What is the $205$th digit after the decimal point when $\frac{14}{360}$ is expressed as a decimal?


How many positive three-digit integers less than 600 have at least two digits that are the same?


Consider the function $$g(x) = \frac{2}{1+\frac{1}{2+\frac{1}{x}}}.$$ Determine the sum of the values of $x$ that make $g(x)$ undefined.


In how many ways can the digits of $47,\!520$ be arranged to form a 5-digit number, where numbers cannot begin with 0?


An isosceles triangle $PQR$ has an area of 144 square inches. A horizontal line $\overline{ST}$ cuts $PQR$ into an isosceles trapezoid and a smaller isosceles triangle at the top. The area of the trapezoid is 108 square inches. If the altitude of triangle $PQR$ from $P$ is 24 inches, find the length of $\overline{ST}$.


When the graph of $y = 2x^2 - x + 7$ is shifted three units to the right and five units up, we get the graph of $y = ax^2 + bx + c$. Find the value of $a+b+c$.


Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 12$, $BC = 6$, $\overline{PA} \perp \overline{AB}$, $\overline{PA} \perp \overline{AD}$, and $PA = 10$, find the volume of $PABCD$. Additionally, compute the area of triangle $ABC$.

Solution: \boxed{36 \text{ square units}}


Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of:
\[
(3 \cos \alpha + 4 \sin \beta - 5)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.
\]


What is the smallest positive integer that must always divide the sum of the first twelve terms of any arithmetic sequence where the terms are positive integers?


Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[
\frac{a^2}{kb} + \frac{b^2}{kc} + \frac{c^2}{ka}.
\]
for some positive constant \( k \).


Five faucets fill a 125-gallon tub in 8 minutes. How long, in minutes, does it take ten faucets to fill a 50-gallon tub? Assume that all faucets dispense water at the same rate.


The Photography club has 24 members: 12 boys and 12 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?


The line \( y = \frac{3}{4}x - 15 \) is parameterized by \( (x,y) = (f(t),20t - 10) \). Determine the function \( f(t) \).

Solution: \boxed{f(t) = \frac{80}{3}t + \frac{20}{3}}


In spherical coordinates, the point $\left( 4, \frac{\pi}{3}, \frac{7 \pi}{4} \right)$ is equivalent to what other point, in the standard spherical coordinate representation? 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( 4, \frac{4 \pi}{3}, \frac{\pi}{4} \right)}


Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and express your answer in simplest form.

Solution: \boxed{\sqrt{10} - \sqrt{6} + \frac{\sqrt{15}}{2} - \frac{3}{2}}


If \(y^2 - 6y + 9 = 0\) and \(y + z = 11\), what are the values of \(y\) and \(z\)?


There exist constants $b_1,$ $b_2,$ $b_3$ such that
\[\cos^3 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta\] for all angles $\theta$. Find \(b_1^2 + b_2^2 + b_3^2.\)


We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively?


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


An amoeba is placed in a controlled environment on day 1. Starting from the initial day, each amoeba splits into two amoebas every day, except that every third day, only half of the amoebas are capable of splitting due to environmental stress. After 10 days, how many amoebas will there be, assuming the controlled environment starts with one amoeba?


For what value of $k$ does the equation $(x+5)(x+2) = k + 3x$ have exactly one real solution? Express your answer as a common fraction.


A function $g(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have:
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 7x + 5\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 3005$. Compute the integer nearest to $T$.


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


Find how many positive integers \( n \), \( 1 \leq n \leq 1500 \), make the polynomial \( x^2 - 2x - n \) factorizable into two linear factors with integer coefficients.


A square pyramid has a base edge of 40 inches and an altitude of 15 inches. A smaller square pyramid, whose altitude is one-third the altitude of the original pyramid, is cut away from the apex. What is the volume of the remaining frustum as a fraction of the volume of the original pyramid?


Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ$. Calculate the area of the parallelogram whose diagonals are represented by the vectors $\mathbf{p} + 2\mathbf{q}$ and $2\mathbf{p} + \mathbf{q}$.


Laura has modified her training routine. Now, she bikes 25 miles at a rate of $3x + 1$ miles per hour, spends ten minutes transitioning to her running gear, and then runs 8 miles at $x$ miles per hour. The entire workout lasts 130 minutes. Determine Laura’s running speed, rounding to the nearest hundredth of a mile per hour. A calculator may be used.

Solution: \boxed{8.00 \text{ mph}}


A circle with a radius of 5 units has its center at $(0, 0)$. A circle with a radius of 3 units has its center at $(10, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


In a right triangle $PQR$ with $\angle R = 90^\circ$, it is known that $3\sin Q = 4\cos Q$. Determine $\sin Q$ and $\tan Q$.

Solution: \boxed{\tan Q = \frac{4}{3}}


Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if two distinct letters each are repeated exactly once, but digits cannot be repeated and the letter 'A' cannot be used?


In a local chess club, there are 5 teams each with 6 members. Teams rotate hosting monthly meetings. At each meeting, each team selects three members to be on the organizing committee, except the host team, which selects four members. Calculate the number of possible 16-member organizing committees.


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


How many prime numbers between 30 and 80 have a prime remainder when divided by 12?


Let $a, b,$ and $t$ be real numbers such that $2a + b = 2t$. Find, in terms of $t$, the minimum value of $a^2 + b^2$.


How many even perfect square factors does $2^6 \cdot 7^{10}$ have?


Calculate the remainder when the sum $1! + 2! + 3! + \cdots + 9! + 10!$ is divided by 7.


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


Find the ordered triplet $(x, y, z)$ that solves the following system of equations:
\begin{align*}
3x + 2y &= z - 1,\\
2x - y &= 4z + 2,\\
x + 4y &= 3z + 9
\end{align*}

Solution: \boxed{\left(-\frac{24}{13}, \frac{18}{13}, -\frac{23}{13}\right)}


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

[asy]
pair A,B,C,D;
A = (0,0);
B = (12*sqrt(5),18);
D = (12*sqrt(5),0);
C = (12*sqrt(5) + 45,0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]

Solution: \boxed{AD = 24}


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
  17x +  by +  cz &= 0 \\
   ax + 29y +  cz &= 0 \\
   ax +  by + 37z &= 0.
\end{align*}
Suppose that $ a \ne 17$ and $ x \ne 0$.  What is the value of
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 37} \, ?\]


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


$\triangle XYZ$ is inscribed inside $\triangle PQR$ such that points $X, Y, Z$ lie on $RQ, QP, PR$, respectively. The circumcircles of $\triangle PXZ, \triangle QYX, \triangle RZY$ have centers $O_4, O_5, O_6$, respectively. $PQ = 26, QR = 29, RP = 25$, and $\stackrel{\frown}{PY} = \stackrel{\frown}{RZ},\ \stackrel{\frown}{QZ} = \stackrel{\frown}{PX},\ \stackrel{\frown}{QY} = \stackrel{\frown}{RX}$. The length of $RQ$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.


Winnie has 15 red balloons, 42 blue balloons, 54 yellow balloons, and 92 purple balloons. She decides to distribute these balloons equally among her 11 closest friends. How many balloons does Winnie end up keeping for herself?


What is the smallest multiple of 7 that is greater than -50?


I have a drawer with 6 shirts, 7 pairs of shorts, and 6 pairs of socks, and 3 hats. If I reach in and randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one hat?


What is the sum of the squares of the lengths of the **medians** of a triangle whose side lengths are $13,$ $14,$ and $15$?


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


What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,-3), G(5,0), and H(5,8)?


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 24 dips?

Solution: \boxed{11.25 \text{ daps}}


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


Let $n$ be a positive integer and $x, y$ be invertible integers modulo $n$ such that $x \equiv 2y \pmod{n}$ and $y \equiv 3x^{-1} \pmod{n}$. What is the remainder when $xy$ is divided by $n$?


How many five-character license plates can be created if the plate must consist of two consonants (from a set of 21 consonants), followed by two vowels (from a set of 5 vowels), and ending with a digit? (Assume an alphabet like Spanish where Y is not considered a vowel.)


Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


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 $1/10000$. Find $n$.


Let $x$, $y$, and $z$ be real numbers such that
\[\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0.\]
Find the sum of all possible values of $\cos 4x + \cos 4y + \cos 4z.$


Find the slope and the $x$-intercept of the line $4x + 7y = 28$.


Find the integer $m,$ $-90 < m < 90,$ such that $\tan m^\circ = \tan 1230^\circ.$


I have four distinct mystery novels, four distinct science fiction novels, and four distinct historical novels. I'm packing for a trip and want to take two books from different genres. How many different pairs of books can I take?


When a water tank is $30\%$ full, it contains 54 gallons less than when it is $10\%$ empty. How many gallons of water does the tank hold when it is full?

Solution: \boxed{90 \text{ gallons}}


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

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


If $a$, $b$, and $c$ are positive numbers such that $ab=45\sqrt[3]{3}$, $ac=75\sqrt[3]{3}$, and $bc=27\sqrt[3]{3}$, then find the value of $abc$.


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


There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.


The terms of the sequence $(b_i)$ are defined by $b_{n + 2} = \frac {b_n + 3001} {1 + b_{n + 1}}$ for $n \ge 1$, where all terms are positive integers. Find the minimum possible value of $b_1 + b_2$.


The Johnson family has 5 sons and 4 daughters. In how many ways can they be seated in a row of 9 chairs such that at least 2 boys are next to each other?


Pirate Pete shares his treasure with Pirate Paul using a modified pattern. Pete initiates by saying, "One for me, one for you," giving himself one coin and starting Paul's pile with one coin. He then says, "Two for me, and two for you," giving himself two more coins and making Paul’s pile two coins in total. This pattern continues with Pete saying, "Three for me, three for you," and so on, until Pete gives himself $x$ more coins but makes Paul’s pile $x$ coins in total. When all coins are distributed, Pirate Pete has exactly five times as many coins as Pirate Paul. How many gold coins do they have in total?


