In right triangle $ABC$ with $\angle C = 90^\circ,$ suppose $AB = 25$ and $BC = 20.$ Find the largest possible value of $\tan A.$


Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$?


Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$?


Let \( c \) be a complex number. Suppose there exist distinct complex numbers \( p \) and \( q \) such that for every complex number \( z \), we have
\[
(z - p)(z - q) = (z - cp)(z - cq).
\]
Compute the number of distinct possible values of \( c \).


If $5u - 6v = 19$ and $3u + 5v = -1$, find the value of $u + v$.


Find the product of the divisors of $36$.


Determine the direction vector of line $m$ which passes through the origin and is associated with the reflection matrix
\[
\begin{pmatrix}
\frac{3}{5} & \frac{4}{5} \\
\frac{4}{5} & -\frac{3}{5}
\end{pmatrix}.
\]
Find the direction vector of line $m$ in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ and $b$ are integers, $a > 0$, and $\gcd(|a|, |b|) = 1$.

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


Suppose a parabola has vertex $\left(\frac{3}{5},-\frac{25}{12}\right)$ and its equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a$.


Alex has 7 pieces of paper, each with a different math problem. In how many ways can he distribute these problems to his 10 friends if each friend can get none, one, or multiple problems?


In a club of 15 members consisting of 8 girls and 7 boys, two members are chosen at random. What is the probability that both are girls, and at least one of them is under 18 years old if 3 of the girls are under 18?


Each day, Jenny ate $25\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 27 jellybeans remained. How many jellybeans were in the jar originally?


Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updating Sam's total to two coins. This continues until Bob says, ``$x$ for me, $x$ for you,'' at which he takes $x$ more coins and makes Sam's total $x$ coins in total. After all coins are distributed, Pirate Bob has exactly three times as many coins as Pirate Sam. Find out how many gold coins they have between them?


How many integers $-10 \leq n \leq 10$ satisfy $(n-3)(n+3)(n+9)>0$?


Factor the expression completely: \[(16y^6 + 36y^4 - 9) - (4y^6 - 6y^4 + 9).\]

Solution: \boxed{6(2y^6 + 7y^4 - 3)}


Find the greatest common divisor of 10,010 and 15,015.


Find the distance between the points \((3, 2, -5)\) and \((6, 9, -2)\) in 3-dimensional space.


In the diagram, circle \(O\) has a radius of 10 units. Chord \(EF\) has a length of 12 units and is parallel to segment \(LB\). Given that \(LA\) = 20 units and points \(L\), \(A\), \(O\), and \(B\) are collinear, what is the area of triangle \(LEF\)? Assume \(E\) and \(F\) are above line \(AB\) and \(LE\) and \(LF\) are segments connecting \(L\) to points on the circle.

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


Jack walks up a hill at a speed of $(x^2 - 11x - 22)$ miles per hour. Jill walks a total distance of $(x^2 - 5x - 36)$ miles in $(x + 4)$ hours. If both Jack and Jill walked at the same speed, what is that speed, in miles per hour?


There are four points that are $3$ units from the line $y=10$ and $10$ units from the point $(3,10)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


The vertical drops of six roller coasters at Fractal Fun Park are listed in the table below:
\begin{tabular}{|l|c|}
\hline
The Spiral Slide & 173 feet \\
\hline
The Velocity Vortex & 125 feet \\
\hline
The Cyclone & 150 feet \\
\hline
The Apex Jump & 310 feet \\
\hline
The Quantum Leap & 205 feet \\
\hline
The Zero Gravity & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values?


Starting with the prime number $3$, add consecutive prime numbers: $3$, $3+5$, $3+5+7$, and so on. How many of the first 15 such sums are also prime?


Take a variable \( c \), triple it, and then add six. From this result, subtract \( 6c \), and then divide the whole expression by three. What is the expression in simplest form?

Solution: \boxed{2 - c}


The function \[f(x) = \left\{ \begin{aligned} 2x + 1 & \quad \text{ if } x < 3 \\ x^2 & \quad \text{ if } x \ge 3 \end{aligned} \right.\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \dots + f^{-1}(4) + f^{-1}(9).$


A rectangular box has a volume of $5000$ cubic inches and a surface area of $1850$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by $3$ inches?

Solution: \boxed{8342 \text{ cubic inches}}


What is the 8th term of an arithmetic sequence of 15 terms where the first term is 5 and the last term is 50?


If $0.75 = \frac{y}{178+x}$, find the smallest value of $x$ given that $x$ and $y$ must be positive integers.


Given a prime \( p = 11 \) and integers \( 1 \leq a \leq p-1 \), determine the sum of integers that are primitive roots modulo 11.


How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w}$, $\mathbf{r}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}$, and $\mathbf{q}$ be the projection of $\mathbf{r}$ onto $\mathbf{v}$. If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{6}{7}$, find $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}$.


Find the distance between the planes \(3x + y - 4z + 3 = 0\) and \(6x + 2y - 8z + 6 = 0.\)


Evaluate the determinant of the matrix:
\[
\begin{pmatrix}
\cos(\alpha + \gamma) \cos \beta & \cos(\alpha + \gamma) \sin \beta & -\sin(\alpha + \gamma) \\
-\sin \beta & \cos \beta & 0 \\
\sin(\alpha + \gamma) \cos \beta & \sin(\alpha + \gamma) \sin \beta & \cos(\alpha + \gamma)
\end{pmatrix}
\]


What is the value of the expression $(35 + 12)^2 - (12^2 + 35^2 - 2\cdot 12 \cdot 35)$?


What is the range of the function \( g(x) = \frac{1}{x^3} \) for \( x \neq 0 \)?

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


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 10?


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 prime numbers have squares between 3000 and 6000?


A frustum of a pyramid with square bases sits on a base with side $a = 6\sqrt{2}$ cm, and the upper base has side $b = 12\sqrt{2}$ cm. The height of the frustum (distance between the bases) is $h = 12$ cm. Determine the volume of this frustum.


There are 7 blue chips and 5 yellow chips in a bag. One chip is drawn from the bag and not replaced. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


A wizard is creating a magical elixir. To complete the elixir, he must select one of four magical herbs and one of six enchanted crystals. However, one of the crystals is incompatible with three of the herbs. How many valid combinations can the wizard choose from to successfully create his elixir?


What is the greatest integer less than 200 for which the greatest common factor of that integer and 30 is 5?


Let $P_1$ be a regular $r$-gon and $P_2$ be a regular $s$-gon $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{68}{67}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?


What is the constant term in the expansion of $\left(\sqrt{x}+\dfrac{3}{x}\right)^{12}$?


A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\pi$.


Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $m$ such that $Q$ is divisible by $3^m$.


What is the sum of the coordinates of the midpoints of two segments, where the first segment has endpoints \((10, 20)\) and \((5, -10)\), and the second segment has endpoints \((-3, 7)\) and \((4, -2)\)?


Consider the quadratic equation $5x^2 - 6x - 12 = 0$. The positive difference between the two roots of this equation can be written as $\frac{\sqrt{p}}{q}$, where $q$ is an integer and $p$ is an integer not divisible by the square of any prime number. Find $p + q$.


Suppose lines $\overline{JK} \| \overline{LM}$ and you are given that $\angle OML$ is $x$, $\angle OJK$ is $3x$, and the supplementary condition holds as these angles are corresponding angles due to parallel lines. Determine the measure of angle $KLM$ if it is directly given as $2x$ in the configuration.


Consider a parallelogram with vertices at $(2,5)$, $(2,23)$, $(7,38)$, and $(7,20)$. A line through the origin cuts this parallelogram into two congruent polygons. Determine the slope of the line, expressed as $m/n$ where $m$ and $n$ are relatively prime positive integers, and find $m+n$.


Compute the determinant of the following matrix:
\[
\begin{vmatrix} 3 & 1 & -2 \\ 8 & 5 & -4 \\ 3 & 3 & 6 \end{vmatrix}.
\]


A frustum of a cone has a small radius of 2 inches, a large radius of 3 inches, and a height of 5 inches. What is the radius of a sphere which has the same volume as this frustum?


For what values of the constant $c$ does the graph of $f(x) = \frac{x^2 - 2x + c}{x^2 + 2x - 8}$ have exactly one vertical asymptote?

Solution: \boxed{-24 \text{ or } 0}


In a right-angled triangle, one leg is twice the length of the other leg, and the sum of the squares of the three side lengths is 2000. What is the length of the hypotenuse of this triangle?


How many integers between 1 and 300 are multiples of both 3 and 5 but not of either 6 or 8?


Let $
g(n) =
\begin{cases}
n^2-1 & \text{if }n\text{ is odd} \\
\dfrac{n}{3} & \text{if }n\text{ is divisible by 3}
\end{cases}.
$

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


Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z=15 \). Determine the maximum value of \[ X\cdot Y\cdot Z + X\cdot Y + Y\cdot Z + Z\cdot X. \]


Mr. and Mrs. Lee have three children. When they get into their family SUV, two people sit in the front, and three sit in the back. Either Mr. Lee or Mrs. Lee must sit in the driver's seat. How many seating arrangements are possible?


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

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


What is the value of $y$ in the equation $\left(\frac{40}{70}\right)^2=\sqrt{\frac{y}{70}}$?


Jessica finds a torn piece of paper with the top part of a monic polynomial of degree \( n \) visible. The terms that can be seen are \( x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + a_{n-3}x^{n-3} \), and it is noted that \( a_{n-1} = -a_{n-2} \). Jessica discerns from the scribbles that her challenge is to find the sum of the cubes of the roots of this polynomial. She is aware that \( a_{n-1} = -2 \) and \( a_{n-3} = 0 \). Determine the sum of the cubes of the roots.


A function $f$ is defined recursively by $f(1)=2$, $f(2)=3$, and for all integers $n \geq 3$, \[f(n)=f(n-1)-f(n-2)+2n.\] What is $f(2023)$?


Tamika selects two different numbers at random from the set $\{7, 8, 11, 13\}$ and multiplies them. Carlos selects two different numbers at random from the set $\{2, 4, 9\}$ and also multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 5?


The graphs of $y = x^3 - 6x + 4$ and $x + 3y = 3$ intersect in the points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$ If $x_1 + x_2 + x_3 = A$ and $y_1 + y_2 + y_3 = B,$ compute the ordered pair $(A,B).$


Expand the following product: $\frac{3}{7}\left(\frac{7}{x^3} - 14x^4\right)$.

