In triangle $ABC$, $AB = 6$, $BC = 8$, and $CA = 10$. Triangle ABC is a right triangle with the right angle at $C$.

Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $C$ than it is to either $A$ or $B$?


For each positive integer $n$, let $n!$ denote the product $1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n$.

What is the remainder when $10!$ is divided by $13$?


If $x$, $y$, and $z$ are positive with $xy=32$, $xz=64$, and $yz=96$, what is the value of $x+y+z$?


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


What is the sum of all integer solutions to $|n| < |n-4| < 10$?


When a polynomial is divided by $3x^3 - 4x^2 + x - 5,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Find the largest prime divisor of $36^2 + 49^2$.


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $15$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


The Lucas sequence is defined by \( L_1 = 1 \), \( L_2 = 3 \), and each subsequent term is the sum of the previous two terms. Calculate the remainder when the $150^\text{th}$ term of the Lucas sequence is divided by 5.


The diameter, in inches, of a sphere with three times the volume of a sphere of radius 9 inches can be expressed in the form \(a\sqrt[3]{b}\) where \(a\) and \(b\) are positive integers, and \(b\) contains no perfect cube factors. Compute \(a+b\).


How many lattice points lie on the graph of the equation \(x^2 - y^2 = 65\)?


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


Milton's equation on his homework paper got a coffee stain, making one of the coefficients unreadable. He remembers that the quadratic had two distinct negative, integer solutions. If the constant term of the quadratic is 48 instead of 36, what is the sum of all distinct possible integers that could be under the coffee stain?

[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,brown);
label("$x + 48 = 0$", (3,1.5),E);
label("$x^2 +$", (0,1.5),W);
[/asy]


In a specific sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. The values of the remaining terms are chosen so that
\[a_n + a_{n + 1} + a_{n + 2} = 2n\]
for all $n \geq 1$. Determine $a_{1000}$.


Let \(a\) and \(b\) be positive real numbers such that \(a + b = 3\). Find the set of all possible values of \(\frac{1}{a} + \frac{1}{b}\).

Solution: \boxed{\left[\frac{4}{3}, \infty\right)}


Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ and $D$ on the surface of a sphere centered at $O$ with radius $r$, and satisfying
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]
the four points $A,$ $B,$ $C,$ and $D$ must lie on a common plane. Determine the value of $k$.


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


Thirty gremlins and twenty imps are at the Interregional Mischief Convention. This year, five of the imps, due to a magical disagreement, refuse to shake hands with ten specific gremlins. The rest of the imps have no issues shaking hands with all gremlins. All gremlins are still friendly and shake hands with each other and with any imp willing to shake hands with them. How many handshakes were at the convention?


Suppose we want to divide 12 rabbits into three groups, one with 4 rabbits, one with 6 rabbits, and one with 2 rabbits. How many ways can we form the groups such that BunBun is in the 4-rabbit group and Thumper is in the 6-rabbit group?


Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2512, \\
pr&=&qs&=&1225.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \).


If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2500$ and $f(i) = 1 + \sqrt{i} + i$?


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & -\frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix}}


What is the area, in square units, of a triangle with sides of $5, 5,$ and $6$ units? Express your answer in simplest radical form.


How many positive 3-digit numbers are multiples of 25, but not of 75?


A point $(x,y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 8$. What is the probability that $x+y \leq 5$? Express your answer as a common fraction.


Find the value of $x$ where $(2015 + x)^2 = x^2$.


Given that 15 is the arithmetic mean of the set $\{8, 12, 23, 17, y\}$, what is the value of $y$?


Determine the tens digit of $13^{2021}$.


How many lattice points lie on the graph of the equation $x^2 - y^2 = 45$?


Consider the geometric sequence starting with $5$, and each subsequent term is multiplied by $5/3$. Calculate the tenth term of this sequence and express your answer as a common fraction.


Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4\}$. How many such polynomials satisfy $P(-1) = 1$?


Let $a$ and $b$ be positive real numbers such that $a > b$. Consider the following five statements:
1. $\frac{1}{a} > \frac{1}{b}$
2. $a^2 > b^2$
3. $a > b$
4. $|a| > 1$
5. $b < 1$

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


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


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


Find the smallest positive integer $x$ that satisfies the congruence $52x + 14 \equiv 6 \pmod {24}$.


Place each of the digits 1, 2, 3 and 4 in exactly one square to make the smallest possible sum. Use the following structure: two digits form one number on the left, and the other two digits form another number on the right. The sum of these two numbers should be minimal. [Diagram explanation: two 2-digit numbers side by side]


The entry fee for a theme park is $20 per adult and $15 per child. Recently, the park collected exactly $1600 in entry fees from both adults and children. Determine the ratio of adults to children at the theme park that is closest to $2$.


Calculate the result of $\begin{pmatrix} 3 \\ -2 \end{pmatrix} - 5 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + 2 \begin{pmatrix} -1 \\ 4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} -9 \\ 21 \end{pmatrix}}


Rationalize the denominator of $\frac{3}{4\sqrt{7} + 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?


Consider the matrix $\mathbf{B} = \begin{pmatrix} 1 & 4 \\ 6 & d \end{pmatrix}$ such that $\mathbf{B}^{-1} = p \mathbf{B}$ for some constant $p$. Additionally, suppose $d \neq 3$. Determine the ordered pair $(d, p)$.

Solution: \boxed{(-1, \frac{1}{25})}


If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 960$ and $\gcd(a,c) = 324$, determine the smallest possible value of $\gcd(b,c)$.


Real numbers $a$ and $b$ satisfy the equation $a^2 + b^2 = 12a - 4b + 20$. What is $a-b$?


A regular pentagon and a square share a common vertex. Let the shared vertex be point $A$, and the adjacent vertices of the pentagon be $B$ and $E$. The square extends outward from $A$ along the sides $AB$ and $AE$. Calculate the degree measure of $\angle BAE$.


In right triangle \( GHI \), we have \( \angle G = 40^\circ \), \( \angle H = 90^\circ \), and \( IH = 12 \). Find \( GH \) to the nearest tenth. You may use a calculator for this problem.


Let $g(x)$ satisfy the equation $g(x - y) = g(x) g(y)$ for all real numbers $x$ and $y$. Also, assume $g(x) \neq 0$ for all real numbers $x$. Determine the value of $g(5)$.

Solution: \boxed{g(5) = 1}


What is the sum of the non-zero digits of the base $8$ representation of $999_{10}$?


Suppose that $b$ is a positive integer greater than or equal to $2.$ When $315$ is converted to base $b$, the resulting representation has $5$ digits. What is the number of possible values for $b$?


Let's consider a seven-digit number $C\,985\,F\,72$, where $C$ and $F$ are unknown digits. If this number is divisible by $9$, find the sum of all possible values of $C+F$.


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


Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $PR=9$. Two ants start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?


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


A right pyramid has a square base, each side measuring 8 cm. Its peak is positioned 15 cm above the center of the square base. Calculate the sum of the lengths of the pyramid's eight edges, rounded to the nearest tenth.

Solution: \boxed{96.1 \text{ cm}}


Determine how many base-2 digits (bits) are present when the base-16 number $A987B_{16}$ is written in base 2.


John has 8 green marbles and 7 purple marbles. He chooses a marble at random, records its color, and then does not put the marble back. He repeats this process 6 times. What is the probability that he chooses exactly three green marbles?


Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 10 centimeters about its vertical line of symmetry. Assume the height of the cylinder is now 20 centimeters. Express your answer in terms of $\pi$.


Let $\star (x)$ be the sum of the digits of a positive integer $x$. Define $\mathcal{S}$ as the set of positive integers such that for all elements $n$ in $\mathcal{S}$, $\star (n)=9$ and $0\le n< 10^{5}$. Compute $\star(m)$ where $m$ is the number of elements in $\mathcal{S}$.


What is the base 4 representation of the base 2 number $10111100_2$?


At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. A sample of gas is initially in a 3.5-liter container at a pressure of 8 kPa. If all of the gas is transferred to a 10.5-liter container at the same temperature, what will the new pressure be?

Solution: \boxed{\frac{8}{3} \text{ kPa}}


A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year?


In convex quadrilateral $ABCD, \angle B \cong \angle D, AB = CD = 200,$ and $AD \neq BC$. The perimeter of $ABCD$ is $720$. Find $\lfloor 1000 \cos B \rfloor.$


For each positive integer $n$, let $n!$ denote the product $1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n$.

What is the remainder when $10!$ is divided by $13$?


The sides of a triangle have lengths $13, 17,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?


If $x$ and $y$ are positive real numbers such that $5x^2 + 15xy = x^3 + 2x^2y + 3xy^2,$ what is the value of $x$?


Given that $b$ is an odd multiple of $7769$, find the greatest common divisor of $4b^2+81b+144$ and $2b+7$.


A right triangle has an area of 180 square units and a leg length of 30 units. What is the perimeter of the triangle, in units?

Solution: \boxed{P = 42 + 2\sqrt{261}}


From this infinite list of numbers, how many are integers? $$\sqrt{6561},\sqrt[3]{6561},\sqrt[4]{6561},\sqrt[5]{6561},\sqrt[6]{6561},\ldots$$


If $x$, $y$, and $z$ are positive with $xy=30$, $xz = 60$, and $yz=90$, what is the value of $x+y+z$?

Solution: \boxed{x+y+z = 11\sqrt{5}}


The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 23,$ $b_7 = 83,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms.  Find $b_2.$


In the diagram, what is the perimeter of polygon $PQRSU$? [asy]
import olympiad;
size(6cm); // ADJUST
pair p = (0, 8);
pair q = (4, 8);
pair r = (4, 4);
pair s = (9, 0);
pair u = (0, 0);
draw(p--q--r--s--u--cycle);
label("$P$", p, NW);
label("$Q$", q, NE);
label("$R$", r, E + NE);
label("$S$", s, SE);
label("$U$", u, SW);
label("$8$", p / 2, W);
label("$4$", p + (q - p) / 2, 2 * N);
label("$9$", s / 2, S);
draw(rightanglemark(p, u, s));
draw(rightanglemark(u, p, q));
draw(rightanglemark(p, q, r));
add(pathticks(p--q, s=6));
add(pathticks(q--r, s=6));
[/asy]


Let $g(x) = \frac{x + 8}{x - 1}$. Define a sequence $(g_n)$ of functions where $g_1 = g$ and
\[ g_n = g \circ g_{n - 1} \]
for all $n \geq 2$. Find the number of distinct real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$.