Solution: \boxed{c = \frac{1}{5^{\frac{9}{5}}}}


The sequence \(b_1, b_2, \ldots\) is geometric with \(b_1 = b\) and a common ratio \(s\), where \(b\) and \(s\) are positive integers. Suppose that \(\log_4 b_1 + \log_4 b_2 + \cdots + \log_4 b_7 = 343\). Determine the number of possible ordered pairs \((b,s)\).


Angle $EAB$ is a right angle, and $BE = 12$ units. If the length $AB$ is $5$ units, what is the sum of the square units in the areas of the two squares $ABCD$ and $AEFG$?


What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper?


If $4$ lunks can be traded for $2$ kunks, and $3$ kunks will buy $6$ oranges, how many lunks are needed to purchase two dozen oranges?


Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$?


For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6?


What is the area, in square units, of triangle $PQR$ given the coordinates $P(-2, 2)$, $Q(8, 2)$, and $R(6, -4)$?

Solution: \boxed{30 \text{ square units}}


Determine the real number value of the expression $3 + \frac{5}{2 + \frac{5}{3 + \frac{5}{2 + \cdots}}}$, where the $2$s and $3$s alternate.

Solution: \boxed{\frac{3 + \sqrt{39}}{2}}


Consider the polynomial $3x^4 + ax^3 + 48x^2 + bx + 12$. It has a factor of $2x^2 - 3x + 2$. Find the ordered pair $(a,b)$.

Solution: \boxed{(-26.5, -40)}


Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f.$ Both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -200),$ what is the value of $a + c$?

Solution: \boxed{a + c = -200}


Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 100$. Express $x$ as a decimal.


The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 8x + 6.$ Find the leading coefficient of $g(x).$


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


What is the largest possible median for the five-number set $\{x, 2x, 4, 1, 7\}$ if $x$ can be any negative integer?


Mary and James are sitting in a row of 10 chairs. They choose their seats randomly. What is the probability that they do not sit next to each other?


Calculate $\sqrt{42q} \cdot \sqrt{7q} \cdot \sqrt{14q}$. Express your answer in simplest radical form in terms of $q$.

Solution: \boxed{14q \sqrt{21q}}


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


Kevin Kangaroo starts hopping on a number line at 0 and aims to reach the point 1. However, he can only hop $\frac{1}{4}$ of the distance to 1 in each hop. Each subsequent hop covers $\frac{1}{4}$ of the remaining distance to 1. Calculate how far Kevin has hopped after six hops. Express your answer as a common fraction.


A spherical ball of cookie dough with a radius of 3 inches is used to make flat cookies. When rolled out, it covers a circular area of 9 inches in radius. Assuming the density of the cookie dough changes such that the resulting volume is doubled, what is the thickness of this layer of cookie dough expressed as a fraction?


Fourteen 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 two girls and throws to the fourth girl, who then skips the next two girls and throws the ball to the seventh girl. If the throwing pattern continues, including Bella's initial throw, how many total throws are necessary for the ball to return to Bella?


Calculate the integer nearest to \(1000\sum_{n=4}^{10005}\frac{1}{n^2-4}\).


Ray climbs a staircase of $n$ steps in two different ways. When he climbs $6$ steps at a time, he has $4$ steps remaining. When he climbs $7$ steps at a time, there are $3$ steps remaining. Determine the smallest possible value of $n$ that is greater than $15$.


Consider a grid where each block is 1 unit by 1 unit. You need to walk from point $A$ to point $B$, which are 9 units apart on the grid. The grid is structured such that point $A$ is at the bottom left and point $B$ is located 5 units to the right and 4 units up from $A$. How many different paths can you take assuming that you need to pass through a checkpoint $C$ located 3 units to the right and 2 units up from $A$?


Rational Woman and Irrational Woman decide to drive around two racetracks. Rational Woman drives on a path parameterized by
\[
x = 2 + \cos t, \quad y = \sin t,
\]
and Irrational Woman drives on a path parameterized by
\[
x = 3 + 3 \cos \frac{t}{2}, \quad y = \sin \frac{t}{2}.
\]
Find the smallest possible distance between any two points, each belonging to one of the tracks.


Ollie's leash is tied to a stake at the center of his yard, which is now in the shape of a square. His leash is exactly long enough to reach the midpoint of each side of his square yard. The leash allows him to cover a circular area centered at the stake where the leash is tied. If the area of Ollie's yard that he is able to reach is expressed as a fraction of the total area of his yard, what is this fraction?


Either increasing the radius of a cylinder by 4 inches or increasing the height by 12 inches results in the same volume. The original height of the cylinder is 3 inches. What is the original radius in inches?

Solution: \boxed{1 + \sqrt{5}}


Determine how many odd whole numbers are factors of 252.


For which positive integer values of $k$ does $kx^2+16x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order.


Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the ninth row?


Consider a monic polynomial of degree \( n \) with real coefficients, where the first two terms after \( x^n \) are \( b_{n-1}x^{n-1} \) and \( b_{n-2}x^{n-2} \), and it is given that \( b_{n-1} = -2b_{n-2} \). Find the absolute value of the greatest lower bound for the sum of the squares of the roots of this polynomial.


How many positive integers less than $201$ are multiples of either $5$ or $7$, but not both at once?


If $a$ and $b$ are integers with $a \neq b$, what is the smallest possible positive value of $\frac{a^2+b^2}{a^2-b^2} + \frac{a^2-b^2}{a^2+b^2}$?


For how many values of $k$ is $36^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, $9^9$, and $k$?


The sum of the first 3000 terms of a geometric sequence is 500. The sum of the first 6000 terms is 950. Find the sum of the first 9000 terms.


Find the matrix $\mathbf{N}$ that doubles the first row and quadruples the second row of a matrix. Specifically,
\[\mathbf{N} \begin{pmatrix} x & y \\ z & w \end{pmatrix} = \begin{pmatrix} 2x & 2y \\ 4z & 4w \end{pmatrix}.\]
If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}}


Consider the function $$g(t) = \frac{1}{(t-2)^2 + (t-3)^2 - 2}.$$ What is the domain of the function $g(t)$? Express your answer in interval notation.

Solution: \boxed{(-\infty, \frac{5 - \sqrt{3}}{2}) \cup (\frac{5 - \sqrt{3}}{2}, \frac{5 + \sqrt{3}}{2}) \cup (\frac{5 + \sqrt{3}}{2}, \infty)}


Michael finds another interesting aspect of quadratic polynomials and focuses on the polynomial $x^2 - sx + q$. He realizes that for certain values of $s$ and $q$, the roots $r_1$ and $r_2$ of the polynomial satisfy $r_1 + r_2 = r_1^2 + r_2^2 = r_1^{10} + r_2^{10}$. He wants to find the maximum value of $\dfrac{1}{r_1^{11}} + \dfrac{1}{r_2^{11}}$. Help Michael determine this maximum value.


Find the least positive integer $x$ that satisfies $x + 7237 \equiv 5017 \pmod{12}$.


An ellipse has a major axis of length 12 and a minor axis of length 10. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle.

Solution: \boxed{6 - \sqrt{11}}


For what value of $k$ does the equation $x^2 + 8x + y^2 + 4y - k = 0$ represent a circle of radius 7?


A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours?


In triangle $ABC$, $AB=15$ and $AC=8$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AB$ and $CM$. The ratio $BP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


The function $g$ is graphed below. Each small box has width and height 1.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-1,9,-1,9);

dot((0,0),red+5bp);
dot((2,2),red+5bp);
dot((4,4),red+5bp);
dot((6,6),red+5bp);
dot((8,8),red+5bp);
dot((1,7),red+5bp);
dot((3,5),red+5bp);
dot((5,3),red+5bp);
dot((7,1),red+5bp);
dot((9,0),red+5bp);

[/asy]

Simon writes the number 2 on his thumb. He then applies $g$ to 2 and writes the output on his index finger. If Simon continues this process of applying $g$ and writing the output on a new finger, what number will Simon write on his tenth finger?


Find $\begin{pmatrix} 5 \\ -3 \end{pmatrix} - (-2) \begin{pmatrix} 3 \\ -4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 11 \\ -11 \end{pmatrix}}


If \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
  H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that minimizes \( J(p) \)?


For what digit $d$ is the five-digit number $5678d$ a multiple of 9?


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


Factor completely: $y^8 - 4y^6 + 6y^4 - 4y^2 + 1$.


What is the smallest positive integer $n$ such that $503n \equiv 1019n \pmod{48}$?


What is $2^{24} \div 8^3$? Write your answer as an integer.


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


What is the median number of moons per celestial body? (Include Pluto and the newly discovered dwarf planet Eris, which has 1 moon.) \begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0\\
Venus & 0\\
Earth & 1\\
Mars & 2\\
Jupiter & 18\\
Saturn & 25\\
Uranus & 17\\
Neptune & 3\\
Pluto & 5\\
Eris & 1\\
\end{tabular}


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


Grace and her sister each spin a spinner once. This spinner has six equal sectors, labeled 1 through 6. If the non-negative difference of their numbers is less than 4, Grace wins. Otherwise, her sister wins. What is the probability that Grace wins? Express your answer as a common fraction.


How many positive integers less than 100 are divisible by 2, 4, and 5?


Four distinct digits $a$, $b$, $c$, and $d$ (none consecutive and all different) are selected to form the four-digit numbers $abcd$ and $dcba$. What is the greatest common divisor (GCD) of all possible numbers of the form $abcd+dcba$, considering $a, b, c, d$ non-consecutive and distinct?


Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$


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


Find all real solutions to $x^3 + (x + 2)^3 + (x + 4)^3 = (x + 6)^3$.


