Piravena has to travel from City $X$ to City $Y$, then from City $Y$ to City $Z$, and finally from City $Z$ to City $X$. The cities form a right-angled triangle as shown, with $Z$ a distance of $4000\text{ km}$ from $X$ and with $Y$ a distance of $4500\text{ km}$ from $X$. The transportation costs are $\$0.20$ per kilometer for taking a train, and for flying it includes a $\$150$ booking fee plus $\$0.12$ per kilometer. Piravena decides to take the train from $X$ to $Y$. Calculate the cost of the train trip from $X$ to $Y$.


Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \geq 1$. Find \[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\]


Define a new operation $x\oplus y = 2xy - 3x + y$. Calculate $(7 \oplus 4) - (4 \oplus 7)$.


A right cylindrical oil tank is $20$ feet tall and its circular bases have diameters of $6$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Express your answer as a decimal to the nearest tenth.


A set of marbles can be divided into equal shares among $4$, $5$, $7$, and $8$ children with no marbles left over. Additionally, the total number of marbles must be a perfect square. What is the least number of marbles that satisfies these conditions?


If $\cos \theta = \frac{3}{5}$, find $\cos 2\theta$ and $\sin 2\theta$.

Solution: \boxed{\cos 2\theta = -\frac{7}{25}, \sin 2\theta = \frac{24}{25}}


Multiply $704_{12}$ by $3_{10}$ and express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


Ethan derives utility according to the relation $$\text{Utility} = 2 \times (\text{hours of coding done}) \times (\text{hours of swimming}) + 1.$$ On Saturday, he swims for $t$ hours and spends $10 - 2t$ hours coding. On Sunday, he gains the same utility while spending $4 - t$ hours swimming and $2t + 2$ hours coding. Find $t$ as a fraction.


What is the largest $5$ digit integer congruent to $18 \pmod{25}$?


Define two operations $\S$ and $\T$ as follows:
- $a \S b = 4a + 6b$
- $a \T b = 5a + 3b$
Calculate the values of $6 \S 3$ and $6 \T 3$.


Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Determine the number of such ordered triples when each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find the remainder when this number is divided by $1000$.


In triangle $XYZ$, the side lengths $XY = 7$, $XZ = 24$, and $YZ = 25$. The medians from the vertices intersect at a centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find the length of $GQ$.


There are positive integers $x$ and $y$ that satisfy the system of equations
\[
\log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) = 24,
\]
\[
\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 156.
\]
Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.


In the Ardmore city parade, there are $25m$ members in a dance troupe, and when they line up in rows of 31, there are 7 members left over. If $m$ is an integer and there are fewer than 1300 members in total, what is the maximum number of people that could be in the dance troupe?


Let $a$ and $b$ be real numbers so that the roots of the quadratic equation
\[ z^2 + (6 + ai)z + (15 + bi) = 0 \]
are complex conjugates. Determine the values of $(a, b)$.

Solution: \boxed{(0, 0)}


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


Let \[g(x) =
\begin{cases}
4x^2 - 5 & \text{if } x \leq 5, \\
bx + 3 & \text{if } x > 5.
\end{cases}
\]
Find $b$ if the graph of $y=g(x)$ is continuous.


What is $5^{-1} + 5^{-2} \pmod{13}$? Express your answer as an integer from $0$ to $12$, inclusive.


Let $T = \{3^0, 3^1, 3^2, \ldots, 3^6\}$. Consider all possible positive differences of pairs of elements in $T$. Let $M$ denote the sum of all these differences. Find $M$.


How many values of $x$ with $0^\circ \le x < 360^\circ$ satisfy $\cos x = -0.6$?


If $x+y=8$ and $xy=12$, what is the value of $x^3+y^3$ and $x^2 + y^2$?


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


Given vectors $\mathbf{a}$ and $\mathbf{b}$ in the plane such that the area of the parallelogram they form is 12, determine the area of the parallelogram formed by the vectors $3\mathbf{a} - 2\mathbf{b}$ and $4\mathbf{a} + \mathbf{b}$.


What is the smallest integer \( k \) such that \( k > 1 \) and \( k \) has remainder \( 1 \) when divided by any of \( 19, 7, \) and \( 3 \)?


Calculate the value of $x$ if $\frac{1}{8}$ of $2^{50}$ equals $4^x$.


Hannah purchases a pack of cookies, a bundle of carrots, a cabbage, and a crate of oranges for $ \$ 24$. If a crate of oranges costs three times as much as a pack of cookies and the price of a cabbage is equal to the price of a pack of cookies minus a bundle of carrots, how much would it cost Hannah to purchase a bundle of carrots and a cabbage?

Solution: \boxed{\$ \frac{24}{5}}


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 1010^\circ.$


Determine all $y \in [0, \pi]$ for which \[\cos(x+y) \leq \cos(x) \cos(y)\] for every $x$ in the interval $[0, \frac{\pi}{2}]$.

Solution: \boxed{[0, \pi]}


How many of the 2401 smallest positive integers written in base 7 use 4, 5, or 6 (or any combination of these) as a digit?


The degree measures of the angles in a convex 15-sided polygon form an increasing arithmetic sequence with integer values. The largest angle is less than $172^\circ$. Find the degree measure of the smallest angle.


Triangle $DEF$ is an equilateral triangle with each interior angle measuring $60$ degrees. If $\angle D$ is decreased by $20$ degrees, what is the measure of the largest interior angle in triangle $DEF$?


The quadratic equation $ax^2 + 30x + c = 0$ has exactly one solution. If $a + c = 45$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{45 - 15\sqrt{5}}{2}, \frac{45 + 15\sqrt{5}}{2}\right)}


Simplify the following expression: $(3y-2+4)\cdot(2y^{12} + 3y^{11} - y^9 - y^8)$. Express your answer as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{6y^{13} + 13y^{12} + 6y^{11} - 3y^{10} - 5y^9 - 2y^8}


If $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 5$, find all integer values of $n$.


For each permutation $b_1, b_2, b_3, \ldots, b_8$ of the integers $1, 2, 3, \ldots, 8$, form the sum
\[|b_1 - b_2| + |b_3 - b_4| + |b_5 - b_6| + |b_7 - b_8|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Suppose a car's brakes are applied, and it travels 11 feet less in each second than the previous second until it fully stops. The car travels 44 feet in the first second after the brakes are applied. Calculate the total distance the car traverses from the time the brakes are applied until the car comes to a complete halt.


An individual invests $\$15,\!000$ in a one-year savings certificate that paid a simple annual interest rate of $5\%$. After one year, they invested the total value of their investment in another one-year certificate. After one more year, the investment was worth $\$16,\!380$. If the annual interest rate of the second certificate is $s\%,$ then what is $s?$


The sandwich shop has five types of bread, seven types of meat, and six types of cheese. A sandwich is made up of one bread, one meat, and one cheese type. Chicken, pepper bacon, Swiss cheese, and rye bread are all available options. If Carl never orders a sandwich with a chicken/Swiss cheese combination, nor a rye bread/pepper bacon combination, and additionally avoids any sandwich containing both chicken and rye bread, how many different sandwiches could Carl order?


A rectangle has a perimeter of 40 units, and one of its dimensions must be an even number. What is the maximum possible area of the rectangle, given that its dimensions are whole numbers?


Points \( A \) and \( B \) are selected on the graph of \( y = -\frac{1}{3}x^2 \) such that triangle \( ABO \) is equilateral. Find the length of one side of triangle \( ABO \).


Find the value of \( 0.\overline{4567} - 0.\overline{1234} - 0.\overline{2345} \). Express your answer as a fraction in the lowest terms.


A \(200 \times 330 \times 360\) rectangular solid is formed by gluing together \(1 \times 1 \times 1\) cubes. An internal diagonal of this solid crosses through the interiors of how many of the \(1 \times 1 \times 1\) cubes?


Let \(x = 2+\frac{\sqrt{3}}{2+\frac{\sqrt{3}}{2+...}}\). Find \(\frac{1}{(x+1)(x-3)}\). When your answer is in the form \(\frac{A+\sqrt{B}}{C}\), where \(A\), \(B\), and \(C\) are integers, and \(B\) is not divisible by the square of a prime, what is \(|A|+|B|+|C|\)?


Simplify $24 \cdot \frac{3}{4} \cdot \frac{2}{11} \cdot \frac{5}{8}$.


What is the first year after 2021 for which the sum of the digits is 15?


Compute $17^{-1} \pmod{1021}$. Express your answer as a residue from $0$ to $1020$, inclusive.


What is the remainder when 876,539 is divided by 7?


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

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


Angle $EAB$ is a right angle, and $BE = 12$ units. Additionally, a point $H$ is placed such that triangle $EAH$ is also a right triangle with $AH = 5$ units. Determine the number of square units in the sum of the areas of the three squares $ABCD$, $AEFG$, and $AHIJ$.


What is the smallest positive integer $n$ such that $623n \equiv 1319n \pmod{32}$?


If 35% of a number is the same as 25% of 40, what is the number?


Solution: \boxed{(5, 5)}


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

Solution: \boxed{\begin{pmatrix} \frac{2\sqrt{2}}{5} \\ -\frac{\sqrt{2}}{2} \\ -\frac{3\sqrt{2}}{10} \end{pmatrix}}


Let $a_n = \frac{7^n - 1}{6}$. Define $d_n$ to be the greatest common divisor (GCD) of $a_n$ and $a_{n+1}$. Determine the maximum possible value that $d_n$ can take on.


Choose the largest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{3} + \frac{1}{6}, \ \ \frac{1}{3} + \frac{1}{7}, \ \ \frac{1}{3} + \frac{1}{5}, \ \ \frac{1}{3} + \frac{1}{9}, \ \ \frac{1}{3} + \frac{1}{8}$$


What is the remainder when 783,245 is divided by 7?


