The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 23, b_{10} = 123,$ and for all $n \ge 3, b_n$ is the arithmetic mean of the first $n-1$ terms. Find $b_2$.


If $a$ and $b$ are positive integers such that $ab - 3a + 7b = 222$, what is the minimal possible value of $|a - b|$?


Let \( x \) and \( y \) be positive real numbers. Find the minimum value of
\[
\frac{\sqrt{(x^2 + y^2)(4x^2 + y^2)}}{xy}.
\]


Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \]


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $1287$, what is the sum of the numbers written on the faces?


What is the remainder when $3012$ is divided by $97$?


Evaluate $(a^b)^a - (b^a)^b$ for $a = 3$ and $b = 4$.


Mateen walks 2 kilometers (2000m) in his rectangular garden. To achieve this, he either walks the length of the garden 40 times or walks its perimeter 16 times. Determine the area of Mateen's garden in square meters.


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


The sum of four positive integers that form an arithmetic sequence is 50. Of all such possible sequences where the third term is even, what is the greatest possible third term?


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


Given that \(5^{-1} \equiv 39 \pmod{89}\), find \(25^{-1} \pmod{89}\), as a residue modulo 89.


What is the remainder when $4x^7 - 2x^6 - 8x^5 + 3x^3 + 5x^2 - 15$ is divided by $3x - 9$?


Points $E$, $F$, $G$, and $H$ lie on a line, in that order. If $EF=3$ units, $FG=8$ units, and $EH=23$ units, what is the ratio of $EG$ to $FH$? Express your answer as a common fraction.


The Brown family has 5 sons and 3 daughters. In how many ways can they be seated in a circle such that at least two boys are next to each other?


A point has rectangular coordinates \((x, y, z)\) and spherical coordinates \((5, \frac{5\pi}{6}, \frac{\pi}{3})\). Find the spherical coordinates of the point with rectangular coordinates \((x, y, -z)\). Enter your answer in the form \((\rho, \theta, \phi)\), where \(\rho > 0\), \(0 \leq \theta < 2\pi\), and \(0 \leq \phi \leq \pi\).

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


A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis.


When the square root of a number $y$ is raised to the fourth power, the result is 256. What is the value of $y$?


In a sports team, athletes can be arranged in two different rectangular formations. When they are arranged in 10 rows, there are 4 positions unoccupied. When they are arranged in 11 rows, there are 5 positions unoccupied. Find the number of athletes in the team if the total number is between 200 and 300.


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


If \(f(x) = f(5 - x)\) for all \(x\), determine the line that necessarily acts as an axis of symmetry for the graph of \(y = f(x)\).

Solution: \boxed{x = 2.5}


What is the sum of $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5 + \left(\dfrac{1}{4}\right)^6$?


Find the sum of the roots of the equation \[(3x^3 - 6x^2 - 9x + 27)(4x^3 - 8x^2 + 16) = 0.\]


Let $f(x)$ be a differentiable function that satisfies the equation
\[ f(x+y) = f(x) + f(y) \]
for all real numbers $x$ and $y$. Find the value of $f'(0)$.


Calculate the sum of the numbers formed by repeating each digit from 2 to 9 such that each digit d forms a number by repeating it d times, i.e., calculate $$2+33+444+5555+66666+777777+8888888+99999999$$ and find the result modulo 9.


The sides of a triangle have lengths of $13$, $14$, and $15$. Find the length of the shortest altitude.


Determine the value of \( x \) at the point of intersection for the lines represented by the equations \( y = 3x + 4 \) and \( 5x - y = 41 \).


A traffic light repeats the following sequence: green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam randomly chooses a four-second interval to observe the light. What is the probability that the color changes during his observation?


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2005} \times 7$?


Solution: \boxed{30x - 10z}


Determine the number of ways to arrange the letters of the word "MISSISSIPPI" such that no two S's are next to each other.


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

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


A point is selected at random from the portion of the number line from 0 to 8. What is the probability that the point is closer to 6 than to 0? Express your answer as a decimal to the nearest tenth.


What is the area, in square cm, of quadrilateral $ABCD$ if $\overline{AB} \perp \overline{BC}$, $\overline{DC} \perp \overline{BC}$, $AB=8$ cm, $DC=3$ cm, and $BC=10$ cm?


Determine the numerical value of:
\[
\frac{\sin 24^\circ \cos 16^\circ + \cos 156^\circ \cos 96^\circ}{\sin 26^\circ \cos 14^\circ + \cos 154^\circ \cos 94^\circ}.
\]


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\[
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z + 1), \\
y &= 2023 - 2024 \operatorname{sign}(x + z - 1), \\
z &= 2023 - 2024 \operatorname{sign}(x + y + 1).
\end{align*}
\]
where $\operatorname{sign}(a)$ is defined as:
\[
\operatorname{sign}(a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.
\]


There are \(30m\) members in the Johnsonville marching band, and when they line up in rows of 34, there are 2 band members left over. If \(m\) is an integer and there are fewer than 1500 band members, what is the maximum number of people that could be in the Johnsonville marching band?


The product of positive integers $a$, $b$, and $c$ equals 1386. What is the minimum possible value of the sum $a + b + c$?


The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation $g(x) = d$ has exactly six solutions.


What is $1254_6 - 432_6 + 221_6$? Express your answer in base $6$.


How many sequences of 8 digits $x_1, x_2, \ldots, x_8$ can we form under the condition that no two adjacent $x_i$ have the same parity until $x_5$, and starting from $x_5$ the digits must alternate starting with the opposite parity of $x_4$? Leading zeroes are allowed.


A hexagon is obtained by joining, in order, the points $(0,0)$, $(2,1)$, $(3,3)$, $(4,2)$, $(5,0)$, $(3,-1)$, and back to $(0,0)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{5}+d\sqrt{10}$, where $a$, $b$, $c$, and $d$ are whole numbers. Find $a+b+c+d$.


A circle with a radius of 3 units has its center at $(0, 0)$. Another 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. Determine the value of $x$. Express your answer as a common fraction.


How many whole numbers lie in the interval between $\frac{7}{2}$ and $3\pi$?


Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $6$ more, she will have a multiple of $7$. If she loses $7$, she will have a multiple of $4$. What is the smallest possible value of $n$?


Find the sum of all integral values of $c$ with $c \leq 15$ for which the equation $y = x^2 - 7x - c$ has two rational roots.

Solution: \boxed{-12 + (-10) + (-6) + 0 + 8 = -20}


A **composite number** is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?


Both roots of the quadratic equation $x^2 - 95x + k = 0$ are prime numbers. Find the number of possible values of $k$.


How many ways are there to put 6 indistinguishable balls into 3 indistinguishable boxes?


Let $T$ be the set of all rational numbers $r$, $0 < r < 1$, that have a repeating decimal expansion in the form $0.ababab\ldots = 0.\overline{ab}$, where the digits $a$ and $b$ are not necessarily distinct. To express the elements of $T$ as fractions in lowest terms, how many different numerators are possible assuming the fraction is reduced to its simplest form?


Find the smallest positive angle $x$ that satisfies \(\tan 3x \cot 2x = 1\) in degrees.


In a right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $DF = 15$. Find $\cos F$.


Find the remainder when the polynomial $x^{1010}$ is divided by the polynomial $(x^3 - 1).$


The sum of the digits of the year 2004 is 6. What is the first year after 2010 for which the sum of the digits is 15?


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?


Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 50$, is $f(n)$ prime?


Let $\mathbf{u}_0$ be a vector. The vector $\mathbf{u}_0$ is first projected onto $\begin{pmatrix} 4 \\ 2 \end{pmatrix},$ resulting in the vector $\mathbf{u}_1.$ The vector $\mathbf{u}_1$ is then projected onto $\begin{pmatrix} 2 \\ 3 \end{pmatrix},$ resulting in the vector $\mathbf{u}_2.$ Find the matrix that takes $\mathbf{u}_0$ to $\mathbf{u}_2.$

Solution: \boxed{\begin{pmatrix} \frac{28}{65} & \frac{14}{65} \\ \frac{42}{65} & \frac{21}{65} \end{pmatrix}}


Consider the decimal function denoted by $\{ x \} = x - \lfloor x \rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\{x\} = \frac{1}{\lfloor x \rfloor}$. Express your answer as a mixed number.


Let \( n \) be a positive integer, and let \( b_0, b_1, \dots, b_n \) be a sequence of real numbers such that \( b_0 = 48 \), \( b_1 = 81 \), \( b_n=0 \), and
\[
b_{k+1} = b_{k-1} - \frac{4}{b_k}
\]
for \( k = 1, 2, \dots, n-1 \). Find \( n \).


You have a rectangular box with length $x^2 + 5$ units, width $2x - 5$ units, and height $x + 25$ units. Determine the number of positive integer values of $x$ such that the volume of the box is less than 1200 units.


What is the least five-digit positive integer which is congruent to 7 (mod 21)?


Factor the following expression: $56b^2a^2 + 168ba$.


What is the sum of the different prime factors of $195195$?


In triangle $ABC$, $BC = 10$. The length of median $AD$ is $6$. Let $N$ be the largest possible value of $AB^2 + AC^2$, and let $n$ be the smallest possible value. Additionally, the altitude from $A$ to $BC$ is $5$. Find $N - n$.


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -4 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{12}{5} \end{pmatrix}}


Find the range of the function $g(x) = \arcsin x - \arccos x + \arctan x$ for $x$ in the interval $[-1,1]$.

Solution: \boxed{\left[-\frac{7\pi}{4}, \frac{3\pi}{4}\right]}


On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?


Determine if the numbers $\sqrt{3v-2}$, $\sqrt{3v+2}$, and $\sqrt{6v}$ can be the side lengths of a triangle. If so, calculate the measure of the largest angle.


Two standard 8-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is less than 10 and is an odd number? Express your answer as a common fraction.


Calculate the degree measure of the smaller angle formed by the hands of a clock at 7 o'clock.


To calculate $48^2$, Emily mentally computes the value $50^2$ then subtracts a certain number. To calculate $46^2$, Emily subtracts another number from $50^2$. What number does she subtract to find $46^2$?