A farmer has a rectangular field with dimensions \( 3m+8 \) and \( m-3 \). If the field has an area of 85 square units, what is the value of \( m \)?

Solution: \boxed{\frac{1 + \sqrt{1309}}{6}}


Given that 12 is the arithmetic mean of the set $\{8, 15, 20, 6, y\}$, what is the value of $y$?


Acme T-Shirt Company now charges a $75 setup fee plus $8 for each shirt printed. Gamma T-shirt Company has no setup fee but charges $16 per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive).  How many lattice points lie on the graph of the equation $x^2-y^2=45$?


The base-10 numbers 356 and 78 are multiplied. The product is then added to 49, and this sum is written in base-8. What is the units digit of the base-8 representation?


A function $g(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 6x + 9\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 3000$. Compute the integer nearest to $T$.


Let $p$ and $q$ be real numbers. One of the roots of the equation
\[ x^3 + px^2 + qx - 6 = 0 \]
is $2 - i$. Determine the ordered pair $(p, q)$.

Solution: \boxed{\left(-\frac{26}{5}, \frac{49}{5}\right)}


A sphere intersects the $xy$-plane in a circle centered at $(3,3,0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,3,-8)$, with radius $r$. Find $r$.


The graph of the function $g(x) = ax^4 + bx^3 + cx^2 + dx + e$ is given. The function intersects the x-axis at $x = -1, 0, 1, 2$ and passes through the point $(0,3)$. Determine the coefficient $b$.

Solution: \boxed{b = -2}


Calculate the sum of the squares of the roots of the equation \[x^{1010} + 22x^{1007} + 6x^6 + 808 = 0.\]


For how many values of the digit $A$ is it true that $49$ is divisible by $A$ and $573{,}4A6$ is divisible by $4$?


Let $f(x) = x^2 - 5x + 12$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(9)$?


Let the ordered triples $(x,y,z)$ of complex numbers satisfy
\begin{align*}
x + yz &= 9, \\
y + xz &= 14, \\
z + xy &= 14.
\end{align*}
Find the sum of all $x$ values from the solutions $(x_1,y_1,z_1), (x_2,y_2,z_2), \dots, (x_n,y_n,z_n)$.


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


If $A\ \diamond\ B$ is defined as $A\ \diamond\ B = 4A + B^2 + 7$, calculate the value of $A$ for which $A\ \diamond\ 3 = 85$.


Compute $\arccos (\sin 3)$. All functions are in radians.

Solution: \boxed{3 - \frac{\pi}{2}}


A frequency distribution for a history class is shown. Calculate the percent of the class that scored in the $70\%$-$79\%$ range.
\begin{tabular}{|c|c|}
Test Scores & Frequencies\\
\hline
$90\% - 100\%$ & IIII I\\
$80\% - 89\%$ & IIII III\\
$70\% - 79\%$ & IIII IIII\\
$60\% - 69\%$ & IIII \\
Below $60\%$ & III
\end{tabular}


Using only the digits 6, 7, and 8, how many positive nine-digit integers can be created that are palindromes?


Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 10 and at least one die shows a 5?


In a different ellipse, the center is at $(2, -1)$, one focus is at $(2, -4)$, and one endpoint of a semi-major axis is at $(2, 3)$. Find the semi-minor axis of this ellipse.


Express $\cos(a + b) + \cos(a - b)$ as a product of trigonometric functions.

Solution: \boxed{2 \cos a \cos b}


Abby, Bart, Cindy, and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, Cindy and Damon weigh 270 pounds, and Abby and Cindy weigh 220 pounds. How many pounds do Abby and Damon weigh together?


A student is solving the expression $\left(7^2 - 6^2\right)^4$. What is the value of the expression?


Let $f(x) = |\{x\}^2 - 2\{x\} + 1.5|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \[n f(xf(x)) = x\] has at least $2023$ real solutions. What is $n$?


What is the 30th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{11}$ and $\frac{1}{13}$?


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

Solution: \boxed{-3x^3 - 4x^2 + 11x - 12}


Jennifer wants to enclose her rectangular vegetable garden using 160 feet of fencing. She has decided that one side of the garden should be exactly 30 feet long. What is the maximum area that she can enclose, assuming the sides of the rectangle are natural numbers?


Calculate both the product and the sum of the least common multiple (LCM) and the greatest common divisor (GCD) of $12$ and $15$.


Let $g(x) = \frac{3x - 2}{x + 4}$. The inverse of $g$ can be written as $g^{-1}(x) = \frac{ax + b}{cx + d}$. Find the value of $\frac{a}{c}$.


Express $\sin 3x + \sin 7x$ as a product of trigonometric functions.

Solution: \boxed{2 \sin 5x \cos 2x}


If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?


A play has three different male roles, three different female roles, and no roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned?


Compute $\arccos(\sin 3)$, where all functions are in radians.

Solution: \boxed{3 - \frac{\pi}{2}}


What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(3, 0)$ and $B(-4,5)$?


In triangle $DEF$, with side lengths $DE=6$, $EF=8$, and $FD=10$, two bugs start simultaneously from point $D$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $G$. What is $EG$?


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


Find the positive value of $x$ that satisfies $ab = x-6i$ given $|a|=3$ and $|b|=5$.


In how many ways can the digits of $70,\!616$ be arranged to form a 5-digit number? (Note, numbers cannot begin with 0.)


Define \( M(x) = x - \frac{x^3}{3} \) for every real number \( x \). If \( n \) is a positive integer, define \( b_n \) by
\[
  b_n = M \Bigl( M \Bigl( M \Bigl( \cdots M \Bigl( \frac{20}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are \( n \) iterations of \( M \). As \( n \) approaches infinity, what value does \( n b_n \) approach?


On a different mathematics test, Mina encountered 80 problems: 20 arithmetic, 35 algebra, and 25 trigonometry problems. This time, she answered $75\%$ of arithmetic, $30\%$ of algebra, and $70\%$ of trigonometry problems correctly. Unfortunately, she missed the pass mark as she got fewer than $55\%$ of the problems correct. Determine how many more questions Mina needed to answer correctly to achieve a $55\%$ pass grade.


If $\sqrt{3\sqrt{s-3}} = \sqrt[4]{9 - s}$, then find $s$.


A projectile is launched, and its height \( h \) in feet after \( t \) seconds is given by the equation \( h = -16t^2 + 64t + 36 \). Determine the maximum height reached by the projectile.


In the diagram, rectangle $ABCD$ is divided into four identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?
[asy]
size(4cm);
pair a = (0, 1); pair b = (4, 1); pair c = (4, 0); pair d = (0, 0);
draw(a--b--c--d--cycle);
draw(shift(1) * (a--d)); draw(shift(2) * (a--d)); draw(shift(3) * (a--d));
label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW);
[/asy]


In triangle $ABC$, the measure of $\angle A$ is $70$ degrees. The measure of $\angle B$ is $30$ degrees more than twice the measure of $\angle C$. Determine the measure, in degrees, of $\angle C$.

Solution: \boxed{\frac{80}{3} \text{ degrees}}


Given a hemisphere of radius 9 cm sitting on a cone of height 12 cm, where the radius of the cone's base is the same as the hemisphere's radius, calculate the total surface area, including the hemisphere's rounded surface, base, and the lateral surface area of the cone. Express your answer in terms of $\pi$.


Determine the value of the following expression as a common fraction:
\[ \left(1-\frac{1}{2}\right) \cdot \left(1-\frac{1}{3}\right) \cdot \left(1-\frac{1}{4}\right) \cdots \left(1-\frac{1}{n+1}\right), \]
where \( n \) is an even number less than 100.


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 10 dips, how many daps are equivalent to 28 dips?

Solution: \boxed{10.5\text{ daps}}


What is the sum of all positive even factors of $1176$?


Find the sum of the first ten terms in the geometric sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$.


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


Pentagon ABCDE has a vertical line of symmetry. What is the new $y$-coordinate of vertex C so that the area of the pentagon is 100 square units? The vertices B and D are now at $(0,6)$ and $(6,6)$ respectively, while A and E are at $(0,0)$ and $(6,0)$. [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,6), C=(3,?), D=(6,6), E=(6,0);
pair[] dots={A,B,C,D,E};

draw(B--C--D--E);
dot(dots);

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(6,0)",E,SE);
label("D(6,6)",D,NE);
label("C",C,N);
label("B(0,6)",B,NW);
[/asy]


The positive four-digit integers that use each of the digits $1,$ $2,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $18^{\text{th}}$ integer in the list?


Find the smallest three-digit palindrome whose product with 102 is not a five-digit palindrome.


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


A geometric sequence has a first term of $12$ and a second term of $-6$. What is the $312^{th}$ term?

Solution: \boxed{a_{312} = -12 \cdot \left(\frac{1}{2}\right)^{311}}


Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat?


Compute $1013^2 - 991^2 - 1007^2 + 997^2$ without using a calculator.


If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^7$ are odd?


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

Let $n$ be the number of possible values of $g(36),$ and let $s$ be the sum of the possible values of $g(36).$  Find $n \times s.$


Ben rolls five fair 10-sided dice, each numbered from 1 to 10. What is the probability that exactly three of the dice show a prime number?


What is the area of the region defined by the equation $x^2 + y^2 + 3 = 6y - 10x + 1$?


The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number?


In a right triangle (not necessarily isosceles), the median to the hypotenuse is $15$ units long. If one leg is $9$ units longer than the other, find the length of the shorter leg. Express your answer in simplest radical form.

Solution: \boxed{a = \frac{-9 + \sqrt{1719}}{2}}


Solution: \boxed{QR = 3\sqrt{74} \text{ units}}


If \(x^{3y} = 8\) and \(x = 2\), what is the value of \(y\)? Express your answer as a common fraction.