Chandra has five bowls, each a different color (red, blue, yellow, green, purple). She also has four glasses, one of each color except purple. If she chooses a bowl and a glass from the cupboard, under the condition that the bowl and glass cannot be the same color, how many valid pairings are possible?


Thirteen girls are standing in a circle. A ball is passed clockwise. The first girl, Bella, starts with the ball, skips the next four girls and throws to the sixth girl. The girl receiving the ball then skips the next four girls and continues the pattern. How many throws are needed for the ball to come back to Bella?


A triangle has an inscribed circle with a radius of 8 cm. Calculate the length of $\overline{AB}$ if the triangle is a right triangle with one angle measuring $30^\circ$. Express your answer in simplest radical form.

Solution: \boxed{\overline{AB} = 16(\sqrt{3} + 1)}


The quadratic $x^2 + 500x + 2500$ can be written in the form $(x + a)^2 + d$, where $a$ and $d$ are constants. Find the value of $\frac{d}{a}$.


A sphere is divided into eight equal parts (octants). The circumference of the sphere is $16\pi$ inches. Find the number of cubic inches in the volume of one of these parts. Express your answer in terms of $\pi$.


A circle centered at $O$ is circumscribed about $\triangle ABC$ as shown. Point $A$ is such that $\angle BOA = 120^\circ$ and $\angle BOC = 140^\circ$. What is the measure of $\angle BAC$, in degrees?


Let $d$ be a positive integer such that when $143$ is divided by $d$, the remainder is $3.$ Compute the sum of all possible values of $d$ between 20 and 100.


Our club has 24 members, 12 boys and 12 girls. In how many ways can we choose a president, a vice-president, and a secretary if the president and the vice-president must be of the same gender, but the secretary can be of any gender and no one can hold more than one office?


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

Solution: \boxed{\frac{5 - \sqrt{33}}{2}}


Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \]


Let $g(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. Note that when $a \neq b$, $(a, b)$ and $(b, a)$ are distinct. What is the smallest positive integer $n$ for which $g(n) = 4$?


The sum of 81 consecutive integers is $9^5$. What is their median?


Simplify the product \[\frac{6}{3}\cdot\frac{9}{6}\cdot\frac{12}{9} \dotsm \frac{3n+3}{3n} \dotsm \frac{3003}{3000}.\]


Let $a_1, a_2, \dots$ be a sequence where $a_1 = 5$, $a_2 = 3$, and $a_n = \frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?


A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 7 black squares, can be drawn on the checkerboard?


For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?


Find the integer $n$, $0 \le n \le 15$, such that \[n \equiv 14567 \pmod{16}.\]


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?

Solution: \boxed{22.5 \text{ daps}}


Bag A has 5 white marbles and 6 black marbles. Bag B has 8 yellow marbles and 6 blue marbles. Bag C has 3 yellow marbles and 9 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C. What is the probability that the second marble drawn is yellow?


Find the quadratic polynomial, with real coefficients, which has $-3 - 4i$ as a root, and where the coefficient of $x$ is $-10$.

Solution: \boxed{-\frac{5}{3}x^2 - 10x - \frac{125}{3}}


Quadrilateral $EFGH$ is 12 cm by 6 cm. $P$ is the midpoint of $\overline{FG}$, and $Q$ is the midpoint of $\overline{GH}$. A diagonal $\overline{EQ}$ is drawn. Calculate the number of square centimeters in the area of the region $EPQH$.

Solution: \boxed{45\text{ cm}^2}


I want to design a 4-character license plate. The requirements are:
1. The first character must be a letter.
2. The second and third characters can each be either a letter or a digit.
3. The fourth character must be a digit.
4. The first character and the third character should be the same. How many ways can I design such a license plate?


Factor $x^6 - 64$ as far as possible, where the factors are monic polynomials with real coefficients.

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


Find the units digit of $7 \cdot 13 \cdot 1957 - 7^4$.


If \(x > 0\) and \(y > 0\), define a new operation \(\Delta\) as follows: $$x \Delta y = \dfrac{x + y}{1 + xy + x^2y^2}.$$
Calculate \(3 \Delta 4.\)


During a discussion about polynomial roots, Sarah considers a quadratic polynomial \[x^2 - tx + q,\] with roots $\alpha$ and $\beta$. She notes that \[\alpha + \beta = \alpha^2 + \beta^2 = \alpha^3 + \beta^3 = \cdots = \alpha^{2010} + \beta^{2010}.\] She aims to find the maximum possible value of \[\dfrac{1}{\alpha^{2011}} + \dfrac{1}{\beta^{2011}}.\]


The area of a triangle is 450 square feet and its perimeter is required. If the length of one side (considered as the base) is 25 feet, find the altitude corresponding to this base and the perimeter, assuming the triangle is equilateral.

Solution: \boxed{75 \text{ feet}}


Calculate \(3^{18} \div 27^2\) and multiply the result by 7. Write your answer as an integer.


Find the product of all positive integer values of $c$ such that $10x^2 + 24x + c = 0$ has two real roots.


There is an unlimited supply of congruent equilateral triangles made of colored paper, each triangle being a solid color with the same color on both sides. A large equilateral triangle is constructed from four smaller triangles as shown. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles match in color. Given that there are eight different colors of triangles and the center triangle can only be one of three specific colors (red, blue, green) chosen from these eight, how many distinguishable large equilateral triangles can be constructed?


Rose wants to fill her new rectangular flower bed, which has been divided into five differently sized rectangular regions, with a unique type of flower per region. The new dimensions, in feet, of the rectangular regions are:
- Region 1: 2 ft by 3 ft
- Region 2: 3 ft by 4 ft
- Region 3: 2 ft by 5 ft
- Region 4: 4 ft by 4 ft
- Region 5: 3 ft by 6 ft
She plants one flower per square foot in each region. The costs of the flowers are as follows: Asters cost $1 each, Begonias $1.75 each, Cannas $2 each, Dahlias $2.25 each, and Easter lilies $3.50 each. What is the least possible cost, in dollars, for planting her garden?


What is the remainder when $1491 \cdot 2001$ is divided by $250$?


Simplify and rationalize the denominator: $$\frac{1}{1+ \frac{1}{\sqrt{5}+2}}.$$

Solution: \boxed{\frac{\sqrt{5} + 1}{4}}


In a survey conducted in Utah, 540 people were asked about their preferred term for carbonated beverages. The results are depicted in a pie chart. The central angle of the "Soda" sector of the graph is $162^\circ$. Determine the number of respondents who favored the term "Soda".


What is the smallest prime whose digits sum to $20$?


The measures of the angles of a pentagon are in the ratio 2:2:3:4:5. Find the measure of the smallest angle.


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


A girl has the following coins in her purse: $3$ pennies, $2$ nickels, $2$ dimes, and $2$ quarters. She draws two coins at random, notes their combined value, and then returns them to her purse. She repeats this process multiple times. What is the maximum number of different sums she can record?


Determine the value of $n$ such that $5 \times 16 \times 2 \times n^2 = 8!$.


In the diagram, what is the perimeter of polygon $ABCDE$? [asy]
import olympiad;
size(6cm);

pair a = (0, 8);
pair b = (4, 8);
pair c = (4, 4);
pair d = (0, 0);
pair e = (9, 0);

draw(a--b--c--e--d--cycle);
label("$A$", a, NW);
label("$B$", b, NE);
label("$C$", c, E + NE);
label("$D$", e, SE);
label("$E$", d, SW);

label("$8$", a / 2, W);
label("$4$", a + (b - a) / 2, 2 * N);
label("$9$", e / 2, S);

draw(rightanglemark(a, d, e));
draw(rightanglemark(d, a, b));
draw(rightanglemark(a, b, c));
add(pathticks(a--b, s=6));
add(pathticks(b--c, s=6));
[/asy]

Solution: \boxed{25 + \sqrt{41}}


A right circular cone is sliced into five pieces by planes parallel to its base, and these pieces have the same height. Calculate the ratio of the volume of the second-largest piece to the volume of the largest piece.


If nine people decide to come to a baseball game, but five of them are only 1/3 sure that they will stay for the entire game (while the other four are sure they'll stay the whole time), what is the probability that at least seven people stayed for the entire game?


In a math test series, four of the following scores belong to Alex and the other two to Jamie: 75, 80, 85, 90, 92, 97. If Alex's mean score is 85.5, what is Jamie's mean score?


Calculate the sum of the squares of the roots of the equation \[x^{12} + 7x^9 + 3x^3 + 500 = 0.\]


A number \( n \) has \( 4 \) divisors. How many divisors does \( n^2 \) have?


A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in simplest radical form.


How many different positive integers can be represented as a difference of two distinct members, where one member is from the set $\{1, 2, 3, \ldots, 9, 10\}$ and the other member is from the set $\{6, 7, 8, \ldots, 14, 15\}$?


Seven distinct integers are picked at random from $\{1, 2, 3, \ldots, 12\}$. What is the probability that, among those selected, the third smallest is $5$?


In how many ways can 8 distinct beads be placed on a necklace? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.)


A bag contains 12 red marbles and 8 blue marbles. Three marbles are selected at random and without replacement. What is the probability that two marbles are red and one is blue? Express your answer as a common fraction.


Find the matrix $\mathbf{M}$ that rotates the elements of a $2 \times 2$ matrix by 90 degrees clockwise. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & a \\ d & b \end{pmatrix}.
\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


Let $\mathcal{T}$ be the set $\lbrace 1, 2, 3, \ldots, 12 \rbrace$. Let $m$ be the number of sets consisting of two non-empty disjoint subsets of $\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$.


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


When a car's brakes are applied, it travels 9 feet less in each second than the previous second until it comes to a complete stop. The car goes 36 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops?


Let \( a, b, c \) be positive real numbers such that \( a + b + c = 1 \). Find the minimum value of
\[
\frac{1}{a^2 + 2b^2} + \frac{1}{b^2 + 2c^2} + \frac{1}{c^2 + 2a^2}.
\]


Let $\alpha$ and $\beta$ be the roots of $x^2 + px + 4 = 0,$ and let $\gamma$ and $\delta$ be the roots of $x^2 + qx + 4 = 0.$  Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\]in terms of $p$ and $q.$

Solution: \boxed{-4(p^2 - q^2)}


The functions $a(x),$ $b(x),$ and $c(x)$ are all invertible. Express the inverse of the function $g = a \circ c \circ b$ in terms of the inverses of $a,$ $b,$ and $c$.