The sum of 19 consecutive positive integers is twice a perfect square. What is the smallest possible value of this sum?


The truncated right circular cone has a large base radius 12 cm and a small base radius of 6 cm. The height of the truncated cone is 10 cm. How many $\text{cm}^3$ are in the volume of this solid?


In how many ways can the digits of $64,\!420$ be arranged to form a 5-digit number, considering that numbers cannot begin with 0?


When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of four consecutive positive integers all of which are nonprime?


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


Both roots of the quadratic equation $x^2 - 58x + k = 0$ are prime numbers. Find the number of possible values of $k.$


The square root of \(3x\) is greater than 4 and less than 5. How many integer values of \(x\) satisfy this condition?


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


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}$. Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a})$ and $\mathbf{v} \times \mathbf{b} = 3(\mathbf{a} \times \mathbf{b})$.

Solution: \boxed{\begin{pmatrix} 8 \\ 9 \\ -4 \end{pmatrix}}


What is the largest four-digit negative integer congruent to $2 \pmod{17}$?


Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$, resulting in the vector $\mathbf{v}_1$. The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$, resulting in the vector $\mathbf{v}_2$. Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2$.

Solution: \boxed{\begin{pmatrix} \frac{24}{85} & \frac{6}{85} \\ \frac{48}{85} & \frac{12}{85} \end{pmatrix}}


Evaluate the determinant:
\[
\begin{vmatrix}
\cos (\alpha + \beta) & \sin (\alpha + \beta) & -\sin \alpha \\
-\sin \beta & \cos \beta & 0 \\
\sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha
\end{vmatrix}.
\]


A unique coin has a probability of $\frac{2}{5}$ of landing on heads and $\frac{3}{5}$ of landing on tails. If you flip this coin, you receive $\$4$ for a head and lose $\$3$ for a tail. What is the expected value, in dollars, of your winnings after one flip?


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


The function $y=\frac{x^3+11x^2+38x+40}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?


A school has between 130 and 210 students enrolled. Every afternoon, all the students gather to participate in a singing session. The students are divided into eight distinct groups. If two students are absent from school, the groups can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Suppose that \(a\) and \(b\) are integers such that \(4b = 10 - 3a\). How many of the first eight positive integers must be divisors of \(2b + 15\)?


A fair, twenty-faced die has $18$ of its faces numbered from $1$ through $18$ and two blank faces. Another fair, twenty-faced die has $17$ of its faces numbered from $3$ through $20$ and three blank faces. When the two dice are rolled, what is the probability that the sum of the two numbers facing up will be $27$? Express your answer as a common fraction.


Contractor Lucy agreed to complete a construction job in 45 days. After 9 days, she found that the 10 people assigned to her team had completed $\frac{1}{5}$ of the job. Assuming that everyone on the team works at the same rate, what is the minimum number of people Lucy must keep on the team to guarantee that the job will be completed on time?


What is the smallest positive integer $n$ such that $5n \equiv 4960 \pmod{31}?$


If $g(x) = g(3-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=g(x)$?

Solution: \boxed{x = 1.5}


Kevin Kangaroo starts at 0 on a number line and aims to reach the point 2, but his hopping strategy alternates. On odd-numbered hops, he covers $\frac{1}{2}$ of the distance to his goal, while on even-numbered hops, he covers $\frac{1}{4}$ of the distance remaining to the goal. Determine the total distance Kevin hops after six hops. Express your answer as a fraction.


Sarah can buy pans, pots, and ladles for $3, 5, and 9$ dollars each, respectively. What is the largest number of ladles she can purchase if she must buy at least two of each item and will spend exactly $100?


The graph of the rational function $\frac{1}{q(x)}$ has vertical asymptotes at $x=-1$, $x=1$, and $x=3$. If $q(x)$ is a cubic polynomial and $q(2) = 12$, find $q(x)$.

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


Suppose that $x$ is a positive multiple of $4$. If $x$ squared is less than $2000$, what is the greatest possible value of $x$?


When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$?


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


The function $f(x)$ satisfies $f(2) = 5$ and
\[f(x + y) = 4^y f(x) + 3^x f(y)\] for all real numbers $x$ and $y.$ Find the function $f(x).$

Solution: \boxed{\frac{5}{7}(4^x - 3^x)}


Each of $a_1, a_2, \dots, a_{150}$ is equal to $1$ or $-1.$  Find the minimum positive value of
\[
\sum_{1 \leq i < j \leq 150} a_i a_j.
\]


Suppose that $a^3$ varies inversely with $b^2$. If $a=5$ when $b=2$, find the value of $a^3$ when $b=8$.


The integer $x$ has 18 positive factors. The numbers $18$ and $20$ are factors of $x$. What is $x$?


Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards of different heights in a row. This time, he wants the tallest and shortest leopards at the ends of the row. How many ways can he line up the leopards?

Solution: \boxed{10080 \text{ ways.}}


If Mark rolls five fair six-sided dice, what is the probability that he rolls more 2's than 4's?


How many integers 1-9 are divisors of the five-digit number 32,481?


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


What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 3:45? Express your answer as a decimal to the nearest tenth.


The combined weight of Alice and Chloe is 200 pounds. If Chloe's weight is subtracted from Alice's weight, the result is one-third of Chloe's weight. How much does Chloe weigh?

Solution: \boxed{\frac{600}{7} \text{ pounds}}


Similarly to the original problem, consider the function $f(x) = 3x^2 - 6x - 8$. By restricting the domain of this function to an interval, we can make it invertible. What is the largest such interval that includes the point $x=3$?

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


The dilation, centered at $-1 + 2i,$ with scale factor 4, takes $3 + 4i$ to which complex number?

Solution: \boxed{15 + 10i}


The sum of the first $30$ positive even integers is also the sum of five consecutive odd integers. What is the largest of these five integers?


What is the smallest prime whose digits sum to $28$?


Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$, and $C(-3, 2)$ is reflected over both the $y$-axis and the $x$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$?


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


When \( 2x^3 - 5x^2 - 12x + 7 \) is divided by \( 2x + 3 \), the quotient is \( x^2 - 4x + 2 \). What is the remainder?

Solution: \boxed{-4x + 1}


Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\]
Among all such vectors $\mathbf{p},$ there exists constants $s$ and $v$ such that $\mathbf{p}$ is at a fixed distance from $s \mathbf{a} + v \mathbf{b}.$ Determine the ordered pair $(s,v).$

Solution: \boxed{\left(\frac{9}{8}, -\frac{1}{8}\right)}


There are two stores offering discounts on Camera $Y$. The list price of Camera $Y$ is $\$59.99$. Store A offers a $15\%$ discount, while Store B offers a flat $\$12$ discount. How much cheaper, in cents, is the cheaper store's price for Camera $Y$ compared to the more expensive store?


What is the remainder when $3021$ is divided by $97$?


Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?


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


Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero.


How many of the 150 smallest positive integers are congruent to 3 (mod 7)?


What is the least positive integer $n$ such that $8820$ is a factor of $n!$?


This pattern is constructed using matchsticks. If the pattern begins with five matchsticks and for each subsequent stage, three matchsticks are added, along with an extra matchstick added after every fifth stage, how many matchsticks are required to form the $20^{th}$ stage?


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


A band has 100 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 4 and 25 students per row, in how many possible row-lengths can the band be arranged?


If $p$ and $q$ are odd integers, how many terms in the expansion of $(p+q)^8$ are odd?


If 20 bahs are equal to 36 rahs, and 12 rahs are equal in value to 20 yahs, how many bahs are equal in value to 1200 yahs?


Find the smallest positive integer $n$ such that there exist positive real numbers $x$ and $y$ for which
\[(x + 2yi)^n = (x - 2yi)^n,\]
and compute $\frac{y}{x}$.


Consider the line $y = 2x - 7$. The line is parameterized by the vector form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -3 \end{pmatrix} + t \begin{pmatrix} 2 \\ n \end{pmatrix}.\] Determine the ordered pair $(s, n)$.

Solution: \boxed{(2, 4)}


The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 4030} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$.


In the diagram, a rectangle has a perimeter of $60$, and the triangle has a height of $30$. If the rectangle and the triangle have the same area, what is the value of $x$?
[asy]
draw((0,0)--(3,0)--(3,1.5)--(0,1.5)--cycle);
draw((4,0)--(7,0)--(7,3)--cycle);
draw((6.8,0)--(6.8,.2)--(7,.2));
label("$x$",(5.5,0),S);
label("30",(7,1.5),E);
[/asy]


An infinite geometric series has a common ratio of $\frac{1}{4}$ and a sum of $40$. What is the second term of the series?


What is the smallest natural number that can be added to 40,317 to make it a palindrome?


In the diagram below, triangle $ABD$ is a right triangle with $AB = 30$ and $\angle ADB = 90^\circ$. We are given that $\cos A = \frac{4}{5}$. Triangle $BCD$ is such that $\angle BDC = 90^\circ$ and $\sin C = \frac{4}{5}$. Find the length of $DC$.

[asy]
pair A,B,C,D;
A = (0,0);
B = (24*sqrt(5),30);
D = (24*sqrt(5),0);
C = (24*sqrt(5) + 75,0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]


Consider a month with 31 days, where the number of the month is a product of two distinct primes (e.g., July, represented as 7). Determine how many days in July are relatively prime to the month number.


If the seven-digit number $945k317$ is divisible by $11$, what is the value of $k$?


Calculate the value of $\frac{10!}{4!}$ given that $10!=3628800$.


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


At Unity Community Club, a survey was conducted to find out the average age of its members. The average age of the female members was found to be 35 years old. The average age of the male members was 30 years old. Additionally, the average age of the child members was 10 years old. The overall average age of all the members combined was 25 years old. What is the ratio of female to male to child members? Express your answer as a common fraction in the form $\frac{f:m:c}$.