Albert continues to make a list, in increasing order, of positive integers that begin with a digit of 1. He writes $1, 10, 11, 12, \ldots$ and aims to know the three-digit number formed by the 1482th, 1483th, and 1484th digits. Determine these digits.


What is the simplified value of $\frac{5}{9}\dagger\frac{4}{6}$, if the operation $\dagger$ is defined as $\frac{m}{n}\dagger\frac{p}{q} = (m^2)(p)(\frac{q}{n})$?


If \( A \diamondsuit B \) is defined as \( A \diamondsuit B = 4A + 3B + 7 \), what is the value of \( A \) when \( A \diamondsuit 5 = 71 \)?


How many 5-letter words can we create using the letters A, E, I, O, and U if we are allowed to repeat letters, and we must use the letter E at least twice?


Compute the exact value of the expression $\left|3 - |9 - \pi^2| \right|$. Write your answer using only integers, $\pi$, and necessary mathematical operations, without any absolute value signs.

Solution: \boxed{12 - \pi^2}


In a circle with center $O$, triangle $ABC$ is inscribed such that $\angle BOC = 110^\circ$ and $\angle AOB = 150^\circ$. Determine the degree measure of $\angle ABC$.


Susan loves strawberry milk and decides to visit the milk bottling plant every day for 6 days. Each day, the plant has a 3/4 chance of bottling strawberry milk. What is the probability that the bottling plant bottles strawberry milk exactly 5 of the 6 days she visits?


Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\angle BCA$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Provide your answer as a simplified fraction.


Milton has another homework paper with a similar issue where ink has covered the coefficient of $x$. He recalls that this quadratic equation also has two distinct negative, integer solutions. What is the sum of all possible integers that could be under the ink stain, given that the constant term is now 24?


Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers?


Determine the volume, in cubic centimeters, of a rectangular box where the areas of its faces are $30$ square centimeters, $18$ square centimeters, and $45$ square centimeters.


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that:
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
Given $\mathbf{A} \mathbf{B} = \begin{pmatrix} 5 & 2 \\ -2 & 4 \end{pmatrix},$
find $\mathbf{B} \mathbf{A}$.

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


Let $w$, $x$, $y$, and $z$ be real numbers with $|w-x|=3$, $|x-y|=2$, and $|y-z|=5$. Introduce an additional condition that $w \geq z$. What is the sum of all possible values of $|w-z|$?


What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 403$ is divided by $6$?


Determine all values of $m$ so that the function
\[ c(x) = \frac{3x^2 - 4x + m}{7x^2 + m - 6} \]
is defined for all real numbers.

Solution: \boxed{(6, \infty)}


If $\log (xy^2)= 1$ and $\log (x^2y) = 1$, what is $\log (xy)$?


How many integers from 1 through 999, inclusive, do not contain any of the digits 0, 2, 5, or 8?


Calculate the expression \(x^3 \cdot x^4\) if \(x = 5\).


How many distinct prime factors does the sum of the positive divisors of $540$ have?


The lengths of the sides of a triangle with positive area are $\log_{10}15$, $\log_{10}81$, and $\log_{10}m$, where $m$ is a positive integer. Find the number of possible values for $m$.


The length of the longer side of rectangle $R$ is $20$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $20$ percent less than the length of a side of square $S.$ Rectangle $R$ is cut along a diagonal to form two right triangles. What is the ratio of the area of one triangle to the area of square $S?$


A basketball is dropped from 150 feet and rebounds two-fifths of the distance it falls each time it bounces. How many feet will the basketball have traveled when it hits the ground the sixth time?


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $42$ faces, $30$ of which are triangular and $12$ of which are quadrilaterals. How many space diagonals does $Q$ have?


Two friends go for a swimming session at a local pool every day. They arrive independently at random times between 3 p.m. and 4 p.m. and each stays for exactly $n$ minutes. The probability that either friend arrives while the other is at the pool is $50 \%,$ and $n = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers, and $r$ is not divisible by the square of any prime. Find $p+q+r.$


What is the remainder when 2,368,297 is divided by 3?


A biased coin has a $\frac{2}{5}$ probability of landing heads, and a $\frac{3}{5}$ probability of landing tails. If it lands on heads, the player wins $\$4$, but if it lands on tails, the player loses $\$3$. Additionally, if the coin has landed tails two times in a row in previous flips (assuming this is at least the third flip), the loss amount doubles to $\$6$. Calculate the expected value in dollars of the third flip, given that the first two flips were tails.


Find the cross product of $\begin{pmatrix} -3 \\ 4 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 21 \\ 22 \\ -5 \end{pmatrix}}


If \( y < 0 \), find the range of all possible values of \( y \) such that \( \lceil{y}\rceil\cdot\lfloor{y}\rfloor = 72 \). Express your answer using interval notation.

Solution: \boxed{(-9, -8)}


A student, Liam, wants to earn a total of 30 homework points. For earning the first four homework points, he has to do one homework assignment each; for the next four points, he has to do two homework assignments each; and so on, such that for every subsequent set of four points, the number of assignments he needs to do increases by one. What is the smallest number of homework assignments necessary for Liam to earn all 30 points?


A stock investment increased by 30% in the first year. In the following year, it increased by an additional 10%. What percent would it have to decrease in the third year to return to its original price at the beginning of the first year?


How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor + 1} = N$ has a solution for $x$?


Circle $P$ has a circumference of $18\pi$ meters, and segment $AB$ is a diameter. If the measure of angle $PAQ$ is $60^{\circ}$, what is the length, in meters, of segment $AQ$?

Solution: \boxed{9 \text{ meters}}


What is the least integer value of $x$ such that $\lvert 3x - 5\rvert \le 22$?


Find a positive integer that is divisible by 27 and whose cube root is a number between 9 and 9.2.


Solution: \boxed{\frac{5}{2019 \cdot 2020}}


Define two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ starting with $\mathbf{v}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.$ For all $n \ge 1,$ let $\mathbf{v}_n$ be the projection of $\mathbf{w}_{n-1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}_n$ be the projection of $\mathbf{v}_n$ onto $\mathbf{w}_0.$ Find the sum $\mathbf{v}_1 + \mathbf{v}_2 + \dotsb.$

Solution: \boxed{\begin{pmatrix} 10 \\ 10 \end{pmatrix}}


Find the minimum value of
\[
\frac{x^2 + 1}{y - 2} + \frac{y^2 + 1}{x - 2}
\]
for real numbers \( x > 2 \) and \( y > 2 \).

Solution: \boxed{4\sqrt{5} + 8}


Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $162\sqrt{3}$ square feet and $288\sqrt{3}$ square feet. The two planes are $6$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


Find all values of $k$ so that the domain of
\[c(x) = \frac{kx^2 - 3x + 7}{-3x^2 - x + k}\] is the set of all real numbers.

Solution: \boxed{\left(-\infty, -\frac{1}{12}\right)}


Bob rolls a fair six-sided die each morning. If Bob rolls an even number, he eats pancakes for breakfast. If Bob rolls an odd number, he eats oatmeal for breakfast. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected value of the difference between the number of days Bob eats pancakes and the number of days Bob eats oatmeal?


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, of which $30$ are triangular, $10$ are quadrilateral, and $4$ are pentagonal. How many space diagonals does $Q$ have?


The numbers $a, b, c, d$ are equal to 1, 3, 5, 7, in some order. Find the largest possible value of
\[(a+b)(c+d) + (a+1)(d+1).\]


Let $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$


Let \( y_1, y_2, \ldots, y_n \) be real numbers which satisfy \( |y_i| < 1 \) for \( i = 1, 2, \ldots, n \), and
\[
|y_1| + |y_2| + \dots + |y_n| = 25 + |y_1 + y_2 + \dots + y_n|.
\]
What is the smallest possible value of \( n \)?


A stock investment went up by 30% in 2006. In 2007, it increased by an additional 20%. What percent would it have to decrease in 2008 to return to its original price at the beginning of 2006?


Find $AB$ in the triangle below.

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0.5,sqrt(3)/2);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18$", (A+C)/2,W);
label("$30^\circ$", (0.3,0),N);
[/asy]


An equilateral triangle has one vertex at $(3,7)$ and another at $(13,7)$. If the third vertex is located in the first quadrant, what is the y-coordinate?

Solution: \boxed{7 + 5\sqrt{3}}


When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$.


Simplify the expression $\frac{5^5 + 5^3 + 5}{5^4 - 2\cdot5^2 + 5}$.


Let \[ g(x) = \left\{
\begin{array}{cl}
-x + 2 & \text{if } x \le 1, \\
2x - 4 & \text{if } x > 1.
\end{array}
\right.\]
How many solutions does the equation $g(g(x)) = 6$ have?


David flips an unfair coin 8 times. The coin has a 1/3 probability of coming up heads and a 2/3 probability of coming up tails. What is the probability that he flips exactly 3 heads?


The first few rows of a sequence are as follows:
- Row 1: $2$
- Row 2: $4, 4$
- Row 3: $6, 6, 6$
- Row 4: $8, 8, 8, 8$

Continue this sequence pattern. What is the value of the $100^{\mathrm{th}}$ number?


For how many digits $C$ is the positive three-digit number $2C4$ a multiple of 4?


When a polynomial is divided by \(3x^2 - 4x + 9,\) what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1}


The sides of a triangle have lengths $12, 18,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?


For an arithmetic sequence $a_1, a_2, a_3, \dots,$ let
\[ S_n = a_1 + a_2 + a_3 + \dots + a_n, \]
and let
\[ T_n = S_1 + S_2 + S_3 + \dots + S_n. \]
Given the value of $S_{2023}$, determine the smallest integer $n$ for which you can uniquely determine the value of $T_n$.


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


A plot of land is represented on a map using a scale where 1 cm equals 3 miles. The land is shaped as a trapezoid with bases of 20 cm and 25 cm, and a height of 15 cm. Given that one square mile is 640 acres, calculate the actual area of the plot in acres.