Solution: \boxed{b^{-1} \circ c^{-1} \circ a^{-1}}


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


In how many ways can the digits of the number $75,\!510$ be arranged to form a 5-digit number that does not begin with the digit '0'?


A coach has a team of $30$ players. He needs to split the players into equal groups of at most $12$ players each. Additionally, one of the groups needs to be exactly half the size of another group. What is the least number of groups the coach needs?


For each positive integer $n$, let $n!$ denote the product $1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n$.

What is the remainder when $10!$ is divided by $13$?


Let $a$ and $b$ be constants. Suppose that the equation \[\frac{(x+a)(x+b)(x+10)}{(x+4)^2} = 0\] has exactly $3$ distinct roots, while the equation \[\frac{(x+2a)(x+4)(x+5)}{(x+b)(x+10)} = 0\] has exactly $1$ distinct root. Compute $100a + b.$


Let $S$ be the sum of all positive integers $n$ such that $n^2 + 8n - 1225$ is a perfect square. Find the remainder when $S$ is divided by $100$.


Consider the given functions:
$$\begin{array}{ccc}
f(x) & = & 5x^3 - \frac{1}{x} + 3\\
g(x) & = & x^2 - kx + c
\end{array}$$
If $f(2) - g(2) = 2$, what are the values of $k$ and $c$ assuming $k=1$?

Solution: \boxed{c = 38.5}


How many different four-digit numbers can be formed using the digits in 2025, given that each digit can only appear as many times as it does in 2025?


The first three stages of a geometric pattern are shown, where each line segment of toothpick forms a growing square grid. Each stage adds layers to the grid in a pattern such that each successive stage squares the number of lines required for a complete layer. If the pattern continues such that at the first stage, one needs 8 toothpicks, that form a shape of a square, then each new stage requires that each side of the square increases by 4 toothpicks more than the extra toothpicks needed for each side in the previous stage, how many toothpicks are necessary to create the arrangement for the 20th stage?


On the game show $\text{\emph{Wheel of Fortune Redux}}$, you see the following spinner. Given that each region is of the same area, what is the probability that you will earn exactly $\$3200$ in your first three spins? The spinner includes the following sections: $"\$2000"$, $"\$300"$, $"\$700"$, $"\$1500"$, $"\$500"$, and "$Bankrupt"$. Express your answer as a common fraction.


Point \(O\) is the center of an ellipse with major axis \(\overline{AB}\) and minor axis \(\overline{CD}\). Point \(F\) is one focus of the ellipse. If \(OF = 8\) and the diameter of the inscribed circle of triangle \(OCF\) is 4, compute the product \((AB)(CD)\).


Let $m = \underbrace{33333}_{\text{5 digits}}$ and $n = \underbrace{666666}_{\text{6 digits}}$.

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


Compute $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.


The side length of square $C$ is 24 cm. The side length of square $D$ is 54 cm. What is the ratio of the area of square $C$ to the area of square $D$? Express your answer as a common fraction.


Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac{1}{6}$ the measure of $\angle 2$. What is the degree measure of $\angle 5$? Refer to the original geometry setup.


Brand Z juice claims, "We offer 30% more juice than Brand W at a price that is 15% less." What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction.


Let $x$, $y$, and $z$ be positive real numbers. What is the smallest possible value of $(x+y+z)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)$?


Triangle $DEF$ has a perimeter of 398. The sides have lengths that are all integer values with $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


How many zeros are in the expansion of $999,\!999,\!999,\!98^2$?


A whole number is defined as "7-light" if the remainder when the number is divided by 7 is less than 3. What is the least three-digit "7-light" whole number?


A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n - 2 & \text{if $n$ is even}, \\
3n & \text{if $n$ is odd}.
\end{array}
\right.\]Suppose $m$ is even and $g(g(g(m))) = 54.$  Find $m.$


At a school cafeteria, Noah wants to buy a meal consisting of one main course, one beverage, and one snack. The table below lists Noah's available choices in the cafeteria. Additionally, Noah avoids having soda with pizza. How many distinct possible meals can Noah buy from these options?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Pizza & Soda & Apple \\ \hline
Burger & Juice & Banana \\ \hline
Pasta & & Cookie \\ \hline
\end{tabular}


Find the equation of the directrix of the parabola $y = 16x^2 + 4.$

Solution: \boxed{y = \frac{255}{64}}


Let $g(x) = x^2 + 4x + d$, where $d$ is a real number. For what values of $d$ does $g(g(x))$ have exactly 3 distinct real roots?

Solution: \boxed{d = 0}


What is the sum of the last two digits of $7^{15} + 13^{15}$?


The number \( 975 \) can be written as \( 23q + r \) where \( q \) and \( r \) are positive integers. What is the greatest possible value of \( q - r \)?


For what real value of \(v\) is \(\frac{-15-\sqrt{409}}{12}\) a root of \(6x^2 + 15x + v\)?


A point \((x,y)\) is randomly selected from inside the rectangle with vertices \((0,0)\), \((4,0)\), \((4,3)\), and \((0,3)\). What is the probability that \(x+1 < y\)?


Ninety percent of adults drink coffee and eighty percent drink tea, while seventy percent drink soda. What is the smallest possible percentage of adults who drink both coffee and tea, but not soda?


Real numbers $x$ and $y$ satisfy the equations $2^x=16^{y+1}$ and $27^y=3^{x-2}$. Find the product $xy$.


Let $f(x) = |x|$ for $-4\leq x \leq 4$. Determine the range of $f(x)-x$ within this interval.

Solution: \boxed{[0, 8]}


If $x$, $y$, and $z$ are positive with $xy=30$, $xz = 60$, and $yz=90$, what is the value of $x+y+z$?


If the current time is 3:00 p.m., what time will it be 1567 minutes later?

Solution: \boxed{5\!:\!07 \text{ p.m.}}


In a classroom, there are 40 students. 18 of them have cool dads, 22 of them have cool moms, and 10 of them have both cool dads and cool moms. Additionally, 8 students have cool siblings. How many students do not have cool parents but have cool siblings?


A steer initially weighs 300 kilograms. After feeding and care, its weight increases by 15%. With approximately 0.4536 kilograms in a pound, to the nearest whole pound, how much does the steer weigh after the weight increase?

Solution: \boxed{761\ \text{pounds}}


How many integers $m \neq 0, -1, 1$ satisfy the inequality $\frac{1}{|m|} \geq \frac{1}{6}$?


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


Find the value of $k$ so that the line $4x + 6y + k = 0$ is tangent to the parabola $y^2 = 32x.$


Given that 12 is the arithmetic mean of the set \(\{8, 15, 24, 9, 12, y\}\), what is the value of \(y\)?


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


Find the greatest common divisor of 1975 and 2625.


Given that $$(x+y+z)(xy+xz+yz)=27$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=9$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Determine the total number of pieces needed to create a seven-row triangle using the pattern where each row's rods increase by three and connectors form a triangle with one extra row than the triangle's rows.


Emma has a quadratic of the form $x^2 + bx + 88$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Emma reshapes this quadratic into the form $(x+n)^2 + 16$. Determine the value of $b$.


A triangle is formed with one vertex at the vertex of the parabola \( y = x^2 + 2 \) and the other two vertices at the intersections of the line \( y = r \) and the parabola. If the area of the triangle is between \( 10 \) and \( 50 \) inclusive, find all possible values of \( r \). Express your answer in interval notation.

Solution: \boxed{[10^{2/3} + 2, 50^{2/3} + 2]}


Let $G$ be the centroid of triangle $XYZ$. If $GX^2 + GY^2 + GZ^2 = 90,$ find the value of $XY^2 + XZ^2 + YZ^2$.


Consider the circle given by the equation $x^2 - 6y - 4 = -y^2 + 6x + 16$. Determine the center $(a, b)$ and radius $r$ of the circle and find the value of $a+b+r$.


Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.\]Find the smallest possible value of $|p| + |q| + |r| + |s|$.


If 3 cards from a standard deck are selected randomly, what is the probability that either three kings or at least 2 aces occur? (There are 4 Aces, 4 kings, and 52 total cards in a standard deck.)


Consider the modified Lucas sequence starting from 2, 3, where the first term is 2, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the $105^{\text{th}}$ term of the sequence is divided by 9?


Find the number of real solutions to the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -1.\]


Starting with the number 200, Shaffiq repeatedly halves the number and then takes the greatest integer less than or equal to that number. He stops when the number goes below 3. How many times must he do this?


Find the greatest possible sum of $x + y$ for integer points $(x, y)$ that lie in the first quadrant and satisfy the equation $x^2 + y^2 = 64$.


A right pyramid has a rectangular base with dimensions 8 cm by 15 cm. Its peak is 10 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges. Round your answer to the nearest whole number.


Factor the following expression: $74a^2 + 222a + 148a^3$.

Solution: \boxed{74a(2a^2 + a + 3)}


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


In square $XYZW$, points $P$ and $S$ lie on $\overline{XZ}$ and $\overline{XW}$, respectively, such that $XP=XS=\sqrt{3}$. Points $Q$ and $R$ lie on $\overline{YZ}$ and $\overline{YW}$, respectively, and points $T$ and $U$ lie on $\overline{PS}$ so that $\overline{QT} \perp \overline{PS}$ and $\overline{RU} \perp \overline{PS}$. If triangle $XPS$, quadrilateral $YQTP$, quadrilateral $WSUR$, and pentagon $YRUTQ$ each has an area of $1.5$, find $QT^2$.
[asy]
real x = 3;
real y = 3*sqrt(3)/2;
real z = 3/sqrt(3);
pair X, Y, Z, W, P, Q, R, S, T, U;
X = (0,0);
Y = (4,0);
Z = (4,4);
W = (0,4);
P = (x,0);
Q = (4,y);
R = (y,4);
S = (0,x);
T = Q + z * dir(225);
U = R + z * dir(225);
draw(X--Y--Z--W--X);
draw(P--S);
draw(T--Q^^U--R);
draw(rightanglemark(Q, T, P), linewidth(.5));
draw(rightanglemark(R, U, S), linewidth(.5));
dot("$X$", X, SW);
dot("$Y$", Y, S);
dot("$Z$", Z, N);
dot("$W$", W, NW);
dot("$P$", P, S);
dot("$Q$", Q, E);
dot("$R$", R, N);
dot("$S$", S, W);
dot("$T$", T, SW);
dot("$U$", U, SW);
[/asy]


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

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


The points $(1,3)$ and $(-4,6)$ are adjacent vertices of a square. What is the area of the square?