Solution: \boxed{\frac{3}{x^3} - 6x^4}


If five people are selected at random from a group of ten men and five women, what is the probability that at least one woman is selected? Express your answer as a common fraction.


In a parade, there are acrobats, elephants, and horses. If there are a total of 72 legs and 20 heads, how many acrobats are in the parade?


In a literature class, students are tasked with reading a 980-page book. Two friends, Daniel and Emma, decide to split the reading. Daniel reads a page in 50 seconds, and Emma reads a page in 40 seconds. They plan to divide the book such that each spends an equal amount of time reading. Determine the last page that Daniel should read to ensure both spend the same amount of time reading.


Compute the product of \( 0.\overline{456} \) and \( 7 \), and express your result as a simplified fraction.


The asymptotes of a hyperbola are $y = x + 2$ and $y = 4 - x.$ The hyperbola passes through the point $(4,4)$. Find the distance between the foci of the hyperbola.


Simplify $\tan \frac{\pi}{12} + \tan \frac{5\pi}{12}.$


Given that positive integers $A, B, C$ form an arithmetic sequence and $B, C, D$ form a geometric sequence with the ratio $\frac{C}{B} = \frac{4}{3}$, find the smallest possible value of $A + B + C + D$.


For some positive integer \( k \), when 90 is divided by \( k^2 \), the remainder is 18. What is the remainder when 130 is divided by \( k \)?


Solve for $t$: $2 \cdot 2^t + \sqrt{4 \cdot 4^t} = 20$.

Solution: \boxed{t = \log_2 5}


Define the operation $\spadesuit$ as $a \spadesuit b = |a - b|$. What is the value of $(3 \spadesuit 5) \spadesuit (2 \spadesuit (1 \spadesuit 6))$?


A bowling ball is a solid ball with a spherical surface and diameter 24 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Jamie has holes drilled that are 10 cm deep and have diameters of 3 cm, 3 cm, and 4 cm. Assuming the three holes are right circular cylinders, find the number of cubic centimeters in the volume of the fitted bowling ball. Express your answer in terms of $\pi$.

Solution: \boxed{2219\pi \text{ cubic cm}}


You have $4$ red shirts, $4$ green shirts, $4$ blue shirts, $6$ pairs of pants, $8$ green hats, $8$ red hats, and $8$ blue hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?


Simplify the following expression: $(3y - 2)\cdot(5y^{12} + 3y^{11} + 6y^{10} + 4 + 2y^{9})$. Express your answer as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{15y^{13} - y^{12} + 12y^{11} - 6y^{10} - 4y^{9} + 12y - 8}


Suppose when $n$ is divided by 5, the remainder is 3. What is the remainder when $7n + 4$ is divided by 5?


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

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


Yan is at a point between his home and a park. To get to the park, he can either walk directly or walk back home and then ride his scooter to the park. He rides 5 times as fast as he walks, and both options take the same amount of time. What is the ratio of Yan's distance from his home to his distance from the park?


A prism has 24 edges. How many faces does the prism have?


What is the sum of all the solutions of \( x = \left|2x - |50 - 2x|\right| \)?


Let vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ satisfy $\|\mathbf{u}\| = 2$, $\|\mathbf{v}\| = 4$, $\|\mathbf{w}\| = 6$, and $\mathbf{u} \times (\mathbf{u} \times \mathbf{v}) = \mathbf{w}$. If $\phi$ is the angle between $\mathbf{u}$ and $\mathbf{v}$, find $\sin \phi$.


A bookstore is deciding the optimal price for a new novel. It has found that if the novel is priced at $p$ dollars, where $p \le 30$, then the number of novels it can sell per month is expressed by the function $150 - 4p$. Determine the price $p$ that maximizes the store's monthly revenue from this novel.


For any positive integer $n$, let $T_n = (1+n)n/2$ be the nth triangular number. Determine the greatest possible value of the greatest common divisor ($\text{gcd}$) of $3T_n + n$ and $n+3$.


Carla's stationery shop charges an additional fee for oversized or undersized envelopes intended for a special craft project. The extra fee is applied if the length of an envelope, in feet, divided by its height, in feet, is less than $1.5$ or greater than $3.0$. Here is a list of four envelopes: \begin{tabular}[t]{ccc}
Envelope & Length in feet & Height in feet\\\hline
X &3 &2\\
Y &10 &3\\
Z &5 &5\\
W &15 &4
\end{tabular}
Determine for how many envelopes the extra fee must be paid?


Determine the number of angles between 0 and \(2\pi\), excluding integer multiples of \(\frac{\pi}{3}\), such that \(\sin \theta\), \(\cos \theta\), and \(\cot \theta\) form a geometric sequence in some order.


If the consecutive integers from $90$ to $1$ were written as $$9089888786...,$$ what would be the $103^{\text{th}}$ digit to be written?


Determine how many positive integers less than 500 are neither 9-nice nor 10-nice.


Find all real values of $x$ that satisfy $\frac{x^3 - 3x^2 + 2x}{x^2 - 2x + 1} \ge 0.$ (Give your answer in interval notation.)

Solution: \boxed{[0, 1) \cup [2, \infty)}


Seven balls are numbered 1 through 7 and placed in a bowl. Jake will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Jake 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 less than 25? Express your answer as a common fraction.


What is the greatest common divisor of $9125$ and $4277$?


Five painters can complete a painting job in 1 work-day. If, due to a shortage of equipment, only two painters can work but they are provided with advanced tools that effectively double their rate, how many work-days will it take them to finish the job?


Below is a portion of the graph of a quadratic function, $y=p(x)=ax^2+bx+c$:

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=12.5,ymin=-6.5,ymax=6.5;

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 4-(x-4)*(x-11)/16;}
draw(graph(f2,-0.99,12.5),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=p(x)$",(12.75,2.5),E);
[/asy]
The graph of $p(x)$ has its axis of symmetry at $x = 7.5$, and it is known that $p(4) = -4$. What is the value of $p(11)$?


If $\frac{1}{8}$ of $2^{36}$ is $4^x$, then what is the value of $x$?


Let $x, y, z$ be real numbers such that:
\begin{align*}
x + y + z &= 6, \\
x^2 + y^2 + z^2 &= 12.
\end{align*}
Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$


Let $f(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2010}.$ There are exactly 1008 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2010}|.\] What is the minimum number of real roots that $f(x)$ can have?


The fencing that enclosed six congruent equilateral triangular corrals is reused to form one large equilateral triangular corral. What is the ratio of the total area of the six small corrals to the area of the new large corral? Express your answer as a common fraction.


Given the right triangles ABC and ABD, determine the length of segment BC, in units, where $AD = 42$ units, $AB = 40$ units, and $BD = 58$ units.
```plaintext
[asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,40);
C=(-30,0);
D=(-58,0);
draw(A--B--D--A);
draw(B--C);
draw((0,3)--(-3,3)--(-3,0));
label("$58$", (B+D)/2, NW);
label("$42$", (C+D)/2, S);
label("$30$", (A+C)/2, S);
label("A", A, SE);
label("B", B, NE);
label("D", D, SW);
label("C", C, S);
[/asy]
```

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


A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(3, \frac{5\pi}{6}, \frac{\pi}{4} \right)$. Find the spherical coordinates of the point with rectangular coordinates $(-x,-y,z)$.

Solution: \boxed{\left(3, \frac{11\pi}{6}, \frac{\pi}{4}\right)}


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


Mia is designing a rectangular flower bed against her house on one side, using 450 feet of fencing to enclose the remaining three sides. If 150 feet of it is planned to be used along the house, what is the maximum area she can achieve for her garden?


How many positive perfect squares less than $10^8$ are multiples of 36?


Laura is training for a marathon and decides to include additional exercises in her training schedule. She walks for $15$ minutes at a speed of $3$ miles per hour, bikes $30$ miles at $3x + 2$ miles per hour, then spends $10$ minutes transitioning to her running phase, and finally runs $10$ miles at $x$ miles per hour. Her entire workout, including transitions, lasts $205$ minutes. Determine how fast Laura runs during her training session, rounded to the nearest hundredth of a mile per hour.

Solution: \boxed{6.35 \text{ mph}}


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


A standard deck of 52 cards is randomly arranged. What is the probability that the Ace of Spades is among the top three cards?


Four of the vertices of a rectangle are located at $(4, 9)$, $(15, 9)$, $(15, -4)$, and $(4, -4)$. Determine the area of the intersection between this rectangle and the region inside the graph of the equation $(x - 4)^2 + (y - 9)^2 = 16$.


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

Then
\[\overrightarrow{J} = x \overrightarrow{P} + y \overrightarrow{Q} + z \overrightarrow{R},\] where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$  Enter the ordered triple $(x,y,z).$

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


How many 3-digit numbers have the property that the units digit is at least twice the tens digit and the tens digit is at least three times the hundreds digit?


The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the $19$ top-rated roller coasters in the world. In the stem-and-leaf plot, $2 \ 15$ represents $2$ minutes, $15$ seconds, which is the same as $135$ seconds. What is the median of this data set? Express your answer in seconds.

\begin{tabular}{c|ccccccc}
0 & 23 & 23 & 45 & 55 & & & \\
1 & 00 & 10 & 30 & 45 & 50 & & \\
2 & 15 & 25 & 28 & 40 & 50 & & \\
3 & 05 & 05 & 15 & & & & \\
6 & 00 & & & & & & \\
\end{tabular}


The arithmetic mean of an even number of consecutive even integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$.


Let $I$ be the incenter of triangle $ABC,$ and $P$ be an arbitrary point. Determine a constant $k$ such that
\[PA^2 + PB^2 + PC^2 = k \cdot PI^2 + IA^2 + IB^2 + IC^2.\]


Compute the remainder when $5^{5^{5^5}}$ is divided by 1000.


Marty wants to paint a box. He can choose to use either blue, green, yellow, black, or white paint. Also, he can style the paint by painting with a brush, a roller, a sponge, or a spray. However, if he chooses white paint, he cannot use a sponge. How many different combinations of color and painting method can Marty choose?


If $x$ and $y$ are positive real numbers such that $3x^2 + 6xy = x^3 + xy^2,$ find the value of $x$.


Points $P$ and $Q$ are selected on the graph of $y = -\frac{1}{3}x^2$ so that triangle $PQO$ is equilateral. Find the length of one side of triangle $PQO$.


