What is the ratio of the area of a square inscribed in a quarter circle with radius $r$ to the area of a square inscribed in a circle with radius $2r$? Express your answer as a common fraction.


Compute the largest integer $m$ such that $2016^m$ divides $2016!$.


Milly has twelve socks, three of each color: red, blue, green, and yellow. She randomly draws five socks. What is the probability that she has exactly two pairs of socks of different colors and one sock of a third color?


If two numbers are randomly chosen without replacement from the set $\{4, 5, 6, 9\}$, what is the probability that their product will be a multiple of 16? Express your answer as a common fraction.


How many total days were there in the years 2005 through 2010?


Let $r$ be a complex number such that $r^7 = 1$ and $r \neq 1.$ Compute
\[(r^2 - 1)(r^3 - 1)(r^4 - 1)(r^5 - 1)(r^6 - 1).\]


When the binary number $110110111101_2$ is divided by 4, what is the remainder (give your answer in base 10)?


How many positive divisors does 8! have?


Three cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$, the second card dealt is a $\clubsuit$, and the third card dealt is a $\spadesuit$?


Let \( T_n \) denote the sum of the first \( n \) terms of an arithmetic sequence with a common difference of 5. If \( \frac{T_{4n}}{T_n} \) is a constant that does not depend on \( n \), for all positive integers \( n \), then find the first term.


In a right-angled triangle, the sum of the squares of the three side lengths is 2500. What is the length of the hypotenuse of this triangle?


A square with sides of 10 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 $\overline{GD}$, what is the area, in square inches, of triangle $APB$? Here, $G$ is the midpoint of side $AD$.
[asy]
pair A, B, C, D, G, P;
A = (0,0); B= (10,0); C = (5,10); D = (10,10); G = (0,10); P = (5,5);
draw(A--B--D--G--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("$G$",G,NW);label("$P$",P,NW);
label("$10''$",(5,0),S);
[/asy]


The hypothetical alphabet for a particular community consists of 14 letters: A, B, C, D, E, F, G, H, I, J, K, L, M, N. Suppose license plates of six letters utilize only the letters in this alphabet. How many license plates of six letters are possible that begin with either B or C, end with N, cannot contain M, and have no letters that repeat?


Find the largest prime factor of $12321$.


Let $A$ be the greatest common factor and let $B$ be the least common multiple of 18, 36, and 72. What is the value of $A + B$?


The numbers $\sqrt{3u-2}$, $\sqrt{3u+2}$, and $2\sqrt{u}$ are given as the side lengths of a triangle. What is the measure of the largest angle in this triangle?


Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$


A kite has diagonals measuring 20 feet and 16 feet. What is the perimeter of the kite?

Solution: \boxed{8\sqrt{41} \text{ feet}}


Rodney uses the following clues to try to guess a secret number:
- It is a two-digit integer.
- The tens digit is odd.
- The units digit is even.
- The number is greater than 75.

What is the probability that Rodney will guess the correct number if he guesses a number that meets all these criteria? Express your answer as a common fraction.


Mr. Lee has 8 children. Assuming that the gender of each child is determined independently but not necessarily equally (probability of a child being male is 0.4 and female is 0.6), what is the probability that Mr. Lee has more sons than daughters or more daughters than sons?

Solution: \boxed{0.59614528 + 0.17161152 = 0.7677568}


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


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


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


The graph of a rational function $\frac{1}{q(x)}$ shows vertical asymptotes at $x = -2$ and $x = 3$. Given that $q(x)$ is a quadratic polynomial and we know that $q(1) = 8$, determine $q(x)$.

Solution: \boxed{-\frac{4}{3}x^2 + \frac{4}{3}x + 8}


Factor $x^2 + 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2 + bx + c)(dx^2 + ex + f)$, with $a < d$.

Solution: \boxed{(-8x^2 + x + 3)(8x^2 + x + 3)}


Cameron writes down the smallest positive multiple of 25 that is a perfect square, the smallest positive multiple of 25 that is a perfect cube, and all the multiples of 25 between them. How many integers are in Cameron's list?


$ABCDEFGH$ is a cube. Determine $\cos \angle BCG$.


Consider the function $y = g(x)$ where $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, and $D, E, F$ are integers. The graph has vertical asymptotes at $x = -3$ and $x = 2$. It is also given that for all $x > 5$, $g(x) > 0.5$. If additionally, $g(0) = \frac{1}{4}$, find $D + E + F$.