Find the quadratic polynomial $q(x)$ such that $q(-1) = 6,$ $q(2) = 1,$ and $q(4) = 20.$

Solution: \boxed{\frac{67}{30}x^2 - \frac{39}{10}x - \frac{2}{15}}


The function \(g(x)\) satisfies
\[c^3 g(a) = a^3 g(c)\] for all real numbers \(a\) and \(c\). If \(g(3) \neq 0\), find
\[\frac{g(6) - g(2)}{g(3)}.\]


Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 3, \\
\log_{10} (ML) + \log_{10} (MR) &= 4, \\
\log_{10} (RA) + \log_{10} (RL) &= 5.
\end{align*}
Compute the value of the product $ARML.$


Convert the decimal number $93_{10}$ to its base $2$ representation.


A club consists of 24 members, split evenly with 12 boys and 12 girls. There are also two classes, each containing 6 boys and 6 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender and from different classes?


Carla rotated point $A$ 750 degrees clockwise about point $B$, and it landed at point $C$. Devon then rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and also landed at point $C$. Afterwards, Devon rotated point $C$ another $45^\circ$ counterclockwise about point $B$. If $x < 360$, what is the new position of $A$ after all rotations, if it does not land back at point $C$?

Solution: \boxed{15^\circ \text{ counterclockwise from its original position}}


An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number itself. Determine how many numbers less than 50 are abundant numbers.


What is the value of the least positive base ten number that requires seven digits for its binary representation?


What is the largest integer that is a divisor of
\[
(n+3)(n+5)(n+7)(n+9)(n+11)
\]
for all positive even integers \( n \)?


Find the largest negative integer $x$ which satisfies the congruence $50x + 14 \equiv 10 \pmod {24}$.


A line \( y = -\frac{2}{3}x + 6 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on the line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?


Four circles of radius 2 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.

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


In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 26$, and $\tan Z = 3 \sin Z$. What is $XZ$?


A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


At the end of the year, the Science Club decided to hold an election for which 4 equal officer positions were available. However, 20 candidates were nominated, of whom 8 were past officers. Of all possible elections of the officers, how many will have at least 2 of the past officers?


Suppose the numbers \[3 - \sqrt{8}, \;5+\sqrt{12}, \;16 - 2\sqrt{9}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


In rectangle $ABCD$, $AB=8$ and $BC=5$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.


In a given diagram, the area of triangle $ABC$ is 45 square units. The point $D$ still lies on the line $AC$, extending beyond $C$ such that $CD = 30$ units. If the length of $AC$ is 10 units, what is the area of triangle $BCD$?


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


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

Solution: \boxed{2 \csc x}


Calculate $\dbinom{7}{5}$ and also find the number of permutations of 7 items taken 5 at a time.


Triangle \(PQR\) is similar to triangle \(XYZ\). If \(PQ = 8\), \(QR = 16\), and \(YZ = 24\) units, what is the length of segment \(XY\) if the perimeter of triangle \(XYZ\) is 60 units?


What is the base $2$ representation of $123_{10}$?


Evaluate the determinant of the following matrix:
\[
\begin{vmatrix}
0 & 2\sin \theta & -\cos \theta \\
-2\sin \theta & 0 & \sin \phi \\
\cos \theta & -\sin \phi & 0
\end{vmatrix}.
\]


A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:
1. The volume of the inscribed sphere.
2. The volume of the inscribed cone.

Express your answer in terms of $\pi$.


If $c$ and $d$ are integers with $c > d$, what is the smallest possible positive value of $\frac{c+2d}{c-d} + \frac{c-d}{c+2d}$?


Let $p$ and $q$ be real numbers so that the roots of
\[3z^2 + (6 + pi) z + (50 + qi) = 0\]are complex conjugates. Enter the ordered pair $(p,q)$.

Solution: \boxed{(0, 0)}


How many integers $-15 \leq n \leq 10$ satisfy $(n-1)(n+3)(n + 7) < 0$?


What integer \( n \) satisfies \( 0 \leq n < 137 \) and
$$ 12345 \equiv n \pmod{137}~? $$


In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$


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


There are four containers:
- Container A holds 5 red balls and 7 green balls.
- Container B holds 7 red balls and 3 green balls.
- Container C holds 8 red balls and 2 green balls.
- Container D holds 4 red balls and 6 green balls.
The probability of choosing container A or D is $\frac{1}{4}$ each, and the probability of choosing containers B or C is $\frac{1}{4}$ each. If a container is selected at random and then a ball is randomly selected from that container, what is the probability that the ball selected is green? Express your answer as a common fraction.


In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the trisection point of side $BC$ (closer to $B$ than to $C$). What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction.


Coach Grunt is preparing a 5-person starting lineup for his basketball team, the Grunters, which includes 15 players. Among them, Ace (a forward) and Zeppo (a guard), who are league All-Stars, will definitely be in the starting lineup. The remaining positions to be filled are one guard, one forward, and one center. If there are 4 additional guards, 5 additional forwards, and 3 centers available, how many different starting lineups are possible?


Find the first term in the geometric sequence $x, y, z, 81, 162$.


Determine all values of \( c \) for which the following system has a solution \( (x, y) \) in real numbers:
\begin{align*}
\sqrt{x^2y^2} &= c^{2c}, \\
\log_c (x^{\log_c y}) + \log_c (y^{\log_c x}) &= 8c^4.
\end{align*}

Solution: \boxed{\left(0, \frac{1}{2}\right]}


Consider two lines: line $r$ parametrized as
\begin{align*}
x &= 2 + 5t,\\
y &=  3 - 2t,
\end{align*}
and the line $s$ parametrized as
\begin{align*}
x &= 1 + 5u,\\
y &=  -2 - 2u.
\end{align*}
Let $C$ be a point on line $r$, $D$ be a point on line $s$, and let $Q$ be the foot of the perpendicular from $C$ to line $s$.

Then $\overrightarrow{QC}$ is the projection of $\overrightarrow{DC}$ onto some vector $\begin{pmatrix} w_1\\w_2\end{pmatrix}$ such that $w_1 - w_2 = 3$. Find $\begin{pmatrix}w_1 \\ w_2 \end{pmatrix}$.

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


What is the units digit of the sum of the squares of the first 2500 odd, positive integers?


Adam and Simon start their bicycle trips from the same point at the same time. Adam travels east at 12 mph and Simon travels south at 16 mph. After how many hours are they 100 miles apart?


I won a trip for six to an exclusive beach resort. I can bring five of my friends. I have 12 friends to choose from. At least one of the friends I choose must be among the 6 who have previously traveled with me. In how many ways can I form my travel group?


Cameron, Dean, and Olivia each have a 6-sided cube. All of the faces on Cameron’s cube have a 6. The faces on Dean's cube are numbered 1, 1, 2, 2, 3, 3. Four of the faces on Olivia’s cube have a 3, and two of the faces have a 6. All three cubes are rolled. What is the probability that Cameron's roll is greater than each of Dean’s and Olivia's?


How many four-digit numbers \( N \) have the property that the three-digit number obtained by removing the leftmost digit is one sixth of \( N \)?


Solution: \boxed{2w^2 - 5w - 3}


Find the minimum value of
\[
\sin^4 x + 2 \cos^4 x,
\]
as \( x \) varies over all real numbers.


Let $O$ be the origin, and let $(ka, kb, kc)$ be a fixed point for some non-zero constant $k$. A plane passes through $(ka, kb, kc)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find
\[
\frac{ka}{p} + \frac{kb}{q} + \frac{kc}{r}.
\]


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 9.$  Find the minimum value of
\[\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.\]


Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2 + y^2 = 20x + 36y$. What is the minimum value of $y$?

Solution: \boxed{18 - 2\sqrt{106}}


Arthur walks 8 blocks east and then 10 blocks north. If each block is one-fourth of a mile, how many miles did Arthur walk in total?


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


Sandy plans to paint her daughter's playhouse including two trapezoidal sides of the roof and the front triangular face, all shaded in the diagram. The paint covers 100 square feet per gallon and costs $\$15$ per gallon. The roof sides are each 8 feet wide at the base and 5 feet tall, tapering linearly to a top width of 4 feet. The front face is an equilateral triangle with a 6 feet side. Calculate the total cost of the paint needed.


Suppose you have a drawer containing 8 forks, 7 spoons, and 5 knives. If you reach in and randomly remove four pieces of silverware, what is the probability that you get two forks, one spoon, and one knife?


Pascal's Triangle starting with row 1 has the sum of elements in row $n$ given by $2^{n-1}$. What is the sum of the interior numbers of the ninth row, considering interior numbers are all except the first and last numbers in the row?


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


In the diagram below, we have \(\sin \angle APB = \frac{3}{5}\). What is \(\sin \angle APC\)?

[asy]
pair A,P,B,C;
C = (-2,0);
P = (0,0);
B = (2,0);
A = rotate(aSin(3/5))*(1.5,0);
dot("$C$",C,S);
dot("$B$",B,S);
dot("$A$",A,N);
dot("$P$",P,S);
draw(B--C);
draw(P--A);
[/asy]


Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$?


At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. Initially, a nitrogen gas is in a 3.4 liter container at a pressure of 7 kPa. If I transfer all the nitrogen to a 4.25 liter container while maintaining the same temperature, what will be the new pressure?

Solution: \boxed{5.6 \, \text{kPa}}


What is the sum of all the positive divisors of 154?


The sum of 22 consecutive positive integers is a perfect cube. What is the smallest possible value of this sum?


If the lengths of two sides of a right triangle are 7 and 24 units, what is the least possible length, in units, of the third side? Express your answer in simplest radical form.


Consider a modified triangular array where each number is obtained similarly by summing two adjacent numbers from the previous row, but now, the numbers on the sides start from 0 and increase by 2, i.e., 0, 2, 4, ..., etc. Calculate the sum of the numbers in the 50th row, assuming each row's sum follows the recursion pattern $f(n) = 2f(n-1) + 4$ with $f(1)=0$.