At a school cafeteria, Sam wants to buy a lunch consisting of one main dish, one beverage, and one snack. The table below lists Sam's choices in the cafeteria. How many distinct possible lunches can he buy if he avoids pairing Fish and Chips with Soda due to dietary restrictions?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Dishes} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Burger & Soda & Apple Pie \\ \hline
Fish and Chips & Juice & Chocolate Cake \\ \hline
Pasta &  &  \\ \hline
Vegetable Salad &  &  \\ \hline
\end{tabular}


What is the sum of the proper divisors of $600$? Additionally, determine if $600$ is a perfect square.


The first three stages of a pattern are shown below, where each line segment represents a toothpick. If the pattern continues such that at each successive stage, four toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 150th stage?


Allie and Carlos play a game where they take turns rolling a standard die. If a player rolls \( n \), she or he is awarded \( g(n) \) points, where
\[
g(n) = \left\{
\begin{array}{cl}
8 & \text{if } n \text{ is a multiple of 3}, \\
3 & \text{if } n \text{ is a multiple of 2 but not of 3}, \\
0 & \text{if } n \text{ is not a multiple of 2 or 3}.
\end{array}
\right.
\]
Allie rolls the die five times and gets a 6, 2, 5, 3, and 4. Carlos rolls and gets 3, 2, 2, 6, and 1. What is the sum of Allie's total points and Carlos' total points?


Assign the numbers 3, 4, 5, 6, 7, and 8 to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, compute a product of three numbers assigned to the three faces that meet at that vertex. Determine the greatest possible value of the sum of these eight products.


A laptop is originally priced at $800. The store offers a $15\%$ discount, followed by another $10\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\%$ discount on the second discounted price. What single percent discount would give the same final price as these three successive discounts?


What is the least prime factor of $11^5 - 11^2$?


Let \(a, b, c, d\) be nonzero integers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 12 & 0 \\ 0 & 12 \end{pmatrix}.
\]
Find the smallest possible value of \(|a| + |b| + |c| + |d|\).


Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)?


Solution: \boxed{x^4 - 2x^2 - 4}


One of the roots of the equation $z^2 = -100 + 75i$ is $5 + 10i$. What is the other root?

Solution: \boxed{-5 - 10i}


As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $15n+4$ and $8n+1$?


Five fair coins are to be flipped. What is the probability that all five will show heads, or at least one shows tails? Express your answer as a common fraction.


Jane is currently 30 years old. Dick is exactly 5 years older than Jane. In $n$ years, where $n$ is a positive integer, both Dick's and Jane's ages will be two-digit numbers, and Jane's age will be obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?


To bake $24$ cookies, I use $1.5$ gallons of milk. There are $4$ quarts in a gallon and $2$ pints in a quart. How many pints of milk are needed to bake $6$ cookies if there is a measurement error that adds $10\%$ more milk than calculated?

Solution: \boxed{3.3 \text{ pints}}


If $x$, $y$, and $z$ are positive integers such that $\gcd(x,y) = 360$ and $\gcd(x,z) = 1176$, find the smallest possible value of $\gcd(y,z)$.


A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. Initially, there are 20 people in total in the club. How many people will be in the club after four years?


A sea turtle shared that the oldest coral reef near his home is $3456_{8}$ years old in octal notation. How old is the coral reef in decimal years?


Find $PQ$ in the triangle below.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$P$",P,S);
label("$Q$",Q,S);
label("$R$",R,N);
label("$18\sqrt{3}$",R/2,W);
label("$30^\circ$",(1.25,0),N);
[/asy]


The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers?


We have a triangle $\triangle DEF$ such that $DE = DF = 10$ and $EF = 12$. Calculate the length of the median $DM$ from vertex $D$ to side $EF$ and also find the length of the altitude $DH$ from $D$ to side $EF$.

Solution: \boxed{DM = 8, DH = 8}


Determine the smallest integer $B$ that is delicious if there exists at least one sequence of several consecutive integers, including $B$, that sums up to 2023.


A tweaked coin results in flipping heads with a probability of $\frac{2}{5}$ and tails with a probability of $\frac{3}{5}$. If winning $\$5$ on heads and losing $\$3$ on tails, calculate the expected value, in dollars, of our winnings for one flip.


For what base is the representation of $500_{10}$ a four-digit number whose final digit is odd?


In a triangle, two of the side lengths are 9 and 8, and the angle between them is \(150^\circ\). Find the length of the third side.

Solution: \boxed{c = \sqrt{145 + 72\sqrt{3}}}


In the game of Zorg, dropping $n$ zorgs gives a score of the sum of the first $n$ positive integers. Eating $n$ zorgs earns $20n$ points. What is the least number of zorgs for which dropping them will earn more points than eating them?

Solution: \boxed{n = 40}


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal unit vectors. Consider the equation:
\[\mathbf{b} = s (\mathbf{b} \times \mathbf{c}) + t (\mathbf{c} \times \mathbf{a}) + u (\mathbf{a} \times \mathbf{b})\]
for some scalars $s,$ $t,$ and $u,$ where $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = 1.$ Find $s + t + u.$


Determine the smallest constant $N$ such that for any triangle with sides $a$, $b$, and $c$, the inequality
\[\frac{a^2 + b^2 + c^2}{ab + bc + ca} > N\]
always holds.


Find the largest value of $m$ such that $5x^2 + mx + 120$ can be factored as the product of two linear factors with integer coefficients.


A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 11 that is not a lucky integer?


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


The sum of the areas of two squares is 130, while the difference in their areas is 58. Find the product of their perimeters.


How many perfect squares less than 500 have a ones digit of 5, 6, or 7?


Find the remainder when $5x^4 - 12x^3 + 3x^2 - 5x + 15$ is divided by $3x - 9$.


What is the value of $x$ for which $(10-x)^2 = x^2 + 6$?

Solution: \boxed{x = 4.7}


Solve the following quadratic equation and find the positive difference between its solutions:
$$x^2 - 5x + m = 13 + (x+5)$$

Solution: \boxed{\Delta x = 2\sqrt{27 - m}}


Calculate the value of $$2+44+666+8888+111110+13131312+1414141414$$ modulo 9, where the result should be within the range of $0 \leq n < 9$.


A rectangular tile measures 5 inches by 6 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 3 feet by 4 feet?


In triangle $XYZ$, $XY = 4$, $XZ = 3$, and $YZ = 5$. The medians $XM$, $YN$, and $ZO$ 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 $GP + GQ + GR$.


How many times does the number $500!$ have the factor 2 in its prime factorization?


A circle with center $O$ has a radius of $10$ units, and a smaller circle with center $P$ has a radius of $3$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is a common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.


How many positive integer multiples of $3003$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$?


A girl has the following seven coins in her pocket: $2$ pennies, $2$ nickels, $2$ quarters, and $1$ half-dollar. She takes out two coins, records the sum of their values, and then puts them back with the other coins. How many different sums can she record at most?


When the product $(5x + 3y + 2)(2x + 6y + 7)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?


In a triangle with base $b = 10$ and height $h = 12$, a circle is inscribed touching the base of the triangle at a single point and the other two sides tangentially. Find the radius $r$ of the circle.


What is $\log_{5}{3100}$ rounded to the nearest integer?


Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles?

Solution: \boxed{\frac{9\pi - 18}{2}}


Express as a common fraction: $(0.\overline{08})(0.\overline{3})$.


Compute $\begin{pmatrix} 1 & 1 \\ -1 & 0 \end{pmatrix}^4.$

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


Determine how many positive integer multiples of $231$ can be expressed in the form $10^j - 10^i$, where $i$ and $j$ are integers and $0 \leq i < j \leq 200$.


Find the smallest positive integer $n$ such that $5n \equiv 850 \pmod{26}$.


Determine how many terms in the expansion of $(m+n)^8$, where $m$ and $n$ are odd integers, are odd.


Find all real numbers \( x \) which satisfy \[\frac{x-2}{x-4} \ge 3.\]

Solution: \boxed{(4, 5]}


Our football team has 12 members, of which only 4 are strong enough to play offensive lineman, while all other positions can be played by anyone. In how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, a wide receiver, and a tight end?


Let $\overline{AD}, \overline{BE}, \overline{CF}$ be the altitudes of acute triangle $ABC$. If
\[7 \overrightarrow{AD} + 3 \overrightarrow{BE} + 8 \overrightarrow{CF} = \mathbf{0},\]
compute $\angle BAC$ in degrees.


Find $y$ such that $\log_y 243 = \log_3 81$.


The function \(g(x)\) satisfies \(g(1) = 2\) and
\[g(x + y) = 5^y g(x) + 3^x g(y)\] for all real numbers \(x\) and \(y.\) Find the function \(g(x).\)

Solution: \boxed{5^x - 3^x}


In isosceles triangle $\triangle DEF$ where $DE = EF = 5$, the altitude from $D$ meets side $\overline{EF}$ at point $J$. If $EJ = 4(JF)$, calculate the length of side $DF$.


What is the range of the function $y = \log_3(\sqrt{\sin x})$ for $0^\circ < x < 180^\circ$?

Solution: \boxed{(-\infty, 0]}


Positive real numbers \(a, b\) satisfy the equations \(a^2 + b^2 = 3\) and \(a^4 + b^4 = \frac{15}{4}\). Find \(ab\).


For a geometric sequence $a_1, a_2, a_3, \dots,$ let
\[ S_n = a_1 + a_2 + a_3 + \dots + a_n, \]
and let
\[ T_n = S_1 + S_2 + S_3 + \dots + S_n. \]
Given the value of $S_{20}$, you can uniquely determine the value of $T_n$ for some integer $n$. What is this integer $n$?

Solution: \boxed{n = 20}


How many total days were there in the years 1996 through 2000?


A rectangular floor measures 12 feet by 15 feet and is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles of radius 1 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total shaded area on the floor.