Find the minimum value of
\[ f(x) = x^2 + \frac{1}{x^2} + \frac{1}{x^2 + \frac{1}{x^2}} \]
for $x > 0$.


If \( x - 5 \) is a factor of \( cx^3 + 23x^2 - 5cx + 55 \), find the constant \( c \).


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

Solution: \boxed{15\sqrt{5} \text{ feet}}


Find the number of solutions to
\[\sin x = \left( \frac{3}{4} \right)^x\]
on the interval \( (0, 50\pi) \).


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

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


Two base-10 numbers, 348 and 27, are multiplied. The product is then written in base-8. What is the units digit of the base-8 representation?


A sample of helium gas is kept in a 3.5 liter container where it exerts a pressure of 8 kPa. If we transfer this gas to a 7 liter container while keeping the temperature constant, what will the new pressure be?


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


The current time is 3 o'clock. On a 12-hour clock, what time will it be 2517 hours from now?


For each positive integer $n$, let $n!$ denote the product $1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$. What is the remainder when $10!$ is divided by $13$?


The median of the set $\{m, m + 2, m + 7, m + 10, m + 12\}$ is 10. What is the mean?


Evaluate the expression \[ (x^2 + c)^2 - (x^2 - c)^2 \] where $x$ represents any real number, and $c$ is a fixed positive constant.


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


Convert the base 2 number \(1011111010_2\) to its base 4 representation.


For what base-7 digit $d$ is $3dd6_7$ divisible by the base 10 number 13? (Here $3dd6_7$ represents a base-7 number whose first digit is 3, whose last digit is 6, and whose middle two digits are both equal to $d$).


The integer $y$ has 18 positive factors. The numbers 18 and 24 are factors of $y$. What is $y$?


All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list?


Suppose three distinct integers are chosen from between 4 and 16, inclusive. What is the probability that their product is odd?


How many positive integers smaller than $500,000$ are powers of $2$, but are not divisible by $5$?


The integer $y$ has 18 positive factors. The numbers 14 and 18 are factors of $y$. What is $y$?


Let $m$ be the product of all positive integers less than $6!$ which are invertible modulo $6!$. Find the remainder when $m$ is divided by $6!$.


Determine all values of $x$ that satisfy the following inequalities:
$$\frac{x}{4} \leq 3 + 2x \quad \text{and} \quad 3 + 2x < -3(2+x).$$
Express your answer in interval notation.

Solution: \boxed{\left[-\frac{12}{7}, -\frac{9}{5}\right)}


Find the equation of the directrix of the parabola \( y = -\frac{1}{4} x^2 \).

Solution: \boxed{y = 1}


A rectangular prism has a length of 4, a width of 3, and a height of 2. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does this rectangular prism have?


Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(x+5) = 26x - 130.$$ If $p > q$, what is the value of $p - q$?


In how many ways can the digits of $64,\!530$ be arranged to form a 5-digit number, given that numbers cannot begin with 0?


The hypotenuse of a right triangle measures 12 inches and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle?


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


The sum of 81 consecutive integers is $3^8$. What is their median?


Let $z$ be a complex number such that
\[ z^2 + |z|^2 = 8 - 2i. \]
Find $|z|^2$.


William is taking the 30-question, multiple-choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last six questions. What is the probability that he will get at least one of these final six questions right?


In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\]


Find the area bounded by the graph of $y = \arccos(\cos x)$ and the $x$-axis on the interval $0 \le x \le 2\pi$.


Solution: \boxed{\frac{7 - x}{3}}


Calculate the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$.


A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors?


Our school's volleyball team has increased to 16 players, including a set of triplets: Alicia, Amanda, and Anna, and a set of twins: Bella and Cindy. In how many ways can we choose 7 starters if exactly one of the triplets and exactly one of the twins must be in the starting lineup?


There are 4 blue chips, 5 yellow chips, and 3 green chips in a bag. One chip is drawn from the bag, and then a second chip is drawn without replacement. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Michael is considering 10 different math classes, out of which 6 are advanced and 4 are introductory. In how many ways can he choose a total of 5 classes such that he must take at least 3 advanced classes?


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


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


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?