Solution: \boxed{2^{51} - 4}


Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x + f(y)) = x + y + k \]
for all real numbers $x$ and $y$, where $k$ is a real constant.


Each term in the sequence $4096, 1024, 256, x, y, 4, 1, \frac{1}{4},...$ is obtained by multiplying the previous term by a constant. What is the value of $x + y$?


Three dice with faces numbered 1 through 6 are stacked as shown. Eight of the eighteen faces are visible, leaving 10 faces hidden (back, bottom, in between). The visible numbers are 1, 2, 3, 4, 4, 5, 6, and 6. What is the total number of dots NOT visible in this view?


Solution: \boxed{(-\infty, 1) \cup (2, 4) \cup (5, 6) \cup (7, \infty)}


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v},$ $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \end{pmatrix}}


Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 15$ and $X$ is an integer, what is the smallest possible value of $X$?


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at the same time?


Yan is situated between his home and the stadium. To get to the stadium, Yan can either walk directly there or return home first and then ride his bicycle to the stadium. He now rides 10 times as fast as he walks, and both routes take the same amount of time. Determine the ratio of the distance from Yan's current position to his home and to the stadium.


How many values of $x$, $-30<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1?$ (Note: $x$ is measured in radians.)


What is the largest integer $n$ for which $\binom{10}{3} + \binom{10}{4} = \binom{11}{n}$?


The sequence \( 9720, 3240, 1080, \ldots \) is created by repeatedly dividing by 3. How many integers are in this sequence?


Eight people can paint a house in four hours. If six people start painting the house and work for two hours, how many more people are needed to finish painting the house in the next two hours?


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{2}{5} \\ -\frac{4}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{22}{5} \\ -\frac{6}{5} \end{pmatrix}}


A triangle has sides of lengths 6, 8, and 10 and is also a right triangle. Each vertex of this triangle is the center of a circle, and each circle is externally tangent to the other two circles. Determine the sum of the areas of these circles.


Linda has 8 daughters and no sons. Some of her daughters have 5 daughters each, and the rest have none. Linda has a total of 43 daughters and granddaughters, and no great-granddaughters. How many of Linda's daughters and granddaughters have no daughters?


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


Let the sequence \(b_1, b_2, b_3, \dots\) be defined such that \(b_1 = 24\), \(b_{12} = 150\), and for all \(n \geq 3\), \(b_n\) is the arithmetic mean of the first \(n - 1\) terms. Find \(b_2\).


A park has two parallel paths 60 feet apart. A crosswalk bounded by two parallel lines crosses the paths at an angle. The length of the path between the lines is 20 feet and each line is 75 feet long. Find the distance, in feet, between the lines.


Let \[f(x) =
\begin{cases}
2x^2 + 5 & \text{if } x \leq 4, \\
bx + 3 & \text{if } 4 < x \leq 6, \\
cx^2 - 2x + 9 & \text{if } x > 6.
\end{cases}
\]
Find values of $b$ and $c$ such that the graph of $y = f(x)$ is continuous.

Solution: \boxed{b = 8.5, c = \frac{19}{12}}


In the diagram, there are two circles. The larger circle has center $O$ and an area of $100\pi$. The smaller circle has its center on the circumference of the larger circle and touches the larger circle internally at one point. Each circle is divided into three equal areas, and we want to find the total area of the shaded regions which cover two of the three parts for each circle.
```asy
size(120);
import graph;
fill(Arc((0,0),2,240,480)--cycle,mediumgray);
fill(Arc((1,0),1,60,300)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
dot((0,0)); label("$O$", (0,0), W);
draw((-2,0)--(2,0));
draw((1,-1)--(1,1));
```


The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{73}$. Find the greatest possible value of $k$.


Find the smallest two-digit prime number with 2 as the tens digit such that reversing the digits of the number produces a composite number.


On a 6 by 6 square grid where each dot is 1 cm apart from its nearest horizontal and vertical neighbors, consider a right triangle with vertices labeled $P$, $Q$, and $R$. The vertices $P$, $Q$, and $R$ are located at coordinates $(1, 5)$, $(5, 5)$, and $(1, 1)$, respectively. What is the product of the area of triangle $PQR$ and the perimeter of triangle $PQR$? Express your answer in simplest form.

Solution: \boxed{64 + 32\sqrt{2} \text{ cm}^3}


A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form a square, and the piece of length $b$ is bent to form a circle. The square and the circle have equal area. What is $\frac{a}{b}$?


Find the reflection of $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ over the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{17}{5} \\ -\frac{6}{5} \end{pmatrix}}


In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?


What is the remainder when \(5x^8 - 3x^7 + 2x^6 - 8x^4 + 3x^3 - 5\) is divided by \(3x - 6\)?


Find the remainder when $7^{1985}$ is divided by $17$.


A biased die has probabilities of rolling a 1, 2, 3, 4, 5, or 6 as \( \frac{1}{12}, \frac{1}{12}, \frac{1}{12}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \) respectively. If rolling a 1, 2, or 3 wins you \$4, but rolling a 4, 5, or 6 loses you \$3, what is the expected value, in dollars, of one roll?


At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants?


Find $101^{-1} \pmod{102}$, as a residue modulo 102.


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{4} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$


From a circular piece of paper with radius $BC = 16$ cm, Jeff removes an unshaded sector. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 15 centimeters and of volume $675\pi$ cubic centimeters. Determine the number of degrees in the measure of angle $ABC$ of the sector that is not used.


A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively.


The function $g(x)$ satisfies
\[ g(xy) = \frac{g(x)}{y} \]
for all positive real numbers $x$ and $y$. If $g(45) = 15$, find $g(60)$.


Given that $(x - y)^2 = 49$ and $xy = -8$, find the value of $x^2 + y^2$.


Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 8x + 12|$ and $y = 6 - x$.


If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 10? Express your answer as a common fraction.


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


The number of gallons of coffee a scientist drinks in a day is inversely proportional to the number of hours he sleeps the previous night. On Wednesday, he slept for 8 hours and consumed 3 gallons of coffee. On Thursday, he slept for 4 hours. On Friday, he slept for 10 hours. How many gallons of coffee did he drink on Thursday and Friday, and what was his average coffee consumption over these three days?


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^3)(x-2^3)(x-3^3)\dotsm(x-50^3)} = 0\] have for \(x\)?


In triangle PQR below, $\cos Q = \frac{5}{13}$, where PQ = 13. What is QR?

[asy]
pair P,Q,R;
P = (0,12);
Q = (0,0);
R = (5,0);
draw(P--Q--R--cycle);
label("$P$",P,NW);
label("$Q$",Q,SW);
label("$R$",R,SE);
label("$13$",(P+Q)/2,W);
[/asy]


Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 24,$ $p(2) = 48,$ $p(3) = 72,$ and $p(4) = 96.$ Find $p(0) + p(5).$


What common fraction is equivalent to $0.4\overline{56}$?


Jane's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 8 rows and at least 12 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?

Solution: \boxed{12 + 15 + 18 + 20 + 24 + 30 + 36 + 40 + 45 = 240}


In the expression $c \cdot b^a - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order, with the restriction that $d = 0$. What is the maximum possible value of this expression?


A $\textit{composite number}$ is a number that has two or more prime factors. The number 105 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?


Determine the ordered pair $(x,y)$ that satisfies the following system of equations:

\[
x + 2y = (7 - x) + (3 - 2y),
\]
\[
x - 3y = (x + 2) - (y - 2).
\]

Solution: \boxed{(9, -2)}


Let $x$ and $y$ be positive real numbers such that $5x + 6y < 90$. Find the maximum value of
\[xy (90 - 5x - 6y).\]


What are the last two digits in the sum of the factorials of the first 15 positive integers?


In triangle $\triangle XYZ$, the medians $\overline{XM}$ and $\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\triangle XYZ$?


Given real numbers $t$, consider the point of intersection of the lines $3x + 4y = 12t + 6$ and $2x + 3y = 8t - 1$. All plotted points from various values of $t$ lie on a line. Determine the slope of this line.


The values of \( a \), \( b \), \( c \), and \( d \) are 1, 3, 5, and 7, but not necessarily in that order. What is the smallest possible value of the sum of the four products \( ab \), \( bc \), \( cd \), and \( da \)?


In triangle $\triangle JKL$, where $\triangle JKL$ is a right triangle at $J$, $\tan K = \frac{4}{3}$. If the length $JK = 3$, what is the length of $KL$?


How many ways are there to arrange the letters of the word $\text{B}_1\text{B}_2\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3$, in which the three A's are considered different, the two B's are identical, and the two N's are different?


The polynomial \(x^{104} + Cx + D\) is divisible by \(x^2 + x + 1\) for some real numbers \(C\) and \(D.\) Find \(C + D.\)


What is the coefficient of $a^2b^3$ in $(a+b)^5\left(c^2+\dfrac{1}{c^2}\right)^3$?


What is the minimum value of the expression $x^2 + y^2 - 8x - 6y + 20$ for real $x$ and $y$?


Integers $a$ and $b$ with $a > b > 0$ satisfy $a + b + ab = 119$. What is $a$?


Compute
\[\cos^2 0^\circ + \cos^2 1^\circ + \cos^2 2^\circ + \dots + \cos^2 180^\circ.\]


The force required to loosen bolts varies inversely with the length of the wrench handle used. Using a wrench with a handle length of 10 inches requires 300 pounds of force to loosen Bolt A. To loosen Bolt B, which is tighter and requires 400 pounds of force with a 10-inch handle, how many pounds of force will be needed using a 20-inch wrench handle?


A point has rectangular coordinates $(3, 8, -6)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$.


Let \(a,\) \(b,\) \(c,\) and \(d\) be nonzero real numbers, and let:
\[
x = \frac{b+d}{c+d} + \frac{c+d}{b+d}, \quad y = \frac{a+d}{c+d} + \frac{c+d}{a+d}, \quad z = \frac{a+d}{b+d} + \frac{b+d}{a+d}.
\]
Simplify \(x^2 + y^2 + z^2 - xyz.\)


Trisha invests \$2,000 in a savings account at an interest rate of 5% per annum, compounded annually. Additionally, she deposits \$300 at the end of each year into the account. How much total interest will Trisha have earned by the end of 5 years?