Solution: \boxed{1944000 \text{ acres}}


In a pentagon $FGHIJ$, angles $F$ and $G$ are $80^\circ$ and $100^\circ$, respectively. Angles $H$ and $I$ are equal, and angle $J$ is $20^\circ$ more than twice angle $H$. Find the measure of the largest angle in the pentagon.


The point $(2,2,2)$ is rotated $180^\circ$ about the $x$-axis, then reflected through the $xz$-plane, reflected through the $yz$-plane, rotated $180^\circ$ about the $z$-axis, and reflected through the $yz$-plane again. Find the coordinates of the final point.

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


Consider a solar system where the planets have the following numbers of moons:

| Planet   | Number of Moons |
|----------|-----------------|
| Mercury  | 0               |
| Venus    | 0               |
| Earth    | 3               |
| Mars     | 4               |
| Jupiter  | 18              |
| Saturn   | 24              |
| Uranus   | 17              |
| Neptune  | 1               |
| Pluto    | 3               |

What is the median number of moons per planet? Additionally, calculate the mean number of moons and compare it to the number of moons Earth has.


A wizard is crafting a magical elixir. For this, he requires one of four magical herbs and one of six enchanted gems. However, one of the gems cannot be used with three of the herbs. Additionally, another gem can only be used if it is paired with one specific herb. How many valid combinations can the wizard use to prepare his elixir?


The lattice is expanded in a way such that each row has numbers listed in a sequence and each row contains $8$ numbers. Determine the fifth number in the $7$th row. Continue the sequence as follows:
\begin{tabular}{rccccc}
Row 1: & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 \\
Row 2: & 18 & 20 & 22 & 24 & 26 & 28 & 30 & 32 \\
Row 3: & 34 & 36 & 38 & 40 & 42 & 44 & 46 & 48 \\
Row 4: & 50 & 52 & 54 & 56 & 58 & 60 & 62 & 64
\end{tabular}


Given the following table, calculate the median salary among the 63 employees at a certain company:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$140{,}000$\\\hline
Senior Vice-President&4&$\$95{,}000$\\\hline
Manager&12&$\$80{,}000$\\\hline
Team Leader&8&$\$55{,}000$\\\hline
Office Assistant&38&$\$25{,}000$\\\hline
\end{tabular}


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal vectors with magnitudes $|\mathbf{a}| = 2$, $|\mathbf{b}| = 3$, and $|\mathbf{c}| = 4$, such that
\[\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a})\]for some scalars $p,$ $q,$ and $r,$ and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 24.$  Find $p + q + r.$


Find the area of the region bounded by the graph of $r = 2\sec \theta$, the graph of $r = 3\csc \theta$, the x-axis, and the y-axis.


Evaluate \(99 \times 105\) in your head.


After Jack takes 30 basketball shots, he has made $60\%$ of them. Following 10 more shots, he raises his shooting percentage to $62\%$. How many of the last 10 shots did he make?


Determine the sum of the first six terms in the geometric sequence $\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots$.


Determine the remainder when $7^{7^{7^7}}$ is divided by 2000.


Find a unit vector that is orthogonal to both $\begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{13\sqrt{3}}{27} \\ \frac{-7\sqrt{3}}{27} \\ \frac{-5\sqrt{3}}{27} \end{pmatrix}}


The decimal expansion of $7/15$ is a repeating decimal. What is the least number of digits in a repeating block of $7/15$?


Calculate the number of terms in the simplified expression of \[(x+y+z)^{2008} + (x-y-z)^{2008}.\]


The product of three different positive integers is equal to \(5^4\). What is the sum of these integers?


Six positive integers from a list of nine positive integers are \(4, 5, 3, 7, 9, 6\). What is the largest possible value of the median of this list of nine positive integers?


Let $M$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that satisfy:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12,13,14\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Calculate $M$.


If \( \begin{vmatrix} p & q \\ r & s \end{vmatrix} = 10 \), then find
\[ \begin{vmatrix} p + 2r & q + 2s \\ r & s \end{vmatrix}. \]


Given the points (4, 12), (7, 26), (13, 30), (17, 45), and (22, 52), determine the sum of the x-coordinates of the points that lie above the line \(y = 3x + 5\) in the coordinate plane.


A triangle in a Cartesian coordinate plane has vertices at points (3, -3), (3, 4), and (8, -3). What is the area of the triangle?


What is the sum of all positive integer values of $n$ such that $\frac{24}{2n - 1}$ is an integer?


After discovering a new group of planets, scientists identify 12 habitable planets. Out of these 12, 6 are considered "Earth-like" (requiring 3 units of colonization each due to their complexity and resource abundance) and the remaining 6 are "Mars-like" (requiring 1 unit of colonization each as they are less resourceful). Given that humanity can mobilize 15 units of colonization in total, how many different configurations of planets can be occupied, considering that all planets are distinct?


Simplify the expression: $$\dfrac{\sqrt{450}}{\sqrt{288}} + \dfrac{\sqrt{245}}{\sqrt{96}}.$$ Express your answer as a common fraction.

Solution: \boxed{\frac{30 + 7\sqrt{30}}{24}}


Determine the number of integer solutions $(x, y)$ to the equation $2^{2x} - 3^{2y} = 85$.


If $\lceil{\sqrt{x}}\rceil=18$, how many possible integer values of $x$ are there?


How many 5-letter words with at least two vowels can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels, and each letter can be used more than once.)


A cylinder has a radius of 5 cm and a height of 10 cm. Determine the length of the longest segment that fits inside the cylinder.


Given that $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 3$, find the value of \[\begin{vmatrix} x & 5x + 4y \\ z & 5z + 4w \end{vmatrix}.\]


What is the smallest integer value of $b$ such that $-6$ is not in the range of $y=x^2+bx+10$?


What is the greatest common factor of all three-digit palindromes where each digit is the same? (Examples of such palindromes include 111, 222, ..., 999.)


Let $P_1$ be a $150$-gon whose vertices are placed in the Cartesian plane. The sum of the x-coordinates of the vertices of $P_1$ is $3024$. Using midpoints of consecutive vertices, we construct a new $150$-gon $P_2$, and similarly, by using midpoints of $P_2$, we construct another $150$-gon $P_3$. Find the sum of the x-coordinates of the vertices of $P_3$.


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

Solution: \boxed{4 + 2\sqrt{3}}


In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 3\angle D$, and $\angle C = 4\angle B$. Find $\angle B$.


If two numbers are randomly chosen without replacement from $\{2, 3, 6, 9\}$, what is the probability that their product will be a multiple of 12? Express your answer as a common fraction.


Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make?


How many odd positive factors of 144 are also multiples of 18?


In a triangle with integer side lengths, one side is twice as long as another side, and the length of the third side is 18. What is the greatest possible perimeter of the triangle?


Given points $D(2, 3)$, $J(3, 7)$, and $H(7, 3)$ in a coordinate system forming triangle $DJH$, the triangle is translated three units to the right and one unit downward to form triangle $D'J'H'$. Determine the midpoint of segment $D'H'$ in $D'J'H'$ and express your answer as an ordered pair.

Solution: \boxed{(7.5, 2)}


Let $x = (3 + \sqrt{8})^{30},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[
x(1 - f).
\]


Calculate the sum $$1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 + 88888888 + 999999999$$ and determine what this total is congruent to modulo 9.


Alice's favorite number is between $30$ and $150$. It is a multiple of $11$, but not a multiple of $2$. The sum of its digits is a multiple of $5$. What is Alice's favorite number?


Start summing consecutive prime numbers beginning with $3$, and continue adding the next prime to each previous sum. Determine how many of the first 10 such sums are prime.


One of the asymptotes of a hyperbola has equation $y = -2x + 4$. The foci of the hyperbola have the same $x-$coordinate, which is $-3$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = 2x + 16}


Let $P$ be the parabola with equation $y = x^2 + 4x + 4$, and let $Q = (10, 24)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s.$ What is $r + s$?


The Euler family consists of twin boys aged $8$, a girl aged $12$, twin girls aged $10$, and a boy aged $16$. What is the mean (average) of the ages of the children?


Find the number of positive integers $n,$ $1 \le n \le 2000,$ for which the polynomial $x^2 + nx - c$ can be factored as the product of two linear factors with integer coefficients, where $c$ is the smallest possible positive integer such that factorization is possible.


How many ways are there to choose 3 cards from a standard deck of 52 cards, if the cards include exactly two suits? (Assume that the order of the cards does not matter.)


A box contains 7 white balls and 5 black balls. Six balls are drawn out of the box at random. What is the probability that at least four of them are white?


In a storage facility, there are $1278$ boxes, each containing $45$ books.

Alice's supervisor orders her to reorganize all the books so that there are $46$ books in each box. After packing as many such boxes as possible, Alice has how many books left over?


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


How many whole integers are between $\sqrt[3]{30}$ and $\sqrt[3]{1000}$?


What is the remainder when $5x^8 - 3x^7 + 4x^6 - 9x^4 + 3x^3 - 5x^2 + 8$ is divided by $3x - 6$?


A biologist wishes to estimate the fish population in a protected area. Initially, on March 1st, she captures and tags 80 fish, then releases them back into the water. Four months later, on July 1st, she captures another 90 fish for a follow-up study, finding that 4 of these are tagged. For her estimation, she assumes that 30% of these fish have left the area by July 1st due to various environmental factors, and that an additional 50% of the fish in the July sample weren't in the area on March 1st due to new arrivals. How many fish does she estimate were in the area on March 1st?


A rectangular room measures 15 feet by 9 feet. How many square yards of carpet, allowing for a 1 foot overlap on each side for fitting, are needed to cover the floor of the room?