Determine the sum of all real numbers $x$ satisfying
\[(x^2-6x+5)^{x^2-3x+1} = 1.\]


Determine the domain of the function $$k(x) = \frac{1}{x+5} + \frac{1}{x^2-9} + \frac{1}{x^3-x+1}.$$

Solution: \boxed{(-\infty, -5) \cup (-5, -3) \cup (-3, 3) \cup (3, \infty)}


Given that $-2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 46 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Let \[f(x) = \left\{
\begin{array}{cl} cx+1, &\text{ if }x>3, \\
2x-7 &\text{ if } -1 \le x \le 3, \\
3x-d &\text{ if } x <-1.
\end{array}
\right.\]Find $c+d$ if the piecewise function is continuous.


A sphere is inscribed in a cube. Given that one edge of the cube is 8 inches, calculate the volume of the inscribed sphere. Express your answer in terms of \(\pi\).

Solution: \boxed{\frac{256}{3} \pi}


What must be the value of the coefficient $c$ in $P(x) = x^3 + 4x^2 + cx - 20$ for $x - 5$ to be a factor of $P$?


Jar C contains exactly six red buttons and ten blue buttons. Carla removes three red buttons and five blue buttons from Jar C and places them in an empty Jar D. Jar C now holds half of its initial total number of buttons. If Carla now were to randomly choose a button from Jar C and a button from Jar D, what is the probability that both chosen buttons will be blue?


At the beginning of each geometry class, Mr. Figueroa selects a random student to assist him with the demonstration. There are 15 students in his class, and the class meets five times a week. How many different sequences of student assistants are possible in one week?


The sum of two numbers \( x \) and \( y \) is 500, and the value of the fraction \( \frac{x}{y} \) is 0.8. What is the value of \( y - x \)?


Find the sum of all positive divisors of $120$ that are also divisors of $45$.


In $\Delta ABC$, $\overline{DE} \parallel \overline{AB}$ and $\overline{FG} \parallel \overline{AB}, CD = 5$ cm, DA = 15$ cm, and CF = 10$ cm. What is the number of centimeters in the length of $\overline{CB}$?

[asy]
pair A,B,C,D,E,F,G;
A = (-3,-6); B = (6,-6); C = (0,0); D = A/4; E = B/4; F=2*D; G=2*E;
draw(E--D--F--G--C--B--A--D);
label("A",A,W); label("B",B,dir(0)); label("C",C,N); label("D",D,W); label("E",E,dir(0)); label("F",F,W); label("G",G,dir(0));
[/asy]

Solution: \boxed{20 \text{ cm}}


Find the integer $n$, $-90 < n < 90$, such that $\tan n^\circ = \tan 240^\circ$.


Contractor Rachel agreed to complete a project in 40 days. After 10 days, she noticed that the 10 workers assigned had completed 40% of the project. If all workers maintain a consistent work rate, what is the least number of workers she should retain on the job to ensure the project is finished on time?


An ellipse has its foci at $(0, -1)$ and $(0, -3).$ Given that it passes through the point $(5, -2),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$


Twelve 6-sided dice are rolled. What is the probability that exactly four of the dice show a 6? Express your answer as a decimal rounded to the nearest thousandth.


Find the sum of the series $2_6 + 4_6 + 6_6 + \cdots + 50_6$. Express your answer in base $6$.


What is the area of the region enclosed by the graph of the equation $(x+2)^2 + (y-10)^2 = 36$ that lies above the line $y=2x-4$?

Solution: \boxed{36 \pi}


Mrs. Luna's math class has 20 boys, and 28 of her students bring their calculators to class. If 15 of the students who brought calculators are girls, then how many boys didn't bring their calculators to class?


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


Let \( g(x) = x^3 + ax^2 + bx + c \) be a polynomial whose roots are all positive odd integers. If \( a + b + c = -11, \) find \( c \).


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

size(6cm);

pair a = (0, 0); pair m = dir(180 - 30); pair d = dir(180 + 30); pair c = 1.3 * dir(30); pair b = 2 * 1.3 * Cos(60) * dir(-30);

label("$M$", m, N); label("$C$", c, N); label("$A$", a, 1.5 * S); label("$D$", d, S); label("$B$", b, SE);

draw(m--b--c--d--cycle);

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

add(pathticks(d--a, s=4));

add(pathticks(c--a, 2, spacing=0.9, s=4));

add(pathticks(c--b, 2, spacing=0.9, s=4));

[/asy]


Calculate the remainder of the binary number $111010110101_2$ when it is divided by 4, and express the answer in base 10.


Find the value of the first term in the geometric sequence $a, b, c, d, 64, 128$.


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. Determine how many different ways are there for some of the books to be in the library and the rest to be checked out if at least two books must be in the library and at least two must be checked out.


What is the greatest common multiple of 10 and 15 that is less than 120?


Find the largest three-digit integer starting with 8 that is divisible by each of its distinct, non-zero digits.


Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 721^\circ$.


In a classroom with 48 chairs arranged in rows and columns, where each row and each column must contain at least two chairs and all chairs must be included, how many possible rectangular arrays can these chairs be arranged?


A point in space $(x,y,z)$ is randomly selected so that $-2 \le x \le 2,$ $-2 \le y \le 2,$ $-2 \le z \le 2$. What is the probability that $x^2 + y^2 + z^2 \le 2.25$?


In a two-day chess tournament, Alpha and Charlie both played games worth a total of 600 points. Alpha scored 180 points out of 360 points attempted on the first day, and scored 120 points out of 240 points attempted on the second day. Charlie, who did not attempt 360 points on the first day, had a positive integer score on each of the two days, and Charlie's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $300/600 = 1/2$.

Find the largest possible two-day success ratio that Charlie could have achieved.


Calculate the value of $(a^3 + b^3) \div (a^2 - 2ab + b^2 + ab)$ for $a=3$ and $b=2$.


Ben selects two random integers between 1 and 60, inclusive (potentially the same number twice). What is the probability that at least one of the numbers is a multiple of 4?


There is a unique two-digit positive integer $s$ for which the last two digits of $13\cdot s$ are $52$.


In $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
label("B", B, E);
label("C", C, NE);
label("P", P, NW);
label("8", 3*dir(A--C), SE);
label("6", B+3*dir(B--C), NE);
label("10", (5,0), S);
[/asy]


Consider a parabola with the vertex at $\left(\frac{3}{5},-\frac{1}{5}\right)$ and the equation $y = ax^2 + bx + c$, where $a < 0$ and $b + 2c$ is an integer. Determine the greatest possible value of $a$.


Let \( m \) be the smallest integer whose cube root is of the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \( 1/100 \). Find \( n \).


Define $L(x) = x - \frac{x^3}{3}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{6}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$. As $n$ approaches infinity, what value does $n a_n$ approach?


The sum of two positive integers $c$ and $d$ is 1089. What is the largest possible value of $\gcd(c, d)$?


Given the expression \[\frac{4^{105} + 3^{105}}{4^{100} + 3^{100}},\] find the greatest integer less than or equal to this expression.


In the diagram, $\angle ABD = 120^\circ$ and $\angle BAE = 60^\circ$. Point $A$ is on a plane with points $B$, $C$, $D$, and $E$ where $D$, $B$, $E$, and $C$ are collinear in that order and $E$ is between $B$ and $C$. Given $\angle BAC = 95^\circ$, find the measure of $\angle ACB$. [asy]
size(250);
draw((-60,0)--(0,0));
draw((0,0)--(64.3,76.6)--(166,0)--cycle);
label("$A$",(64.3,76.6),N);
label("$95^\circ$",(64.3,73),S);
label("$120^\circ$",(0,0),NW);
label("$B$",(0,0),S);
label("$D$",(-60,0),S);
label("$C$",(166,0),S);
label("$E$", (83,0), S);
[/asy]


Consider a $4$-digit number geometric if it has $4$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers with these properties.


In the diagram, $D$ and $E$ are points on the line segments $AB$ and $BC$, respectively, such that $AD = \frac{1}{4}AB$ and $BE = \frac{1}{4}BC$. If $A(0,8)$, $B(0,0)$, and $C(16,0)$ in a coordinate plane, find the area of $\triangle DBC$.

[asy]
size(200); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E;
A=(0,8);
B=(0,0);
C=(16,0);
D=(0,6);
E=(12,0);
draw(A--B--C--cycle);
draw(E--A);
draw(B--D);
draw((0,8)--(0,-1));
draw((-1,0)--(17,0), EndArrow);
label("$A(0,8)$", A, W);
label("$B(0,0)$", B, SW);
label("$C(16,0)$", C, S);
label("$D$", D, W);
label("$E$", E, S);
[/asy]


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


Find the smallest composite number that has no prime factors less than 15.


Let $ f(x) = x^3 - 2x^2 + x - 2$. Suppose $ g$ is a cubic polynomial such that $ g(0) = 2$, and the roots of $ g$ are the cubes of the roots of $ f$. Find $ g(8)$.


Let $b_n = \frac{10^n - 9}{3}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


The diagonal lengths of a rhombus are 30 units and 8 units. Calculate both the area of the rhombus and the length of a side.

Solution: \boxed{\sqrt{241} \text{ units}}


Let \(k, a_2, a_3\) and \(k, b_2, b_3\) be nonconstant geometric sequences with different common ratios. If
\[
a_3 - b_3 = 3(a_2 - b_2) - k,
\]
then what is the sum of the common ratios of the two sequences?


Consider the function $f(x) = x^3 + 3\sqrt{x}$. Evaluate $3f(3) - 2f(9)$.

Solution: \boxed{-1395 + 9\sqrt{3}}


In triangle $XYZ$, $XY = 9$, $XZ = 17$, and the length of median $XM$ is 12. Find the area of triangle $XYZ$.


Find the sum of all positive integer solutions less than or equal to $15$ to the congruence $7(5x-3) \equiv 35 \pmod{12}$.


Find two pairs $(x,y)$ of real numbers such that $x + y = 3xy = 6$. Given that the solutions for $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$?


Consider a polynomial
\[ a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 \]
with integer coefficients and integer roots \( r_1, r_2, \ldots, r_6 \). Additionally, the roots of the polynomial
\[ a_0 x^6 + a_1 x^5 + a_2 x^4 + a_3 x^3 + a_4 x^2 + a_5 x + a_6 = 0 \]
are exactly \( r_1, r_2, \ldots, r_6 \). Determine the number of possible multisets \( S = \{r_1, r_2, \ldots, r_6\} \).


