arithmetic sequence where all terms are positive, such that $abcd = 256$. Find the smallest possible value of $b$.


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


A garden can be watered by any of three sprinklers X, Y, or Z. Sprinklers X and Y together take 5 hours to water the garden. Sprinklers X and Z together take 6 hours to water the garden. Sprinklers Y and Z together take 7 hours to water the garden. How many hours does it take sprinklers X, Y, and Z working together to water the garden? Express your answer as a decimal to the nearest hundredth.


Let $m$ be the integer such that $0 \le m < 41$ and $4m \equiv 1 \pmod{41}$. What is $\left(3^m\right)^4 - 3 \pmod{41}$?


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


James walks up a hill at a speed of $(x^2 - 13x - 30)$ miles per hour. Meanwhile, Jane travels a total distance of $(x^2 - 5x - 66)$ miles in $(x+6)$ hours. If James and Jane walked at the same speed, what is their speed in miles per hour?

Solution: \boxed{-4 + 2\sqrt{17}}


What is the least natural number that can be added to 250,000 to create a palindrome?


Find the positive value for the slope of an asymptote of the hyperbola defined by:
\[
\sqrt{(x-1)^2+(y+1)^2} - \sqrt{(x-7)^2+(y+1)^2} = 4
\]


For how many real numbers $a$ does the quadratic equation $x^2 + ax + 9a = 0$ have only integer roots for $x$?


Four vertices of a cube are \( A=(1,4,2) \), \( B=(2,0,-7) \), and \( C=(5,-5,1) \). What is the surface area of this cube?


The complete graph of $y=g(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)

What is the sum of the $x$-coordinates of all points where $g(x) = x - 2$?


Let $x$ and $y$ be nonzero real numbers. Find the minimum value of
\[ x^2 + y^2 + \frac{4}{x^2} + \frac{2y}{x}. \]


How many positive integers, not exceeding 100, are multiples of 5 or 7 but not 8?


Complex numbers $x, y, z$ form an equilateral triangle with side length 24 in the complex plane. If $|x + y + z| = 72$, find $|xy + xz + yz|$.


Let \[f(x) = \left\{
\begin{array}{cl}
ax+6, & \text{if } x > 3, \\
x-7 & \text{if } -3 \le x \le 3, \\
3x-b & \text{if } x < -3.
\end{array}
\right.\]
Find $a+b$ if the piecewise function is continuous.


How many distinct four-digit numbers can be formed using the digits $1$, $2$, $3$, $4$, $5$ such that exactly one digit repeats once?


Let \( a \) and \( b \) be real numbers. One of the roots of \( x^3 + ax + b = 0 \) is \( 2 + i \). Find \( a + b \).


The Dunkers are coached by Coach Loudvoice. The Dunkers have 15 players, but three of them, Tom, Dick, and Harry, refuse to play together in any combination of two or more. How many starting lineups (of 5 players) can Coach Loudvoice make, if the starting lineup can't contain any two of Tom, Dick, and Harry together?


Find all real $x$ such that \[\left\lfloor 2x \lfloor x \rfloor \right\rfloor = 58.\]

Solution: \boxed{[5.8, 5.9)}


If $r$ and $s$ vary inversely, and when $r$ is $1500$, $s$ is $0.25$, determine the value of $s$ when $r$ is $3000$. Following that, find the value of $r$ if $s = 0.15$.


Determine the units digit of the quotient when the number $2^{2023} + 3^{2023}$ is divided by $7$.


How much greater, in square inches, is the area of a circle of radius 30 inches than a circle whose circumference is 60π inches? Express your answer in terms of $\pi$.


In how many ways can 8 people be seated in a row of chairs if two of the people, Wilma and Paul, refuse to sit next to each other, and another pair, Adam and Eve, also refuse to sit next to each other?


Solution: \boxed{\log_{13} 75}


Find the greatest common divisor of $15893$ and $35542$.


How many three-digit positive integers are congruent to 2 (mod 4)?


Given that four real numbers $a$, $b$, $c$, and $d$ satisfy the following equations:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= -5, \\
a+c+d &= 10, \text{ and} \\
b+c+d &= -1,
\end{align*}
determine the value of $ab + cd$.


Circle \( D \) has a radius of 8 cm. Determine the area of the largest possible inscribed triangle having one side as a diameter of circle \( D \).