Solution: \boxed{21\text{ square yards}}


The parabola $y = ax^2 + bx + c$ has vertex $(p, -p)$ and $y$-intercept $(0, p)$, where $p \neq 0$. What is the value of $b$?


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that:
\[f(x^2 + 2f(y)) = xf(x) + yf(z)\]
for all real numbers $x,$ $y,$ and $z.$

Find the number $n$ of possible functions $f$, and determine the sum $s$ of $f(3)$ among all possible functions. Calculate $n \times s.$


Lena selects two different numbers at random from the set $\{7,8,9\}$ and multiplies them. Jonah takes three different numbers at random from the set $\{2,4,6\}$ and takes the sum of the first two numbers then multiplies by the third. What is the probability that Lena's result is greater than Jonah's result? Express your answer as a common fraction.


Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$ respectively. Find $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$.


Solution: \boxed{\frac{-13x + 26}{12}}


Five concentric circles are drawn with radii of 1, 4, 6, 8, and 10. The innermost circle is painted black, the next ring is white, the following ring is black, the next ring is white, and the outermost ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


Consider a "Modulo $m$ graph paper," with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \leq x, y < 17$. We graph the congruence $$5x \equiv 3y + 2 \pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < 17$. Determine $x_0 + y_0$.


Compute the exact value of the expression
\[\left| |\pi - |\pi - 10| | - |\pi - 12|\right|.\]
Write your answer using only integers and $\pi$, without any absolute value signs.

Solution: \boxed{2 + \pi}


How many positive four-digit integers are divisible by both 13 and 7?


In a modified bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize and the winner bowls #2. The loser of this game receives $3^{rd}$ prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes?


How many perfect square factors are there for the number 450?


There are approximately 0.9072 kilograms in 2 pounds. To the nearest whole pound, how many pounds does an object that weighs 500 kg weigh?

Solution: \boxed{1102\text{ pounds}}


The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = 2x, \quad g(x,y) = g(y,x), \quad \text{and} \quad (x + y)g(x,y) = yg(x, x + y).
\]
Calculate $g(18, 45)$.


Find the focus of the parabola \(y = -4x^2 - 8x + 1.\)

Solution: \boxed{(-1, \frac{79}{16})}


Consider the ellipse \[25(x+3)^2 + 4y^2 = 100.\] Let $C$ be one of the endpoints of its major axis, and let $D$ be one of the endpoints of its minor axis. Find the distance $CD$.


The mean of $7, 9, 14$, and $23$ is equal to the mean of $18$ and $y$. What is the value of $y$?


Triangles $PIQ$ and $PQR$ are right triangles in the quadrilateral $PIQR$ with $IP = 16$ units, $PQ = 20$ units, and $PR = 25$ units. Calculate the area of quadrilateral $PIQR$.

Solution: \boxed{246 \text{ square units}}


The eight horizontal and eight vertical lines on a $7 \times 7$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


The function $g(x)$ satisfies
\[ 4g(x) - 3g\left(\frac{1}{x}\right) = x^2 - 2 \]
for all $x \neq 0$. Find $g(2)$.


Write the expression $\frac{5 - 4c}{9} - 3$ as a single fraction.

Solution: \boxed{\frac{-22 - 4c}{9}}


Find $x$, given that $x$ is nonzero and the numbers $\{x\}$, $\lfloor x \rfloor$, and $x$ form a geometric sequence in that order.

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


Simplify the expression $i^0 + i^1 + \cdots + i^{2013}$.

Solution: \boxed{1 + i}


A circular floor pattern includes a central point from which 12 equally spaced rays emerge. One ray points due North. Determine the measure in degrees of the smaller angle formed between the ray pointing due North and the ray pointing South-Southeast.


Find $\tan \left( \frac{7 \pi}{4} \right)$.


When three positive integers are divided by $30$, the remainders are $11$, $5$, and $14$, respectively.

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


Suppose $\{a_n\}$ is an arithmetic sequence where $$a_1 + a_2 + \cdots + a_{50} = 150 \quad \text{and} \quad a_{51} + a_{52} + \cdots + a_{100} = 250.$$ What is the value of $a_{52} - a_{51}$?


Two circles intersect at points $C$ and $D$. One circle is centered at $(3, -2)$ and has a radius of $5$, and the other circle is centered at $(3, 4)$ with a radius of $3$. Find $(CD)^2$.


A rubber ball is released from a height of 120 feet and rebounds to three-quarters of the height it falls each time it bounces. How far has the ball traveled when it strikes the ground for the fifth time?


Determine how many ordered 11-tuples $(a_1, \dots, a_{11})$ of integers exist such that the cube of each number in the tuple equals the sum of the other 10 numbers in the 11-tuple.


In triangle $ABC$ with circumcenter $O$, we know that $\angle BOC = 100^\circ$ and $\angle AOB = 160^\circ$. Determine the degree measure of $\angle ABC$.


Suppose $507\cdot 873 \equiv n \pmod{77}$, where $0 \le n < 77$.

What is the value of $n$?


Find the sum of the coefficients in the polynomial \(3(x^8 - 2x^5 + 4x^3 - 6) - 5(x^4 - 3x + 7) + 2(x^6 - 5)\).


Find the sum of the coefficients in the polynomial $-3(x^8 - 2x^5 + 4x^3 - 6) + 5(2x^4 + 3x^2 - x) - 2(3x^6 - 7)$.


How many ways are there to put 7 balls in 3 boxes if the balls are distinguishable but the boxes are not?


Suppose $h(x) = \sqrt[4]{\frac{x+5}{5}}$. Determine the value of $x$ for which $h(3x) = 3(h(x))$.


Find all real numbers $x$ such that:
\[
4 \leq \frac{x}{2x - 7} < 10.
\]
(Give your answer in interval notation.)

Solution: \boxed{(\frac{70}{19}, 4]}


What is the sum of the squares of the coefficients of $5(x^5 + 4x^3 + 2)$?


How many ways are there to put 4 distinguishable balls into 4 distinguishable boxes if one particular ball cannot be placed in one specific box?


How many numbers can you get by multiplying any two or more distinct members of the set $\{2, 3, 5, 7, 13\}$ while ensuring the product is not divisible by 10?


A right triangle with legs \(12\) and \(9\) is similar to another right triangle with corresponding legs \(x\) and \(6\), respectively. What is the value of \(x\) and what is the length of the hypotenuse of the second triangle?

Solution: \boxed{x = 8, \quad \text{hypotenuse} = 10}


The mean (average) of 8, 14, and 24 is equal to the mean (average) of 16 and $z$. What is the value of $z$?


If $y < 0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor = 210$. Express your answer using interval notation.

Solution: \boxed{(-15, -14)}


Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 63 \times 73 \times 83 \times 93$ is divided by $6$.


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


Let $d$ be a positive number such that when $157$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$.


Contractor Steve agreed to complete a job in 40 days. After 10 days he found that the 10 people assigned to the work had already done $\frac{1}{4}$ of the job. If everyone works at the same rate, what is the least number of people he must keep on the job to ensure that the job will be completed on time?


Find the domain of the function:
\[ g(x) = \sqrt{4 - \sqrt{6 - \sqrt{7 - x}}}.\]

Solution: \boxed{[-29, 7]}


Of the numbers 1, 2, 3, ..., 20, which number has the greatest number of divisors and what is the sum of these divisors?


In a triangle, two of the side lengths are 9 and 10, and the angle between them is \(135^\circ\). Find the length of the third side.

Solution: \boxed{\sqrt{181 + 90\sqrt{2}}}


A park has shaped an equilateral triangle flowerbed, which has an area of $36\sqrt{3}$ square meters. The park plans to reduce each side of the flowerbed by 3 meters. Calculate how many square meters the area of the flowerbed will decrease.


If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row contains no more than one pawn?


In 2003, the average monthly rainfall in Clouddale was $37.5\text{ mm}.$ Each year, the average monthly rainfall increases by $3\text{ mm}$ more than the previous year's average. What was the total amount of rain that fell in Clouddale in 2005?

Solution: \boxed{522\text{ mm}}


Yan is at a point between his house and a park. He has two options to reach the park: He can either walk directly to the park or he can walk back to his house and use his scooter to reach the park. He scoots 10 times faster than he walks, and both routes take the same amount of time. Determine the ratio of Yan's distance from his house to his distance to the park.


A triangle has vertices at coordinates (1,2), (4,7) and (7,2). What is the number of units in the length of the longest side of the triangle?


In how many ways can the digits of $405,\!220$ be arranged to form a 6-digit number where numbers cannot begin with 0?


The ratio of sunflowers to tulips in Carl's garden is 3:4. He currently has 28 tulips. He plans to add 16 more tulips and wants to maintain the same ratio by adding enough sunflowers. How many sunflowers will Carl need in total after his addition?


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


Let $g(x) = \frac{1}{2ax + b}$ where $a$ and $b$ are nonzero constants. Find $x$ for which $g^{-1}(x) = 2$.

Solution: \boxed{\frac{1}{4a + b}}


Compute the sum:
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]


Suppose there are two congruent triangles $\triangle ABC$ and $\triangle ABD$ such that $AB = AC = AD$, and $\angle BAC = 30^\circ$. Find the measure of $\angle BDA$.


In a right triangle, find $\cos A$ if side $AB = 3$ and hypotenuse $AC = 5$.


Say that an integer $B$ is yummy if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest yummy integer?


Let \( a\), \( b\), \( c\), \( x\), \( y\), and \( z\) be real numbers that satisfy the three equations:
\[
\begin{align*}
  17x +  by +  cz &= 0, \\
   ax + 29y +  cz &= 0, \\
   ax +  by + 53z &= 0.
\end{align*}
\]
Assuming \( a \ne 17 \) and \( x \ne 0 \), find the value of
\[
\frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53}.
\]


In right triangle $PQR$ where $\angle Q = 90^\circ$, it is known that $3\tan P = 4$. What is $\sin P$?