Solution: \boxed{(0, 3)}


Add a condition to the previous problem: Determine how many numbers less than 50 are abundant numbers.


An isosceles triangle has two sides measuring 15 cm and one side measuring 24 cm. If a similar triangle has its longest side measuring 72 cm, find the perimeter of this larger triangle.

Solution: \boxed{162 \text{ cm}}


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


Ms. Lee's class has each student on average absent for two days out of fifty school days. What is the probability that among any three randomly selected students, exactly two students are present and one is absent? Express your answer as a percent rounded to the nearest tenth.


Isosceles triangle $PQR$ with an area of 144 square inches is cut by line segment $ST$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 108 square inches. If the altitude of triangle $PQR$ from $P$ is 24 inches, determine the length of segment $ST$.

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


Find $87^{-1} \pmod{88}$, as a residue modulo 88. (Give an answer between 0 and 87, inclusive.)


What is the perimeter of quadrilateral $WXYZ$ where $W = (0, 0)$, $X = (1, 0)$, $Y = (1, \sqrt{3})$, and $Z = (0, 2)$?

Solution: \boxed{3 + \sqrt{3} + \sqrt{8 - 4\sqrt{3}}}


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


Suppose that \( b \) is a positive integer greater than or equal to \( 2 \). When \( 256 \) is converted to base \( b \), the resulting representation has \( 4 \) digits. What is the number of possible values for \( b \)?


The sequence 800,000; 400,000; 200,000 and so on, is generated by continuously dividing by 2. What is the last integer in this sequence?


A regular tetrahedron with edge length 2 cm has a dot marked in the centre of one of its faces. It is sitting on a flat table. The tetrahedron is rolled, without lifting or slipping, over its edges so that at least two of its vertices are always touching the table. The tetrahedron is rolled until the dot is again on the top face. The length, in centimeters, of the path followed by the dot is $d\pi$, where $d$ is a constant. What is $d$?


Let \( y = \frac{\sum\limits_{n=1}^{30} \cos (2n)^\circ}{\sum\limits_{n=1}^{30} \sin (2n)^\circ} \). What is the greatest integer that does not exceed \( 50y \)?


What is the base-10 integer 875 when expressed in base 7?


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 8?


What are the last two digits in the sum of factorials of the first 50 positive integers?


Find how many integers satisfy the following inequalities:
\[
\begin{array}{cccc}
(1) & -5x & \geq & 2x + 10 \\
(2) & -x & \leq & 14 \\
(3) & -3x & \geq & x + 8
\end{array}
\]


A box contains 4 blue, 3 orange, and 5 green marbles. If the 12 marbles are randomly drawn from the box, one at a time and without replacement, what is the probability that the marbles are drawn in such a way that the 4 blue marbles are drawn consecutively, the 3 orange marbles are drawn consecutively, and the 5 green marbles are drawn consecutively, but not necessarily in the blue-orange-green order? Express your answer as a common fraction.


Two adjacent sides of a rhombus form a $45$-degree angle. If each side of the rhombus measures $3$ cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form.


If $x$ and $y$ are positive integers less than $15$ for which $x + y + xy = 119$, what is the value of $x + y$?


Point \(Q\) lies on the line \(x = -2\) and is 8 units from the point \((4, -1)\). Find the product of all possible \(y\)-coordinates that satisfy the given conditions.


Consider a grid where movements are restricted to the right or down across drawn segments. How many different routes are there from point $A$ to point $B$ through a grid set up as a 3x2 rectangle (meaning 3 rows and 2 columns)?

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


Determine the number of roots of unity that are also roots of the polynomial equation $z^3 + az^2 + bz + c = 0$ for some integers $a$, $b$, and $c$.


What is the period of $y = \tan x + \cot x - \sec x$?


What is the base 8 representation of the base 2 number $110110110_2$?


How many natural numbers greater than 9 but less than 100 are relatively prime to 18?


Let $d$ be a complex number. Suppose there exist distinct complex numbers $a$, $b$, and $c$ such that for every complex number $z$, we have:
\[
(z - a)(z - b)(z - c) = (z - d^2a)(z - d^2b)(z - d^2c).
\]
Compute the number of distinct possible values of $d$.


An investor placed $\$15,000$ in a one-year bond that paid a simple annual interest rate of $10\%$. After one year, they reinvested the total amount into another one-year bond. At the end of the second year, the investment had grown to $\$17,325$. If the annual interest rate of the second bond is $s\%$, what is $s$?

Solution: \boxed{s = 5}


Tom's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 10 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?


How many 3-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, $E$, and $F$? Assume that each letter may be used more than once in a word.


A line passing through the points $(4, 20)$ and $(-6, -2)$ intersects the $y$-axis at what point?

Solution: \boxed{(0, 11.2)}


A box contains eight cards. Four of the cards are blue on both sides, two cards are blue on one side and green on the other, and two of the cards are green 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 green, what is the probability that the other side is also green?


The line \( y = \frac{3}{4} x + 3 \) is parametrized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -9 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -7 \end{pmatrix}.
\]
Enter the ordered pair \( (s,l) \).

Solution: \boxed{\left( -\frac{15}{4}, -\frac{28}{3} \right)}


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


Six bowlers participated in a professional bowling tournament. In round one, bowlers #4, #5, and #6 compete in a sequence of two matches: first #6 bowls #5, the winner bowls #4. The top two among these three then advance, with the loser of the second game taking the 6th prize, and the second last of the first match taking 5th prize. The next rounds follow the playoff format with a slight change: the final winner of the initial round bowls against #3, and continue as previously described with the loser of each game receiving the next available descending prize, and the final match winner and loser receiving 1st and 2nd prizes respectively. In how many orders can bowlers #1 through #6 receive the prizes?


Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 60^\circ$, and $\angle ADC = 150^\circ$. What is the degree measure of $\angle BAD$?


Math City now has ten streets, each straight and no two streets are parallel. Additionally, two involvements are such that these streets cross each other precisely at an angle of 180 degrees, effectively not creating an intersection. How many police officers are needed to manage each intersection, assuming each intersection is manned by one officer?


A 2% late charge was added to John's bill on the 30th day past its due date. The same late charge was applied again every subsequent 30 days for a total of three times. Additionally, John's bill was subject to an annual interest rate of 5% compounded yearly, applied at the end of one year. John's original bill was $500. What is the total amount due on his bill at the end of one year?


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


**Given:** Triangle $PQR$ has side-lengths $PQ = 14, QR = 28,$ and $PR = 21$. The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. **Find:** the perimeter of $\triangle PXY$.


How many integers, $x$, satisfy $|7x - 4| \le 10$?


The mean (average) of 8, 10, and 22 is equal to the mean (average) of 16 and $z$. What is the value of $z$?


**How many integer values of $n$ satisfy $-125 < n^3 < 125$?**


Find the number of positive divisors of the number 2310, and then find the sum of these divisors.


Calculate the simplified form of the expression \[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2\].

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


Emily rolls 6 fair 8-sided dice. What is the probability that at least two dice show the same number?


Given a circle with center $O$, and triangle $ABC$ inscribed in the circle such that $\angle BOC = 130^{\circ}$ and $\angle AOB = 150^{\circ}$. Determine the degree measure of $\angle ABC$.


Compute $(2 \cos 30^\circ + 2i \sin 30^\circ)^4.$ Enter your answer in rectangular form.

Solution: \boxed{-8 + 8i \sqrt{3}}


Let \[q(x,y) =
\begin{cases} x + 2y &\quad \text{if } x > 0 \text{ and } y \ge 0, \\
x - 3y &\quad \text{if } x < 0 \text{ and } y \le 0, \\
4x + 2y &\quad \text{otherwise}.
\end{cases}
\]
Calculate $q(q(2,-2), q(-3,1))$.


If the three points $(2, p, q), (p, 3, q), (p, q, 4)$ are collinear, what is the value of $p + q$?


Thirteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Bella, starts with the ball, skips the next five girls, and throws to the seventh girl, who then continues to skip five girls and proceeds with the throw. If this pattern continues, including Bella's initial throw, what is the total number of throws necessary for the ball to return to Bella?


Find the remainder when \( 3x^7 - 2x^5 + 5x^3 - 6 \) is divided by \( x^2 + 3x + 2 \).

Solution: \boxed{354x + 342}


A line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}.
\]
A second line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \end{pmatrix} + u \begin{pmatrix} 4 \\ -5 \end{pmatrix}.
\]
Find the point where the lines intersect.

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


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


Factor the following expression: $75x^{19} + 165x^{38}$.

Solution: \boxed{15x^{19}(5 + 11x^{19})}


Points $P$, $Q$, $R$, and $S$ are in space, such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = 15$, $SQ = 15$, and $SR = 12$, compute the volume of pyramid $SPQR$.


Football tickets now cost $\$15$ each. Jane has $\$180$. If Jane buys more than 10 tickets, she receives a discount of $\$2$ per ticket for every ticket. What is the maximum number of tickets Jane can buy?


When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \(a, b, c\) are obtained respectively where \(a\) and \(b\) are from the standard dice and \(c\) is from the 8-sided die. Find the probability that \((a-1)(b-1)(c-1) \neq 0\).


Find the number of integers $n$ that satisfy
\[16 < n^2 < 121.\]


Marius is entering a wildlife photo contest and decides to arrange his nine snow leopards of different heights in a row. The two shortest leopards still have inferiority complexes and demand to be placed at the ends of the row. Additionally, the tallest leopard insists on being positioned in the middle of the row. How many ways can Marius line up the leopards?

Solution: \boxed{1440 \text{ ways.}}


Calculate $\begin{pmatrix} 3 \\ -8 \end{pmatrix} - (-5) \begin{pmatrix} 4 \\ 6 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 23 \\ 22 \end{pmatrix}}


In a championship 200-meter race, there are 10 sprinters competing, including 4 Americans. The gold and silver medals are awarded to the first and second places, respectively. How many ways can the medals be awarded if at most one American sprinter wins a medal?


Simplify the following expression:

$$7x + 8 - 3x + 15 - 2y$$

Solution: \boxed{4x - 2y + 23}


What is the greatest common factor of 36, 54, and 81?


A rectangular box has a volume of $3000$ cubic inches and a surface area of $1380$ square inches. The sum of the lengths of its $12$ edges is $160$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{4548 \text{ cubic inches}}


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


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


The graph of the equation \( y = \frac{x}{x^3 + Dx^2 + Ex + F} \), where \( D, E, F \) are integers, has three vertical asymptotes at \( x = -3, 0, 4 \). Determine the sum \( D + E + F \).