Solution: \boxed{180 - 45\pi \text{ square feet}}


A point is chosen at random on the number line between 0 and 2, and the point is colored blue. Then, another point is chosen at random on the number line between 0 and 2, and this point is colored red. What is the probability that the number of the red point is greater than the number of the blue point, but less than twice the number of the blue point?


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


What is the remainder when \(5x^6 - 3x^4 + 6x^3 - 8x + 10\) is divided by \(3x - 9\)?


In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?


What number results when 80 is increased by $150\%$ and then decreased by $30\%$?


Suppose that \(a\), \(b\), and \(c\) are three positive numbers that satisfy the equations \(abc = 1\), \(a + \frac{1}{c} = 7\), and \(b + \frac{1}{a} = 35\). Find \(c + \frac{1}{b}\).


A $270^\circ$ rotation around the origin in the counter-clockwise direction is applied to $8 - 4i$. What is the resulting complex number?

Solution: \boxed{-4 - 8i}


Construct a triangular array of numbers starting with initial values 1 on the sides and interior numbers obtained by adding the two adjacent numbers from the previous row. Rows 1 through 4 are shown:

\[
\begin{array}{ccccccccc}
& & & 1 & & 1 & & & \\
& & 1 & & 2 & & 1 & & \\
& 1 & & 3 & & 3 & & 1 & \\
1 & & 4 & & 6 & & 4 & & 1 \\
\end{array}
\]

Find the sum of the numbers in the 50th row.


The graph of the function $y = g(x)$ is displayed. For all $x > 3$, $g(x) > 0.5$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A$, $B$, and $C$ are integers, determine $A+B+C$. The vertical asymptotes of $g$ are at $x = -3$ and $x = 2$, and the horizontal asymptote is at $y = 1/A$ with $1/A < 1$.


Triangle ABC has a horizontal line of symmetry through vertex B. The coordinates of vertex A are (0,0) and vertex C is (8,0). We need to find the y-coordinate of vertex B such that the area of the triangle ABC is 32 square units.


John decides to make the challenge a bit more complex by drawing a regular six-pointed star (a hexagram) in the sand, placing different unique objects at each point where lines intersect. Each point of the six outward-pointing tips and each point where two lines intersect internally (a total of twelve points) has a unique stone from John's collection of twelve stones. How many distinct ways can he place the stones given that arrangements considered as equivalent include rotations and reflections of the hexagram?


Let \( Q(x) = 0 \) be the polynomial equation of least possible degree, with rational coefficients, having \( \sqrt[3]{8} + \sqrt[3]{64} \) as a root. Compute the product of all of the roots of \( Q(x) = 0 \).


Jennifer has 20 marbles, including three red, three green, and two blue marbles. How many ways can Jennifer choose 5 marbles such that exactly two of the chosen marbles are either red, green, or blue?


Two rectangles have integer dimensions, and both have a perimeter of 180 cm. What is the greatest possible difference between the areas of two such rectangles?

Solution: \boxed{1936 \text{ square cm}}


A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards include exactly two $\spadesuit$s and one $\heartsuit$?


The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, $z$ are $10$, $7$, and $4$ respectively. What is the value of $x^2+y^2+z^2$?


The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{145}$. Find the greatest possible value of $k$.


Solve for the sum of the cubes of the roots of the equation:
\[(x - \sqrt[3]{27})(x - \sqrt[3]{64})(x - \sqrt[3]{125}) = \frac{4}{3}.\]


A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\triangle DEF$.


Suppose we have 12 dogs and we want to divide them 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 sum of the squares of the lengths of the medians of a triangle whose side lengths are 13, 14, and 15, and what is the area of this triangle?


Given that the volume of a parallelepiped determined by the three-dimensional vectors $\mathbf{p}$, $\mathbf{q}$, and $\mathbf{r}$ is 12, find the volume of the parallelepiped determined by the vectors $2\mathbf{p} - \mathbf{q}$, $\mathbf{q} + 4\mathbf{r}$, and $\mathbf{r} - 3\mathbf{p}$.


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


Let $c$ and $d$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. What is the value of $(3c-4)(6d-8)$?


Given a positive integer $n = 4$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the "base" $-4+i$ using the integers $1,2,\ldots,n^2$ as "digits." That is, the equation \[r+si = a_m(-4+i)^m + a_{m-1}(-4+i)^{m-1} + \cdots + a_1(-4+i) + a_0\] is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,16\}$, with $a_m \ne 0$. We write \[r+si=(a_ma_{m-1}\ldots a_1a_0)_{-4+i}\] to denote the base $-4+i$ expansion of $r+si$.

There are only finitely many integers $k+0i$ that have three-digit expansions
\[k=(a_2a_1a_0)_{-4+i} \quad (a_2 \neq 0).\] Find the sum of all such $k$.


Shari walks at a constant rate of 4 miles per hour. She walks for 2 hours but takes a 30-minute break during her walk. How many miles did Shari walk? Express your answer as a decimal to the nearest tenth.


Consider a right triangle $XYZ$ where the right angle is at $Y.$ Let $XZ = \sqrt{85}$ and $XY = 7.$ A circle is drawn with its center on $XY$ such that the circle is tangent to $XZ$ and $YZ.$ If $Q$ is the point where the circle and side $XZ$ meet, determine the length of $ZQ.$


A shaded region consists of 25 congruent squares arranged in a pentagonal shape with point P at one vertex and point Q at another. If the distance $PQ = 10$ cm, which represents the diagonal of a larger conceptual square composed of 16 of these shaded squares, find the area of the entire shaded region.

Solution: \boxed{78.125\text{ square cm}}


In an equilateral triangle $ABC$, lines $DE$, $FG$, and $HI$ are drawn parallel to side $BC$, with $AD = DE = EG = GI$. Find the ratio of the area of trapezoid $HIAB$ to the area of triangle $ABC$.


Josef, Timothy, and Anna play a game. Josef picks an integer between 1 and 1440 inclusive. Timothy states whether the quotient of 1440 divided by Josef's integer is an integer. Finally, Anna adds a twist by stating whether Josef's integer is a multiple of 5. How many integers could Josef pick such that both Timothy's and Anna's conditions are satisfied?


The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?


Samantha intends to create two different solutions in her Chemistry class. For the first solution, she mixes 0.05 liters of chemical A with 0.03 liters of chemical B to make 0.08 liters of solution. She needs 0.64 liters of this solution for her experiment. For the second solution, she mixes chemical B with water at a ratio by volume of 1:2 (chemical B:water) and needs 0.75 liters of this solution. Calculate how many liters of water she will use in making the second solution.

Solution: \boxed{0.5 \text{ liters}}


A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element?


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


Given
\[
\cot x = \frac{a^2 - b^2}{2ab},
\]
where \( a > b > 0 \) and \( 0^\circ < x < 90^\circ \), find \( \sin x \) in terms of \( a \) and \( b \).

Solution: \boxed{\frac{2ab}{a^2 + b^2}}


Express the quotient $2213_4 \div 21_4$ in base 4.


**Martin's Coin Collection**

Martin organizes his coin collection from different regions and by the decade in which they were minted. The values of coins from different regions he collected are: USA and Mexico, $8$ cents each; Canada $5$ cents each; and Brazil $7$ cents each. (USA and Canada are North American countries and Mexico and Brazil are in Central and South America, respectively.) He has the following number of coins from different decades:

- **1950s**: USA: 5, Canada: 9, Mexico: 7, Brazil: 3
- **1960s**: USA: 11, Canada: 6, Mexico: 4, Brazil: 8
- **1970s**: USA: 9, Canada: 10, Mexico: 14, Brazil: 11
- **1980s**: USA: 8, Canada: 12, Mexico: 20, Brazil: 14

In dollars and cents, how much did his Central and South American coins minted before the $80\text{'s}$ cost him?


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $100$, and $BC=12$. Find the volume of the tetrahedron.


Consider the quadratic equation $x^2 - 2x + 8 = 24$. If $a$ and $b$ are the solutions to this equation, with $a \geq b$, find the sum of $3a + 2b$.

Solution: \boxed{5 + \sqrt{17}}


How many integers between 2000 and 3000 have all three of the numbers 18, 24, and 32 as factors?


Let \[g(x) =
\begin{cases}
x/3 &\quad \text{if } x \text{ is divisible by } 3, \\
5x+1 &\quad \text{if } x \text{ is not divisible by } 3.
\end{cases}
\]
What is $g(g(g(g(2))))$?


Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\prod_{i=1}^6 h(y_i)$.


Suppose $17^7\equiv m \pmod {11}$, where $0 \leq m < 11$.

What is the value of $m$?


Given \( w \) and \( z \) are complex numbers such that \( |w+z|=2 \) and \( |w^2+z^2|=8, \) find \( |w^4+z^4| \).

Solution: \boxed{|w^4+z^4| = 56}


A point $P$ is chosen at random in the interior of square $ABCD$. What is the probability that triangle $ABP$ has a greater area than each of triangle $BCP$, $CDP$, and $DAP$?


If $x$ is a real number and $\lfloor x \rfloor = -5,$ how many possible values are there for $\lceil x^3 \rceil$?


Consider the polynomials \[ f(x) = 2 - 6x + 4x^2 - 7x^3 + 8x^4 \] and \[ g(x) = 5 - 3x - 9x^3 + 12x^4. \] Find $d$ such that the polynomial $f(x) + dg(x)$ has degree 3.


Define the sequence $(b_i)$ by $b_{n+2} = \frac{b_n + 2011}{1 + b_{n+1}}$ for $n \geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$.


The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000?


Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{T}$ is the set of positive integers such that for all elements $n$ in $\mathcal{T}$, we have that $\star (n)=15$ and $0\le n< 10^{8}$. If $p$ is the number of elements in $\mathcal{T}$, compute $\star(p)$.