Consider triangle $DEF$ with vertices $D(1, 6)$, $E(3, -2)$, $F(7, -2)$. A line passing through $E$ bisects the area of triangle $DEF$. Determine the sum of the slope and the $y$-intercept of this line.


In the expansion of $(x+3)^{40}$, what is the coefficient of the $x^2$ term?

Solution: \boxed{780 \cdot 3^{38}}


The positive integers $A, B$ and $C$ form an arithmetic sequence while the integers $B, C$ and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{4}$, what is the smallest possible value of $A + B + C + D$?


If $a$, $b$, and $c$ are positive integers satisfying $ab+c = ac+b = bc+a = 50$, what is the value of $a+b+c$?


In a modified deck of 60 cards with 30 black and 30 red cards, the cards are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black? Express your answer as a common fraction.


Let $m$ be a scalar such that for any vectors $\mathbf{p}$, $\mathbf{q}$, and $\mathbf{r}$ satisfying $\mathbf{p} + \mathbf{q} + \mathbf{r} = \mathbf{0}$, the equation
\[m (\mathbf{q} \times \mathbf{p}) + 2\mathbf{q} \times \mathbf{r} + 2\mathbf{r} \times \mathbf{p} = \mathbf{0}\] holds. Find the scalar $m$.


Determine the last two digits of the sum: $$3! + 6! + 9! + 12! + 15!$$


In triangle $PQR$, two of the altitudes have lengths of $3$ and $9$. If the length of the third altitude is also an integer, what is the largest possible value of the third altitude?


If 9 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are exactly equal to 80 knocks?

Solution: \boxed{192 \text{ knicks}}


Factor $x^2 - 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a < d$.

Solution: \boxed{(-8x^2 + x - 3)(8x^2 + x - 3)}


We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 16?


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


Find the largest value of $n$ such that $3x^2 + nx + 108$ can be factored as the product of two linear factors with integer coefficients.


Two congruent cylinders each have a radius of 10 inches and a height of 4 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. What is the increase in inches? Express your answer as a common fraction.


Find the maximum difference between the $y$-coordinates of the intersection points of the graphs of $y=4-x^2+x^3$ and $y=2+x^2+x^3$.


Find the area of the triangle with vertices $(2,-3),$ $(0,1),$ and $(4,-4).$


Julie works for 60 hours per week for 8 weeks during the summer, making \$6000. If she plans to work for 40 weeks during the school year at the same rate of pay and needs to make another \$10000, how many hours per week must she work?


Expand and simplify the following product: $\frac{3}{4}\left(\frac{8}{y} + 6y^3\right)$.

Solution: \boxed{\frac{6}{y} + \frac{9y^3}{2}}


Find the $50^{\text{th}}$ term in the sequence of all positive integers which are either powers of 3 or sums of distinct powers of 3.


A triangle with sides of 8, 15, and 17 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?


The Rotokas of Papua New Guinea have twelve letters in their alphabet: A, E, G, I, K, O, P, R, T, U, V, and B. Suppose license plates of five letters utilize only these letters. How many license plates of five letters are possible that begin with either G or K, end with B, cannot contain V, and have no letters that repeat?


Express $325_{6}-215_{9}$ as a base 10 integer.


A sequence consists of $1500$ terms. Each term after the first is $2$ larger than the previous term. The sum of the $1500$ terms is $12000$. When every third term is added up starting with the first term, what is the sum?


What is the smallest odd number with four different prime factors where the smallest factor must be greater than $3$?


What is the smallest positive integer $n$ such that $1023n \equiv 2147n \pmod{30}?$


How many nonnegative integers can be represented in the form \[
a_7 \cdot 4^7 + a_6 \cdot 4^6 + a_5 \cdot 4^5 + a_4 \cdot 4^4 + a_3 \cdot 4^3 + a_2 \cdot 4^2 + a_1 \cdot 4^1 + a_0 \cdot 4^0,
\]
where $a_i \in \{0, 1, 2\}$ for $0 \leq i \leq 7$?


Five of the following test scores are Timothy's and the other four are Anna's: 88, 90, 92, 95, 96, 98, 100, 102, 105. Timothy's mean score is 95. What is Anna's mean score?


It is currently $10\!:\!15\!:\!30 \text{ a.m.}$ What time will it be in $9999$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)


A function $f$ is defined by $f(z) = -i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Determine how many values of $z$ satisfy both $|z| = 7$ and $f(z) = z$.