Bertha has 8 daughters and no sons. Each of her daughters has 4 daughters. Bertha has a total of 40 daughters and granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 317^\circ.$


Point $A$ has coordinates $(x, 7)$. When Point $A$ is reflected over the x-axis, it lands on Point $B$. What is the sum of the four coordinate values of points $A$ and $B$?


The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$?

[asy]
size(4cm,4cm);

pair a1, b1, c1, d1;
a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);

pair e1, f1, g1, h1;
e1=(0,0);
f1=c1-(a1-d1);
g1=b1+(c1-a1);
h1=e1+(g1-b1);

draw(a1--d1--e1--b1--a1);
draw(b1--g1--c1--a1);
draw(c1--f1--d1);
draw(g1--h1--e1,dotted+1pt);
draw(h1--f1,dotted+1pt);

label("$A$",e1,SW);
label("$B$",c1,NE);
[/asy]


Mr. and Mrs. Lopez, their two children, and their babysitter are going on a trip in their family car, which has a driver's seat, one front passenger seat, and three back seats. Either Mr. or Mrs. Lopez must be the driver. How many different seating arrangements are possible under these conditions?


Let
\[
\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -8 & -3 \end{pmatrix}.
\]
Compute $\mathbf{C}^{50}$.

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


How many positive integer multiples of $1001^2$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are even integers and $1\leq i < j \leq 99$?


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


A triangle in a Cartesian coordinate plane has vertices at (2, 3), (2, -4), and (9, 3). What is the area of this triangle, expressed in square units?


Suppose two distinct numbers are chosen from between 6 and 20, inclusive. What is the probability that their product is even, or exactly one of the numbers is a prime?


Find the distance between the vertices of the hyperbola \( 16x^2 + 64x - 4y^2 + 8y + 36 = 0 \).


Let $\theta$ be the angle between the planes $3x - 2y + z - 4 = 0$ and $9x - 6y - 4z + 12 = 0$. Find $\cos \theta$.

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


The number $1023$ can be written as $17n + m$ where $n$ and $m$ are positive integers. What is the greatest possible value of $n - m$?


Find the sum of all possible positive integer values of \( b \) such that the quadratic equation \( 3x^2 + 7x + b = 0 \) has rational roots.


Suppose $x,$ $y,$ and $z$ are real numbers such that
\[
\frac{xz}{x + y} + \frac{yx}{y + z} + \frac{zy}{z + x} = -5
\]
and
\[
\frac{yz}{x + y} + \frac{zx}{y + z} + \frac{xy}{z + x} = 7.
\]
Compute the value of
\[
x + y + z.
\]


A point has rectangular coordinates $(8, 6, -3)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi)$.

Solution: \boxed{(8, -6, -3)}


One of the roots of \( z^2 = -63 + 16i \) is \( 5 + 4i \). What is the other root?

Solution: \boxed{-5 - 4i}


The length of each of the equal sides of an isosceles triangle is $20$ inches, and the base is $24$ inches. This triangle is circumscribed by a circle. Determine the radius of the circle and the height of the triangle from the base to the apex.

Solution: \boxed{16 \text{ inches}}


What is $3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} \pmod{7}$?

Express your answer as an integer from $0$ to $6$, inclusive.


Two wheels, one larger outside the other and concentric, with radii $2 \text{ m}$ and $1 \text{ m}$ respectively, are rolled in a straight line on a flat horizontal surface. Each makes one complete revolution. How many meters did the center of each wheel travel horizontally from their starting locations?

Solution: \boxed{2 \pi}


Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.


Find the remainder when $5x^5 - 12x^4 + 3x^3 - 7x + 15$ is divided by $3x - 6.$


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


Palindromes are numbers that read the same backwards and forwards, like 1221. What is the least possible four-digit palindrome that is divisible by 5?


Determine if there is a scalar \(d\) such that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) + \mathbf{v} = d \mathbf{v}\]
for all vectors \(\mathbf{v}\).


In the given diagram, there are two regular polygons, a pentagon and a square. Find the sum of the measures of angles \(ABC\) and \(ABD\) in degrees.

[asy]
draw(10dir(0)--10dir(72)--10dir(144)--10dir(216)--10dir(288)--cycle,linewidth(2));
draw(10dir(216)--10dir(288)--10dir(288)+(0,-10)--10dir(216)+(0,-10)--10dir(216)--cycle,linewidth(2));

label("A",10dir(216),W);
label("B",10dir(288),E);
label("C",10dir(0),E);
label("D",10dir(288)+(0,-10),E);
[/asy]


Let three positive integers be \( a \), \( b \), and \( c \). The product \( N \) of these integers is \( 8 \) times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of \( N \).


John is training for a duathlon but skips the swimming part. He cycles 15 miles at a speed of $3x - 2$ miles per hour, then takes a 10-minute break to switch gear, and finally runs 3 miles at $x$ miles per hour. His total exercise time, including the break, is 130 minutes. Determine John's running speed, rounding to the nearest hundredth of a mile per hour.

Solution: \boxed{4.44 \text{ mph}}


Solve the inequality:
\[\frac{x + 5}{x^2 + 3x + 9} \ge 0.\]
Enter your answer using interval notation.

Solution: \boxed{[-5, \infty)}


The function \( g(x) \) satisfies
\[ g(3^x) + xg(3^{-x}) = 2 \] for all real numbers \( x \). Find \( g(3) \).


Given that $3+\sqrt{5}$ is a root of the equation \[x^3 + cx^2 + dx + 20 = 0\] and that $c$ and $d$ are rational numbers, compute $d.$


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


The first term of a geometric sequence is 250. If the sum of the first 50 terms is 625 and the sum of the first 100 terms is 1225, find the sum of the first 150 terms.


The graph of an equation \[\sqrt{(x-1)^2 + (y-3)^2} + \sqrt{(x+7)^2 + (y+2)^2} = 24.\] is an ellipse. What is the distance between its foci?


A triangle has vertices at $(1, 3), (-2, 4), (4, -1)$. Calculate the area of this triangle.


Expand $(4x + 3y + 2)(2x + 5y + 3)$ and find the sum of the coefficients of the terms which contain a nonzero power of $y$.


The vertical drops of six roller coasters at Fermat Fun World are shown in the table.
\begin{tabular}{|l|c|}
\hline
The Loop & 180 feet \\ \hline
The Cyclone & 150 feet \\ \hline
The Giant Drop & 210 feet \\ \hline
The Sky Scream & 195 feet \\ \hline
The Thunder & 170 feet \\ \hline
The Ultra Twister & 220 feet \\ \hline
\end{tabular}
What is the positive difference between the mean and the median of these values?


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

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


Let \( a \), \( b \), \( c \), and \( d \) be real numbers such that \( |a-b|=2 \), \( |b-c|=4 \), and \( |c-d|=5 \). Determine the sum of all possible values of \( |a-d| \).


Half of the people in a room are seated in two-thirds of the chairs. The rest of the people are standing. If there are 8 empty chairs, how many people are in the room?


Given the function $f(x)$ values in the table below:

\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $f(x)$ & 4 & 3 & 2 & 5 & 1 \\ \hline
\end{tabular}

If $f^{-1}$ exists, then what is $f^{-1}(f^{-1}(f^{-1}(3)))$?


Determine the number of positive integers \( a \) less than \( 14 \) such that the congruence \( ax \equiv 1 \pmod{14} \) has a solution in \( x \).


Find the largest value of $n$ such that $6x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.


Find the value of $b$ if $b$ is positive, and the numbers $10, b, \frac{2}{3}$ are the first, second, and third terms, respectively, of a geometric sequence.


Alice throws five identical darts. Each hits one of three identical dartboards on the wall. After throwing the five darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible?


Mrs. Lee has 12 grandchildren. Assuming that each grandchild is male with a probability of $\frac{2}{3}$ and female with a probability of $\frac{1}{3}$, independently of the others, what is the probability that Mrs. Lee has more grandsons than granddaughters or vice versa?


If $x$ is a real or complex number and $x^3 = 64$, what is the sum of all real values of $x$?


Six balls are numbered 1 through 6 and placed in a bowl. Sarah will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Sarah will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and less than 20? Express your answer as a common fraction.


Determine the number of solutions to
\[3\cos^3 x - 7 \cos^2 x + 3 \cos x = 0\] in the range $0 \le x \le 2\pi.$


Determine the minimum value of $y$ if $y = 5x^2 + 20x + 25$.


Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[
\{-6, -4, -3, -1, 1, 3, 5, 7\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


Find the integer $m$, where $0 \le m \le 360$, such that $\cos m^\circ = \cos 970^\circ$.


For the set $\{1, 2, 3, \ldots, 8\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, starting with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 7, 8\}$ is $8-7+3-2+1=3$. Calculate the sum of all such alternating sums for $n=8$.


Determine how many distinct prime factors the sum of the positive divisors of $450$ has and find the greatest common divisor (GCD) of this sum with $450$.


Evaluate the expression: $\sqrt{10 - 2\sqrt{21}} + \sqrt{10 + 2\sqrt{21}}$.


Determine the product of all possible values for $c$ if the length of the segment between the points $(3c, c+5)$ and $(1, 4)$ is $5$ units.


In the diagram, $AB$ is a line segment, and $C$, $D$, and $E$ are points such that $CD$ is perpendicular to $AB$ and $CE$ is a diagonal line making an angle with $CD$. What is the value of $x$ if the angle $DCE$ measures $60^\circ$?

[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ$",(4.75,2.25));
label("$60^\circ$",(5.5,0.75));
label("$C$",(4,0),S);
label("$D$",(4,8),N);
label("$E$",(9,7),NE);
[/asy]


In the given configuration, a regular pentagon and a regular triangle are attached at one vertex. Find the sum, in degrees, of the measures of angles \(PQR\) and \(PQS\), where \(P\) is the shared vertex, \(Q\) is a vertex on the pentagon, and \(S\) is a vertex on the triangle.

[asy]
draw(10dir(18)--10dir(90)--10dir(162)--10dir(234)--10dir(306)--cycle,linewidth(2)); // Pentagon
draw(10dir(234)--10dir(306)--10dir(306)+(0,-10)--cycle,linewidth(2)); // Triangle

label("P",10dir(234),W);
label("Q",10dir(306),E);
label("R",10dir(18),E);
label("S",10dir(306)+(0,-10),E);
[/asy]


Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 58^\circ$ and $\angle ABC = 69^\circ$, then what is $\angle CHX$?