Define a function $g$ from nonnegative integers to real numbers such that $g(2) = 4$ and
\[ g(m + n) + g(m - n) = \frac{g(2m) + g(2n)}{2} \]
for all nonnegative integers $m \ge n$. Find the sum of all possible values of $g(10)$.


Suppose in triangle $PQR$, the angles satisfy:
\[\cos 4P + \cos 4Q + \cos 4R = 1.\]
Two sides of the triangle have lengths 7 and 24. Find the maximum length of the third side.


For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \leq 50$.
(2) the quadratic equation $x^2 + (2a+2)x + (a+1)^2 = 0$ has two integer solutions?


Alice has some 5 cent stamps and some 7 cent stamps. What is the least number of stamps she can combine so the total value of the stamps is 48 cents?


Let $p,$ $q,$ $r,$ $s,$ $t,$ $u$ be positive real numbers such that $p + q + r + s + t + u = 8.$  Find the minimum value of
\[\frac{1}{p} + \frac{9}{q} + \frac{16}{r} + \frac{25}{s} + \frac{36}{t} + \frac{49}{u}.\]


What is the smallest whole number that has a remainder of 3 when divided by 5, a remainder of 2 when divided by 4, and a remainder of 2 when divided by 6?


The sum of two numbers $x$ and $y$ is 540, and the value of the fraction $\frac{x}{y}$ is $\frac{7}{8}$. What is the value of $y - x$?


Daniel works at an electronics store and suggests that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost, but with a promotional twist: if a toaster's cost decreases by 25%, sales increase by an additional 10% beyond the usual increase due to the price drop. If 15 customers buy a toaster that costs $\$600$, how many customers would buy the same model when it is on promotion and costs $\$450$?


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


Find the quotient when \(x^6 + 5x^4 + 3\) is divided by \(x - 2.\)

Solution: \boxed{x^5 + 2x^4 + 9x^3 + 18x^2 + 36x + 72}


The Badgers play the Cougars in a series of seven basketball games. The probability of the Badgers winning an individual game is $\frac{2}{3}$. Determine the probability that the Badgers win at least four games. Express your answer as a common fraction.


A number $x$ is equal to $11 \cdot 36 \cdot 54$. Find the smallest positive integer $y$ such that the product $xy$ is a perfect cube.


What is the slope of the line determined by any two solutions to the equation $\frac{4}{x} + \frac{6}{y} = 0$? Express your answer as a common fraction.


What is the base 4 representation of the base 2 number $101010101_2$?


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


Let $\overline{AD},$ $\overline{BE},$ $\overline{CF}$ be the altitudes of acute triangle $ABC.$ If
\[8 \overrightarrow{AD} + 3 \overrightarrow{BE} + 5 \overrightarrow{CF} = \mathbf{0},\]
then compute $\angle ACB,$ in degrees.


The coefficients of the polynomial
\[ b_{8} x^8 + b_7 x^7 + b_6 x^6 + \dots + b_2 x^2 + b_1 x + b_0 = 0 \]
are all integers, and its roots $s_1, s_2, \dots, s_8$ are all integers. Furthermore, the roots of the polynomial
\[ b_0 x^8 + b_1 x^7 + b_2 x^6 + \dots + b_6 x^2 + b_7 x + b_8 = 0 \]
are also $s_1, s_2, \dots, s_8$. Find the number of possible multisets $T = \{s_1, s_2, \dots, s_8\}$.


How many rows of Pascal's Triangle contain the number $53$?


Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$?


What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-2|\le1$?


What is the greatest integer less than 150 for which the greatest common factor of that integer and 30 is 5?


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[4]{\sqrt{\frac{1}{15625}}}}$$


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25,$ $b_9 = 121,$ and for all $n \geq 3, b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2$.


Chloe wants to buy a pair of shoes that costs $\$45.50$. She empties her wallet and finds she has four $5 bills, ten $1 coins, and a pile of dimes. What is the minimum number of dimes that must be in her pile so she can pay for the shoes?


Find the number of real solutions to the equation
\[
\frac{2}{x - 1} + \frac{4}{x - 2} + \frac{8}{x - 3} + \dots + \frac{2^{100}}{x - 100} = x.
\]


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


Kim earned scores of 89, 85, 91, and 87 on her first four mathematics examinations. If Kim receives a score of 82 on the fifth exam, what will be the change in her average score?


What is the largest three-digit integer whose digits are distinct and form a geometric sequence, with the hundreds digit being less than or equal to 8?


There are two spinners, $A$ and $B$. Spinner $A$ has numbers 1 through 5, and Spinner $B$ has numbers 1 through 4, with each number equally likely to be chosen. What is the probability that the product of the two spinners' numbers is even?


An organization initially consists of 10 leaders and a certain number of regular members. Each year, all leaders are replaced, and each regular member recruits two new members. After this, 10 new leaders are elected from outside the organization. Initially, there are 30 people in total in the organization. How many people will be in the organization after 10 years?


Simplify $20 \cdot \frac{9}{14} \cdot \frac{1}{18}$.


Find the solutions to \(z^6 = 64i\) and calculate the sum of their angles in degrees. Express each solution \(z_k\) in the form
\begin{align*}
z_k = r_k (\cos \theta_k + i \sin \theta_k),
\end{align*}
where \(r_k > 0\) and \(0^\circ \leq \theta_k < 360^\circ.\) Then find the sum \(\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6.\)


One caterer charges a basic fee of \$50 plus \$18 per person. A second caterer charges a basic fee of \$180 plus \$15 per person, but offers a \$30 discount on the basic fee for groups of 30 or more people. What is the least number of people for which the second caterer is cheaper?


Define $f(n)$ to be the sum of all divisors of a positive integer $n$. If $f(f(n)) = n + 3$, then call $n$ superdeficient. Determine how many superdeficient positive integers exist.


A very large number $y$ is equal to $2^43^35^47^26^78^39^{10}$. What is the smallest positive integer that, when multiplied with $y$, produces a product that is a perfect square?


Let $\mathbf{p} = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} \sqrt{3} \\ 5 \\ -2 \end{pmatrix}$, and $\mathbf{r} = \begin{pmatrix} -7 \\ 3 \\ 10 \end{pmatrix}$. Find the angle between the vectors $\mathbf{p}$ and $(\mathbf{p} \cdot \mathbf{r}) \mathbf{q} - (\mathbf{p} \cdot \mathbf{q}) \mathbf{r}$, in degrees.


A conference room includes seating for twelve members: six armchairs, four benches, and two stools. Each type of seat is indistinguishable from others of the same type. The seats need to be arranged in a row for a meeting. How many distinct ways can the chairs be arranged?


Let \( x_1, x_2, \ldots, x_n \) be real numbers such that each \( |x_i| < 1 \) and \( |x_i| \neq |x_j| \) for \( i \neq j \). Determine the smallest possible value of \( n \) for which the equation
\[
|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|
\]
holds.


Determine the simplified form of the polynomial expression \[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2 + 2x^3 - 4x^3\].

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


Liam was given some coins by his grandparents. He lost one-third of them during a bike ride. After searching, he found five-sixths of the coins he had lost. Before finding these coins, however, he accidentally spent one-fourth of the total coins he initially had. How many fractions of the original total coins is Liam still missing after all these events?


Expand and simplify the product \((x+6)(x+8+y)\).

Solution: \boxed{x^2 + 14x + xy + 48 + 6y}


Daniel derives utility according to the relation $$\text{Utility} = \text{hours of reading} \times \text{hours of soccer played}.$$ On Wednesday he plays \( t \) hours of soccer and spends \( 10 - t \) hours reading. On Thursday, he ends up getting the same amount of utility as Wednesday while spending \( 4 - t \) hours playing soccer and \( t+4 \) hours reading. Find \( t \). Answer as a fraction in lowest terms.


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


Calculate $$\frac{4 \cdot 7! + 28 \cdot 6!}{8!}$$


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


How many multiples of 12 are there between 24 and 250?


The expression \(2y^2 - 5y - 12\) can be written as \((2y + a)(y + b)\), where \(a\) and \(b\) are integers. What is \(a - b\)?


In a triangle, two of the side lengths are 9 and 12, and the angle between them is $150^\circ$. Find the length of the third side.

Solution: \boxed{\sqrt{225 + 108\sqrt{3}}}


Find the coefficient of the $x^2$ term in the expansion of the product \((3x^3 + 4x^2 + 5x)(2x^2 - 9x + 1)\).


Determine the sum of all integral values of $c$ with $c \le 30$ for which the equation $y = x^2 - 9x - c$ has two rational roots.


What is the remainder of $17^{1999}$ divided by 26?


How many times does the digit 8 appear in the list of all integers from 1 to 700?


You are given a $2 \times 2$ grid and digits 1, 2, 3, and 4. In how many distinct ways can you fill the grid with these digits under the condition that digit 4 must always occupy one of the top boxes?


Altitudes $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at $H$. If $\angle BAC = 58^\circ$ and $\angle ABC = 67^\circ$, then what is $\angle AHB$?

[asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (1.8,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",P,S); label("$E$",Q,E); label("$H$",H,NW);
draw(rightanglemark(C,P,H,3.5));
draw(rightanglemark(H,Q,C,3.5));
[/asy]


The perimeter of a rectangle is 60 meters. The ratio of its length to its width is 5:2. What is the length in meters of a diagonal of the rectangle?


What is the greatest common factor and the least common multiple of 360 and 210?


What are the last two digits when $7^{10}$ is multiplied out?


A square EFGH has its center at $(5, 3)$ and an area of 16 square units. The top side of the square is horizontal. The square is then dilated from a center at $(0,0)$ with a scale factor of 3. Determine the coordinates of the vertex of the image of square EFGH that is closest to the origin. Provide your answer as an ordered pair.