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


In a city modeled by a Cartesian plane, Daniel is located at $(10, -15)$ and Eliza is at $(-4, 16)$. They decide to meet at a point equidistant from their current locations and then proceed vertically to reach Fiona's location at $(3, 8)$. How many units do Daniel and Eliza walk upwards together from their meeting point to reach Fiona?


Given that $f(1)=7$, $f(2)=12$ and $f(x) = ax + bx + c$, where $c$ is a constant, find the value of $f(3)$.


Write the expression $\frac{5+4d}{8}+3$ as a single fraction.


**What is the probability of drawing three cards in sequence from a standard deck of 52 cards such that the first card is a Jack, the second card is a Queen, and the third card is the King of Hearts, assuming the cards are dealt without replacement?**


Let $n$ be the integer such that $0 \le n < 29$ and $5n \equiv 1 \pmod{29}$. What is $\left(2^n\right)^3 - 3 \pmod{29}$?

Express your answer as an integer from $0$ to $28$, inclusive.


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads?


Calculate the product $\left(\frac{2}{4}\right)\left(\frac{4}{6}\right)\left(\frac{6}{8}\right)\cdots\left(\frac{1000}{1002}\right)$. Express your answer as a common fraction.


Suppose a city has bicycle license plates with three letters. The first letter is chosen from $\{B, F, J, N, T\}$, the second from $\{E, U\}$, and the third from $\{G, K, R, Z\}$. To increase the number of unique plates, the city decides to add three new letters. One new letter will be added to each set. Determine the largest possible number of ADDITIONAL license plates that can be made.


Find the integer \( n \), \( 0 \leq n \leq 8 \), such that \[ n \equiv -4982 \pmod{9} \].


Find the area of the triangle with vertices $(-2,3)$, $(6,-1)$, and $(12,6)$.


Hannah can buy pins, pens, and pads for $\$2$, $\$3$, and $\$9$ each, respectively. What is the largest number of pads she can purchase if she must buy at least one of each item and will spend exactly $\$60$?


How many three-digit positive integers \( x \) satisfy \( 5462x + 729 \equiv 2689 \pmod{29} \) and \( x \) is even?


Without using a calculator, find the largest prime factor of $18^4 + 2 \times 18^2 + 1 - 17^4$.


Determine the domain of the function $f(x) = \sec(\arcsin(x^3)).$

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


Our club now has 30 members, 15 boys and 15 girls. Each gender has equal numbers of juniors and seniors (7 juniors and 8 seniors each). In how many ways can we choose a president and a vice-president if they must be of the same gender and should be from different grades?


For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?


What is the remainder when 3042 is divided by 29?


Jessica borrowed money from her friend to buy concert tickets. Her friend agreed to let her repay the debt by helping organize her garage sales under the following conditions: her first hour of help is worth $\$2$, the second hour $\$4$, the third hour $\$6$, the fourth hour $\$8$, the fifth hour $\$10$, the sixth hour $\$12$, the seventh hour $\$2$, the eighth hour $\$4$, etc. If Jessica repays her debt by working for 45 hours, how many dollars did she borrow?


A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. The altitude of the frustum is 15 centimeters, the area of its larger base is $64\pi$ square centimeters, and the area of its smaller base is $16\pi$ square centimeters. Calculate the altitude of the smaller cone that was removed.


If $p$, $q$, and $r$ are positive integers such that $\gcd(p, q) = 540$ and $\gcd(p, r) = 1080$, what is the smallest possible value of $\gcd(q, r)$?


Let \( a \), \( b \), \( c \), and \( d \) be real numbers satisfying:
\begin{align*}
a + b + c &= 5, \\
a + b + d &= -1, \\
a + c + d &= 8, \text{ and} \\
b + c + d &= 12.
\end{align*}
Calculate \( ab + cd \).


Determine the mean of the set $\{m, m + 6, m + 8, m + 14, m + 21\}$ if the median of the set is 16.


A pyramid with a volume of 60 cubic inches has a rectangular base. If the length of the base is tripled, the width quadrupled, and the height doubled, what is the volume of the new pyramid, in cubic inches?


Let $S$ be the set of positive real numbers. Define $f : S \to \mathbb{R}$ such that
(i) $f(1) = 0,$
(ii) $f \left( \frac{1}{x} + \frac{1}{y} \right) = f(x) + f(y)$ for all $x, y \in S,$
(iii) $\log(x + y) f(x+y) = \log(x)f(x) + \log(y)f(y)$ for all $x, y \in S.$

Find the number of possible functions $f(x).$


In a gathering, there are 25 witches and 18 zombies. Witches refuse to interact with other witches, but they willingly interact with all the zombies. The zombies are more sociable and interact with everyone, including other zombies. Each pair of creatures interacts at most once. Calculate how many interactions happened during this event.

Solution: \boxed{603 \text{ total interactions.}}


One hundred cards, numbered 1-100, are placed in a box. One card is randomly selected. What is the probability that the number on the card is both a prime and a multiple of 11?


You recently obtained over 200 oranges, organized into containers capable of holding exactly 15 oranges each. However, you found out that three containers were short by 2 oranges each. All the other containers were completely filled. What is the smallest number of oranges you could possess at this moment?


Jason is attempting to unlock a safe but has forgotten his six-digit combination. He only used digits 1 through 6 (possibly repeated), ensuring that every even number is followed by an odd number, and every odd number is followed by an even number. How many different combinations could Jason possibly have used?


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


A rectangular array of chairs in a classroom is to be arranged such that each row has the same number of chairs as every other row, and each column has the same number of chairs as every other column. There must be at least two chairs in every row and column and all of the chairs in the room must be included. How many distinct arrays are possible for a classroom containing 48 chairs?


How many three-digit positive integers $x$ satisfy $4253x + 527 \equiv 1388 \pmod{29}$?


In right $\Delta XYZ$, $\angle XYZ$ is a right angle. Point $N$ is the midpoint of $\overline{XZ}$. Length $\overline{XY}$ is 5 cm and $\overline{YZ}$ is 12 cm. Calculate the lengths of the median $\overline{YN}$ and the hypotenuse $\overline{XZ}$. Express your answers as decimals to the nearest tenth.

Solution: \boxed{6.5 \text{ cm}}


Consider the product $(2x^3 - 3x^2 + 5x - 1)(8 - 3x)$. If this product is written in the form $ax^4 + bx^3 + cx^2 + dx + e$, where $a, b, c, d, e$ are real numbers, find the value of $16a + 8b + 4c + 2d + e$.


The surface area of a sphere is $256\pi\text{ cm}^2$ and it is circumscribed around a cube. Find the volume of the sphere in cubic centimeters, and additionally find the side length of the cube. Express your answers in terms of $\pi$.

Solution: \boxed{s = \frac{16\sqrt{3}}{3}\text{ cm}}


Calculate three-fifths of the result after subtracting 3 from 10 times 7.


The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-6 & \mbox{if } n \geq 500 \\ g(g(n+7)) & \mbox{if } n < 500. \end{cases} \] Find $g(49)$.


Let $a$ and $b$ be real numbers. Consider the following five statements:
1. $\frac{1}{a} > \frac{1}{b}$
2. $|a| > |b|$
3. $a > b$
4. $a < 0$
5. $b > 0$

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


In a pre-game press conference, nine baseball All-Stars are to be seated in a row. Three are Cubs, three are Red Sox, and three are Yankees. If teammates insist on sitting together, how many ways can the nine All-Stars be seated?


What is the units digit of the product of all the odd positive integers between 20 and 120?


A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is either less than 3 or greater than 7? Express your answer as a common fraction.


A ball was floating in a river when it was trapped by rapidly forming ice. After the ball was carefully removed, a hole $30$ cm across at the top and $10$ cm deep was left. Determine the radius of the ball (in centimeters).


What is the base-10 integer 947 when expressed in base 7?


In Ms. Johnson's class, each student averages one day absent out of fifteen. What is the probability that out of any three students chosen at random, exactly one student will be absent while the other two are present? Express your answer as a percent rounded to the nearest tenth.


Jori starts with 3 gallons of distilled water. She uses 5/4 gallons for a biology experiment and then uses another 1/3 gallon for water testing. How much water does she have left?


How many even abundant numbers are there less than 50?


When a positive integer is expressed in base 8, it is $AB_8$, and when it is expressed in base 6, it is $BA_6$. What is the positive integer in decimal?


For each positive integer $n$, the mean of the first $n$ terms of a sequence is $2n$. What is the 1010th term of the sequence?


Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$.

Solution: \boxed{\sqrt{\sqrt{6} + 6}}


Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\circ$.

Solution: \boxed{\sqrt{289 + 120\sqrt{3}}}


What is the remainder when the sum of the first 150 counting numbers is divided by 5600?


Consider a quadratic of the form $x^2+bx+\frac14$, where $b$ is a specific negative number. George, using his skills in completing the square, rewrites this quadratic in the form $(x+n)^2+\frac{1}{16}$. What is $b$?


A rectangular prism has dimensions 12-inches by 30-inches by 15-inches. Determine the length, in inches, of the diagonal connecting the vertex at one corner to the opposite vertex at the far end. Express your answer in simplest radical form.

Solution: \boxed{3\sqrt{141} \text{ inches}}


What is the smallest prime divisor of $3^{15} + 5^{21}$?


Solution: \boxed{2012 - \frac{1}{2013}}


An infinite geometric series has a first term of $540$ and a sum of $4500$. What is its common ratio, and what is the second term of the series?


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


Let \( p_1, p_2, \) and \( p_3 \) be the roots of the cubic equation
\[ x^3 - 4x^2 + 5x - 3 = 0. \]
Find the monic polynomial in \( x \) whose roots are \( 3p_1, 3p_2, \) and \( 3p_3. \)

Solution: \boxed{x^3 - 12x^2 + 45x - 81}


Find the distance between the points (0,10) and (24,0).


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


Calculate the angle \(\theta\) for the expression \[ e^{11\pi i/60} + e^{23\pi i/60} + e^{35\pi i/60} + e^{47\pi i/60} + e^{59\pi i/60} \] in the form \( r e^{i \theta} \), where \( 0 \leq \theta < 2\pi \).