Compute the value of \[M = 120^2 + 119^2 - 118^2 - 117^2 + 116^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2,\]where the additions and subtractions alternate in pairs.


A fair 8-sided die is rolled. If I roll \(n\), then I win \(n^3\) dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


Each day, Jenny ate 30% of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 28 remained. How many jellybeans were in the jar originally?


Let $r(x)$ be a monic quartic polynomial such that $r(1) = 0,$ $r(2) = 3,$ $r(3) = 8,$ and $r(4) = 15$. Find $r(5)$.


Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her three stamp books. One of her stamp books has a total of 924 stamps, another has 1260 stamps, and the third has 1386 stamps. What is the largest number of stamps that Jenna could be putting on each page?


Let $p,$ $q,$ and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\]is true if and only if either $x < -6$ or $3 \le x \le 8.$ Given that $p < q,$ find the value of $p + 2q + 3r.$


Triangle $PQR$ has vertices $P(1, 6)$, $Q(3, -2)$, and $R(9, -2)$. A line through $R$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.


Let $\mathbf{p} = \begin{pmatrix} -4 \\ 6 \\ 2 \end{pmatrix},$ $\mathbf{q} = \begin{pmatrix} 3 \\ e \\ -1 \end{pmatrix},$ and $\mathbf{r} = \begin{pmatrix} -1 \\ -3 \\ 5 \end{pmatrix}.$ Compute
\[(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].\]


Consider the sum
\[\text{cis } 20^\circ + \text{cis } 30^\circ + \text{cis } 40^\circ + \dots + \text{cis } 160^\circ.\]
Express this sum in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


A triangle with side lengths in the ratio 5:12:13 is inscribed in a circle of radius 6.5. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth.


Find the remainder when $x^4 - x^3 + 1$ is divided by $x^2 - 4x + 6.$

Solution: \boxed{6x - 35}


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


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


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


If the least common multiple of two 7-digit integers has 11 digits, then their greatest common divisor has at most how many digits?


Roberto now owns five pairs of trousers. He can wear any of his seven shirts with any outfit, but two specific pairs of his trousers must be paired with either one of two specific jackets from his collection of three jackets. How many different outfits can Roberto put together under these constraints?


Consider two lines: line \( l' \) parametrized as
\[
\begin{align*}
x &= 2 + 5t,\\
y &=  3 + 4t,
\end{align*}
\]
and line \( m' \) parametrized as
\[
\begin{align*}
x &=-7  + 5s,\\
y &=  8 + 4s.
\end{align*}
\]
Let \( A' \) be a point on line \( l' \), \( B' \) be a point on line \( m' \), and let \( P' \) be the foot of the perpendicular from \( A' \) to line \( m' \).

Then \( \overrightarrow{P'A'} \) is the projection of \( \overrightarrow{B'A'} \) onto some vector \( \begin{pmatrix} v_1\\v_2\end{pmatrix} \) such that \( v_1 + v_2 = 7 \). Find \( \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \).

Solution: \boxed{\begin{pmatrix} -28 \\ 35 \end{pmatrix}}


When \( y \) is divided by each of \( 3 \), \( 7 \), and \( 8 \), remainders of \( 2 \), \( 6 \), and \( 7 \) (respectively) are obtained. What is the smallest possible positive integer value of \( y \)?


Factor the quadratic expression $16x^2 - 40x + 25$.

Solution: \boxed{(4x - 5)^2}


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


Determine the remainder when the sum $1! + 2! + 3! + \cdots + 99! + 100!$ is divided by $30$.


A grocer makes a display of cans where the top row has two cans and each lower row has three more cans than the row above it. If the display contains 225 cans, how many rows does it contain?

Solution: \boxed{n = 12}


What is the modulo $7$ remainder of the sum $1+2+3+4+ \ldots + 152+153+154+155+156?$


What is the smallest positive five-digit integer equivalent to 4 mod 9?


Suppose $a$, $b$, $c$, and $d$ are even integers satisfying: $a-b+c=8$, $b-c+d=10$, $c-d+a=4$, and $d-a+b=6$. What is the value of $a+b+c+d$?


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


A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$


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


For how many values of $c$ in the interval $[0, 2000]$ does the equation \[10 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for $x$?


What are the last two digits in the sum of the factorials of the first 15 positive integers?


Find the remainder when $x^4 + x^2 + 1$ is divided by $x^2 - 2x + 4.$

Solution: \boxed{-6x - 3}


Let $A = (2,0)$ and $B = (7, 6)$. Let $P$ be a point on the parabola $y^2 = 8x$. Find the smallest possible value of $AP + BP$.


Expand the following expression: $(6x + 8 - 3y) \cdot (4x - 5y)$.

Solution: \boxed{24x^2 - 42xy + 32x - 40y + 15y^2}


Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king?


The Cookie Monster encounters another cookie with the equation $x^2 + y^2 + 35 = 6x + 22y$. He needs help finding out the radius of this cookie to decide if it should be kept for dessert.


Let $f(x) = 4x^2-3$ and $g(f(x)) = x^2 + x + 1$. Find the sum of all possible values of $g(35)$.


A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For instance, July 19 is relatively prime since 7 and 19 have no common divisors other than 1. Identify the month with the fewest relatively prime dates when February has 29 days (leap year). How many relatively prime dates are in that month?


A circle is tangent to the lines $3x + 4y = 24$ and $3x + 4y = -6$. The center of the circle lies on the line $3x - y = 0$. Find the center of the circle.

Solution: \boxed{\left(\frac{3}{5}, \frac{9}{5}\right)}


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


What is the greatest common divisor of $7!$ and $\frac{12!}{5!}?$ Express your answer as an integer.


My five friends and I gather every weekend for a game night. Every time, three of us set up the games and events, while the remaining three handle cleaning and arranging the food. How many different ways can we choose the groups for setup and others for food and cleaning?


The Grunters play the Screamers 6 times. The Grunters are a strong team with an $80\%$ probability of winning any given game. What is the probability that the Grunters will win exactly 5 out of these 6 games? Express your answer as a common fraction.


A lattice is structured such that each row contains 7 consecutive integers. If the pattern continues and the lattice has 10 rows, what is the fifth number in the 10th row?


Eleven positive integers from a list include $5, 6, 7, 2, 4, 8, 3$. What is the largest possible value of the median of this list of eleven positive integers if one of the remaining numbers must be at least 10?


On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$.


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

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


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


The circles given by the equations \(x^2 + y^2 - 6x + 4y - 20 = 0\) and \(x^2 + y^2 - 8x + 18y + 40 = 0\) intersect at points \(C\) and \(D\). Determine the slope of \(\overline{CD}\).


Jasmine hiked the Sierra Crest Trail last week. It took five days to complete the trip. The first three days she hiked a total of 36 miles. The second and fourth days she averaged 15 miles per day. The last two days she hiked a total of 38 miles. The total hike for the first and fourth days was 32 miles. How many miles long was the trail?


Six identical rectangles are arranged to form a larger rectangle $PQRS$, where the width of each smaller rectangle is $\frac{2}{5}$ of the length of the larger rectangle. If the area of $PQRS$ is $6000$, what is the length $x$, rounded off to the nearest integer?


Find the matrix $\mathbf{X}$ such that
\[
\mathbf{X} \begin{pmatrix} 2 & -1 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix} = \mathbf{I}.
\]

Solution: \boxed{\begin{pmatrix} \frac{5}{7} & \frac{1}{7} & 0 \\ \frac{3}{7} & \frac{2}{7} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix}}


In the diagram, $ABCD$ is a trapezoid with an area of $30.$ $CD$ is three times the length of $AB.$ Determine the area of $\triangle ABC.$

[asy]
draw((0,0)--(1,4)--(9,4)--(16,0)--cycle);
draw((9,4)--(0,0));
label("$C$",(0,0),W);
label("$A$",(1,4),NW);
label("$B$",(9,4),NE);
label("$D$",(16,0),E);
[/asy]


Find the greatest common divisor of 9242, 13863, and 34657.


Define a $\textit{great word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many nine-letter great words are there?


What is the value of $n$ in the equation $n + (n + 1) + (n + 2) + (n + 3) = 20$?


Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. If $n$ is a multiple of 24, what is the smallest value of $n$ such that $g(n) > 24$?

Solution: \boxed{n = 696}


On a certain day, I worked \( t+2 \) hours and earned \( 4t-4 \) dollars per hour. My friend Bob worked \( 4t-7 \) hours but only earned \( t+3 \) dollars an hour. At the end of the day, I had earned three dollars more than Bob. What is the value of \( t \)?


Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[
\frac{(x^2 + 4x + 1)(y^2 + 4y + 1)(z^2 + 4z + 1)}{xyz}.
\]


A regular dodecagon \( Q_1 Q_2 \ldots Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (2,0) \) and \( Q_7 \) at \( (4,0) \). If \( Q_n \) is the point \( (x_n, y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\]


Tom has a red marble, a green marble, a blue marble, three identical yellow marbles, and two identical white marbles. How many different groups of two marbles can Tom choose?


Define $a * b$ as $2a - b^2$. If $a * 7 = 16$, what is the value of $a$?


A school has eight identical copies of a specific textbook. Every day, some number of these textbooks are available in the classroom and some are taken by students for homework. Find the number of ways in which this distribution can occur if at least two books must be available in the classroom and at least three books must be taken by students.


In $\triangle ABC$, the sides are such that $AC:CB = 4:5$. The bisector of the interior angle at $C$ intersects $AB$ at point $Q$. Determine the ratio $AQ:QB$.


Factor the expression $5y(y-2) + 10(y-2) - 15(y-2)$.


A sea turtle claims that the mysteriously inscribed number on an ancient coral reef is $732_{8}$. First, determine the number of years this represents in base sixteen and then convert it to base ten.


Find the sum $m + n$ where $m$ and $n$ are integers, such that the positive difference between the two roots of the quadratic equation $2x^2 - 5x - 12 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, and $m$ is not divisible by the square of any prime number.


If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$


How many integers \( n \neq 0 \) satisfy the inequality \( \frac{1}{|n|} \geq \frac{1}{5} \)?