Solution: \boxed{(9, 3)}


The table below gives the air distance in miles between selected world cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London & New York \\ \hline
Bangkok & & 6300 & 6609 & 5944 & 8650 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 & 7800 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 & 4980 \\ \hline
London & 5944 & 5989 & 7240 & & 3470 \\ \hline
New York & 8650 & 7800 & 4980 & 3470 & \\ \hline
\end{tabular}


Spinners $C$ and $D$ are spun. On each spinner, the arrow is equally likely to land on each number. Spinner $C$ has numbers 1, 2, 3, 4, 5 and Spinner $D$ has numbers 1, 2, 3, 4. What is the probability that the product of the two spinners' numbers is a multiple of 3? Express your answer as a common fraction.


What is the coefficient of \( x^4 \) in the expansion of \( (3x + 4)^8 \)?


If \(c\) is the smallest positive five-digit multiple of 7, and \(d\) is the smallest positive four-digit multiple of 9, then find \(c - d\).


Consider a matrix for reflecting over a line $\ell$ through the origin given by:
\[
\begin{pmatrix} \frac{3}{5} & -\frac{4}{5} \\ -\frac{4}{5} & -\frac{3}{5} \end{pmatrix}
\]
Find the direction vector of line $\ell$. The vector should be 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}}


Fifty teams play a tournament where each team plays every other team exactly once, and each game results in a win or loss with equal probability. Determine the probability that no two teams win the same number of games, expressed as $\frac{p}{q}$ where $p$ and $q$ are coprime. Find $\log_2 q$.


Find the product of all values of $t$ such that $t^3 = 125$.


Examine the square of $9,999,999$ and determine the number of zeros in the expansion of $9,999,999^2$.


When three standard dice are rolled, the numbers $a, b, c$ appear on the top faces. Find the probability that $a + b + c = 3$.


If \( x \) is real, compute the maximum integer value of the expression
\[
\frac{4x^2 + 8x + 19}{4x^2 + 8x + 5}.
\]


Phoenix hiked the Mountain Valley Trail last week. It took five days to complete the trip. The first two days she hiked a total of 30 miles. The second and fourth days she averaged 15 miles per day. The last three days she hiked a total of 45 miles. The total hike for the first and third days was 33 miles. How many miles long was the trail?


The Sharks play the Dolphins in a series of nine basketball games. Each team has an equal chance of winning each game. What is the probability that the Sharks will win at least five games? Express your answer as a common fraction.


In a survey, it was found that $92.3\%$ of the respondents believed that cats are generally friendly. Of the respondents who believed this, $38.4\%$ mistakenly thought that cats could not scratch. This mistaken belief was held by 28 people. How many people did the survey include in total?


There are 7 blue chips, 5 yellow chips, and 4 red chips in a bag. One chip is drawn from the bag and then 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.


Below is the graph of $y = a \sin(bx + c)$ for some constants $a > 0$, $b > 0,$ and a phase shift $c$. If the graph achieves a maximum at $x = \frac{\pi}{3}$ and the maximum value of $y$ is 3, find $a$ and $c.$

Solution: \boxed{c = \frac{\pi}{6}}


A $120\times 280\times 360$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


Any seven points are taken inside or on a circle with a diameter of $1$ unit. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$. Then $b$ is?


What is the probability that Kevin gets fewer heads than tails if he flips 12 coins?


Let $d$ and $e$ be the solutions of the quadratic equation $3x^2 + 4x - 7 = 0$. What is the value of $(d-2)(e-2)$?


How many distinct four-digit numbers are divisible by 5 and end with 45?


The angles of pentagon $PQRST$ satisfy $\angle P = 2\angle Q = 4\angle R = 3\angle S = 6\angle T$. What is the degree measure of $\angle P$?


A rhombus has an area of 150 square units. The lengths of its diagonals have a ratio of 5 to 3. What is the length of the shortest diagonal, in units?


Mark collects baseball cards. He organizes them equally on each page and then places each page into one of his three albums. The first album contains a total of 1,080 baseball cards, the second one contains 1,620 cards, and the third contains 540 cards. What is the largest number of baseball cards that Mark could be organizing on each page?


If 100 cards are dealt to 15 people as evenly as possible, how many people will end up with fewer than 7 cards?


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


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


A $3\times 5$ rectangle and a $4\times 6$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?


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

Solution: \boxed{-2 + 17 = 15}


In a triangle $XYZ$, $G$ and $H$ are the midpoints of $\overline{XY}$ and $\overline{YZ}$ respectively. If $X=(0,8)$, $Y=(0,0)$, and $Z=(10,0)$, find the sum of the $x$ and $y$ coordinates of $J$, the point of intersection of $\overline{XH}$ and $\overline{YG}$.


What is the largest digit $N$ for which $5789N$ is divisible by 6?


The executive board at a company has 12 members, of which 5 are executives. A five-person task force, which must include at least two executives, needs to be formed from the board members. How many distinct task forces are possible?


Let \[ \mathbf{C} = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}. \] Compute $\mathbf{C}^{50}.$

Solution: \boxed{\begin{pmatrix} 101 & 50 \\ -200 & -99 \end{pmatrix}}


For all real numbers $r$ and $s$, define the mathematical operation $\#$ such that the following conditions apply: $r \# 0 = r$, $r \# s = s \# r$, and $(r + 1) \# s = (r \# s) + s + 1$. What is the value of $9 \# 7$?


A card is chosen at random from a standard deck of 52 cards, and then it is replaced and a second card is chosen and replaced, followed by a third card. What is the probability that at least one of the cards is a heart or a king?


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


Of the five points (4, 15), (8, 25), (14, 42), (19, 48), and (22, 60), what is the sum of the $x$-coordinates of the points that lie in the region above the line $y = 3x + 4$ in the coordinate plane?


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

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


Determine the 25th digit after the decimal point in the sum of the decimal equivalents for the fractions $\frac{1}{8}$ and $\frac{1}{4}$.


The equation $x^2 + ax = -20$ has only integer solutions for $x$. What is the greatest possible value of $a$, given that $a$ is a positive integer?


Calculate the product $\left(\frac{3}{6}\right)\left(\frac{6}{9}\right)\left(\frac{9}{12}\right)\cdots\left(\frac{2505}{2508}\right)$. Express your answer as a common fraction.


Parallelogram $ABCD$ with $A(3,4)$, $B(5,8)$, $C(9,4)$, and $D(7,0)$ is reflected across the $y$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x-1$. Determine the ordered pair of $D''$ in the coordinate plane after these transformations, where $D'$ is the image of $D$ and $D''$ is the image of $D'$.


Al is eating candies over a week, starting with a certain number on the first day and increasing his consumption by four more candies each subsequent day than the previous day. After a week, he ate a total of 140 candies. How many candies did Al eat on the sixth day?


BoatCrafters started the year by making 5 canoes in January. Each month thereafter, they produced three times the canoes produced the previous month. However, each month, one canoe is taken out from production for quality checks and is not included in the available fleet. How many total canoes did BoatCrafters have available by the end of April of the same year?


Suppose I have 8 shirts, 5 ties, 3 pairs of pants, and 2 belts. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, and can either include a belt or not, how many outfits can I make?


A region is bounded by semicircular arcs constructed on the sides of a square, where each side of the square has a length of $4/\pi$. Calculate the perimeter of this region.


Let $M$ be the smallest integer whose square has exactly $3$ digits when written in base 7.

What is $M$, expressed in base 7?


Consider an AC circuit where $V$ represents the voltage, $I$ the current, and $Z$ the impedance. If $V=2+3i$ and $Z=2-4i$, calculate the current $I$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers.

Solution: \boxed{-\frac{2}{5} + \frac{7}{10}i}


Find the greatest common divisor of $8!$ and $(6!)^2.$


Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$?


Given a trapezoid where each vertex has integer coordinates, the coordinates are (2,3), (7,3), (9,7), and (0,7). Calculate the perimeter of this trapezoid and its area. What is the sum of the perimeter and area?

Solution: \boxed{42 + 4\sqrt{5}}


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


The line joining $(2,3)$ and $(5,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The vertices of the square are $(2, 0), (5, 0), (5, 3), (2, 3)$.


The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $50^{\mbox{th}}$ term of this sequence.


A circle with center $O$ has a radius of $8$ units, and a circle $P$ with center $C$ has a radius of $4$ units. Both circles are externally tangent to each other at point $Q$. Another circle, also centered at point $C$ and also with a radius of $4$ units, is externally tangent to circle $P$. Segments $TS$ is the common external tangent of circle $O$ and circle $C$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.


If $x^{3y} = 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


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


A whole number is said to be "9-heavy" if the remainder when the number is divided by 9 is greater than 5. What is the least four-digit 9-heavy whole number?


Each letter of the alphabet is assigned a random integer value, with $H=8$. Calculate the values of words based on the sum of their assigned letter values. If $MATH = 32$ points, $TEAM = 40$ points, and $MEET = 36$ points, what is the value of $A$?


How many four-character license plates can be created if each consists of a consonant, followed by a vowel, followed by another consonant, and ending with a digit? (For this problem, consider Y as a consonant.)


What is the greatest product obtainable from two integers whose sum is 280, if one of the integers must be even?


Given \( a \geq b > 2 \), what is the largest possible value of \( \log_a (a^2/b^2) + \log_b (b^2/a^2) \)?


Daniel attempts 30 basketball throws and successfully makes 60% of them. After 10 more throws, his successful shot percentage increases to 62%. How many of the last 10 shots did he make?


$PQRS$ is a trapezoid where the measure of base $\overline{PQ}$ is three times the measure of the base $\overline{RS}$. Point $T$ is the point of intersection of the diagonals. The measure of diagonal $\overline{PR}$ is 15. Find the length of segment $\overline{RT}$.