Apply a $270^\circ$ rotation around the origin in the counter-clockwise direction to the complex number $-7 - 4i$. What is the resulting complex number?

Solution: \boxed{-4 + 7i}


Find \(AX\) in the triangle if \(CX\) bisects \(\angle ACB\).

[asy]
import markers;

real t=.6;
pair A=(0,0);
pair B=(4,2);
pair C=(1,2);
pair X=t*A+(1-t)*B;

draw(C--A--B--C--X);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SE);

label("$32$",.5*(B+X),SE);
label("$36$",.5*(B+C),N);
label("$27$",.5*(A+C),NW);

[/asy]


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 330^\circ & -\sin 330^\circ \\ \sin 330^\circ & \cos 330^\circ \end{pmatrix}^n = \mathbf{I}.\]


Determine the value of $y$ that satisfies $\sqrt[3]{y\sqrt[4]{y^6}}=5$.


Determine the real numbers $A$ and $B$ such that the polynomial $x^{205} + Ax + B$ is divisible by $x^2 + x + 1$. Find $A + B$.


Suppose the point $(2,3)$ is on the graph of $y=\frac{f(x)}{2}$. Determine the sum of the coordinates of the one point which must be on the graph of $y=2f^{-1}(x)$.


Let \( z = re^{i\theta} \) be a complex number such that \( |z| = r \neq 1 \) and \( r > 0 \). Find the real part of \( \frac{1}{1 - z^2} \).

Solution: \boxed{\frac{1 - r^2 \cos(2\theta)}{1 - 2r^2 \cos(2\theta) + r^4}}


Each of the numbers \( b_1, b_2, \dots, b_{100} \) is \( \pm 1 \). Find the smallest possible positive value of
\[
\sum_{1 \leq i < j \leq 100} b_i b_j.
\]


A club has 18 members with varying lengths of membership ranging from 1 to 18 years. The club needs to choose 2 members to be co-presidents on the condition that the two selected members must have at least 5 years difference in their membership length. How many ways can the club choose its co-presidents under this condition?

Solution: \boxed{153 - 62 = 91}


Given the equation $a + b = 40$, where $a$ and $b$ are positive even integers, how many distinct ordered-pair solutions $(a, b)$ exist?


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


For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor (GCD) of $8!$, $10!$, and $11!$.


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 $C$ and $D,$ such that $PC < PD.$ If $PC = 4$ and $PT = CD - PC,$ what is $PD$?


Find the smallest possible value of $A + B + C + D$ where $A, B, C$ form an arithmetic sequence and $B, C, D$ form a geometric sequence. Given $\frac{C}{B} = \frac{4}{3}$, what is the smallest possible value of the sum?


Given the equation $z^2 - 6z + 34 = 0$, how many possible values are there for $|z|$?


Given invertible functions s(x), t(x), and u(x), and defining g = t ∘ s ∘ u, determine the correct expression for g^(-1).

Solution: \boxed{u^{-1} \circ s^{-1} \circ t^{-1}}


Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 3,$ and $x \ge -1,$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of
\[
\sqrt{4x + 4} + \sqrt{4y + 8} + \sqrt{4z + 12}.
\]


Let $r$ and $s$ denote the two real roots of $x^2 - 2x \sqrt{3} + 1 = 0.$  Then determine $r^6 + s^6.$


Phil has 8 green marbles and 4 purple marbles in a bag. He removes a marble at random, records its color, puts it back, and then repeats this process until he has withdrawn 8 marbles. What is the probability that exactly four of the marbles that he removes are green? Express your answer as a decimal rounded to the nearest thousandth.


Let $p$, $q$, and $r$ be integers such that:
\[\mathbf{B} = \frac{1}{4} \begin{pmatrix} -1 & p \\ q & r \end{pmatrix}\]
and $\mathbf{B}^2 = \mathbf{I}.$ Find the largest possible value of $p + q + r$.


In Ms. Chen's class, 120 students took a test. What is the largest number of students who can be awarded extra credit if extra credit is given to students who score above the class mean?


Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 5BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$? Illustrate the locations of the points on the line.


Let $\theta$ be the angle between the line
\[\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z + 5}{5}\]
and the plane $4x + 5y - 2z = 6.$ Find $\sin \theta.$


Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]
over all real numbers $x$.


A basketball team has 12 players. The coach needs to select a team consisting of one captain and 5 other players to start a game. How many different starting teams can the coach form?


Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 5 dogs, and one with 3 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 5-dog group?


What is the probability that Fatima gets more heads than tails if she flips 12 coins?


Using only the digits 1, 2, and 3, how many positive nine-digit integers can be made that are palindromes?


The sum of three numbers $a$, $b$, and $c$ is 108. If we increase $a$ by 8, decrease $b$ by 4, and multiply $c$ by 6, the three resulting numbers become equal. What is the value of $b$?

Solution: \boxed{b = \frac{724}{13}}


Find the smallest positive multiple of 30 that is a perfect square and the smallest positive multiple of 30 that is a perfect cube. Then, list all the multiples of 30 between these two numbers, inclusive. How many integers are in this list?


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

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


We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


There are four points that are $7$ units from the line $y=15$ and $15$ units from the point $(9,15)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


Consider dividing 12 dogs into three groups, with group sizes of 4, 6, and 2. How many ways can you organize these groups such that Fluffy is in the 4-dog group and Nipper is in the 6-dog group?


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


A $\frac{1}{p}$-array is structured where each row forms a sequence such that the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row, and each succeeding term in a row is $\frac{1}{p^2}$ times the previous term. Construct and find the sum of all terms in a $\frac{1}{2008}$-array, written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Then, provide the remainder when $m+n$ is divided by 2008.


What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome?


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


What is the smallest four-digit integer $n$ that satisfies $$75n\equiv 225\pmod{345}~?$$


Given vectors $\mathbf{u}$ and $\mathbf{z}$ such that $\|\mathbf{u}\| = 5,$ $\|\mathbf{z}\| = 8,$ and $\mathbf{u} \cdot \mathbf{z} = 20,$ find the magnitude of the projection of $\mathbf{u}$ onto $\mathbf{z}$.


John draws a regular seven-pointed star in the sand, and at each of the 7 outward-pointing points and 7 inward-pointing points, he places one of fourteen different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?


A geometric sequence has terms $81, b, \frac{8}{27}$ as the first, second, and third terms respectively. Determine the value of $b$ if $b$ is positive.


Let the numbers $p$, $q$, $r$, $s$ be equal to 2, 3, 5, 6, in some order. Find the largest possible value of
\[pq + qr + rs + sp.\]


If \( f(x) = f(5-x) \) for all \( x \), what line is necessarily an axis of symmetry of the graph of \( y = f(x) \)?

Solution: \boxed{x = 2.5}


Evaluate the product $\left(1 - \frac{1}{2}\right)^2\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{4}\right)^2\left(1-\frac{1}{5}\right)^2 \dotsm \left(1-\frac{1}{100}\right)^2$. Express your answer as a common fraction.


Consider two concentric circles where the larger circle's radius is $b$ and the smaller circle's radius is $a$. The area of the shaded region between the two circles is four times the area of the smaller circle. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.


A jar contains quarters (worth $0.25$ each), nickels (worth $0.10$ each), and pennies (worth $0.02$ each). The value of the quarters is $12.50$, the value of the nickels is $15.00$, and the value of the pennies is $8.00$. If Lena randomly chooses one coin from the jar, what is the probability that it is a quarter?


Compute $\arcsin \left( \frac{1}{2} \right)$ and $\arccos \left( \sqrt{3}/2 \right)$. Express your answers in radians.


In triangle $XYZ$, suppose $\angle Z = 90^\circ$ and $\angle X = \angle Y = 45^\circ$, with $XZ = YZ = 5$ cm. What is the area of $\triangle XYZ$?

Solution: \boxed{12.5 \text{ cm}^2}


What is the largest integer less than $\log_3 \frac{3}{2} + \log_3 \frac{4}{3} + \cdots + \log_3 \frac{1005}{1004} + \log_3 \frac{1006}{1005}$?


In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\).


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the minimum value of
\[\frac{9}{x} + \frac{16}{y} + \frac{25}{z}.\]


What is the remainder when $2011 \cdot 2012 \cdot 2013 \cdot 2014 \cdot 2015$ is divided by 23?


If
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^m,\]
then find $m.$


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


A number is divisible by $3$ if the sum of its digits is divisible by $3.$ Consider the number $2A6B8,$ where $A$ and $B$ are digits. If $2A6B8$ is divisible by $3,$ what is the sum of all possible values of $A+B?$


How many rows of Pascal's Triangle contain the number $47$?


Add 74.6893 to 23.152 and round to the nearest hundredth.


The set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 6 \\ -24 \\ 12 \end{pmatrix}\]
forms a solid in space. Find the volume of this solid.

Solution: \boxed{756 \sqrt{21} \pi}


Oakridge Middle School runs an intense chess team tournament with twelve players, including Anna and Bob. Among these, Anna and Bob have been selected as the team captains and are required to play at least once per week together. Each week, the twelve players participate in two chess matches, each consisting of six players in no particular order. Over the academic year, each possible combination of six players plays together exactly once. How many times do Anna and Bob play in the same match?


In a similar problem, Bert receives a triangular prism-shaped container that holds 150 marbles when filled to capacity. The container has a base area of 10 square units and a height of 3 units. A few weeks later, Carrie gets a larger triangular prism-shaped container that is twice as long, twice as wide, and twice as tall as Bert's. How many marbles does Carrie's container hold when filled to capacity?


Let $p,q,r$ be the roots of $x^3 - 6x^2 + 8x - 1 = 0$, and let $t = \sqrt{p} + \sqrt{q} + \sqrt{r}$. Find $t^4 - 12t^2 - 8t$.


Find the maximum difference between the $y$-coordinates of the intersection points of the graphs $y = 5 - 2x^2 + x^3$ and $y = 1 + x^2 + x^3$.


What is 150% of the median of the first fifteen positive integers? Express your answer as a decimal.


One of the roots of the cubic polynomial
\[ ax^3 + 4x^2 + bx - 100 = 0, \]
is $-3 - 4i$, where $a$ and $b$ are real numbers. Find the real root of this cubic polynomial.