Samantha in her Chemistry class has a different task where she needs to prepare a solution by mixing 0.08 liters of chemical A with 0.04 liters of water, alongside 0.02 liters of chemical B. She plans to create 0.84 liters of this new solution. How many liters of water should she use?


In a given diagram, point $O$ is the center of the circle that circumscribes $\triangle DEF$, where $\angle DOF = 150^\circ$ and $\angle EOD = 130^\circ$. Determine the degree measure of $\angle DEF$.


Six horizontal lines and five vertical lines are drawn in a plane. Three of the horizontal lines are red, and two of the vertical lines are blue. In how many ways can four lines be chosen such that they form a rectangle, and the rectangle has sides from at least one red horizontal line and one blue vertical line?


Compute $3^{-1} \pmod{1013}$, and express your answer as a residue from $0$ to $1012$, inclusive.


Find all values of $x$ which satisfy $$\frac{x}{4} \leq 3 + 2x < -3(1 + 2x).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.

Solution: \boxed{\left[-\frac{12}{7}, -\frac{3}{4}\right)}


The Dunkers are coached by Coach Loudvoice. The Dunkers have 15 players, but three of them, Bob, Yogi, and Zack, refuse to play together in pairs. How many starting lineups (of 5 players) can Coach Loudvoice make, if the starting lineup can't contain any two of Bob, Yogi, and Zack at the same time?


Each point in an octagonal lattice is one unit from its nearest neighbor. How many equilateral triangles have all vertices in this lattice? Assume the shape is a regular octagon with a center point.


Find the angle $\theta$ in degrees where
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 140^\circ\]
is expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$.


Charlie and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where
\[g(n) = \left\{
\begin{array}{cl}
7 & \text{if } n \text{ is a multiple of 3 and 5}, \\
3 & \text{if } n \text{ is only a multiple of 3}, \\
0 & \text{if } n \text{ is not a multiple of 3}.
\end{array}
\right.\]
Charlie rolls the die four times and gets a 6, 2, 3, and 5. Dana rolls and gets 5, 3, 1, and 3. What is the product of Charlie's total points and Dana's total points?


If $x = \frac{2}{3}$ and $y = \frac{3}{2}$, find the value of $\frac34 x^8 y^9$.


When $\sqrt[4]{2^{10} \cdot 5^2}$ is fully simplified, what is the form $a \sqrt[4]{b}$, where $a$ and $b$ are positive integers? Additionally, find the sum $a+b$.


What is the median number of moons per celestial body? (Include Pluto and introduce a hypothetical planet X)
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0\\
Venus & 0\\
Earth & 1\\
Mars & 3\\
Jupiter & 18\\
Saturn & 25\\
Uranus & 17\\
Neptune & 2\\
Pluto & 4\\
Planet X & 3\\
\end{tabular}


If $\sqrt{3} \sin 20^\circ$ can be written as $\cos \theta + 2 \sin \theta$ for some acute angle $\theta$, what is $\theta?$ (Provide your answer in degrees, not radians.)


Consider a guessing game involving a two-digit number. Here are the clues given to guess the number:
1. It is a two-digit integer.
2. The tens digit is odd.
3. Both digits add up to an even number.
4. The number is between 40 and 80.
What is the probability that a guess following these clues will be the correct number? Express your answer as a common fraction.


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


Determine the smallest even abundant number greater than 12.


What is the coefficient of \( x^2 \) when
$$x^5 - 2x^4 + 4x^3 - 5x + 2$$
is multiplied by
$$3x^4 - x^3 + x^2 + 4x - 1$$
and like terms are combined?


Find the remainder when the polynomial $x^{1012}$ is divided by the polynomial $(x^2 + 1)(x - 1)$.


Let $M$ be the largest integer whose square has exactly $4$ digits when written in base 7.

What is $M$, expressed in base 7?


Let \( x \) be a real number. Consider the following five statements:

1. \( 0 < x^2 < 1 \)
2. \( x^2 > 1 \)
3. \( -1 < x < 0 \)
4. \( 0 < x < 1 \)
5. \( 0 < 2x - x^2 < 2 \)

What is the maximum number of these statements that can be true for any value of \( x \)?


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


A square contains six circles arranged in two rows and three columns. Each circle has a radius of 3 inches. What is the area, in square inches, of the square?


Determine the sum of consecutive integers $a$ and $b$ such that $\log_{10}{475728}$ is between $a$ and $b$.


What is the probability of rolling five standard, five-sided dice and getting five distinct numbers? Express your answer as a common fraction.


How many positive 3-digit numbers are multiples of 30, but not of 75?


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


Bag X has 4 white marbles and 5 black marbles. Bag Y has 7 yellow marbles and 3 blue marbles. Bag Z has 3 yellow marbles and 6 blue marbles. A marble is drawn at random from Bag X. If it is white, a marble is drawn at random from Bag Y, otherwise, if it is black, a marble is drawn at random from Bag Z. What is the probability that the second marble drawn is yellow?


Given five positive integers from a list of seven positive integers as $3, 5, 9, 7, 8$. What is the smallest possible value of the median of this list of seven positive integers if any additional numbers added must be less than 3?


What is the value of the expression $(50 + 12)^2 - (12^2 + 50^2)$?


Compute
\[
\sin^2 3^\circ + \sin^2 6^\circ + \sin^2 9^\circ + \dots + \sin^2 177^\circ.
\]


Determine the greatest common divisor of $g(x)=(5x+3)(8x+2)(12x+7)(3x+10)$ and $x$, where $x$ is a multiple of 46800.


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


Mark has $\frac{5}{6}$ of a dollar, Carolyn has $\frac{4}{9}$ of a dollar, and David has $\frac{7}{12}$ of a dollar. How many dollars do they have altogether? (Give your answer as a decimal.)


Alice's favorite number is between $80$ and $130$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Alice's favorite number?


A truck travels 300 miles using 10 gallons of gas. How far can it travel on 15 gallons of gas? Additionally, calculate the fuel efficiency of the truck.


Among all triangles $ABC$, find the maximum value of $\sin A + \sin B \sin C + \cos B \cos C.$


The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. Initially, an astronaut, Lucy, is 5,000 miles from the center of Earth where the gravitational force is 480 Newtons. Due to a mission, she travels to another celestial body located 300,000 miles from the center of the Earth. What is the new gravitational force (in Newtons) on her? Express your answer as a fraction.


What is the sum of all integer solutions to the inequality $4 < (x-3)^2 < 49$?


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) + g(y) - g(x)g(y) - xy\] for all $x,$ $y.$ Find the sum of all possible values of $g(1).$


Let
\[\mathbf{A} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.\]
Compute $\mathbf{A}^{98}.$

Solution: \boxed{\mathbf{A}^{98} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}}


How many four-digit perfect cubes are divisible by $16$?


In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 75^\circ,$ and $AC = 6.$ Find $BC$.

Solution: \boxed{6\sqrt{3} - 6}


Three of the four vertices of a rectangle are at $(4, 10)$, $(15, 10)$, and $(15, -3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 4)^2 + (y + 3)^2 = 16$? Express your answer in terms of $\pi$.


What is the smallest odd number with five different prime factors?


Let $Q$ be a point on the line segment $\overline{CD}$ such that the ratio $CQ:QD = 4:1$. Find the position vector $\overrightarrow{Q}$ in terms of $\overrightarrow{C}$ and $\overrightarrow{D}$, and determine the ordered pair $(t, u)$ such that
\[\overrightarrow{Q} = t \overrightarrow{C} + u \overrightarrow{D}.\]

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


A ball is dropped from a height of 20 meters and on each bounce, it rises to 80% of the height of the previous high point. The ball is caught after it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it traveled?


Solve for \( x \), where \( x \) is an integer and \( 2x^3 + 5x^2 - 9x - 18 = 0 \).


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


Acme Corporation has upgraded their alphabet soup to include the vowels A, E, I, O, U, and Y, each appearing five times in the soup, except for A, which appears only three times. How many five-letter words can be formed under these constraints, where the words do not have to be actual words in English?


Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$.


In quadrilateral $ABCD$, let $AB = BC = CD = DA$ and $\angle ABC = 90^\circ$. What is the measure of $\angle BCD$?


Suppose you have $7$ blue shirts, $6$ green shirts, $7$ pairs of pants, $10$ blue hats, and $9$ green hats, all distinct. How many different outfits can you create consisting of one shirt, one pair of pants, and one hat, where the shirt and hat are not the same color?


Let $\triangle XYZ$ be a right triangle with $Y$ as a right angle. A circle with diameter $YZ$ intersects side $XZ$ at point $W$. If the area of $\triangle XYZ$ is $195$ and $XZ = 30$, find the length of $YW$.


In right triangle $ABC$, $AB=12$, $BC=15$, and $\angle B = 90^\circ$. Points $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively; $\overline{CD}$ and $\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ to the area of triangle $BXC$.


Find the sum $m + n$ of the reduced fraction $\frac{s}{r}$, where $s$ is the number of squares and $r$ is the number of rectangles that can be formed in a $7\times7$ checkerboard grid formed by the 8 horizontal and 8 vertical lines.


A circle with radius 5 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of \(\pi\).

Solution: \boxed{7.5\pi \text{ cm}}


Alice can buy pens, pencils, and erasers for $\$3$, $\$4$, and $\$8$ each, respectively. What is the maximum number of erasers she can purchase if she must buy at least one of each item and will spend exactly $\$60$?


What is the base-10 integer 784 when expressed in base 7?


Let $p$ and $q$ be real numbers. One of the roots of $x^3 + px + q = 0$ is $2 + i$. Find $p + q$.


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$, where the graph of $y=g(x)$ consists of the three points forming a triangle of area $15$ square units. You now consider the graph of $y = 3g(\frac{x}{3})$. Determine the area of the triangle formed by the points of this new graph.


Lucy bought a new mountain bike and paid $20\%$ of the cost upfront, which was $\$240$. After enjoying the bike, she received a $10\%$ post-purchase cashback on the total cost. What was the effective price Lucy paid after accounting for the cashback?


The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1)?


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


How many integer values of $n$ satisfy $-60 < n^3 < 100$?