Find the area of the parallelogram generated by vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}$.

Solution: \boxed{2 \sqrt{381}}


Assuming the observed pattern holds true, how many zeros are in the decimal expansion of $(999,999,999)^2$?


If $x\%$ of five-digit numbers have at least one repeated digit, where the first digit cannot be zero, what is $x$? Express your answer as a decimal to the nearest tenth.


Simplify the expression
\[\frac{(\sqrt{2} + 1)^{1 - \sqrt{3}}}{(\sqrt{2} - 1)^{1 + \sqrt{3}}},\]
writing your answer in the form $a + b \sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime.

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


A circle has a radius of 5 inches. The distance from the center of the circle to chord $PQ$ is 4 inches. How many inches long is chord $PQ$? Additionally, calculate the area of the circle. Express all answers in simplest radical form.


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


A special deck of cards has 60 cards divided into 5 suits, with each suit having 12 cards. The cards in the deck are randomly ordered. How many ways can we pick three different cards where the order in which the cards are picked matters?


An 8-sided die is rolled. If the roll is even, then you win that amount of dollars (for example, if you roll 6, then you win \$6). If the roll is odd, you win nothing. What is the expected value of your winnings?


How many integers are between $(10.1)^3$ and $(10.4)^3$?


A teacher wants to arrange 4 copies of Introduction to Geometry and 5 copies of Introduction to Number Theory on a bookshelf. However, the first book on the left must always be a copy of Introduction to Geometry. In how many ways can he do this?


Find the least positive integer \( n \) such that
$$ \frac{1}{\sin 15^\circ \sin 16^\circ} + \frac{1}{\sin 17^\circ \sin 18^\circ} + \cdots + \frac{1}{\sin 173^\circ \sin 174^\circ} = \frac{1}{\sin n^\circ}. $$


Three vertices of a cube are $P = (6, 11, 11)$, $Q = (7, 7, 2)$, and $R = (10, 2, 10)$. What is the surface area of this cube?


Suppose $c$ and $d$ are positive integers such that $\gcd(c, d)$ is divisible by exactly 11 distinct primes, and $\mathop{\text{lcm}}[c, d]$ is divisible by exactly 44 distinct primes.

If $c$ has fewer distinct prime factors than $d$, then $c$ has at most how many distinct prime factors?


In how many ways can you arrange 4 different math books and 6 different history books on a bookshelf, if a math book must be placed at both ends and all history books must be grouped together?


Suppose $x$ is directly proportional to $y^2z$, and $y$ is inversely proportional to $\sqrt{z}$. If $x = 4$ when $z = 16$, find the value of $x$ when $z = 50$.


The diagonal of a square is 10 inches and the diameter of a circle is also 10 inches. Calculate:
1. By how many square inches is the area of the circle greater than the area of the square?
2. By how many inches is the circumference of the circle greater than the perimeter of the square? Express your answers as decimals to the nearest tenth.


Solution: \boxed{b = 12 \pm \sqrt{19}}


Simplify $((6p+2)-3p\cdot5)^2 + (5-2\div4)(8p-12)$ to a simpler expression of the form $ap^2 + bp + c$.

Solution: \boxed{81p^2 - 50}


The bacteria in a jar triple every 30 seconds. After five minutes, there are 1,239,220 bacteria in the jar. However, halfway through the experiment, half of the bacteria were killed by an antibiotic. How many were in the jar at the beginning of the experiment?


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


The lengths of the sides of a triangle with positive area are $\ln 20$, $\ln 150$, and $\ln(n!)$, where $n$ is a positive integer. Find the number of possible values for $n$.


The planning committee at school has 12 members. Exactly five of these members are teachers. A five-person subcommittee with at least two members who are teachers must be formed from the members of the planning committee. How many distinct subcommittees are possible?


Consider the parabola given by $x = 3y^2 - 9y + 5$. Determine the $x$-intercept $(a, 0)$ and the $y$-intercepts $(0, b)$, $(0, c)$, and find the value of $a + b + c$.


What is the $100^{\mathrm{th}}$ odd positive integer and what is the sum of the first 100 odd positive integers?


If two distinct numbers are selected at random from the first eight prime numbers (2, 3, 5, 7, 11, 13, 17, 19), what is the probability that their sum is an even number? Express your answer as a common fraction.


An arithmetic sequence with first term $3$ has a common difference of $8$. A second sequence begins with $5$ and has a common difference of $9$. In the range of $1$ to $150$, what is the largest number common to both sequences?


Laura has three different types of vegetable plants and three different types of flower plants. In how many ways can she arrange the plants in a row if she puts all the flower plants next to each other?


Ten stones are arranged in a straight line. They are counted from left to right as $1,2,3, \ldots, 10$, and then from right to left, so that the stone previously counted as 9 is counted as 11. The pattern is continued to the left until the stone previously counted as 1 is counted as 19. The pattern then reverses so that the stone originally counted as 2 is counted as 20, 3 as 21, and so on. The counting continues in this manner. Which of the original stones is counted as 145? Express your answer as a single digit which corresponds to the first digit assigned to that stone.


Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 100) < 0.\]


How many positive divisors does the number $360$ have? Also, calculate the sum of all positive divisors of $360$ that are greater than $30$.


Consider a pentagon \(ABCDE\) where the angles satisfy the condition \(\angle A = 3\angle B = 4\angle C = 5\angle D = 6\angle E\). Determine the degree measure of \(\angle A\), rounding to the nearest whole number.


Let $N$ denote the number of permutations of the $18$-character string $AAAAABBBBBBCCCCCC$ such that:
1. None of the first five letters is an $A$.
2. None of the next six letters is a $B$.
3. None of the last seven letters is a $C$.
Find the remainder when $N$ is divided by $500$.


Calculate $\frac{4 \cdot 6! + 24\cdot 5!}{7!}$


For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^6$, $9^9$, and $k$?


A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 7 units. Calculate the total volume, in cubic units, of the space inside the sphere and outside the cylinder, expressed as \( V\pi \). Find \( V \) as a common fraction.

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


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


For some complex number \( \omega \) with \( |\omega| = 3, \) there is some real \( \lambda > 1 \) such that \( \omega, \omega^2, \text{ and } \lambda \omega \) form an equilateral triangle in the complex plane.  Find \( \lambda. \)

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


Determine the units digit of the product $3 \times (1! + 2! + 3! + 4! + \cdots + 12!)$.


When a polynomial is divided by $3x^3 - 2x^2 + ix + 6,$ where $i$ is the imaginary unit, what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


What is the sum of all integers between -15.7 and 6.4?


If 2 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 1 jack occurs? (There are 4 Jacks, 4 Queens, and 52 total cards in a standard deck.)


Winnie has 60 cherry lollipops, 105 wintergreen lollipops, 5 grape lollipops, and 230 shrimp cocktail lollipops. Ignoring the flavors, Winnie decides to give each of her 13 friends an equal number of lollipops, distributing as many as possible. How many lollipops does Winnie retain for herself?


What is the smallest prime whose digits sum to $23$?


A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along an edge of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.


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


Let $T = \{7^k | k \in \mathbb{Z}, 0 \le k \le 1000 \}$. Given that $7^{1000}$ ends with the digits $0001$ and has $202$ digits, how many elements of $T$ end with the digit $1$?


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


Evaluate the polynomial \[ y^3 - 3y^2 - 9y + 3, \] where \( y \) is the positive number such that \( y^2 - 3y - 9 = 0 \).


The line \(y = \frac{3}{4} x + 5\) is parametrized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -5 \end{pmatrix}.\]Determine the ordered pair \((s,l).\)

Solution: \boxed{\left( -\frac{1}{4}, -\frac{20}{3} \right)}


A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ is a line segment that connects two non-adjacent vertices. Calculate how many diagonals a convex polygon with 25 sides has. Additionally, calculate the sum of all the interior angles in this polygon.


What is the value of $n$ such that $10^n = 10^{-2}\times \sqrt{\frac{10^{45}}{0.0001}}$?


Six different numbers are 1873, 1997, 2021, 2028, 2117, and 2127. Find the mean of two numbers when it's known that the mean of four of these numbers is 2012.


How many three-digit integers are divisible by 6 if the ones digit must be even?


Either increasing the radius of a cylinder by 4 inches or the height by 10 inches results in the same volume. The original height of the cylinder is 5 inches. What is the original radius in inches?

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


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


Find the greatest common divisor of 12222, 18333, and 36666.


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 110^{\circ}$ and $\angle AOB = 150^{\circ}$. Determine the degree measure of $\angle ABC$.


The image of the point with coordinates $(0,1)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(4, 5)$. Find $m+b$.


An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. Find the smallest abundant number that is not a multiple of 10.


A cylindrical billboard post with a circumference of 6 feet has a ribbon tied around it spiraling from bottom to top. The ribbon completes six loops evenly, starting from the bottom and finishing at the top. The post stands 18 feet high. What is the total length of the ribbon?

Solution: \boxed{18\sqrt{5} \text{ feet}}


Find integers $c$ and $d$ such that $\log_{10}50$ is between these integers, and calculate $c+d$.


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


In how many ways can the digits of $40,\!555$ be arranged to form a 5-digit number where none of the numbers can begin with either 0 or 4?


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

[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.25,xmax=4.25,ymin=-4.25,ymax=4.25;

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

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

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

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return 2*x+4*sin(x*pi/4);}
draw(graph(f2,-4.25,4.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]

What is the exact value of $v(-3.42)+v(-1.27)+v(1.27)+v(3.42)$ ?


What is the largest integer less than $\log_3 \frac{2}{1} + \log_3 \frac{3}{2} + \cdots + \log_3 \frac{1004}{1003} + \log_3 \frac{1005}{1004}$?


How many terms of the arithmetic sequence 105, 100, 95, ... appear before the number 10 appears?