Astrid rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number she obtains? Express your answer as a decimal.


Solution: \boxed{3x^{13} + 7x^{12} + 9x^{11} + 3x^9 + 8x^7 + 6x^4 - 7x + 9}


Let \( O \) be the origin, and \( (a, b, c) \) be a fixed point. A plane passes through \( (a,b,c) \) and intersects the \( x \)-axis, \( y \)-axis, and \( z \)-axis at \( A, B, \) and \( C \), respectively, all distinct from \( O \). Let \( (d, e, f) \) be another distinct point not lying on the axes or the origin. Let \( (p, q, r) \) be the center of the sphere passing through \( A, B, C, O, \) and \( (d, e, f) \). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]


Without using a calculator, find the largest prime factor of $16^4 + 2 \times 16^2 + 1 - 13^4$.


What is the sum of the digits of the base-2 expression for $345_{10}$?


To transmit a positive integer less than 200, the Networked Number Node offers two options.

Option 1: Pay $\$$d to send each digit d. For example, 187 would cost $\$$1 + $\$$8 + $\$$7 = $\$$16 to transmit.

Option 2: Encode integer into binary first, and then pay $\$$d to send each digit d. For example, 187 becomes 10111011 in binary and would cost $\$$1 + $\$$0 + $\$$1 + $\$$1 + $\$$1 + $\$$0 + $\$$1 + $\$$1 = $\$$6.

What is the smallest integer less than 200 that costs the same whether using Option 1 or Option 2?


Given that $11^{-1} \equiv 5 \pmod{23}$, find the modular inverse of $17 \pmod{23}$, expressed as a residue between 0 and 22, inclusive.


Ms. Johnson holds an art competition with 200 participants. Each participant's work is awarded points. To encourage competition, extra credit points are awarded to those whose scores exceed the average score of all participants. What is the maximum number of participants who can receive extra credit points?


Berengere and her Canadian friend Lucas are at a bakery in Paris that accepts both euros and Canadian dollars. They want to buy a pastry, but neither of them has enough money alone. If the pastry costs 8 euros and Lucas has a Canadian ten-dollar bill, how many euros does Berengere need to contribute to the cost of the pastry if 1 euro = 1.5 CAD?

Solution: \boxed{\frac{4}{3} \text{ euros}}


On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in the same lane is one car length for every 10 kilometers per hour of speed or fraction thereof. Suppose each car is 5 meters long, and a photoelectric eye at the side of the road detects the number of cars that pass by in one hour. Determine the whole number of cars passing the eye in one hour if the speed in the right lane is 50 kilometers per hour. Calculate $M$, the maximum result, and find the quotient when $M$ is divided by 10.


Find the largest possible value of $k$ for which $5^7$ is expressible as the sum of $k$ consecutive positive integers.


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


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


Let \(a,\ b,\ c,\ d\) be real numbers such that \(a + b + c + d = 10\) and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of \(d\).

Solution: \boxed{\frac{5 + \sqrt{105}}{2}}


Suppose that $p$ is a prime number and $2034_p + 405_p + 12_p + 212_p + 7_p = 1315_p + 541_p + 222_p$. How many possible values of $p$ are there?


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


According to the table below, how many dollars are in the median value of the 75 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$135{,}000$\\\hline
Vice-President&5&$\$95{,}000$\\\hline
Director&12&$\$80{,}000$\\\hline
Associate Director&8&$\$55{,}000$\\\hline
Administrative Specialist&33&$\$28{,}000$\\\hline
Entry Level Analyst&16&$\$18{,}000$\\\hline
\end{tabular}


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that \( g(2) = 2 \) and
\[ g(xy + g(x)) = xg(y) + g(x) \]
for all real numbers \( x \) and \( y \). Determine the number of possible values of \( g \left( \frac{1}{3} \right) \), denote it by \( m \), and the sum of all possible values of \( g \left( \frac{1}{3} \right) \), denote it by \( t \). Find \( m \times t \).