How many integers $n$ satisfy the condition $200 < n < 300$ and the condition that $n$ has the same remainder whether it is divided by $7$ or by $9$?


What is the product of the three largest one-digit primes and the second largest two-digit prime?


A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up?


Consider a regular octagon where each side is 3 units long except for three consecutive sides which are formed by halving their original lengths. Calculate the perimeter of this modified octagon.


In triangle $XYZ$, the sides are tangent to a circle with center $O$. If $\angle XYZ = 75^\circ$ and $\angle YZO = 40^\circ$, find $\angle YXZ$.


A comet follows an elliptical orbit around a star with the star situated at one focus. At its closest approach (perihelion), the comet is 3 astronomical units (AU) from the star, and at its farthest point (aphelion), it is 15 AU away. Calculate the distance of the comet from the star when it is exactly halfway along its orbit.


What is the sum of the different prime factors of $204204?$


A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students sit in the last row. When the students try to sit in rows of 6, 3 students are left in the last row. How many students are in this class?


A right triangle with integer leg lengths is termed "unique'' if the area in square units is three times the sum of the lengths of its legs. Determine the sum of all different possible areas of such unique right triangles.


Calculate $7\cdot7! + 6\cdot6! + 5\cdot5! + 5!$.


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


Compute the distance between the vertices of the parabolas represented by the graph of the equation \[\sqrt{x^2+y^2} + |y-2| = 5.\]


The product of the digits of 1423 is 24. Find how many distinct four-digit positive integers have a product of their digits equal to 18.


Alice is comparing two internet data plans. Plan X has no initial fee and charges $15$ cents per MB of data. Plan Y has an initial fee of $25$ and thereafter charges $10$ cents per MB of data. What is the minimum whole number of MBs Alice must use for Plan Y to become the cheaper option?


Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the sixth row is 30. What is the sum of the interior numbers of the eighth row?


A right triangle has sides of 8, 15, and 17. Find the distance between the incenter and the circumcenter of this triangle.


In $\triangle ABC$, $AB=AC=24$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?


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


A sphere is cut into eight congruent wedges. The circumference of the sphere is $16\pi$ inches. What is the volume of one wedge in cubic inches? Express your answer in terms of $\pi$.


Solution: \boxed{-1 + 1 = 0}


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


Let \( A, B, C \) be the angles of a triangle where \( A = 45^\circ \) and \( A + B + C = 180^\circ \). Compute:
\[
\begin{vmatrix}
\tan A & 1 & 1 \\
1 & \tan B & 1 \\
1 & 1 & \tan C
\end{vmatrix}.
\]


Calculate the sum of $45_{10}$ and $52_{10}$ in base 4.


Henry is planning to build a fence around a rectangular playground. He has exactly 400 feet of fencing material. He needs to ensure the length of the fence enclosure is at least 100 feet and the width at least 50 feet. Henry wants the maximum possible area for play equipment and benches. What is the maximum possible area, in square feet?


Let $m$ be the smallest positive integer that is a multiple of $45$ and has exactly $45$ positive integral divisors, including $1$ and itself. Find $\frac{m}{45}$.


A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 12 units and an altitude of 15 units. The cylinder's height is three times its radius. The axes of the cylinder and cone coincide. Determine the radius of the cylinder. Express your answer as a common fraction.


For real numbers $t,$ the point of intersection of the lines $2x + 3y = 9t + 4$ and $3x - 2y = 5t + 3$ is plotted. All the plotted points lie on a line. Find the slope of this line.


Solution: \boxed{7x^{12} + 2x^{10} + 9x^9 + 2x^8 + x^7 + 6x^4 + 9}


Suppose \( b \) and \( n \) are positive integers with \( b \leq 20 \) and \( n \leq 20 \). What is the greatest number of positive factors \( b^n \) can have?


In the diagram below, $AB = 80$. Find the length of $AX$.

[asy]
import markers;

real t = 0.75;
pair A = (0,0);
pair B = (3, -2);
pair C = (1.5, 1.5);
pair X = t*A + (1-t)*B;

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

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

markangle(n=1, radius=15, A, C, X, marker(markinterval(stickframe(n=1), true)));
markangle(n=1, radius=15, X, C, B, marker(markinterval(stickframe(n=1), true)));

label("$72$", .5*(B+C), E);
label("$36$", .5*(A+C), NW);
[/asy]


Mr. and Mrs. Lopez now have three children, making a family of five. They have a larger family car with two seats in the front and three seats in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?


A list of five positive integers has all of the following properties:

$\bullet$ The only integer in the list that occurs more than once is $6,$

$\bullet$ its median is $7,$ and

$\bullet$ its average (mean) is $11.$

What is the largest possible integer that could appear in the list?


Using only the digits 0, 7, 8, and 9, how many positive nine-digit integers can be made that are palindromes?


A $6$-foot by $9$-foot rectangular piece of plywood will be cut into $3$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?


Let $\mathbf{N}$ be a matrix such that
\[\mathbf{N} \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} -2 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.\]Compute $\mathbf{N} \begin{pmatrix} 3 \\ 11 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 7.4 \\ 17.2 \end{pmatrix}}


Allison, Brian, and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 6. The faces on Brian's cube are numbered 2, 2, 4, 4, 6, and 6. Three of the faces on Noah's cube have a 4 and three of the faces have an 8. All three cubes are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction.


A meeting includes ten employees consisting of six men and four women. However, one man and one woman are not available for selection. If three people are selected at random from the available employees, what is the probability that at least one woman is selected? Express your answer as a common fraction.


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


Alice buys three burgers, two sodas, and one fry for \$4.20, and Bill buys two burgers, one soda, and two fries for \$3.40. Determine the cost of one fry if a burger costs 80 cents and a soda costs 60 cents.


If $A65B2C1$ is divisible by $9,$ where $A$, $B$, and $C$ each represent a single digit, what is the sum of all possible values of the sum $A+B+C?$


Calculate $\frac{3}{4} \cdot \frac{5}{9} \cdot \frac{8}{13} \cdot \frac{3}{7}$.


How many digits does the smallest repeating block in the decimal expansion of $\frac{11}{13}$ contain?


The lengths of the sides of a triangle with positive area are \( \log_2 20 \), \( \log_2 90 \), and \( \log_2 m \), where \( m \) is a positive integer. Find the number of possible values for \( m \).


If the price of a notebook is 37 cents, what is the maximum number of notebooks that can be purchased with \$58?


A pentagon has consecutive angles that form an arithmetic sequence. If the largest angle measures $150^\circ$, what is the measure of the smallest angle?


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 12, determine the area of $\triangle ABC$.


There are three real numbers $x$ that are not in the domain of $$ f(x) = \frac{1}{1+\frac{1}{1+\frac{2}{x}}}.$$ What is the sum of those three numbers?


Let $\mathcal P$ be the parabola given by $y = x^2 + 2$, and let $V_1$ and $F_1$ be its vertex and focus, respectively. Let $A = (a, a^2 + 2)$ and $B = (b, b^2 + 2)$ be points on $\mathcal P$ such that $\angle AV_1B = 90^\circ$ and $ab = -1$. Let $\mathcal Q$ be the locus of the midpoint of $\overline{AB}$. Determine the coordinates of $V_1$, $F_1$, $V_2$, and $F_2$ and calculate the ratio $\frac{F_1F_2}{V_1V_2}$.


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0), (2010,0), (2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$?


Find the smallest positive integer that is an integer power of 13 and is not a palindrome.


How many ways are there to distribute 5 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?


Solution: \boxed{\log_{13} 339}


The graph of the parabola defined by the equation \( y = (x-2)^2 + 3 \) is rotated 180 degrees about its vertex, then shifted 4 units to the left, and then shifted 3 units down. Find the sum of the zeros of the resulting parabola.


What is the units digit of the sum of the squares of the first 2025 odd positive integers?


In a new race setup, all runners must start at point A, touch any part of the 1500-meter wall, and stop at point B. Point A is 400 meters vertically from the wall, and point B is 600 meters above the wall and 150 meters horizontally to the right of point A. Determine the minimum distance a participant must run, rounded to the nearest meter.

Solution: \boxed{1011 \text{ meters}}


How many of the integers between 1 and 1200, inclusive, can be expressed as the difference of the squares of two nonnegative integers?


A student must complete homework assignments to earn points. For each of the first four points, one assignment is required per point. For the next four points, two assignments are needed per point, and this pattern continues, increasing the number of assignments per point by one for every new set of four points. How many assignments are required in total to earn 20 points?


What is the remainder when $4x^8 - 2x^6 + 5x^4 - x^3 + 3x - 15$ is divided by $2x-6$?


For an upcoming weekend, the weather forecast predicts that the probability of rain on Saturday is $30\%$. If it rains on Saturday, the probability of rain on Sunday increases to $70\%$, otherwise, it remains at $60\%$. What is the probability that it rains on at least one day during the weekend? Express your answer as a percentage.


In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?


Find the maximum number of digits the greatest common divisor (GCD) of two 7-digit integers can have if their least common multiple (LCM) is a 12-digit number.


In the diagram, the area of triangle $ABC$ is 36 square units. The length of segment $AC$ is 9 units and segment $CD$ is 33 units. What is the area of triangle $BCD$?


Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.\]
Find the smallest possible value of $|a| + |b| + |c| + |d|$.


Sue owns 15 pairs of shoes: eight identical black pairs, four identical brown pairs, and three identical gray pairs. If she picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


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

Let $p$ be the number of possible values of $f(3),$ and let $t$ be the sum of all possible values of $f(3)$. Find $p \times t$.


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


Define $g(x) = 7x - 2$. If $g(x) = f^{-1}(x) - 1$ and $f^{-1}(x)$ is the inverse of the function $f(x) = ax + b$, find $7a + 3b$.