Solution: \boxed{64 \text{ square centimeters}}


The fictional country of Isoland uses an alphabet with ten unique letters: A, B, D, E, I, L, N, O, R, U. License plates in Isoland are structured with five letters. How many different license plates are possible if they must begin with A or I, end with R, cannot include the letter B, and no letters may repeat?


Triangle $ABC$ has vertices $A(0, 7)$, $B(3, 0)$, and $C(9, 0)$. A horizontal line with equation $y = t$ intersects line segment $\overline{AB}$ at $T$ and line segment $\overline{AC}$ at $U$, forming $\triangle ATU$ with area 18. Compute $t$.

Solution: \boxed{7 - \sqrt{42}}


In triangle $PQR$, $PQ = 5$, $PR = 8$, and $\cos \angle P = \frac{3}{5}$. Find the length of the median $\overline{PM}$ from $P$ to side $QR$.


Initially, there were twenty states that could opt to change their primary energy source to solar. If fifteen states decide to go ahead, and two later revert to their previous energy sources, what will be the new ratio rounded to the nearest tenth?


How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-150)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-12^2)}
\]
have for \(x\)?


The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = x, \quad
g(x,y) = g(y,x), \quad
(x + y) g(x,y) = yg(x, x + y).
\]
Calculate $g(18,63)$.


What is the largest three-digit multiple of 8 whose digits' sum is 24?


On a one-way single-lane road with a speed limit of 80 kilometers per hour, vehicles must maintain a safe following distance calculated as one vehicle length for each 10 kilometers per hour of speed or fraction thereof. There's a monitoring device alongside the road that counts the number of vehicles passing within one hour. Assuming each vehicle is 5 meters long, determine the maximum number of complete vehicles passing the monitoring device in an hour if no vehicles exceed the speed limit. Calculate the quotient when this number is divided by 10.


When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(6, 8)$, and $H(6, 2)$. What is the area of trapezoid $EFGH$?


A club has 5 members from each of 4 different schools, for a total of 20 members. Determine how many possible ways there are to arrange a presidency meeting under the following conditions:
i. The club must choose one of the 4 schools at which to host the meeting.
ii. The host school sends 3 representatives to the meeting, and two of the other three schools send 1 representative each.


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, $30$ of which are triangular and $14$ of which are quadrilaterals. How many space diagonals does $Q$ have?


A solid sphere has a diameter of 24 cm. To customize it for a project, three cylindrical holes are drilled into the sphere. The holes have depths of 10 cm, 10 cm, and 5 cm with diameters of 3 cm, 3 cm, and 4 cm respectively. Calculate the remaining volume of the sphere in cubic centimeters, expressed in terms of \(\pi\).


Three dice with faces numbered 1 through 6 are stacked. Five of the eighteen faces are visible. The visible numbers are 1, 2, 3, 3, and 4. What is the total number of dots NOT visible in this view?


Find the sum of all angles $\theta_1, \theta_2, \dots, \theta_5$ where each $\theta_k$ is a solution in degrees to the equation
\[ z^5 = -\frac{1}{2} - \frac{\sqrt{3}i}{2}. \]
The roots are expressed as $\text{cis } \theta_k$, with $0^\circ \leq \theta_k < 360^\circ$ for all $1 \leq k \leq 5$.


Suppose $a$, $b$, $c$, $d$, and $e$ are integers satisfying: $a-b+c=7$, $b-c+d=8$, $c-d+e=9$, $d-e+a=4$, and $e-a+b=3$. What is the value of $a+b+c+d+e$?


The average years of experience of three employees, David, Emma, and Fiona, at a company is 12 years. Five years ago, Fiona had the same years of experience as David has now. In 4 years, Emma's experience will be $\frac{3}{4}$ of David's experience at that time. How many years of experience does Fiona have now?


A $120\times 270\times 300$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


Let \( S \) be the set of all sides and diagonals of a regular hexagon. A pair of elements of \( S \) are selected at random without replacement. What is the probability that the two chosen segments have the same length?


If 18 bahs are equal to 30 rahs, and 12 rahs are equal in value to 20 yahs, how many bahs are equal in value to 800 yahs?


What is the sum of the different prime factors of $300560$?


Tamika selects two different numbers at random from the set $\{11, 12, 13\}$ and multiplies them. Carlos takes two different numbers at random from the set $\{4, 6, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.

Solution: \boxed{\frac{9}{9} = 1}


Let \[\mathbf{M} = \begin{pmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{pmatrix}\] be a matrix with complex entries such that $\mathbf{M}^3 = \mathbf{I}$. If $ab = 1$, find the possible values of $a^3 + b^3$.


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