A rectangular pyramid has a base with a length of 20 cm and a width of 10 cm, and its altitude is 12 cm. A cut is made parallel to the base of the pyramid 6 cm from the apex, creating a smaller pyramid and a frustum. The area of the top rectangle of the frustum is one-fourth the area of the base of the original pyramid. Calculate the volume of the frustum.

Solution: \boxed{700 \text{ cm}^3}


Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. Write the simplified expression in the form $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Determine $A+B+C+D$ if the greatest common divisor of $A$, $C$, and $D$ is 1.


Convert $157_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 1, \\
a+b+d &= 5, \\
a+c+d &= 14, \text{ and} \\
b+c+d &= 9,
\end{align*}
what is $ab + cd$?


Call a positive integer $M$ an 8-10 match if the digits of the base-$8$ representation of $M$ form a base-$10$ number that equals $M$. For example, consider if $M=65$ whose base-$8$ representation is $101$. What is the largest 8-10 match?


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{250}$?


What is the base $2$ representation of $95_{10}$?


When a polynomial is divided by $3x^2 - 5x + 12,$ what are the possible degrees of the remainder?

Solution: \boxed{0, 1}


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


In the diagram, the smaller circles touch the larger circle externally and touch each other externally. The radius of the larger circle is $10$ units. If the smaller circles are identical and their centers are located on the diameter of the larger circle, what is the area of the shaded region?

[asy]
size(100);
import graph;
filldraw(Circle((0,0),5),mediumgray);
filldraw(Circle((-3,0),2),white);
filldraw(Circle((3,0),2),white);
[/asy]


Let $M$ be the largest integer whose square has exactly $4$ digits when written in base 7.

What is $M$, expressed in base 7?


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


What is the largest value among $\operatorname{lcm}[20,2]$, $\operatorname{lcm}[20,4]$, $\operatorname{lcm}[20,6]$, $\operatorname{lcm}[20,8]$, $\operatorname{lcm}[20,10]$, and $\operatorname{lcm}[20,12]$? Express your answer as an integer.


Consider a hyperbola for which the equations of the asymptotes are $y = 3x + 2$ and $y = -3x + 8$. It is given that this hyperbola passes through the point $(1, 6)$. The standard form for the equation of the hyperbola is:
\[
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,
\]
where $a,$ $b$, $h,$ and $k$ are constants, and $a, b > 0.$ Determine the value of $a + h$.


Solution: \boxed{6z^7 + 8z^6 - 23z^5 - 10z^4 + 27z^3 + 5z^2 - 14z + 2}


Joe now consumes fruits for breakfast, lunch, snack, and dinner, where he randomly chooses either an apple, an orange, or a banana with equal probabilities at each meal. What is the probability that Joe will eat at least two different kinds of fruit in a given day?


In a right triangle with integer-length sides, the hypotenuse is 65 units. Find the length of the shorter leg.


Let a function \( f \) be defined by \( f(z) = (5 + 2i) z^2 + \alpha z + \gamma \) for all complex numbers \( z \), where \( \alpha \) and \( \gamma \) are complex numbers, and \( i^2 = -1 \). Assume \( f(1) \) and \( f(-i) \) are both real numbers. Determine the smallest possible value of \( | \alpha | + |\gamma | \).


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


The digits of a four-digit positive integer add up to 16. The sum of the two middle digits is ten, and the thousands digit minus the units digit is two. If the integer is divisible by 11, what is the integer?


Solution: \boxed{\log_3 \left(\frac{55}{3}\right)}


Let \(x\) and \(y\) be positive real numbers. Find the minimum value of
\[
\left( x + \frac{1}{y^2} \right)\left( x + \frac{1}{y^2} - 500 \right) + \left( y + \frac{1}{x^2} \right)\left( y + \frac{1}{x^2} - 500 \right).
\]


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.2\overline{34}$?


In a set of five consecutive even integers, the largest integer is less than thrice the average of these five integers. What is the smallest integer that could be in this set?


When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 216$.


Let $x, y, z$ be nonnegative real numbers such that $x + y + z = 7.$ Find the maximum value of
\[
\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.
\]


What is the least common multiple of the numbers 504, 630, and 980?


Suppose that $b$ is a positive integer greater than or equal to $2$. When $256$ is converted to base $b$, the resulting representation has exactly $3$ digits. What is the number of possible values for $b$?


If I roll 3 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a prime number?


If $5x + 9 \equiv 3 \pmod{19}$, then $3x + 14$ is congruent $\pmod{19}$ to what integer between $0$ and $18$, inclusive?


Let $a$ and $b$ be positive real numbers such that $6a + 5b = 75.$ Find the maximum value of $ab.$


The Eiffel Tower in Paris is 324 meters tall. A scale model of the tower is designed with a height of 50 centimeters. Determine how many meters of the tower corresponds to one centimeter of the model?


Let $C$ equal the number of four-digit numbers that are both odd and multiples of 3. Let $D$ equal the number of four-digit multiples of 10. Find $C + D$.


Calculate the sum of the series \( \left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5 \).


If $\displaystyle\frac{a}{b} = 5$, $\displaystyle\frac{b}{c} = \frac{1}{2}$, $\displaystyle\frac{c}{d} = 4$, and $\displaystyle\frac{d}{e} = \frac{1}{3}$, then what is $\displaystyle\frac{e}{a}$?

Solution: \boxed{\frac{e}{a} = \frac{3}{10}}


Let $a,$ $b,$ and $c$ be the roots of the polynomial
\[x^3 - 5x + 6 = 0.\] Find the monic polynomial, in $x,$ whose roots are $a - 3,$ $b - 3,$ and $c - 3.$

Solution: \boxed{x^3 + 9x^2 + 22x + 18}


When $\frac{7}{5000}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


What is the largest four-digit number that is two more than a multiple of 11 and four more than a multiple of 7?


Calculate the volume in cubic feet of a round swimming pool with a diameter of 20 feet, where the depth varies linearly from 3 feet at one end to 6 feet at the other end. Express your answer in terms of $\pi$.


A day can be evenly divided into 36,000 periods of 2.4 seconds; 18,000 periods of each 4.8 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers and the total number of seconds in a day is 86,400?


For how many integer values of $n$ between 1 and 529 inclusive does the decimal representation of $\frac{n}{530}$ terminate?


Determine the value of $n$ if $n \cdot n! + n! = 5040$.


Suppose two distinct integers are chosen from between 1 and 20, inclusive. What is the probability that their product is odd and at least one of the integers is a prime number?


A book with 63 pages is numbered from 1 to 63. The pages are then renumbered in reverse, resulting in the last page (previously page 63) being numbered as 1, and so forth. Determine how many pages have the same units digit in both the original and the new numbering.


In a regular tetrahedron, if you scale down the coordinates of its vertices by half and then determine the vertices of the tetrahedron formed by connecting these points, what is the ratio, $p/q$, of the volume of this smaller tetrahedron to the larger regular tetrahedron, where $p$ and $q$ are relatively prime integers? Find $p+q$.


A sports conference has 12 teams arranged into two divisions of 6 teams each. Calculate the total number of games in a complete season for the conference if each team plays every other team in its own division three times and every team in the other division twice.


Given the five digits 1, 3, 5, 8, and 9, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer and the integer must be odd?


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

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


Consider the expression given by partial fractions:
\[
\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4} + \frac{F}{x + 5}
\]
for constants \(A, B, C, D, E, F\). Find \(A + B + C + D + E + F\).


Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket?


Evaluate the sum of the series: $2 - 5 + 8 - 11 + \cdots - 47 + 50 - 53 + 56 - 59$.


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


Find the product of all positive integral values of \( n \) such that \( n^2 - 47n + 552 = p \) for some prime number \( p \). Note that there is at least one such \( n \).


A square with sides 8 inches is shown. If $P$ is a point such that the segments $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{FD}$, find the area, in square inches, of triangle $APB$. Assume that point $C$ is at the midpoint of $\overline{FD}$. [asy]
pair A, B, C, D, F, P;
A = (0,0); B= (4,0); C = (2,4); D = (4,4); F = (0,4); P = (2,2);
draw(A--B--D--F--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$P$",P,NW);label("$F$",F,NW);
label("$8''$",(2,0),S);
[/asy]


John is conducting an experiment in his Chemistry class where he needs to create a solution by mixing 0.05 liters of Solution X with 0.03 liters of Solution Y to make a new solution. He wants to prepare a total of 0.64 liters of this mixture. How many liters of Solution Y will John need to use?


Determine the value of $x$ for which $\frac{1}{\log_5 x} + \frac{1}{\log_7 x} + \frac{1}{\log_8 x} = 1$.


A conference hosts 25 engineers, each assigned their own room numbered consecutively from 1 to 25. If engineers in rooms 15, 16, and 17 do not show up, what is the median room number of the present engineers?


Let $g(x) = \cos{x} + 3\sin{x} + 4\cot{x}$, using radian measure for the variable $x$. Let $s$ be the smallest positive value of $x$ for which $g(x) = 0$. Find $\lfloor s \rfloor$.


Let $b_1, b_2, \ldots$ be a sequence defined by $b_n = \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3 and $b_n = 2b_{n-1} + 1$ if $b_{n-1}$ is not divisible by 3. Determine for how many positive integers $b_1 \leq 2500$ it is true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$.


The digits 0, 3, 5, and 7 are arranged randomly to form a four-digit number. What is the probability that the number formed is an odd number and at least four digits long (non-zero leading digit)?


Determine the measure of angle $x$ in degrees in the triangle below:
[asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(70),Sin(70)), C = (Cos(70) + Sin(70)*sqrt(3),0);
draw(A--B--C--cycle);
label("$50^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy]


The number 1320 is the product of two consecutive integers and is also the product of three consecutive integers. What is the sum of those five integers?


After Jack makes 30 shots, he has scored $60\%$ of them. He then takes 10 more shots, and his shooting percentage increases to $62\%$. How many of the last 10 shots did Jack make?


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


A quadratic equation $x^2 - 6x + 15 = 51$ has two solutions, $p$ and $q$, with $p\geq q$. Calculate the value of $3p + 2q$.