A circular spinner for a game has a radius of 15 cm. The probability of winning on one spin of this spinner has two possibilities with each winning sector having an equal chance, and each winning probability is $\frac{1}{6}$. Calculate the total area, in sq cm, of the WIN sectors combined. Express your answer in terms of $\pi$.


In an isosceles right triangle, the altitude to the hypotenuse has a length of 4 units. What is the area of this triangle?


A function $f$ from the integers to the integers is defined as:
\[f(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $k$ is odd and $f(f(f(k))) = 57.$ Find $k.$


Convert the point $\left( 10, \frac{\pi}{4}, -3 \right)$ in cylindrical coordinates to rectangular coordinates.

Solution: \boxed{(5 \sqrt{2}, 5 \sqrt{2}, -3)}


How many non-empty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ consist entirely of odd numbers and do not include the subset $\{1, 5\}$?


What integer is closest to the value of $\sqrt[3]{5^3+7^3}$?


What is the product of the solutions of the equation $-20 = -2x^2 - 6x?$


Consider two lines: line $l$ parametrized as
\[
\begin{align*}
x &= 2 + 5t,\\
y &=  3 + 2t,
\end{align*}
\]
and the line $m$ parametrized as
\[
\begin{align*}
x &= -7 + 5s,\\
y &=  9 + 2s.
\end{align*}
\]
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.

Then $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that $v_2-v_1 = 7$. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.

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


What is the largest four-digit negative integer congruent to $1 \pmod{17}?$


A sphere has a volume of \( 288\pi \) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \( \pi \).


The matrix $\mathbf{B} = \begin{pmatrix} 4 & 5 \\ 3 & m \end{pmatrix}$ satisfies
\[\mathbf{B}^{-1} = j \mathbf{B}\]
for some constant $j$. Determine the ordered pair $(m,j)$.

Solution: \boxed{(-4, \frac{1}{31})}


Find all values of $x$ such that $\arctan x > \arccot x.$

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


Find all positive integers \(n\) such that the difference when the product of 3 and \(n\) is subtracted from 25 is at least 4.

Solution: \boxed{1, 2, 3, 4, 5, 6, 7}


Express $\frac{0.\overline{142857}}{2.\overline{857142}}$ as a common fraction.


Simplify $\sqrt{50x^3} \cdot \sqrt{18x^2} \cdot \sqrt{35x}$.

Solution: \boxed{30x^3 \sqrt{35}}


In the diagram, $\triangle ABC$ is right-angled at $A$, with $AB=45$ and $AC=60.$ The point $D$ is on $BC$ such that $AD$ is perpendicular to $BC.$ Determine the length of $BD.$


Let $h(x) = x - 3$ and $k(x) = x / 4$. Also denote the inverses to these functions as $h^{-1}$ and $k^{-1}$. Compute \[h(k^{-1}(h^{-1}(h^{-1}(k(h(27)))))).\]


Find all 6-digit multiples of 18 of the form $6d6,\!28e$ where $d$ and $e$ are digits. What is the maximum value of $e$?


In Mrs. Vale's class, there are 18 students, and each student's initials are unique with the format YY where both the first name and the last name start with "Y", considered as a vowel for this scenario. The available initials in the class are only from the subset of letters that includes B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, Y. What is the probability of randomly picking a student whose initials are right from the vowel list (where Y is a vowel)? Express your answer as a common fraction.


Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + b^2 + 1) = [g(a)]^2 + [g(b)]^2\] for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $g(26),$ and let $s$ be the sum of the possible values of $g(26).$ Find $n \times s.$


The product of two consecutive page numbers is $15,300.$ What is the sum of the two pages numbers?


In the arithmetic sequence $11, d, e, f, g, 39$, what is the value of $e$?


A point $Q$ is located inside $\triangle DEF$ such that lines drawn through $Q$ parallel to the triangle's sides result in smaller triangles with areas of $16$, $25$, and $36$, respectively. Compute the area of $\triangle DEF$.


Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$.


The function $f(x)$ satisfies
\[f(x) - 3 f\left(\frac{1}{x}\right) = 3^x\]
for all $x \neq 0$. Find $f(2)$.

Solution: \boxed{f(2) = -\frac{9 + 3\sqrt{3}}{8}}


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 $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$


A certain integer has $5$ digits when written in base $9$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?


Ten points are marked on the circumference of a circle. How many different chords can be drawn by connecting two of these ten points? Additionally, how many regions are formed inside the circle by these chords?


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{40}$ and $a^{-1}$ is also undefined $\pmod{90}$?


Find the number of real solutions to the equation
\[\frac{3x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -\frac{5}{3}.\]


Given that the three points $(2,a,b)$, $(a,3,b)$, and $(a,b,4)$ are collinear, find the value of $a + b$.


The least common multiple of two numbers is 4620, and the greatest common divisor is 33. Given that one of the numbers is 231, what is the other number?


Express \(\frac{0.\overline{72}}{1.\overline{24}}\) as a common fraction.


Determine the number of solutions for the following equation:
\[
\frac{(x-1)(x-2)(x-3)\cdots(x-50)}{(x-1^3)(x-2^3)(x-3^3)\cdots(x-10^3)} = 0
\]
for real $x$.


After having coffee together, Amy, Ben, and Carl start walking from the same point but in different directions. Amy walks north at a rate of 1 mile every 20 minutes, Ben walks east at a rate of 3 miles every 40 minutes, and Carl walks south at 2 miles every 30 minutes. How far apart are Amy and Carl after 2 hours?

Solution: \boxed{14 \text{ miles}}


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


A monic polynomial of degree $n$, with real coefficients, has its first two terms after $x^n$ as $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. It’s known that $a_{n-1} = 2a_{n-2}$. Determine the absolute value of the lower bound for the sum of the squares of the roots of this polynomial.


If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{a} + \mathbf{a} \times \mathbf{b},$ and $\mathbf{b}.$


The number of students in a school competition is more than 100 but fewer than 200. The number of students is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. Find how many students were in the competition.


In a school club of 80 students, 50 take mathematics, 40 take physics, 25 students take both mathematics and physics, and 10 students take chemistry but neither mathematics nor physics. How many students take neither mathematics, physics, nor chemistry?


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

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-6.25,ymax=7.25;

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 f1(real x){return (x-1)*(x+1)*(x+2)/6+x+2;}
draw(graph(f1,-3.25,3.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=g(x)$",(3.5,0.6),E);
[/asy]

If $g(c)=3$ and $g(3)=c$, then what is the value of $c-3$?


Let $x$, $y$, and $z$ be real numbers such that
\[\tan x + \tan y + \tan z = 0\quad \text{and} \quad \sec x + \sec y + \sec z = 3.\]
Find the sum of all possible values of $\sec 2x + \sec 2y + \sec 2z$.


A standard deck of 52 cards has 13 ranks and 4 suits, with each suit equally represented among the ranks. The suits are categorized into two colors: red ($\heartsuit$, $\diamondsuit$) and black ($\spadesuit$, $\clubsuit$). What is the probability that in a random arrangement of this deck, the first card is red, the second card is black, and the third card is red?


Evaluate the expression $305\cdot 20 - 20\cdot 9 + 5 \pmod{19}$.


In Mr. Thompson's math class, 50 of the 60 students took the Chapter 5 test and their mean score was $68\%$. The next day, 8 students took the test and their mean score was $75\%$. Two days later, the remaining 2 students took the test and scored an average of $82\%$. What is the new class mean? Express your answer as a percent.


Consider the lines:
\[
y = 4x + 6, \quad 2y = 6x + 7, \quad 3y = 12x - 3, \quad 2y = 3x - 4, \quad 4y = 2x - 6.
\]
Determine how many pairs of lines are either parallel or perpendicular.


For any positive integer \( n \), \( n! \) is the product of the first \( n \) positive integers. What is the greatest common divisor of \( 8! \) and \( 10! \)?


In the seven-digit integer $4B8,\!3942$, what is the smallest digit $B$ so that the seven-digit integer will be divisible by 3?


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


All squares have the same value, and all circles have the same value. What is the sum of four circles? \begin{align*}
\square + 2\bigcirc + \square + \bigcirc &= 26\\
2\square + \bigcirc + \square + \bigcirc &= 23\\
\bigcirc + \bigcirc + \bigcirc + \bigcirc &= \ ?
\end{align*}


Consider that \( p \) is a prime number and the equation \( 2006_p + 404_p + 215_p + 222_p + 9_p = 433_p + 572_p + 321_p \). How many possible values of \( p \) are there?


Solve $x|x| = 3x + 4$ for the smallest value of $x$.


In a right triangle, one leg measures 40 units and the other leg measures 30 units. Calculate the measure of the hypotenuse and the area of this triangle.

Solution: \boxed{600 \text{ square units}}


If \(x\) and \(y\) are positive integers less than \(30\) for which \(x + y + xy = 119\), what is the value of \(x + y\)?


Let $m$ be the smallest integer whose cube root is of the form $n + r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/5000$. Find $m$.


What is the greatest divisor of 620 that is smaller than 100 and also a factor of 180?


Solve for $x:\ \log_2 x + \log_4 x + \log_8 x = 9.$


Let $x$ and $y$ be angles such that $\cos x + \cos y = 1$ and $\sin x + \sin y = \frac{1}{2}$. Find $\tan \left( \frac{x + y}{2} \right)$.


Find the quadratic polynomial \( p(x) \) such that \( p(-6) = 0 \), \( p(3) = 0 \), and \( p(1) = -24 \).

Solution: \boxed{p(x) = \frac{12}{7}x^2 + \frac{36}{7}x - \frac{216}{7}}


Let's suppose that $x^2$ is inversely proportional to $y^4$. If $x = 6$ when $y = 2$, determine the value of $x^2$ when $y = 4$.

Solution: \boxed{x^2 = 2.25}


The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 25,$ $b_{10} = 105,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$


The diameter, in centimeters, of a sphere with three times the volume of a sphere of radius 6 centimeters can be expressed in the form \(a\sqrt[3]{b}\) where \(a\) and \(b\) are positive integers, and \(b\) contains no perfect cube factors. Compute \(a+b\).


The equation $y = -8t^2 - 12t + 72$ describes the height (in feet) of a ball thrown downward at 12 feet per second from a height of 72 feet on the moon. Determine how many seconds will pass before the ball hits the ground. Express your answer as a decimal rounded to the nearest hundredth.