Given $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 3,$ find $\begin{vmatrix} 3x & 3y \\ 6z & 6w \end{vmatrix}.$


Among all the roots of
\[z^{12} - z^9 + z^6 - z^3 + 1 = 0,\]
the maximum imaginary part of a root can be expressed as $\sin \theta,$ where $-90^\circ \le \theta \le 90^\circ.$ Find $\theta.$


Below is a portion of the graph of a function, \( y = F(x) \):

\[ F(x) = \sqrt{|x+2|} + \frac{12}{\pi}\tan^{-1}(\sqrt{|x-1|}) \]

Determine the value of \( F(4) \).

Solution: \boxed{\sqrt{6} + 4}


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

Solution: \boxed{16.25 \text{ cm}}


Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)


The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?


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


At a gathering, there are eight individuals—four men and four women—of differing heights. Each man decides only to shake hands with individuals shorter than himself and doesn’t shake hands with any of the women. Each woman decides to shake hands only with women shorter than herself. How many handshakes take place if all participants follow these rules?


A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(4, -3),$ $(0, -2),$ and $(2, t).$ Find \( t^2 \).


A hemisphere-shaped bowl with radius 2 feet is filled with jelly. The jelly is then evenly distributed into 64 smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet?


Consider a new quadratic equation $$25x^2 + 30x - 35 = 0.$$ Rewrite this equation in the form $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. Determine the value of $a + b + c$.


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


Crestview's school is introducing a new color, green, in addition to purple and gold. The students are tasked with designing a new flag using four solid-colored horizontal stripes. How many different flags are possible if the first and the last stripes can only use either purple or gold, whereas the remaining stripes can use any of the three colors, and the middle second stripe must be a color different from the first?


Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 15$, what is $b_2b_7$?


In the game of CodeCracker, secret codes are created by placing pegs of any of six different colors into five slots. Colors may be repeated, however, the first slot must always contain a color that is not used in any of the other four slots. How many secret codes are possible?


Let $Q$ be the point on line segment $\overline{CD}$ such that $CQ:QD = 3:5.$ Then
\[\overrightarrow{Q} = s \overrightarrow{C} + v \overrightarrow{D}\]for some constants $s$ and $v.$ Enter the ordered pair $(s, v).$

Solution: \boxed{\left( \frac{5}{8}, \frac{3}{8} \right)}


Compute the sum of the squares of the roots of the equation \[x^{2023} - 50x^{2020} + 5x^4 - 505 = 0.\]


Consider two lines: line $l$ parametrized as
\[
\begin{align*}
x &= 2 + 3t,\\
y &=  5 + 2t
\end{align*}
\]
and the line $m$ parametrized as
\[
\begin{align*}
x &= 1 + 2s,\\
y &=  3 + 2s.
\end{align*}
\]
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.

Then $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ such that $2v_1 - v_2 = 3$. Find $\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$.

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


Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 60^\circ$, find $\angle CED$, in degrees.


How many three-digit numbers remain valid if we exclude all three-digit numbers of the form $ABC$ where the sum of the first two digits equals the third digit ($A + B = C$), and $A \neq 0$?


The side lengths of a triangle are 9 cm, 40 cm, and 41 cm. Calculate both the area of the triangle and the height from the hypotenuse to the right angle vertex.

Solution: \boxed{\frac{360}{41} \text{ cm}}


Consider three distinct prime numbers as the sides of a scalene triangle where the perimeter must be a prime number. Determine the second smallest possible perimeter for such a triangle.


Calculate the absolute difference between the two sequences:
\[ A = 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 41 \times 42 + 43 \]
\[ B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 38 \times 39 + 40 \times 41 + 42 \]


Solution: \boxed{202x + 70}


Determine the tens digit of $11^{2045}$.


A store offers an initial discount of 25% on all items. A week later, the store applies an additional 15% discount to the already discounted items and advertises that the cumulative discount equals 40%. What is the actual total discount as a percentage of the original price, and how much does this differ from the store’s claim?


A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are \( 210, 336, 294, 252, x, \) and \( y \). Find the greatest possible value of \( x+y \).


Calculate the number of "hyper-prime dates" in 2008. A "hyper-prime date" is a date where both the month and the day are prime numbers, but the months only include prime-numbered months if the last two digits of the year are divisible by that prime month.


Find the length of side $XZ$ in the right triangle below, where hypotenuse $XY = 13$ and angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (0.8,0);
R = (0,0.95);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,W);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$13$",P--Q,N);
label("$60^\circ$",R--P,E);
[/asy]


For what values of $x$ is $x^2 - 50x + 575 \le 25$?