Solution: \boxed{15 + 3\sqrt{5}}


A cylindrical tank with a circumference of 6 feet has a rope wrapped around it spiraling from the bottom to the top. The rope makes six complete loops around the tank, starting at the bottom edge and finishing at the top edge. The tank's height is 18 feet. Determine the length of the rope.

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


How many prime numbers between 40 and 85 have a prime remainder when divided by 12?


At a school with 2500 students, each student studies either Italian or German, and some study both languages. The number who study Italian is between 70% and 75% of the student population, while the number who study German is between 35% and 45%. Determine the difference between the maximum and minimum number of students who could study both languages.


Compute the value of \[M = 150^2 + 148^2 - 146^2 - 144^2 + 142^2 + \dots + 6^2 + 4^2 - 2^2.\] Here the additions and subtractions alternate in pairs as before, but now the pairing starts with addition for the first two and subtraction for the next two.


If I roll 6 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?


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

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


Five packages are delivered to five different houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction.


What is the smallest even abundant number?


Find the matrix $\mathbf{N}$ that doubles the first row and triples the second row of a matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.\]If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}}


Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 10$. Find the maximum possible value of $x^2 + 2xy + 3y^2$ and express it in the form $\frac{a + b \sqrt{c}}{d}$ where $a, b, c, d$ are positive integers. Determine $a + b + c + d$.


In triangle $ABC$, $P$ and $Q$ are points on sides $\overline{AC}$ and $\overline{AB}$ respectively, such that $AP = \frac{1}{3}AC$ and $AQ = \frac{1}{3}AB$. The area of $\triangle ABC$ is given as 36 square units. Determine the area of $\triangle CPQ$.


In triangle $ABC$, $\cot A \cot C = \frac{1}{3}$ and $\cot B \cot C = \frac{2}{9}$. Find $\tan C$.

Solution: \boxed{\tan C = \sqrt{6}}


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

Solution: \boxed{\left(\frac{27}{13}, \frac{9}{13}\right)}


Determine the remainder when $(x^5-1)(x^3-1)$ is divided by $1+x+x^2$.


What is the area enclosed by the graph of $|5x| + |2y| = 20$?


Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x+y)f(x-y) = (f(x) + f(y))^2 - 4x^2y^2 \]
for all real numbers $x$ and $y$.

Solution: \boxed{f(x) = -x^2}


An association begins with twenty people, including six leaders. Each year, the current leaders leave the association. Every remaining regular member then finds three new people to join as regular members. Subsequently, six new leaders are elected from outside the association. How many total people will be in the association after four years?


When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point?


Let $b \equiv (4^{-1}+6^{-1}+9^{-1})^{-1} \pmod{13}$. What is the remainder when $b$ is divided by 13?


Let $f(x) = x^2 - 9x + 20$, and let $g(f(x)) = 3x - 4$. What is the sum of all possible values of $g(9)$?


What is the value of $x + y$ if the sequence $3, ~7, ~11, \ldots, ~x, ~y, ~31$ is an arithmetic sequence?


If the product \((5x^2 - 3x + 2)(9 - 3x)\) can be written in the form \(ax^3 + bx^2 + cx + d\), where \(a, b, c, d\) are real numbers, then find \(27a + 9b + 3c + d\).


Fifteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Bella, starts with the ball, skips the next five girls and throws to the seventh girl, who then skips the next five girls and throws the ball to the following seventh girl. If the throwing pattern continues, how many total throws are necessary for the ball to return to Bella?


The value of
\[
\frac{n}{3} + \frac{50}{n}
\]
is smallest for which positive integer \( n \)?


Determine how many integer values of \(b\) are possible for which there are exactly three integers \(x\) satisfying the inequality
\[x^2 + bx + 6 \leq 0.\]


What is the base-10 integer 600 when expressed in base 7?


Determine the smallest positive integer $n$ such that
\[
\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}^n = \mathbf{I}.
\]


Let $Q(x) = x^4 - 4x^3 - 20x^2 + 44x + 120$. A real number $x$ is chosen at random from the interval $3 \le x \le 10$. Calculate the probability that $\lfloor \sqrt[4]{Q(x)} \rfloor = \sqrt[4]{Q(\lfloor x \rfloor)}$.


Find the 6-digit repetend in the decimal representation of $\frac{7}{29}$.


An ordinary 8-sided die has a number on each face from 1 to 8 (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces do not add up to a prime number?


What is the smallest positive integer $n$ such that $\frac{n}{n+50}$ is equal to a terminating decimal?


Find the smallest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{7x}{5}-2=\frac{4}{x}$, where $a, b, c,$ and $d$ are integers. What is $\frac{acd}{b}$?


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

Solution: \boxed{\left( 2\sqrt{2}, \frac{7\pi}{4} \right)}


Two circles are drawn in a 15-inch by 16-inch rectangle. Each circle has a diameter of 7 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{\sqrt{145}\text{ inches}}


Determine the number of solutions to the equation
\[3\cos^3 x - 9\cos^2 x + 6\cos x = 0\]
in the range $0 \leq x \leq 2\pi$.


A cone is formed from a 240-degree sector of a circle with a radius of 24. Calculate the result when the volume of the cone is divided by $\pi$.


On a Cartesian plane, where each unit represents one foot, a dog is tied to a post at point $(6, -2)$ with a $12$ foot rope. What is the greatest distance the dog can be from the origin?

Solution: \boxed{12 + 2\sqrt{10}}


Points $P$, $Q$, $R$, and $S$ are located consecutively on line segment $\overline{PQ}$ such that $PQ = 4PR = 8RS$. If a point is selected at random on $\overline{PQ}$, what is the probability that it is between $R$ and $S$?


Find the coefficient of the $x^2$ term in the expansion of the product $$(3x^2 +4x +5)(6x^2 +7x +8).$$


How many positive perfect squares less than $10^4$ are multiples of 36?


What is the median number of moons per celestial body? (Include dwarf planets Pluto and Eris, but exclude planets with no moons)
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Earth &1\\
Mars &2\\
Jupiter&20\\
Saturn&27\\
Uranus&17\\
Neptune&2\\
Pluto&5\\
Eris&1\\
\end{tabular}


Determine all values of the constant $d$ such that the function $g(x) = \frac{x^2 - 2x + d}{x^2 - 5x + 6}$ has exactly one vertical asymptote.

Solution: \boxed{0 \text{ or } -3}


Calculate the simplified form of $\sqrt[4]{2^8 \cdot 3^5}$. Express your answer in the form $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers, and find $a+b$.


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


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

Solution: \boxed{\left(-\frac{1}{4}, \frac{195}{4}\right)}


Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[ a - \frac{1}{b} = b - \frac{1}{c} = c - \frac{1}{a}. \]
Find $|abc|$.


A parallelogram has adjacent sides of lengths $3s$ units and $4s$ units forming a 30-degree angle. The area of the parallelogram is $18\sqrt{3}$ square units. What is the value of $s$?


Solution: \boxed{\frac{N^2 + N + 1}{N^3+3N^2+2N}}


Find the integer $n$, $5 \le n \le 9$, such that \[n \equiv 12345 \pmod{6}.\]


What is the probability of spinning the spinner shown and getting either a prime number or a multiple of 4? Express your answer as a common fraction. [asy]
import olympiad; defaultpen(linewidth(0.8)); size(100); dotfactor=4;
draw(Circle((0,0),1));
string[] labels = {"3","6","1","4","5","2","8"};
for(int i = 0; i < 7; ++i){

label(labels[i],0.7*dir(360/7*i));

draw((0,0)--dir(360/7*i - 360/14));
}
draw((0,0)--0.7*dir(360/7*2 - 360/14),linewidth(2.0),EndArrow(size=6));
[/asy]


At the grocery store, I bought 5 different items. I have 3 identical bags, and I give them to the cashier. How many ways can the cashier place these items into the 3 bags, given that some bags may remain empty?


Add $5_6 + 21_6.$ Express your answer in base $6.$


A line through the points $(1, 4)$ and $(m, -3)$ is parallel to the line $5x + 4y = 20$. What is the value of $m$?


A square has sides of length 12, and a circle centered at one of its vertices has radius 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.

Solution: \boxed{144 + 108\pi}


How many integers are there between $\sqrt{28}$ and $\sqrt{65}$ on a number line?


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


What is the value of \(x\) for which \((2012 + x)^2 = x^2\)?


What is the least four-digit positive integer, with all different even digits, that is divisible by each of its digits and also divisible by $5$?


Solution: \boxed{\sqrt{\sqrt{6} - 1}}


For how many four-digit positive integers is the sum of the digits equal to $9$?


Alice and Bob modify their game with a focus on improving their passing skills. Now, if Alice has the ball, there is a 2/3 chance she will pass it to Bob and a 1/3 chance she will keep it. If Bob has the ball, there is a 1/2 chance he will pass it back to Alice, otherwise, he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?


When the two-digit integer \( XX \), with equal digits, is multiplied by the one-digit integer \( X \), the result is the three-digit integer \( PXQ \). What is the greatest possible value of \( PXQ \) if \( PXQ \) must start with \( P \) and end with \( X \)?


Determine the coefficient of $x$ when simplifying the expression $2(x - 5) + 5(8 - 3x^2 + 4x) - 9(3x - 2)$.


Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a}
\]
where \(a,\ b,\) and \(c\) are positive real numbers.


What is the modulo $7$ remainder of the sum $1+2+3+\ldots+1000$?


A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 8 cm. Calculate the volume of the truncated cone and its lateral surface area.

Solution: \boxed{15\pi\sqrt{89} \text{ cm}^2}


Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5?


Both roots of the quadratic equation $x^2 - 79x + k = 0$ are prime numbers. Find the number of possible values of $k$.


How many of the 512 smallest positive integers written in base 8 use 6 or 7 (or both) as a digit?


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?


Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 6$ and $X$ is an integer, what is the smallest possible value of $X$?


Consider the polynomial \( Q(x) = x^3 + dx^2 + ex + f \) which has the property that twice the mean of its zeros, the product of its zeros taken two at a time, and the sum of its coefficients, all equal. If the \( y \)-intercept of the graph of \( y = Q(x) \) is 4, determine \( e \).