The Cookie Monster now encounters another cookie with a boundary described by the equation $x^2 + y^2 + 10 = 6x + 12y$. Determine the radius of this cookie.


Two spinners are labeled differently. The first spinner is divided into fifths, and the second into sixths. The numbers on the first spinner are 2, 4, 5, 7, and 9. The numbers on the second spinner are 1, 3, 4, 6, 8, and 10. If each spinner is spun once, what is the probability that the product of the results of the two spins is an even number? Express your answer as a common fraction.


Evaluate $\left\lceil -\sqrt{\frac{64}{9} - 1} \right\rceil$.


Let $g$ be a function defined by $g(x)= \frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are nonzero real numbers. Given that $g(13)=13$, $g(31)=31$, and $g(g(x))=x$ for all values except $\frac{-d}{c}$, find the unique number that is not in the range of $g$.


Ryan now has 4 red lava lamps and 4 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 4 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost two lamps are both red and on?


Suppose the function $g(x)$ is defined on domain $\{x_1,x_2,x_3\}$ such that the graph of $y=g(x)$ consists of just three points, forming a triangle of area $48$. Consider the graph of $y = 3g(\frac{x}{4})$. What is the area of the triangle formed by these three points?

Solution: \boxed{576 \text{ unit}^2}


Determine the value of $k$ such that
\[\frac{9}{x + y + w} = \frac{k}{x + z + w} = \frac{12}{z - y}.\]

Solution: \boxed{k = 21}


Jenny jumps on a number line, starting at 0, aiming to reach a point at 1. Initially, she jumps $\frac{1}{4}$ of the distance towards her target with each leap. Each subsequent jump covers $\frac{1}{4}$ of the remaining distance to the target. Determine the total distance Jenny has jumped after seven hops. Express your answer as a common fraction.


One of the asymptotes of a hyperbola has the equation $y=5x$. The foci of the hyperbola have the same $x-$coordinate, which is $3$. Determine the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = -5x + 30}


In triangle $ABC$, $AB = 5$, $AC = 5$, and $BC = 6$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.


What is the value of $a$ for which $\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} + \frac{1}{\log_5 a} = \frac{7}{4}$?


A ball was submerged in water in a cylindrical container when the water froze. The ball was then removed (without breaking the ice), leaving a hole $30$ cm across at the top and $10$ cm deep. What was the radius of the ball (in centimeters)?


Suppose six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Calculate the value of $b$.


In triangle $ABC$, where $\angle A = 90^\circ$, $BC = 25$, and $\tan C = 3\sin C$. What is $AC$?


Let $n=4$ be the given power for which there exist positive real numbers $a$ and $b$ such that:
\[(a + bi)^4 = (a - bi)^4.\]
Compute $\frac{b}{a}$.


In 1970, there were 480,000 cases of measles reported in the U.S. By 2000, the number had decreased to 600 cases reported. How many cases of measles would have been reported in 1990 if the number of cases decreased linearly from 1970 to 2000?


The positive integers $X, Y$ and $Z$ form an arithmetic sequence, while the integers $Y, Z$ and $W$ form a geometric sequence. If $\frac{Z}{Y} = \frac{7}{4},$ what is the smallest possible value of $X + Y + Z + W$?


A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria?


Completely factor the expression: $$x^8 - 81.$$

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


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 $A \cap B \cap C = \emptyset$. Consider each set to be a subset of $\{1,2,3,4,5,6,7,8\}$. Let $M$ be the number of such minimally intersecting ordered triples. Find the remainder when $M$ is divided by $1000$.


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


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


Determine the positive integer $M$ such that $$ 18^3 \times 50^3 = 30^3 \times M^3 \ ? $$


Solution: \boxed{2016 + \frac{1}{2017}}


Find the number of solutions to the equation
\[\tan (3 \pi \cos \theta) = \cot (3 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$


Anna Lisa bought three dozen apples for $\$$24.00. At the same rate, how much would five dozen of these apples cost if a 10% discount is applied to purchases exceeding four dozen?


Let $x$ and $y$ be real numbers such that $4x + 3y \leq 10$ and $3x + 5y \leq 12$. Find the largest possible value of $2x + y$.


Consider the partial fractions decomposition:
\[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 5)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 5}\]
for some constants $A$, $B$, $C$, $D$, and $E$. Find the sum $A + B + C + D + E.$


The admission fee for an exhibition is $30 per adult and $15 per child. On a certain day, the exhibition collected $2700 in admission fees, with at least one adult and at least one child attending. Among all possible ratios of adults to children on that day, which one is closest to 1?


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


Let set $\mathcal{C}$ be a 75-element subset of $\{1,2,3,\ldots,120\}$, and let $U$ be the sum of the elements of $\mathcal{C}$. Find the number of possible values of $U$.


Simplify $\sqrt{5 \cdot 3} \cdot \sqrt{3^4 \cdot 5^2}$.


Determine all real numbers $a$ such that the inequality $|x^2 + 2ax + 4a| \leq 4$ has exactly one solution in $x$.


What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form.


In the six-digit integer $5B6,\!584$, what is the largest digit $B$ so that the six-digit integer will be divisible by 3?


Let \(x,\) \(y,\) and \(z\) be positive real numbers such that \(x + y + z = 2.\) Find the minimum value of
\[
\frac{x + y}{xyz}.
\]


Calculate the expression: $-6 \cdot 5 - (-4 \cdot -2) + (-12 \cdot -6) \div 3$.


Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 3,$ and $a \ge -\frac{1}{2},$ $b \ge -2,$ and $c \ge -\frac{7}{3}.$  Find the maximum value of
\[\sqrt{4a + 2} + \sqrt{4b + 8} + \sqrt{4c + 10}.\]


Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 4x + 3|$ and $y = 6 - x$.


Consider a quadratic function \( g(x) = 3x^2 - 9x + 2 \). By restricting the domain of this function to an interval, it can be made invertible. What is the largest such interval that includes the point \( x = 1 \)?

Solution: \boxed{(-\infty, \frac{3}{2}]}


Determine how many integer values of $b$ are possible such that the inequality
\[x^2 + bx + 6 \le 0\]
has exactly three integer solutions for $x$.


A sea turtle claims her shell is $3754_{8}$ years old. How many years old is her shell in decimal (base-ten)?


Calculate how many terms of the arithmetic sequence starting at 102, decreasing by 6 each time, appear before the number 0 appears.


In a coordinate plane, points $A$, $B$, and $C$ are given as $A(0,5)$, $B(0,0)$, and $C(10,0)$ respectively. $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Determine the area of $\triangle DBC$.


What is the largest multiple of $11$ which is smaller than $-85$?


Determine the values of constants \(A\), \(B\), and \(C\) such that the equation \[\frac{(x+B)(Ax+40)}{(x+C)(x+5)} = 3\] has infinitely many solutions for \(x\), and calculate the sum of those values of \(x\) which are not solutions to this equation.


Under a revised proposal, a year that ends in double zeroes will be considered a leap year only if the year leaves a remainder of 300 or 600 when divided by 800. How many leap years will there be that end in double zeroes from the year 1500 to 3500?


How many integers, $x$, satisfy the inequality $|7x + 5| \le 9$?


Find the value of $a$ such that the remainder is constant when $10x^3 - 7x^2 + ax + 10$ is divided by $2x^2 - 5x + 2$.


What is the smallest possible median for the five number set {$x, 4x, 8, 1, x+1$} if $x$ is a non-negative integer?


Find $88^{-1} \pmod{89}$, as a residue modulo 89. (The answer should be between 0 and 88, inclusive.)


Four individuals, each with distinct weights, are at a gathering. Each individual agrees to shake hands only with others lighter than themselves. How many handshakes occur?


If $\lceil{\sqrt{x}}\rceil=18$, what is the sum of all possible integer values of $x$?


Given two vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 3$ and $\|\mathbf{w}\| = 5$, and the angle between them is $\frac{\pi}{3}$, find $\mathbf{v} \cdot \mathbf{w}$.


Find all $x$ such that $\lfloor \lfloor 3x \rfloor - 1/2 \rfloor = \lfloor x + 3 \rfloor.$

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


Determine the number of ordered quadruples \((a, b, c, d)\) of nonnegative real numbers such that:
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81,
\]
\[
a + b + c + d = 6.
\]


The sum of two positive integers $x$ and $y$ is 780. What is the largest possible value of $\gcd(x,y)$?


At the end of a basketball tournament game, each of the seven members of the three participating teams shakes hands with each member of the other two teams, and all of the players shake hands with each of the three referees. How many handshakes occur?


Find the number of positive integers $n \le 2000$ such that $21n$ is a perfect square.


How many integers fall between $\sqrt{5}$ and $\sqrt{50}$ on a number line?


In triangle $XYZ$, $XY = 4$, $XZ = 3$, and $YZ = 5$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.


Mr. Abraham's class consists of 30 students, of whom 20 received an 'A' on the latest exam. Mrs. Berkeley, who teaches a different class, uses the same grading standards. If the proportion of students receiving an 'A' in Mr. Abraham's class holds for Mrs. Berkeley's class, and she has 18 students, how many students in Mrs. Berkeley's class received an 'A'? Furthermore, if Mrs. Berkeley's students continue to maintain this proportion of 'A's on a subsequent test, how many would you expect to receive an 'A' if her class size increases to 27 students?


In right triangle $ABC$ where $\angle BAC = 90^\circ$, $AB = 40$, and $AC = 41$. Find $\tan C$ and $\sin C$.


The quadratic $8x^2 - 48x - 320$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?


In parallelogram \(EFGH\), angle \( F \) measures \( 120^\circ \). What is the number of degrees in the measure of angle \( H \)? Additionally, if side \( EF \) is \( 10 \) cm, what is the length of side \( GH \)?

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


Let $a,$ $b,$ and $c$ be distinct real numbers. Simplify the expression
\[
\frac{(x + 2a)^2}{(a - b)(a - c)} + \frac{(x + 2b)^2}{(b - a)(b - c)} + \frac{(x + 2c)^2}{(c - a)(c - b)}.
\]


A right triangle has sides $ka$, $lb$, and hypotenuse $c$. If $k$ and $l$ are positive constants, find the largest possible value of
\[\frac{ka + lb}{c}.\]


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

Solution: \boxed{-b + 2}


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

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


A regular polygon has an exterior angle that measures $18^\circ$. How many sides does the polygon have, and what is the sum of its interior angles?