A circular cylindrical post with a circumference of 6 feet has a string wrapped around it, spiraling from the bottom to the top. The string makes three full loops, starting at the bottom and finishing at the top, over a height of 15 feet. What is the length, in feet, of the string?

Solution: \boxed{3\sqrt{61} \text{ feet}}


Express the sum as a common fraction: $.2 + .04 + .006 + .0008 + .00010.$


At a point between his home and the stadium, Yan faces two choices to reach the stadium: he can either walk directly to the stadium, or walk back home first and then take his bicycle to the stadium. His bicycle speed is 10 times his walking speed. If both options take the same total time, what is the ratio of Yan’s distance from his home to his distance from the stadium?


What is the least positive integer $n$ such that $2200$ is a factor of $n!$?


The first three stages of a toothpick pattern are shown below. In each stage, the number of toothpicks used to create the next arrangement increases by four from the previous stage. If this pattern continues, how many toothpicks are necessary to create the arrangement for the 250th stage? Assume the first stage starts with 5 toothpicks.


Consider the matrix $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ modified by adding scalar $k$ times the identity matrix to it. Given that $(\mathbf{A} + k\mathbf{I})^T = (\mathbf{A} + k\mathbf{I})^{-1}$, with $\mathbf{I}$ being the identity matrix, find $a^2 + b^2 + c^2 + d^2$ assuming $k = 1$ and $\mathbf{A}$ is orthogonal.


Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}.$


Marie baked 30 pies; one-third of them contained blueberries, three-fifths contained raspberries, five-sixths contained blackberries, and one-tenth contained walnuts. What is the largest possible number of pies that had none of these ingredients?


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 26 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. What is the area of quadrilateral $EFCD$ in square units?


Let $\mathcal{S}$ be the set $\{1, 2, 3, \ldots, 12\}$. Determine the number of sets containing two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder when this number is divided by $1000$.


What is the smallest integer \( n \), greater than \( 2 \), such that \( n^{-1} \mod{735} \) is defined?


Determine the smallest positive integer $n$ such that $7^n \equiv n^7 \pmod{5}$.


Find the integer $n$, $6 \le n \le 12$, such that \[n \equiv 10\,403 \pmod{7}.\]


Consider the equation \[\frac{(x-3)^2}{49} + \frac{(y-4)^2}{25} = 1.\] Determine the largest possible $y$-coordinate for any point on this ellipse.


Let $m$ and $n$ be any two even numbers, with $n$ less than $m$. Find the largest integer which divides all possible numbers of the form $m^2 - n^2$.


In triangle $ABC$, $AB=AC$. Point $D$ is on $\overline{AC}$ such that $\overline{BD}$ bisects angle $ABC$. Point $E$ is on $\overline{AB}$ such that $\overline{CE}$ bisects angle $ACB$. If $BD=BC$ and $CE=CB$, find the measure, in degrees, of angle $A$.


The three-digit positive integer $M$ has a ones digit of 5. What is the probability that $M$ is divisible by 3? Express your answer as a common fraction.


James is nine years older than Louise. Five years from now, James will be three times as old as Louise was three years ago. What is the sum of their current ages?


Let $x, y, z$, and $w$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
x^2+y^2&=&z^2+w^2&=&2500,\\
xz&=&yw&=&1200.
\end{array}
\]
If $T = x + y + z + w$, compute the value of $\lfloor T \rfloor$.


Determine the range of the function
\[g(x) = \frac{3(x + 5)(x - 4)}{x + 5}.\]

Solution: \boxed{(-\infty,-27) \cup (-27,\infty)}


Given the function $t(x)$ with a domain $\{0, 1, 2, 3\}$ and a range $\{1, 3, 5, 7\}$. Another function $u(x)$ has a domain $\{2, 3, 4, 5\}$ and is defined by $u(x) = x - 1$. Compute the sum of all possible values of $u(t(x))$.


In the diagram, $BP$ and $BQ$ trisect $\angle ABC$. $BM$ bisects $\angle ABP$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$.


Points \((2, -3)\) and \((14, 9)\) are the opposite vertices of a parallelogram. A third vertex of the parallelogram is located at \((5, 7)\). What are the coordinates of the point where the diagonals of the parallelogram intersect?

Solution: \boxed{(8, 3)}


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$, meaning that if the length of the object was originally $x$, then it is now $x + \frac{1}{3} x$. On the second day he increased the object's longer length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day of performing this trick, Barry will make the object's length exactly 50 times its original length. What is the value of $n$?


What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 5?


What is the smallest positive integer $n$ such that $3300$ is a factor of $n!$?


In triangle $XYZ$, points $M$ and $N$ are on $\overline{XY}$ and $\overline{XZ}$, respectively, and angle bisector $\overline{XU}$ intersects $\overline{MN}$ at $P$. If $XM = 2$, $MY = 6$, $XN = 3$, and $NZ = 9$, compute $\frac{XP}{XU}$.


You are given $4$ red shirts, $4$ green shirts, $4$ blue shirts, $10$ pairs of pants, $6$ red hats, $6$ green hats, and $4$ blue hats, all distinct. How many different outfits can you make consisting of one shirt, one pair of pants, and one hat such that the hat doesn't match the shirt in color?


The sequence consists of all those positive integers which are either powers of 4 or sums of distinct powers of 4. Find the $50^{\text{th}}$ term of this sequence.


A sports conference has 16 teams split into two divisions of 8. Each team must play every other team in its own division three times and every team in the other division twice. How many games are there in a complete season for this conference?


The line \(y = c - x\) with \(0 < c < 6\) intersects the \(y\)-axis at point \(P\) and the line \(x = 6\) at point \(S\). If the ratio of the area of triangle \(QRS\) to the area of triangle \(QOP\) is 4:16, what is the value of \(c\)? Express the answer as a decimal to the nearest tenth.


The sum of 64 consecutive integers is $2^{12}$. What is their median?


Determine the sum of the tens digit and the units digit in the decimal representation of $(11^{2004} - 5)$.


In the diagram below, $ABCD$ is a square piece of paper with a side length of 8 cm. Corner $C$ is folded over so that it coincides with $E$, which is one-third of the way along $\overline{AD}$ from $D$. If $\overline{GF}$ represents the crease created by the fold where $F$ is on $\overline{CD}$, what is the length of $\overline{FD}$?


Define a $\textit{strong word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$, where:
- $A$ is never immediately followed by $B$ or $C$,
- $B$ is never immediately followed by $A$ or $C$,
- $C$ is never immediately followed by $A$ or $B$.
How many eight-letter $\textit{strong words}$ are there?


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


Define a function $g(z) = (3 + i)z^2 + \alpha z + \gamma$ for all complex $z$, where $\alpha$ and $\gamma$ are complex numbers. Assume that $g(1)$ and $g(i)$ both yield real numbers. Determine the smallest possible value of $|\alpha| + |\gamma|$.


For his birthday, Kevin receives a box that holds $216$ marbles when it is full. The box has dimensions are $3 \times 3 \times 8$. Two months later, Laura receives a larger box of marbles which is three times as high, three times as wide, and three times as long as Kevin’s box. How many marbles does Laura get if the boxes are filled to capacity?


Automobile license plates for a state consist of four letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly one letter is repeated exactly once, and digits cannot be repeated?


A map shows the distance from Stockholm to Uppsala as 45 cm. If the scale of the map is 1 cm : 10 km, and for the first 15 cm, the scale changes to 1 cm : 5 km, how far is it between the two city centers in kilometers?

Solution: \boxed{375 \text{ km}}


Genevieve wants to further promote her school's annual Kite Olympics by making yet another large kite. This time, she designs it by quintupling both the height and width dimensions of the original small kite on a grid where points are one inch apart. Specifically, the quadrilateral extends for a base length of 35 inches (quintupling the original 7 inches) and a height of 15 inches from the mid-point (quintupling the original 3 inches). What is the area in square inches of this new large kite?


Given $J_m = 10\ldots 032$, where there are $m$ zeros between the $1$ and the $3$, determine the maximum value of $M(m)$, where $M(m)$ is the number of factors of $2$ in the prime factorization of $J_m$.


The value $b^n$ has both $b$ and $n$ as positive integers. If $b$ is less than or equal to 15, and $n$ is less than or equal to 20, what is the greatest number of positive factors $b^n$ can have?


Let $a,$ $b,$ $c$ be real numbers such that $a + b + 2c = 0.$  Determine the set of all possible values of $ab + ac + bc.$

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


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


Find the real roots for the polynomial equation
\[ x^n - x^{n-1} + x^{n-2} - \dots + (-1)^{n-1}x + (-1)^n = 0, \]
where $n$ is a positive integer.


Find the ones digit of \(34^{34 \cdot 17^{17}}\).


Acme Corporation has now released a new version of its vowel soup, where each of the vowels (A, E, I, O, U) appears six times. How many six-letter sequences can be formed from this new bowl of Acme Vowel Soup?


How much greater, in square inches, is the area of a circle of radius 30 inches than a circle of diameter 15 inches? Express your answer in terms of \(\pi\).


The values of $a$, $b$, $c$, and $d$ are 2, 3, 4, and 5, but not necessarily in that order. What is the largest possible value of the sum of four products $ab$, $bc$, $cd$, and $da$?


To calculate $51^2$, Maria mentally computes the value of $50^2$ and adds 101. Maria subtracts a number from $50^2$ to calculate $49^2$. What number does she subtract?


What is the probability of rolling five distinct eight-sided dice and getting all distinct outcomes? Express your answer as a common fraction.


What is the remainder when 385,857 is divided by 6?


What is the area, in square units, of a triangle that has sides of $5, 3,$ and $3$ units? Express your answer in simplest radical form.


In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge between the losing teams on Saturday to decide the starting server for Sunday's consolation match. One possible ranking of the team from first to fourth at the end of the tournament is EGHF. Determine the total number of possible four-team ranking sequences at the end of the tournament.