In how many ways can the digits of $36,\!720$ be arranged to form a 5-digit number, where numbers cannot begin with $0$?


Find the sum of the distinct prime factors of $7^7 - 7^4$.


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


In a right-angled triangle, one leg is twice the length of the other leg. The sum of the squares of the three side lengths is 1450. What is the length of the hypotenuse of this triangle?

Solution: \boxed{5 \sqrt{29}}


In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Determine the sum of the $x$ and $y$ coordinates of $F$, the point of intersection of $\overline{AE}$ and $\overline{CD}$. Given:
- Point $A$ is at $(0,8)$
- Point $B$ is at $(0,0)$
- Point $C$ is at $(10,0)$ [asy]
size(200); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,8);
B=(0,0);
C=(10,0);
D=(0,4);
E=(5,0);
F=(10/3,16/3);
draw(E--A--C--D);
draw(D--B, dashed);
draw((-1,0)--(12,0), EndArrow);
draw((0,-1)--(0,9), EndArrow);
label("$A(0,8)$", A, NW);
label("$B(0,0)$", B, SW);
label("$C(10,0)$", C, SE);
label("$D$", D, W);
label("$E$", E, S);
label("$F$", F, NE);
label("$x$", (12,0), dir(0));
label("$y$", (0,9), dir(90));
[/asy]

Solution: \boxed{\frac{10}{3} + \frac{8}{3} = \frac{18}{3} = 6}


Find the range of $g(x) = \sin^4 x + \cos^4 x + 2\sin x \cos x$.

Solution: \boxed{\left[ -\frac{1}{2}, \frac{3}{2} \right]}


Given the matrix
\[\mathbf{B} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\]
compute $\mathbf{B}^{101}$.

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


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

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


Given values \(f\), \(g\), \(h\), and \(j\) correspond to numbers 3, 4, 5, and 6, but not necessarily in that order. Find the largest possible sum of the four products \(fg\), \(gh\), \(hj\), and \(fj\).


What is the smallest four-digit palindrome that is divisible by 5?


Billy chose a positive integer less than 500 that is a multiple of 30, and Bobbi selected a positive integer less than 500 that is a multiple of 45. What is the probability that they selected the same number? Express your answer as a common fraction.


Rationalize the denominator of the fraction $\frac{\sqrt{50}}{\sqrt{25} - 2\sqrt{2}}$. Write your answer in the form $\frac{A\sqrt{B} + C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Find the minimum possible value of $A+B+C+D$.


How many distinct four-digit numbers can be written with the digits $1$, $2$, $3$, $4$, and $5$ if each digit can be used only once and the number must include the digit $5$?


I have 6 shirts, 5 pairs of pants, and 6 hats. Each piece of clothing comes in six possible colors: tan, black, blue, gray, white, and yellow. I refuse to wear an outfit in which all 3 items are the same color, and I also refuse to wear pants and hats of the same color. How many outfit selections, consisting of one shirt, one pair of pants, and one hat, do I have?


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


A team is divided into two subgroups of 3 members each. In how many ways can a President and a Vice-President be chosen from this team of 6, such that they are not from the same subgroup?


The polynomial $x^4 - ax^3 + bx^2 - cx + 2520$ has four positive integer roots. What is the smallest possible value of $a$?


A rectangle $ABCD$ contains four small squares within it. Three squares are shaded and have non-overlapping interiors, as shown in the diagram. The side length of each smaller shaded square is $2$ inches. Calculate the area of rectangle $ABCD$.

[asy]
size(150);
pair A = (0,0), D = (6,0), C = (6,4), B = (0,4);
draw(A--B--C--D--cycle);
draw(A--(2,0)--(2,2)--(0,2)--cycle);
filldraw(B--(2,4)--(2,2)--(0,2)--cycle, gray(0.7), black);
filldraw((2,0)--(4,0)--(4,2)--(2,2)--cycle, gray(0.7), black);
filldraw((4,0)--(6,0)--(6,2)--(4,2)--cycle, gray(0.7), black);
label("$A$",A,WSW);
label("$B$",B,WNW);
label("$C$",C,ENE);
label("$D$",D,ESE);
[/asy]


The statue of Liberty is 305 feet tall. A scale model of the statue in a museum is 10 inches tall. How many feet of the statue does one inch of the model represent?


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 15 respectively, and the height (distance between the bases) is 6. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?

Solution: \boxed{22.5 \text{ daps}}


Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.
[asy]
/* Modified AMC8-like Problem */
draw((0,0)--(10,0)--(10,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10));
draw((0,0)--(0,-10)--(10,-10)--(10,0));
draw((0,0)--(-10,10)--(0,20)--(10,10));
draw((9,0)--(9,1)--(10,1));
label("100", (5, 5));
label("64", (15, 5));
label("100", (5, -5));
[/asy]
Assume the triangle is a right isosceles triangle.


Rosie can make three pies out of 12 apples and 6 pears. How many pies can she make if she has 36 apples and 18 pears?


In square $PQRS$, point $X$ is the midpoint of side $PQ$ and point $Y$ is the midpoint of side $QR$. If point $Z$ is the midpoint of segment $PY$, find the ratio of the area of triangle $PXZ$ to the area of square $PQRS$.


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


Determine the value of $b$ for which $\frac{1}{\log_3 b} + \frac{1}{\log_5 b} + \frac{1}{\log_6 b} = 1$.


A math teacher asks Liam to undertake a challenging homework regimen. For earning each of the first seven homework points, Liam needs to complete three homework assignments; for each of the next seven points, he needs to complete four homework assignments; and so on, such that the number of homework assignments increases by one for every new set of seven points. Calculate the smallest number of homework assignments that Liam needs to complete in order to earn exactly 40 homework points.


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


Find the modular inverse of $34$, modulo $35$.

Express your answer as an integer from $0$ to $34$, inclusive.


A matrix reflects points over a line $\ell,$ passing through the origin, represented by the following matrix:
\[\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} x \\ y \end{pmatrix},$ where $x,$ and $y$ are integers, $x > 0,$ and $\gcd(|x|,|y|) = 1.$

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


Calculate the area of a rhombus $EFGH$ where the circumradii for triangles $EFG$ and $EHG$ are $15$ and $30$ respectively.


Is there a number that is congruent to 0 modulo every positive integer less than 10?


Solution: \boxed{3x^{18} - 12x^{11} - 27x^3 + 6}


Consider the equation \[\sqrt{(x-2)^2 + (y+3)^2} - \sqrt{(x-8)^2 + (y+3)^2} = 4.\] Find the positive value for the slope of an asymptote of this hyperbola.


Let \[g(x) = \left\{
\begin{array}{cl}
\frac{x}{28} & \text{ if }x\text{ is a multiple of 4 and 7}, \\
4x & \text{ if }x\text{ is only a multiple of 7}, \\
7x & \text{ if }x\text{ is only a multiple of 4}, \\
x+4 & \text{ if }x\text{ is not a multiple of 4 or 7}.
\end{array}
\right.\]If $g^a(x)$ denotes the function nested $a$ times, find the smallest value of $a$ greater than 1 such that $g(2)=g^a(2)$.


Given that $b$ is an even multiple of $97$, find the greatest common divisor of $3b^2 + 41b + 74$ and $b + 19$.


Determine $\cos B$ in the following right triangle:

[asy]
pair A, B, C;
A = (0,0);
B = (8,0);
C = (0,6);
draw(A--B--C--A);
draw(rightanglemark(C,A,B,10));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$10$", (A+C)/2, NW);
label("$8$", (A+B)/2, S);
[/asy]


Consider the functions:
$$\begin{array}{ccc}
f(x) & = & 4x^2 - 3x + 5\\
g(x) & = & x^2 - mx - 8
\end{array}$$
If $f(5) - g(5) = 20,$ what is the value of $m$?


If $128^7 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.

Solution: \boxed{2^{-x} = \frac{1}{2^{49/4}}}


A $8\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $8\times 1$ board in which at least two different colors are used. Find the remainder when $N$ is divided by $1000$.


Three standard dice are rolled, yielding numbers $a, b, c$. Find the probability that the sum of these numbers is $a+b+c=15$.


Find the vector $\mathbf{u}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 1 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{45}{10} \\ \frac{15}{10} \end{pmatrix}\]
and
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 2 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{36}{5} \\ \frac{72}{5} \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} -\frac{6}{5} \\ \frac{93}{5} \end{pmatrix}}


Let $f(x) = \frac{4x^2 + 6x + 9}{x^2 - 2x + 5}$ and $g(x) = x + 2$. Calculate $f(g(x)) + g(f(x))$, evaluated when $x=0$.


Gage skated 1 hour each day for 6 days and 2 hours each day for 2 days. How many minutes would he have to skate the ninth day in order to average 100 minutes of skating each day for the entire time?

Solution: \boxed{300 \text{ minutes}}


Piravena must make a trip from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally from $Z$ back to $X$. The cities are arranged in a right-angled triangle, with $XZ = 4000$ km and $XY = 5000$ km. Travel costs are as follows: by bus, the cost is $\$0.20$ per kilometer; by airplane, there is a $\$150$ booking fee plus $\$0.15$ per kilometer. Determine the least expensive travel configurations and calculate the total minimum cost for the trip.


Suppose I have 8 shirts, 5 ties, 4 pairs of pants, and the choice of either wearing or not wearing one of 2 different jackets. If an outfit requires a shirt and pants, can optionally include a tie, and can also include a jacket or not, how many different outfits can I create?


Let \( x \) and \( y \) be two positive real numbers such that \( x + y = 28 \). Find the ordered pair \( (x, y) \) for which \( x^5 y^3 \) is maximized.

Solution: \boxed{(17.5, 10.5)}


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


At a school award ceremony, a class won 23 out of a possible 150 awards. Which one of the following fractions is closest to the fraction of awards that they won?
$$ \frac{1}{5} \quad \frac{1}{6} \quad \frac{1}{7} \quad \frac{1}{8} \quad \frac{1}{9} $$


Compute
\[
\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.
\]


A line is parameterized by a parameter $t$, such that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -7 \end{pmatrix}.$ Find the vector on the line at $t = -2.$

Solution: \boxed{\begin{pmatrix} -4 \\ 17 \end{pmatrix}}


What is the greatest common divisor of $12345$ and $6789$?


Add 34.789, 15.2467, and 9.0056. Then, round to the nearest hundredth.