There are four distinct positive integers $a, b, c, d$ less than $12$ which are invertible modulo $12$. Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $12$.


Simplify the expression $\frac{6x^2}{4y^{-2}} \cdot \frac{2y^3}{3x^{-1}}$.


Consider the function \( g(x) = 3x^2 - 6x - 8 \). Determine the largest interval containing the point \( x = 2 \) on which \( g \) can be restricted to make it invertible.

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


Six couples are at a gathering. Each person shakes hands exactly once with everyone else except their own spouse and except one person who is not their spouse. How many handshakes are exchanged?


Tom is attempting to enclose a rectangular area of at least 150 sq. ft using the minimum length of fencing. The length of the rectangle should be twice the width. What should the width, in ft, be?


In a diagram, triangle $PQR$ has points $S$ and $T$ on line segment $QR$, dividing $QR$ into three segments $QS$, $ST$, and $TR$ such that $QS:ST:TR = 2:5:3$. The area of $\triangle PQS$ is $60$ square centimeters. What is the area of $\triangle PTR$?

Solution: \boxed{90 \text{ square centimeters}}


If $g(x) = 3x^3 + 3x^2 - x + 1$, what is the value of $g(g(3))$?


A butcher needs to know the weight in pounds of a steer that weighs 350 kg for record-keeping, using the conversion rate of 0.454 kilograms per pound. Round the result to the nearest whole pound.

Solution: \boxed{771 \text{ pounds}}


For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(6,6)--(14,2)--(8,-9)--cycle,linewidth(0.7));
draw((0,0)--(14,2),dashed);
label("9",(3,3),NW);
label("12",(10,4),NE);
label("20",(11, -3.5),SE);
label("15",(4,-3.5),SW);
[/asy]


A circle centered at $(0,k)$ with $k > 4$ is tangent to the lines $y=x$, $y=-x$, and $y=4$. Determine the radius of this circle.

Solution: \boxed{r = 4(1+\sqrt{2})}


A right cylinder has a height of 10 inches and a radius of 3 inches. Calculate both the area of the curved surface and the total surface area of the cylinder, in square inches. Express your answers in terms of \(\pi\).


Determine the domain of the function $g(x) = \log_5(\log_3(\log_2(x)))$.

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


The equation $x^2 - 6x + 13 = 25$ has two solutions, $c$ and $d$, with $c \geq d$. What is the value of $c + 2d$?

Solution: \boxed{9 - \sqrt{21}}


Tamika selects two different numbers at random from the set $\{6, 7, 8\}$ and multiplies them. Carlos takes two different numbers at random from the set $\{2, 3, 5\}$ and also multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.

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


What is the largest four-digit number whose digits add up to 20?


For $-1 < r < 1$, let $T(r)$ represent the sum of the geometric series \[20 + 20r + 20r^2 + 20r^3 + \cdots.\] Determine $a$ between $-1$ and $1$ such that $T(a)^2 \cdot T(-a)^2 = 640000$. Find $T(a) + T(-a)$.


Find the remainder when $2 \times 13 \times 22 \times 23 \times 32 \times 33 \times \ldots \times 82 \times 83 \times 92 \times 93$ is divided by $5$.


Given that $f(x) = x^k + 5$ where $k < 0$, what is the range of $f(x)$ on the interval $[2, \infty)$?

Solution: \boxed{(5, 2^k + 5]}


The side of a square has the length $(x - 4)$, while a rectangle has a length of $(x - 5)$ and a width of $(x + 6)$. If the area of the rectangle is three times the area of the square, what is the sum of the possible values of $x$?


Consider an isosceles triangle with two sides of 15 cm and 30 cm. If the smallest side of a similar triangle is 75 cm, find the perimeter of this larger triangle.

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


How many non-similar quadrilaterals have angles whose degree measures are distinct positive integers in an arithmetic progression?


How many three-digit numbers are multiples of neither 6 nor 8?


In triangle $PQR,$ points $J$ and $K$ are on $\overline{PQ}$ and $\overline{PR},$ respectively, and the angle bisector $\overline{PS}$ intersects $\overline{JK}$ at $L.$ If $PJ = 2,$ $JQ = 6,$ $PK = 3,$ and $KR = 9,$ compute $\frac{PL}{PS}.$

Solution: \boxed{\frac{5}{20} = \frac{1}{4}}


Find the maximum value of
\[\cos^2 \frac{\theta}{2} \cdot (1 - \sin \theta)\] for $0 \leq \theta \leq \frac{\pi}{2}.$


Square $ABCD$ has sides of length 6, and $M$ is the midpoint of $\overline{CD}$. A circle with radius 3 and center $M$ intersects a circle with radius 6 and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?


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


Find the intersection of the lines $6x - 5y = 15$ and $8x + 3y = 1.$ Express your answer as an ordered pair $(x,y).$

Solution: \boxed{\left(\frac{25}{29}, -\frac{57}{29}\right)}


Two classes each consisting of 15 students are participating in a quiz event where each student from one class is paired with a student from the other class. If Margo from class A wants to be paired with Irma from class B, what is the probability that this occurs if the pairing is done randomly? Express your answer as a common fraction.


Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x - \cos x = \sqrt{2}.$


If $m$ and $n$ are even integers, how many terms in the expansion of $(m+n)^8$ are odd?


Suppose $\sqrt{3 + \sqrt{4y-5}} = \sqrt{8}$; find $y$.


What is the diameter in centimeters of a circle whose area is $400\pi \text{cm}^2$, if the diameter is defined as four times the square of the radius?

Solution: \boxed{1600 \text{ cm}}


Michael is preparing a report on foreign language classes for a school project. His school offers French and Spanish. There are 30 students enrolled in at least one of these language classes. Specifically, 20 students are enrolled in the French class, and 24 students are enrolled in the Spanish class. Michael will randomly select two students from his list to interview. What is the probability that he obtains enough information to write about both the French and Spanish classes after the interviews? Express your answer as a fraction in simplest form.


An athlete's heart rate starts at 140 beats per minute at the beginning of a race and increases by 5 beats per minute for each subsequent mile. How many times does the athlete's heart beat during a 10-mile race if the athlete runs at a pace of 6 minutes per mile?


What is the area enclosed by the graph of $|2x| + |3y| = 6$?


How many positive 4-digit numbers are divisible by 9?


Let $a \star b = \frac{\sqrt{a+2b}}{\sqrt{a-b}}$. If $y \star 10 = 5$, find $y$.


The numbers from 1 to 200, inclusive, are placed in a bag, and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1)?


Find the point on the line
\[y = \frac{x - 3}{3}\]
that is closest to the point $(4,-2)$.

Solution: \boxed{\left( \frac{33}{10}, \frac{1}{10} \right)}


The graph of $y = \frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determine $\frac{p(-1)}{q(-1)}$ if $q(x) = (x+4)(x-1)$ and $p(x) = kx + m$.


What is the area, in square units, of a trapezoid bounded by the lines \(y = x + 1\), \(y = 12\), \(y = 7\) and the \(y\)-axis?


In the diagram, two concentric circles are drawn with $O$ as their common center. The outer circle is divided into two equal areas, and the inner circle has two-thirds of its area shaded. If the area of the larger circle is $81\pi,$ determine the total area of the shaded regions. The diagram is as follows:
[asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,0),1,0,240)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((0,0),1));

dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,0)--(cos(240*pi/180),sin(240*pi/180))); //indicates 240 degrees, approximately.
[/asy]


A square is inscribed in the ellipse
\[\frac{x^2}{5} + \frac{y^2}{10} = 1,\]
so that its sides are parallel to the coordinate axes. Find the area of the square.


There exists a positive real number $x$ such that $\sin (\arctan (x)) = \frac{1}{x}$. Find the value of $x^2$.

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


How many prime numbers have squares between 2500 and 5500?


Let $0 \le a, b, c \le 2$. Find the maximum value of
\[\sqrt{a^2b^2c^2} + \sqrt{(2 - a)(2 - b)(2 - c)}.\]


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Find the intersection of the lines $8x - 5y = 40$ and $6x + 2y = 14.$ Express your answer as an ordered pair $(x, y).$

Solution: \boxed{\left(\frac{75}{23}, \frac{-64}{23}\right)}


Grandpa now has 12 pieces of art, including 4 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all four pieces by Escher will be placed consecutively?


Find the matrix $\mathbf{N}$ that doubles the first row and triples the second row of a matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.\]
If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.


**
Let $n$ be a positive integer, and let $b_0, b_1, \dots , b_n$ be a sequence of real numbers such that $b_0 = 41$, $b_1 = 68$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{4}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$.

**


Hazel has forgotten her safe code which is a sequence of three numbers, each in the range from 1 to 40, inclusive. She remembers that the first number is a multiple of 4, the second number is odd, and the third number is a multiple of 5. How many combinations could possibly be Hazel's?


How many different six-digit integers can be formed using the digits 2, 2, 2, 5, 5, and 9?


Calculate how many distinct numbers are present in the list:
\[\left\lfloor \frac{1^2}{2000} \right\rfloor, \left\lfloor \frac{2^2}{2000} \right\rfloor, \left\lfloor \frac{3^2}{2000} \right\rfloor, \dots, \left\lfloor \frac{1000^2}{2000} \right\rfloor.\]


Let $x$ and $y$ be two positive real numbers such that $x + y = 50.$ Determine the ordered pair $(x,y)$ for which $x^4 y^3$ is maximized.

Solution: \boxed{\left(\frac{200}{7}, \frac{150}{7}\right)}


How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$? Assume $B$, $C$, and $D$ are consonants, words are valid irrespective of their meaning in the English language, and letters may be reused.


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


Three tour guides are leading eight tourists. The guides decide to split up, and each tourist must choose to join one of the three guides, but under the condition that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?


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


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


When $g(x) = cx^3 - 8x^2 + dx - 7$ is divided by $x - 2,$ the remainder is $-7.$ When $g(x)$ is divided by $x + 3,$ the remainder is $-80.$ Find the ordered pair $(c,d).$

Solution: \boxed{\left(-\frac{47}{15}, \frac{428}{15}\right)}