Solution: \boxed{[25 - 5\sqrt{3}, 25 + 5\sqrt{3}]}


Determine how many divisors of \(9!\) are multiples of 3.


Consider the lines:
\[
y = 4x + 3, \quad 2y = 8x + 6, \quad 3y = -12x + 2, \quad y = -\frac{1}{4}x + 7, \quad 4y = -x + 4.
\]
Determine the number of pairs of lines that are either parallel or perpendicular.


Calculate the sum of all binomial coefficients $\dbinom{n}{k}$ for $k$ ranging from 0 to 3, where $n$ is a positive integer greater than or equal to 3.

Solution: \boxed{S = 1 + n + \frac{n(n-1)}{2} + \frac{n(n-1)(n-2)}{6}}


How many of the integers from 50 to 300 are congruent to 3 modulo 7?


Ben throws six identical darts onto four identical dartboards. He lists the results in descending order based on the number of darts that hit each board. How many different lists can Ben possibly create?


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


If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 211$, what is the minimal possible value of $|a - b|$?


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


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

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


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


Consider integers $x$, $y$, and $z$ such that:
\[\mathbf{B} = \frac{1}{7} \begin{pmatrix} -4 & x \\ y & z \end{pmatrix},\]
and $\mathbf{B}^2 = 2\mathbf{I}.$ Find the largest possible value of $x + y + z$.


A positive real number $x$ is such that
\[
\sqrt{1-x^2} + \sqrt{1+x^2} = 2.
\]
Find $x^4$.


A spherical balloon deflates slightly and forms a hemisphere on a flat surface. The hemisphere's volume is half that of the original sphere. If the radius of the hemisphere is $4\sqrt[3]{2}$ cm, find the initial radius of the sphere.

Solution: \boxed{4\sqrt[3]{2} \text{ cm}}


Let $\mathbf{u}$ and $\mathbf{v}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$  Find the area of the parallelogram whose diagonals correspond to $\mathbf{u} + 3\mathbf{v}$ and $3\mathbf{u} + \mathbf{v}.$


A cylindrical container of milk has a diameter of 5 inches and is 10 inches tall, selling for $\$$2.50. At the same rate, what would be the price for a cylindrical container that is 10 inches in diameter and 15 inches tall?


A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From vertex A, the ant then randomly selects one of the five adjacent vertices, called vertex B. What is the probability that vertex B is the bottom vertex opposite the top one?


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


There are several positive integers that are divisors of each number in the list $$50, 100, 150, 200.$$ Find the sum of these positive integers.


What is the minimum number of equilateral triangles needed to cover an equilateral triangle of side length 9 units, assuming you have triangles with side lengths 1 unit and 2 units available?


We define two sequences of vectors $(\mathbf{u}_n)$ and $(\mathbf{z}_n)$ as follows: First, $\mathbf{u}_0 = \begin{pmatrix} 2 \\ 1 \end{pmatrix},$ $\mathbf{z}_0 = \begin{pmatrix} 3 \\ -1 \end{pmatrix}.$ Then for all $n \ge 1,$ $\mathbf{u}_n$ is the projection of $\mathbf{z}_{n - 1}$ onto $\mathbf{u}_0,$ and $\mathbf{z}_n$ is the projection of $\mathbf{u}_n$ onto $\mathbf{z}_0.$ Find
\[\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dotsb.\]

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


For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$


Quadrilateral $ABCD$ is an isosceles trapezoid with $AB \parallel CD$. On extending side $DC$, an external angle of $70^\circ$ is formed. What is the degree measure of angle $A$?


Determine the number of degrees in the measure of the smaller angle that is formed between the hour-hand and minute-hand of a clock at 7:30.


What is the greatest common divisor of 2873 and 1233?


How many positive perfect squares less than $10^6$ are multiples of 36?


Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$.


An arithmetic sequence with first term $2$ has a common difference of $8$. A second sequence starts with $5$ and has a common difference of $9$. In the range from $1$ to $200$, what is the largest number common to both sequences?


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


After a meeting, Anna and Mark start walking in opposite directions. Anna walks at a speed of 1 mile every 20 minutes, while Mark jogs at a speed of 3 miles every 40 minutes. How far apart are they after 2 hours?