In 2007, the average monthly rainfall in Clouddale was 45.2 mm. In 2008, which was a leap year, the average monthly rainfall in Clouddale was 3.5 mm more than in 2007. Calculate the total amount of rain that fell in Clouddale in 2008.

Solution: \boxed{584.4 \text{ mm}}


A four-inch wooden cube is painted blue on all six faces. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least two faces?


If $x$ and $y$ are positive integers such that $7x + 5y = 200$, what is the greatest possible value of $xy$?


In the coordinate system, trapezoid $WXYZ$ has vertices $W(2, 3)$, $X(6, 3)$, $Y(7, -1)$, and $Z(3, -1)$. Determine the area of trapezoid $WXYZ$.


Let $\omega$ be a nonreal root of $x^3 = 1$. Compute
\[
(1 - \omega + \omega^2)^6 + (1 + \omega - \omega^2)^6.
\]


Calculate $\left(-7\right)^7 \div 7^4 + 2^8 - 10^1$ as an integer.


Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition?


Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates can Sunland issue than Moonland.


Find the remainder when $3x^5 + 2x^3 - 5x + 8$ is divided by $x^2 - 2x + 1$.

Solution: \boxed{16x - 8}


In a right triangle, the ratio of the areas of two squares constructed on the legs of the triangle is $64:1$. What is the measure of the largest angle of this triangle?


A cuboid has a surface area of 150 square units, and each of its dimensions is a whole number. What is the maximum possible volume of this cuboid in cubic units?


If $\det \mathbf{A} = -5$ and $\det \mathbf{B} = 8,$ then find $\det (\mathbf{A} \mathbf{B} \mathbf{B}^{-1}).$


For how many different digits $n$ is the three-digit number $24n$ divisible by $n$?


In a community group of 25 people, 10 are left-handed. 18 people like rock music. 3 are right-handed and do not like rock music. How many group members are left-handed and like rock music? Assume all individuals are either left-handed or right-handed.


Find the minimum value of the expression \[(\tan x + \cot x)^2 + (\sin x + \cos x)^2\] for \(0 < x < \frac{\pi}{2}\).


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

Solution: \boxed{\frac{4 \sqrt{6}}{3}}


How many four-digit numbers are multiples of neither 3 nor 4?


The pages of a book are numbered from 1 to n. When the page numbers were summed, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 2550. What was the page number that was added twice?


Roberto has 5 pairs of trousers, 6 shirts, 4 jackets, and 3 pairs of shoes. He can pair any shoes with trousers and jackets but can only wear two specific pairs of shoes with one particular jacket due to style mismatch. How many different outfits can he put together if an outfit consists of a pair of trousers, a shirt, a jacket, and a pair of shoes?


How many rows of Pascal's Triangle contain the number $31$?


It is now 3:25:15 PM, as shown on a 12-hour digital clock. After adding 137 hours, 59 minutes, and 59 seconds to the current time, what is the sum of all digits shown on the clock at the new time?


How many positive integers smaller than $500,000$ are powers of $3$, but are not powers of $9$?


Elena has two numbers, \(x\) and \(y\). When she divides \(x\) by 60, she gets a remainder of 53. When she divides \(y\) by 45, she gets a remainder of 28. What remainder does she get when she divides \(3x - 2y\) by 30?


For how many values of the digit $A$ is it true that $45$ is divisible by $A$, and $357{,}1A6$ is divisible by $4$ and $5$?


A box contains 8 white balls and 7 black balls. Six balls are drawn out of the box at random. What is the probability that they all are white?


Consider the functions:
$$\begin{array}{ccc}
f(x) & = & 4x^2 + 5x + 6\\
g(x) & = & 2x^3 - mx + 8
\end{array}$$
If $f(5) - g(5) = 15,$ what is the value of $m$?


Given that \( b \) is an even multiple of \( 431 \), find the greatest common divisor of \( 8b^2 + 63b + 143 \) and \( 4b + 17 \).


A right, rectangular prism has three faces with areas of $12, 18,$ and $24$ square inches. What is the volume of the prism, in cubic inches?


"Modulo $17$ graph paper" consists of a grid of $17^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<17$. Given the graphs of $$y\equiv 7x+3\pmod{17}$$ and $$y\equiv 13x+14\pmod{17}$$ on modulo $17$ graph paper, find the sum of the $x$-coordinates of the points they have in common.


How many positive multiples of $9$ that are less than $800$ end with the digit $7$?


The real function $f$ has the property that whenever $c,$ $d,$ $m$ are positive integers such that $c + d = 3^m,$ the equation
\[ f(c) + f(d) = m^2 \] holds. What is $f(243)$?


If altitude $CD$ is $\sqrt2$ centimeters in triangle $ABC$, where $\angle BAC = 45^\circ$, what is the number of square centimeters in the area of $\triangle ABC$?

Solution: \boxed{2 \text{ square centimeters}}


A square and a regular pentagon are coplanar and share a common side \( \overline{AD} \). Determine the degree measure of the exterior angle \( BAC \).


In triangle $ABC,$ $AB = 30$ and $BC = 18.$ Find the largest possible value of $\tan A.$


Evaluate the complex number sum $15 e^{3 \pi i/13} + 15 e^{24 \pi i/26}$ and express as $r e^{i \theta}.$ Enter the ordered pair $(r, \theta).$

Solution: \boxed{\left(30 \cos \left(\frac{9 \pi}{26}\right), \frac{15 \pi}{26}\right)}


Let \(x\) and \(y\) be positive real numbers such that
\[
\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.
\]
Find the minimum value of \(x + 3y.\)

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


Given that both $b$ and $n$ are positive integers where $b, n \leq 20$ and $b$ must be a power of a product of two different primes, determine the greatest number of positive factors that $b^n$ can have.


Let $a$ and $b$ be real numbers. One of the roots of the polynomial
\[x^3 + ax^2 + 2x + b = 0\]
is $2 - 3i$. Determine the ordered pair $(a, b)$.

Solution: \boxed{\left(-\frac{5}{4}, \frac{143}{4}\right)}


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


Simplify $\displaystyle\frac{2-3i}{4+5i+3i^2}$, where $i^2 = -1.$

Solution: \boxed{-\frac{1}{2} - \frac{1}{2}i}


Four positive integers, when divided by $53$, have remainders $33$, $26$, $18$, and $6$ respectively. After each of these integers is multiplied by $2$, what is the remainder when the sum of the resulting products is divided by $53$?


Tracy had a bag of candies, and none of the candies could be broken into pieces. 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 brother took somewhere from two to six candies, leaving Tracy with four candies. How many candies did Tracy have at the start?


The lottery game in our town features two draws. Firstly, a MegaBall is chosen from 30 uniquely numbered balls. Secondly, six WinnerBalls are chosen from a set of 45 differently numbered balls. To win this lottery, one must correctly guess the MegaBall number and also correctly guess the numbers of the six WinnerBalls regardless of the order they are drawn in. What is the probability that the ticket I hold matches all the winning numbers?


A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical layer was $2$ feet and the thickness was $1$ foot. Calculate the total volume of sand delivered, expressing your answer in terms of $\pi$.


Compute \[
\left\lfloor \frac{2017! + 2014!}{2016! + 2015!}\right\rfloor.
\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)


If I have nine boxes arranged in a $3$ x $3$ grid, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit, and digit $1$ must be placed in one of the corner boxes?


Find the value of \( n \) such that the equation \( 16x^2 + nx + 4 = 0 \) has exactly one solution in \( x \).

Solution: \boxed{\pm 16}


Laura has $x$ candies. Mark has four times as many candies as Laura. Nina has twice as many candies as Mark, and Oliver has six times the number of candies Nina has. If, in total, Laura, Mark, Nina, and Oliver have 360 candies, what is the value of $x$?


What must be the value of the coefficient \(c\) in \(P(x) = x^3 + 2x^2 + cx + 8\), in order for \(x-3\) to be a factor of \(P\)?


What is the smallest three-digit number in Pascal's triangle that is also a multiple of 5?


What is the area, in square units, of a trapezoid bounded by the lines $y = 2x$, $y = 8$, $y = 2$, and the $y$-axis?


Marissa constructs a large spherical snow sculpture by placing smaller snowballs inside it with radii of 4 inches, 6 inches, and 8 inches. Assuming all snowballs are perfectly spherical and fit exactly inside the sculpture, and the sculpture itself is a sphere whose radius is such that the sum of the volumes of the smaller snowballs equals the volume of the sculpture, find the radius of the sculpture. Express your answer in terms of \( \pi \).


Simplify $8 \cdot \frac{15}{4} \cdot \frac{-28}{45}$.


Find the equation of a parabola with vertex at $(3, -2)$, a vertical axis of symmetry, and passing through the point $(5, 6)$. Also, determine the new vertex of the parabola if the entire graph is shifted by the vector $(2, 3)$. Express your initial parabola equation in the form "$ax^2 + bx + c$".

Solution: \boxed{(5, 1)}


Let $g$ be a function taking the integers to the integers such that
\[g(m + n) + g(mn + 1) = g(m) g(n) + 1\] for all integers $m$ and $n.$

Let $n$ be the number of possible values of $g(2),$ and let $s$ be the sum of all possible values of $g(2).$ Find $n \times s.$


Define the operation $\diamond$ as $a \diamond b = a^2 + \frac{a^2}{b}$. What is the value of $5 \diamond 2$?


What is the sum of all of the positive even factors of $630$?


What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 9?


Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number?


Express $\frac{0.\overline{72}}{1.\overline{81}}$ as a common fraction.


Find the interval of all $x$ such that both $3x$ and $5x$ are in the interval $(1,2)$.

Solution: \boxed{\left(\frac{1}{3}, \frac{2}{5}\right)}


Every week, Alice goes to the supermarket and buys the following: $4$ apples at $\$2$ each, $2$ loaves of bread at $\$4$ each, $3$ boxes of cereal at $\$5$ each, $1$ chocolate cake at $\$8$, and a $\$6$ package of cheese. This week the store has a sale where all cereals are $\$1$ off and breads are buy-one-get-one-free. Alice also has a coupon for $\$10$ off on spending of $\$40$ or more. How much does Alice spend on this shopping trip?


A square is divided into 4 congruent rectangles. If the perimeter of the square is 160 units, what is the perimeter of one of the four congruent rectangles?

Solution: \boxed{100 \text{ units}}


In a basketball series of nine games between the Badgers and the Cougars, where each team has an equal probability of winning each game, what is the probability that the Badgers win at least five games?