How many values of $x$, $-30<x<150$, satisfy $\cos^2 x + 3\sin^2 x = 1?$ (Note: $x$ is measured in radians.)


How many integers are common solutions to these three inequalities?
\[
\begin{array}{cccc}
(1) & 2y & \leq & -y + 4 \\
(2) & 5y & \geq & -10 \\
(3) & 3y & \leq & -2y + 20
\end{array}
\]


The real function \( g \) has the property that, whenever \( x, \) \( y, \) \( m \) are positive integers such that \( x + y = 5^m, \) the equation
\[ g(x) + g(y) = 3m^2 \]holds. What is \( g(625) \)?


Triangle $DEF$ has a perimeter of 2010 units. The sides have lengths that are all integer values with $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


What is the least natural number that can be added to 54,321 to create a palindrome?


In triangle $XYZ$, $YZ = 10$. The length of median $XM$ is 7. Let $M$ be the largest possible value of $XY^2 + XZ^2$, and let $m$ be the smallest possible value. Find $M - m$.


Calculate
\[
\cos\left(4 \arccos \frac{2}{5}\right).
\]


In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100\sqrt{2},$ and $AC = 100.$ Find the sum of all possible values of $BC.$


A point has rectangular coordinates $(x, y, z)$ and spherical coordinates $\left(5, \frac{5\pi}{6}, \frac{\pi}{4}\right)$. Find the spherical coordinates of the point with rectangular coordinates $(-x, y, z)$. Enter your answer in the form $(\rho, \theta, \phi)$, where $\rho > 0$, $0 \le \theta < 2\pi$, and $0 \le \phi \le \pi$.

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


If 3124 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


A sphere is cut into three equal wedges. The circumference of the sphere is $18\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\pi$.


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^2 + yg(z)) = xg(x) + zg(y)\] for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $g(4),$ and let $s$ be the sum of all possible values of $g(4).$ Find $n \times s.$


A license plate in a certain state consists of 4 digits and 3 letters. The digits are not necessarily distinct, and the letters, also not necessarily distinct, must appear consecutively. How many distinct license plates are possible if the characters can appear in any order but the three letters must appear together?


Alice and Bob each flip three coins: two fair coins and a third coin that comes up heads with a probability of $\frac{5}{8}$. Determine the probability that Alice and Bob get the same number of heads. Express your answer as a sum of the numerator and denominator of the reduced fraction representing this probability.


Alice, Beth, Cathy, Daisy, Eve, and Fiona are the participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties, and Alice must finish in the top 3?


Find the domain of the function \( g(x) = \tan(\arcsin(x^3)) \).

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


Mrs. Crabapple has tweaked her class routine a bit. In her British Literature class, which still consists of 11 students and meets four times a week, she now alternates between giving out a crabapple and a rose every other class. She starts the week with a crabapple. Given this new setup, how many different sequences of gift recipients are possible in one week?


The value of $x$ is one-third the value of $y$, and the value of $y$ is one-fourth the value of $z$, and then $z$ is doubled. If $z$ is originally 48, what is the value of $x$ after $z$ is doubled?


The value of $\log_{3}{850}$ is between which two consecutive integers $c$ and $d$? Find $c+d$.


Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles.


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 $\mathbf{a},$ $\mathbf{a} + \mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$


The volume of a rectangular solid is $1000 \text{ cm}^3$, its total surface area is $600 \text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.


Arc $DE$ is a semi-circle with center $F$. The shaded region $DEF$ is "rolled" along a straight board $RS$ until it reaches its original orientation for the first time with point $F$ landing at point $F'$. If $EF = \frac{4}{\pi}$ cm, what is the length of the path that point $F$ travels?

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


Sasha has $\$4.80$ in U.S. coins. She has four times as many dimes as she has nickels and the same number of quarters as nickels. What is the greatest number of quarters she could have?


Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) and total to 15?


A sorcerer is crafting a magical elixir. He must choose one of four magical herbs and one of six enchanted crystals to complete his elixir. However, two of the crystals are incompatible with three of the herbs each. How many valid combinations can the sorcerer use to create his elixir?


The graph of the equation \[\sqrt{x^2+y^2} + |y-2| = 5\] consists of portions of two different parabolas. Compute the distance between the vertices of these parabolas.


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}}


One hundred more than the square of a number is the same as the square of the quantity that is $20$ less than the number. What is the number?


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


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

Solution: \boxed{0, 1}


In triangle $XYZ$, $XY=15$ and $XZ=9$. The angle bisector of $\angle X$ intersects $YZ$ at point $D$, and point $M$ is the midpoint of $XD$. Let $P$ be the point of intersection of $XZ$ and $BM$. The ratio of $ZP$ to $PX$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Let $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$


Define the operation $a\nabla b = 3 + b^a$. What is the value of $(2\nabla 3) \nabla 4$?


In a new game, I walk outdoors where, on move 1, I do nothing, but on move $n$ where $2 \le n \le 30$, I take two steps forward if $n$ is prime and three steps backwards if the number is composite. After all 30 moves, I stop. How many steps away from my starting point am I?


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


How many positive divisors does 8! have?


What is the sum of all the solutions of $x = \left|3x - |80 - 3x|\right|$?


If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\]


Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]


Determine the number of ways to arrange the letters of the word BANANAS.


What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.


How many positive four-digit integers are divisible by both 13 and 7?


How many integers satisfy the inequality $(x+3)^2 \leq 4$?


The product $N$ of three positive 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 $N$.

Solution: \boxed{272 + 160 + 128 = 560}


How many four-digit numbers satisfy the property that the second digit is the average of the first and the third digits?


Express $0.\overline{2}+0.\overline{03}+0.\overline{0004}$ as a common fraction.


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


Pirate Pete and Pirate Polly decide to share a treasure. Pete starts by saying, "One for me, one for you," giving himself one coin and starting Polly's pile with one coin. The pattern continues with Pete saying, "Two for me, two for you," giving himself two more coins but making Polly's pile two coins in total. This continues until Pete says "$x$ for me, $x$ for you," where he has given himself $x$ more coins, and Polly's pile is $x$ coins in total. At this distribution point, Pirate Pete has exactly three times as many coins as Pirate Polly. How many gold coins do they have in total?


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


Find the smallest composite number that has no prime factors less than 20.


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


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


In how many different ways can five students stand in a straight line if two of the students refuse to stand next to each other?


Express the sum as a common fraction: $.2 + .04 + .006 + .0008 + .00010.$


Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).


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


A nine-digit integer is formed by repeating a three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?


What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$?


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


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

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


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


Let $x$ and $y$ be positive real numbers such that $x + y = 2.$ Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}.$

Solution: \boxed{[2, \infty)}


A number is chosen at random from the set of consecutive natural numbers $\{1, 2, 3, \ldots, 120\}$. What is the probability that the number chosen is a factor of $5!$? Express your answer as a common fraction.


What is the units digit of the sum of the squares of the first 2011 odd, positive integers?


How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'?


How many three-digit numbers are multiples of neither 3 nor 11?


A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,2)$, and $(0,2)$. What is the probability that $x < 2y$?


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


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


Find $x$ such that $\lfloor x \rfloor + x = \dfrac{17}{4}$. Express $x$ as a common fraction.


In the expansion of $(x+1)^{50}$, what is the coefficient of the $x^3$ term?


How many positive multiples of 7 that are less than 200 have a units digit of 7?


If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn?


How many three-digit whole numbers have at least one 8 or at least one 9 as digits?


What is the area, in square units, of a region bounded by the curve \(y = x^2\), the lines \(y = 10\), \(y = 5\), and the \(y\)-axis?

Solution: \boxed{\frac{20\sqrt{10} - 10\sqrt{5}}{3} \text{ square units}}


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


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


How many positive factors of 450 are perfect squares?


Solve the equation \[\frac{4x^2 + 3x + 2}{x-2} = 4x + 2\] for \(x\).


My friend Sara only likes numbers that are divisible by 6. How many different ones digits are possible in numbers that Sara likes?


Find the least positive integer \( x \) that satisfies both \( x + 7219 \equiv 5305 \pmod{17} \) and \( x \equiv 4 \pmod{7} \).


The mean of $7, 10, 15$ and $23$ is equal to the mean of $18, y$, and $30$. What is the value of $y$?


Find the smallest abundant number that is not a multiple of 5.


A triangle is made of wood sticks of lengths 5, 12, and 13 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks to make this happen?


Solution: \boxed{2xz + 2yz + 10x + 10y + 20z + 100}


An ellipse has foci at $(0, 3)$ and $(4, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair.

Solution: \boxed{\left(\frac{56}{11}, 0\right)}


Given an acute angle $\theta$ in a right triangle, determine the smallest possible positive value of $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} + \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta}$.


What is the base-10 integer 789 when expressed in base 7?


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


The graph of $y = \frac{p(x)}{q(x)}$ is shown, where $p(x)$ and $q(x)$ are quadratic polynomials. The horizontal asymptote is $y = 2$, and the vertical asymptote is $x = -3$. There is a hole in the graph at $x=4$.

Find $\frac{p(5)}{q(5)}$ if the graph passes through $(2,0)$.


In rectangle $ABCD$, $AB = 6$ cm, $BC = 8$ cm, and $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.

Solution: \boxed{EF = 8 \text{ cm}}


Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v}$, $\mathbf{P} \mathbf{v}$ 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}}


Compute without using a calculator: $12! - 11!$.


When $\sqrt[3]{8000}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$?


If $x$, $y$, and $z$ are positive integers satisfying $xy+z = yz+x = zx+y = 53$, what is the value of $x+y+z$?


Find all solutions to the inequality \[\frac{(x-1)^2}{(x-5)^2} \ge 0.\] (Give your answer in interval notation.)

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


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


MATHCOUNTS is again spelled out with 10 magnets, but now the N's are also identical to each other, making both T's and N's indistinguishable. Three vowels and two consonants fall off and are put away in a bag. How many distinct possible collections of letters could be put in the bag?


How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one eleventh of $N$?


There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if exactly two Americans get a medal?


An amoeba is placed in a large tank and on the same day it splits into two amoebas. For the next six days, each amoeba splits into two new amoebas every day. Starting from the eighth day, each amoeba begins to split into three new amoebas every day. How many amoebas will there be in the tank after two weeks?


What is the largest value of $x$ if $\frac{3x}{7} + \frac{2}{9x} = 1$?

Solution: \boxed{\frac{63 + \sqrt{2457}}{54}}


The number of rounds of golf played by each golfer of an amateur golf association is shown in the chart below. What is the average number of rounds played by each golfer, rounded to the nearest whole number?
- 0 rounds: 3 golfers
- 1 round: 4 golfers
- 2 rounds: 3 golfers
- 3 rounds: 6 golfers
- 4 rounds: 2 golfers
- 5 rounds: 4 golfers


Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 7,$ $\|\mathbf{b}\| = 9,$ and $\|\mathbf{a} + \mathbf{b}\| = 13.$ Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.


How many integers between $200$ and $250$ have three different digits in decreasing order?


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


Below is the graph of \(y = a \cos (bx + c)\) for some positive constants \(a\), \(b\), and \(c\). The graph reaches its maximum at \(x = -\frac{\pi}{4}\). Find the smallest possible value of \(c\).


Corn costs 99 cents per pound, and beans cost 55 cents per pound. If Jordan buys 24 total pounds of corn and beans, and it costs $\$20.70$, how many pounds of corn did Jordan buy? Express your answer as a decimal to the nearest tenth.


If 13,254,839 pencils are packaged in sets of 7, how many pencils will be left over?


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

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


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


What is the sum of the greatest common divisor of $120$ and $3507$ and the least common multiple of $120$ and $3507$?


Let $\triangle XYZ$ be a right triangle with $Y$ being the right angle. A circle with diameter $YZ$ intersects side $XZ$ at point $W$. If the area of $\triangle XYZ$ is $200$ and $XZ = 30$, find the length of $YW$.


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


Let $n$ be the number of ordered quadruples $(x_1, x_2, x_3, x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 100$. Find $\frac{n}{100}$.


Determine the positive integer $N$ such that $$ 15^4 \times 28^2 = 12^2 \times N^2. $$


What is the coefficient of \( x^4 \) in the expansion of \( (x + 3\sqrt{2})^9 \)?

Solution: \boxed{122472 \sqrt{2}}


The ten horizontal and ten vertical lines on a $9\times9$ chessboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^2 + yg(z)) = xg(x) + yg(z)\]
for all real numbers $x,$ $y,$ and $z.$

Let $m$ be the number of possible values of $g(4),$ and let $t$ be the sum of all possible values of $g(4).$  Find $m \times t.$


In a certain hyperbola, the center is at \((1, -1)\), one focus is at \((1, 5)\), and one vertex is at \((1, 1)\). The equation of this hyperbola can be written in the form
\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.
\]
Find \(h + k + a + b.\)

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


How many perfect squares less than 500 have a ones digit of 6, 7, or 8?


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$. Compute $ \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.


Determine the number of different complex numbers \( z \) that satisfy \( |z| = 1 \) and \( z^{10} - z^{8} \) is a real number.


What is the sum of the tens digit and the units digit in the decimal representation of $9^{2002}$?


Lucas bought 50 items, each priced at 50 cents, 4 dollars, or 5 dollars. If his total purchase price was $\$$100.00, how many 50-cent items did he purchase?


How many ways are there to divide a 12-hour period into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?


Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p  +q + r + s = 10$ and
\[pq + pr + ps + qr + qs + rs = 22.\]
Find the largest possible value of $s.$

Solution: \boxed{\frac{5 + \sqrt{93}}{2}}


What is the total number of digits used when the first 1500 positive even integers are written?


A courier company charges an extra $\$0.15$ in postage if the length of a package, in inches, divided by its width, in inches, is less than $1.5$ or greater than $3.0$. Additionally, any package weighing more than $5$ ounces incurs the extra charge. Determine how many of these four packages must pay the extra $\$0.15$ in postage:
\begin{tabular}[t]{cccc}
Package & Length in inches & Width in inches & Weight in ounces\\\hline
X &8 &6 &4\\
Y &12 &4 &6\\
Z &7 &7 &5\\
W &14 &4 &3
\end{tabular}


Calculate the product \(\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{3004}{3008}\right)\). Express your answer as a common fraction.


The second and fourteenth terms of an arithmetic sequence are $\frac{8}{11}$ and $\frac{9}{13}$, respectively. What is the eighth term?


Given that $-6 \leq x \leq -3$ and $3 \leq y \leq 6$, what is the smallest possible value of $\frac{x+y}{x^2}$?


Mrs. Crabapple now teaches two different classes of British Literature, each consisting of 8 students. She continues her tradition of picking a random student at the beginning of every class to receive a crabapple. Both classes meet three times a week. How many different sequences of crabapple recipients are possible in a week for both classes combined?


If $x$ and $y$ are inversely proportional and when $x+y=60$, $x$ is three times $y$, what is the value of $y$ when $x=12$?


Using the digits $1, 2, 3, 4, 5, 6$, how many different $6$-digit positive integers can be formed such that the digit $1$ is to the left of the digit $2$ and the digit $3$ is to the left of the digit $4$? The digits must all be distinct.


$ABCDEFGH$ is a cube with volume 8. Find the volume of pyramid $ACDH$.


The sum of three numbers $a$, $b$, and $c$ is 72. If we increase $a$ by 4, decrease $b$ by 8, and multiply $c$ by 4, the three resulting numbers are equal. What is the value of $a$?


For how many integers $n$ between 1 and 200 is the greatest common divisor of 35 and $n$ equal to 7?


Alice is thinking of a number. She gives the following 3 clues. "My number has 243 as a factor. My number is a multiple of 36. My number is between 1000 and 3000." What is Alice's number?


For how many different negative values of \( x \) is \( \sqrt{x + 196} \) a positive integer?


If $h(x)$ is a monic quartic polynomial such that $h(-2) = -4$, $h(1) = -1$, $h(-4) = -16$, and $h(3) = -9$, find $h(0)$.


Let $PQRS$ be a tetrahedron such that edges $PQ$, $PR$, and $PS$ are mutually perpendicular. Let the areas of triangles $PQR$, $PRS$, and $PQS$ be denoted by $a$, $b$, and $c$, respectively. In terms of $a$, $b$, and $c$, find the area of triangle $QRS$.


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


Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be non-zero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Find \(\sin \phi\).


What is the modular inverse of $13$, modulo $1728$?

Express your answer as an integer from $0$ to $1727$, inclusive.


Determine the value of $m$ for which the seven-digit number $856m248$ is divisible by $11$.


In a survey of 1500 people, $25\%$ do not like television, and $15\%$ of the people who do not like television also do not like sports. How many people surveyed do not like both television and sports?


In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XZ = \sqrt{41}$. What is $\tan X$?


A marine biologist interested in monitoring a specific fish species population in a coastal area. On January 15, he captures and tags 80 fish, then releases them back into the water. On June 15, he captures another sample of 100 fish, finding that 6 of them are tagged. He assumes that 20% of the tagged fish have died or migrated out of the area by June 15, and also that 50% of the fish in the June sample are recent additions due to birth or migration. How many fish were in the coastal area on January 15, based on his assumptions?


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


Suppose $x$, $y$, and $z$ are positive numbers satisfying:
\[
\frac{x^2}{y} = 2, \quad \frac{y^2}{z} = 3, \quad \text{and} \quad \frac{z^2}{x} = 4.
\]
Find $x$.

Solution: \boxed{x = 24^{2/7}}


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 $X$?


For a list of six positive integers, none greater than 200, the mean is 2 times the mode. If 30, 60, 70, 150, $x$, and $x$ are the six integers, what is the value of $x$?


The perimeter of a rectangular playground is 90 meters. If the length of the playground is three times the width, what is the area of the playground, in square meters?


Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$.


Determine the real number $k$ such that the equation
\[\begin{pmatrix} 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 5 \\ -3 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} -2 \\ k \end{pmatrix}\]does not have any solutions in $t$ and $s$.


Interior numbers begin in the third row of Pascal's Triangle. Given the sum of the interior numbers in the fifth row is 14, and the sum of the interior numbers of the sixth row is 30, what is the sum of the interior numbers of the eighth row?


In a new shooting competition, ten clay targets are set up in four hanging columns with four targets in column $A$, three in column $B$, two in column $C$, and one in column $D$. A shooter must continue following the sequence:
1) The shooter selects one of the columns.
2) The shooter must then hit the lowest remaining target in that chosen column.
What are the total possible sequences in which the shooter can break all the targets, assuming they adhere to the above rules?


Express \( 7.\overline{036} \) as a common fraction in lowest terms.


Define a $\it{good\ word}$ 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 eight-letter good words are there?


Let $R = (10, 8)$. The lines whose equations are $9y = 18x$ and $15y = 6x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. Find the length of $PQ$ expressed as a reduced fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Then find $m + n$.


Let $p$, $q$, and $r$ be solutions of the equation $x^3 - 6x^2 + 11x = 14$.

Compute $\frac{pq}{r} + \frac{qr}{p} + \frac{rp}{q}$.


If $\frac{120}{2^4 \cdot 5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


How many perfect squares less than 20,000 can be represented as the difference of the squares of two integers that are two apart?


Solution: \boxed{(5, -2.5, 7.5)}


Suppose that a parabola has vertex $\left(-\frac{1}{3}, \frac{5}{9}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a$.