The roots of the equation $3x^2 - kx + p = 0$ sum to -3 and multiply to -6. What is the value of $k + p$?


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices equals $357$, what is the sum of the numbers written on the faces?


John hypothesizes that the amount of study time he spends before a test and his score on that test are directly proportional. On his first exam, he studied for 3 hours and scored 60. To the nearest tenth, how many hours does John believe he must study the night before his second exam so that the average of his two exam scores is a 75?


A circular cylindrical pole has a circumference of 6 feet and stands 18 feet tall. A string is wrapped evenly around it, making three full loops from bottom to top. Calculate the total length of the string.

Solution: \boxed{18\sqrt{2} \text{ feet}}


Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c^2+1$, find the value of the product of all possible values of $c$.


A matrix $\mathbf{B} = \begin{pmatrix} 4 & 7 \\ 8 & d \end{pmatrix}$ satisfies the condition
\[\mathbf{B}^{-1} = k \mathbf{B}\]
for some constant $k$. Determine the ordered pair $(d, k)$.

Solution: \boxed{(-4, \frac{1}{72})}


Solution: \boxed{2^{567} + 64}


Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + 2x + 3$. Find the sum of all possible values of $g(64)$.


Find the smallest value of $y$ that satisfies the equation $|5y - 9| = 55$.


In Zuminglish, all words consist only of the letters $M, O, P$. $O$ is a vowel and $M$ and $P$ are consonants. A valid word in Zuminglish requires that between any two $O's$ there must be at least two consonants. Define $N$ as the number of 8-letter words in Zuminglish. Find the remainder when $N$ is divided by 1000.


Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \theta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \theta - \delta \theta^2.\]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is $\frac{1}{4}$ length from $R$. Given $LM = 4$, and $LO = MO = 3$. Find the area of quadrilateral $PRSQ$.


A line passes through the points $(1,3,2)$ and $(4,3,-1)$. Find the $y$-coordinate when the $x$-coordinate of a point on this line is 7. Also, determine where this line intersects with the $z=0$ plane.

Solution: \boxed{(3, 3, 0)}


Five concentric circles are drawn with radii of 1, 3, 5, 7, and 9. The innermost circle is painted black, the next ring is white, followed by black, white, and then the outermost ring is black again. Calculate the ratio of the black area to the white area. Express your answer as a common fraction.


In a triangle, two of the side lengths are 9 and 10, and the angle between them is $150^\circ$. Find the length of the third side.

Solution: \boxed{\sqrt{181 + 90\sqrt{3}}}


Suppose that $a$ and $b$ are digits, not both five and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?


There are 8 books on a shelf, which include a trilogy that must be selected together. In how many ways can 5 books be selected from this shelf if the order in which the books are selected does not matter?


Let \( f(x) = 9x + 2 \). Find the sum of all \( x \) that satisfy the equation \( f^{-1}(x) = f((3x)^{-1}) \).


Let $x_1, x_2, \dots, x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{50}.\]
Additionally, ensure that the cubic sum $x_1^3 + x_2^3 + \dots + x_n^3 \le \frac{1}{150}.$ Find the smallest possible value of $n.$


What is the shortest distance between the circles defined by \(x^2 - 6x + y^2 - 8y + 4 = 0\) and \(x^2 + 8x + y^2 + 12y + 36 = 0\)?

Solution: \boxed{\sqrt{149} - \sqrt{21} - 4}


Find the greatest common divisor of $9!$ and $(7!)^2$.


Given that $x = \frac{5}{7}$ is a solution to the equation $49 x^2 - 88x + 40 = 0,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction.


What is the largest whole number value of $n$ that makes the following inequality true? $$\frac{1}{4} + \frac{2n}{5} < \frac{7}{8}$$


In right triangle $XYZ$ with $\angle Y = 90^\circ$, we have $$4\sin X = 5\cos X.$$ What is $\sin X$?


Determine the residue of $-1235 \pmod{29}$. Your answer should be an integer in the range $0,1,2,\ldots,27,28$.


What is the minimum value of the expression $7x^2 - 28x + 1425$, where $x$ is a real number?


Let $f(x)=25x+4$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-2})$.


Consider sequences of positive real numbers of the form \( x, 1000, y, \dots \) in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of \( x \) does the term 1001 appear somewhere in the sequence?


How many nonnegative integers can be written in the form \[b_8\cdot3^8 + b_7\cdot3^7 + b_6\cdot3^6 + b_5\cdot3^5 + b_4\cdot3^4 + b_3\cdot3^3 + b_2\cdot3^2 + b_1\cdot3^1 + b_0\cdot3^0,\] where \( b_i \in \{0, 1, 2\} \) for \( 0 \leq i \leq 8 \)?


A club has 25 members, consisting of 12 boys and 13 girls. How many ways can we choose a president, a vice-president, and a secretary such that the president and the vice-president are of different genders, and the secretary is of the same gender as the president?


Determine the domain of the function $y=\dfrac{\sqrt{x^2-16}}{x-7}$.

Solution: \boxed{(-\infty, -4] \cup [4, 7) \cup (7, \infty)}


Let $a \oslash b = (\sqrt{3a+b})^4$. If $5 \oslash x = 256$, find the value of $x$.


In triangle $PQR,$ the side lengths $p,$ $q,$ and $r$ satisfy $p + 3q + 3r = p^2$ and $p + 3q - 3r = -1.$ Find the largest angle of the triangle, in degrees.

Note: $p$ is the side length opposite $\angle P,$ etc.


A meeting is scheduled with a committee of 12 women and 3 men. In this meeting, they use indistinguishable rocking chairs for the women and indistinguishable stools for the men. Furthermore, it's decided that a particular man, Mr. X, must sit at the very left end of the row. How many distinct ways can the chairs and stools be arranged for this meeting?


A space explorer discovers $215_{7}$ different types of alien species on a newly explored planet. How many types are there in base 10?


Convert the point $(-4, 4\sqrt{3}, 2)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2\pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left( 2\sqrt{17}, \frac{2\pi}{3}, \cos^{-1}\left(\frac{1}{\sqrt{17}}\right) \right)}


How many different four-letter arrangements can be formed using the letters $A, B, C, D, E, F, G$, if the first letter must be $C$, one of the other letters must be $B$, a second one of the other letters must be $D$, and no letter can be used more than once in the arrangement?


How many prime numbers are there between 50 and 70 that are also greater than 55?


Simplify the expression \[\frac{1}{\dfrac{3}{\sqrt{2}+2} + \dfrac{4}{\sqrt{5}-2}}.\]

Solution: \boxed{\frac{1}{11 + 4\sqrt{5} - \frac{3\sqrt{2}}{2}}}


If an integer is divisible by $8$ and the sum of its last two digits is $15$, what is the product of its last two digits?


What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$?


An equilateral triangle is divided into two 30-60-90 triangles by drawing a line from one vertex to the midpoint of the opposite side. If the length of the median to the longest side (hypotenuse in the 30-60-90 triangle) is $15$ units, what is the length of the shortest side of one of the 30-60-90 triangles, in units?

Solution: \boxed{15 \text{ units}}


Solution: \boxed{\sqrt{\frac{1 + \sqrt{33}}{2}}}


Given three vertices of a cube are $P=(10, 15, 14)$, $Q=(11, 11, 5)$, and $R=(14, 6, 13)$, determine the surface area of the cube.


What is the remainder when $4509$ is divided by $98$, and what is the result when this remainder is doubled?


Olivia rewrites the quadratic $16x^2 - 40x - 24$ in the form of $(dx + e)^2 + f,$ where $d,$ $e,$ and $f$ are all integers. What is $de$?


How many integers between 500 and 800 have the sum of their digits equal to 18?


The line \( y = \frac{1}{3}x + 3 \) is parameterized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -9 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -4 \end{pmatrix}.
\]
Enter the ordered pair \( (s, l) \).

Solution: \boxed{\left(0, -12\right)}


Let $h(x)$ be a monic quartic polynomial such that $h(-2) = -4$, $h(1) = -1$, $h(-4) = -16$, and $h(3) = -9$. Find $h(2)$.


Find the minimum value of
\[\sqrt{x^2 + (x-2)^2} + \sqrt{(x-2)^2 + (x+2)^2}\] over all real numbers $x$.


A repair crew needs to place planks to walk to the center $C$ of a circular fountain, due to maintenance. They initially place a 20-foot plank from $A$ to $B$ and then a 12-foot plank from $D$ to $C$ where $D$ is the midpoint of $\overline{AB}$. Calculate the area of the circular base of the fountain. Express your answer in terms of $\pi$.

Solution: \boxed{244\pi \text{ square feet}}


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $729x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2 + q^2 + r^2 + s^2 + t^2 + u^2$?


In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 15$, and $\tan C = 3\sin C$. What is $AC$?

Solution: \boxed{AC = 5}


Determine the value of the expression
\[
\log_3 (64 + \log_3 (64 + \log_3 (64 + \cdots))),
\]
assuming it is positive.


Three distinct vertices of a decagon are chosen at random. What is the probability that all three vertices form a side of the decagon?


If $f(x)$ is defined for all integers $x \geq 0$, $f(1) = 3$, and for all integers $a, b \geq 0$,
\[
f(a + b) = f(a) + f(b) - 2f(a \cdot b) + 1
\]
compute $f(1987)$.

Solution: \boxed{f(1987) = 3}


What is the sum of the 2010 fractions of the form $\frac{3}{n(n+3)}$ if the values of $n$ range from 1 to 2010? Express your answer as a decimal to the nearest thousandth.


Find the number of integers $n$ that satisfy
\[50 < n^2 < 200.\]


Find the greatest common divisor of 14658 and 11241.

Solution: \boxed{\gcd(14658, 11241) = 3}


A right triangle has legs measuring 15 inches and 21 inches. What is the length of the hypotenuse, in inches?


Find the sum of all positive integer values of $b$ such that the quadratic equation $3x^2 + 6x + b = 0$ has rational roots, and $b$ is a factor of 18.


The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $s$. Four vertical edges connect the base to a point directly above the center of the base at height $h$. Given that $s = 8$ and $h = 6$, what is the volume of the solid?


What is the smallest whole number that is divisible by 9, but leaves a remainder of 1 when divided by any integer 2 through 6?


Find the sum of the rational roots of the polynomial $h(x) = x^3 - 8x^2 + 15x - 6$.