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


Let \( t \) be the positive real solution to \( x^3 - \frac{1}{4}x - 1 = 0 \). Find the exact numerical value of
\[
t^2 + 2t^5 + 3t^8 + 4t^{11} + \dotsb.
\]


Simplify $\cot 20^\circ + \tan 10^\circ.$

Enter your answer as a trigonometric function evaluated at an integer, such as "sin 15".

Solution: \boxed{\csc 20^\circ}


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

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


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

Solution: \boxed{25x^2 + x - 3}


Compute \(\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}^{2023}.\)

Solution: \boxed{\begin{pmatrix} 1 & 0 \\ 4046 & 1 \end{pmatrix}}


A train departs from a station with 100 passengers. At the first stop, one-third of the passengers get off. At the second stop, one-quarter of the remaining passengers get off. At the third stop, one-fifth of the remaining passengers get off. How many passengers are left on the train after the third stop?


Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2+y^2=10x+60y$. What is the maximum value of $y$?

Solution: \boxed{30 + 5\sqrt{37}}


Calculate the sum of the digits and find the remainder when the sum is divided by both 3 and 5 for the sequence \(2, 12, 22, \dots, 92\).


A number in the set $\{50, 54, 58, \ldots, 998\}$ is randomly selected. What is the probability that it is divisible by 6?


$A, B, C, D,$ and $E$ are collinear in that order such that $AB = 1, BC = 3, CD = 4,$ and $DE = 5$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?


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


Express $345_{13} + 4CD_{14}$ as a base 10 integer. Assume $C = 12$ and $D = 13$ in base 14.


The positive four-digit integers that use each of the four digits $2,$ $3,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $17^{\text{th}}$ integer in the list?


Consider two functions defined as \( p(x) = x^2 - 4 \) and \( q(x) = -x \) plotted on the same set of axes. Calculate \( q(p(x)) \) at \( x = -3, -2, -1, 0, 1, 2, 3 \) and find the sum of these seven values.


Evaluate
\[\begin{vmatrix} \sin \alpha \sin \beta & \sin \alpha \cos \beta & \cos \alpha \\ \cos \beta & -\sin \beta & 0 \\ \cos \alpha \sin \beta & \cos \alpha \cos \beta & \sin \alpha \end{vmatrix}.\]

Solution: \boxed{\cos 2\alpha}


At the Pawsitive Pup Training Center, dogs can learn to do three tricks: jump, fetch, and shake. Of the dogs at the center: \begin{tabular}{l@{\qquad}l}
70 dogs can jump & 30 dogs can jump and fetch \\
40 dogs can fetch & 20 dogs can fetch and shake \\
50 dogs can shake & 25 dogs can jump and shake \\
15 dogs can do all three & 15 dogs can do none
\end{tabular} How many dogs are in the center?


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


In right $\Delta DEF$, $\angle DFE$ is a right angle. Point $N$ is the midpoint of $\overline{EF}$. If $DE = 5$ cm and $DF = 12$ cm, find the length of median $\overline{DN}$. Express your answer as a decimal to the nearest tenth.


Determine all solutions to $z^5 = 32i$ in the form:
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 &= r_5 (\cos \theta_5 + i \sin \theta_5),
\end{align*}
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$ Find $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5,$ in degrees.


A triangle has two vertices at $(0,7)$ and $(10,7)$ and is equilateral. If the third vertex is in the first quadrant, what is the y-coordinate?

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


A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many chairs should be removed?


How many positive integers smaller than $500{,}000$ are powers of $3$, but are not powers of $9$? You may find it useful to consider that $3^6 = 729$.


What is the smallest integer $k$ such that $k>2$ and $k$ has remainder $2$ when divided by any of $19,$ $7,$ and $4?$


Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = 2,$ $\|\mathbf{v}\| = 3,$ and $\|\mathbf{w}\| = 4,$ and
\[\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}.\] Compute $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}.$


Let $p(x)$ be a monic polynomial of degree 3, such that $p(1) = 7$, $p(2) = 14$, and $p(3) = 21$. Find $p(0) + p(5)$.


Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?


A frustum of a right circular cone is formed by cutting a small cone from a larger cone. The frustum has an altitude of 30 cm, the area of its lower base is $324\pi$ sq cm, and the area of its upper base is $36\pi$ sq cm. Determine the altitude of the small cone that was removed.

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


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


Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 30$ and
\[((x - y)^2 + (x - z)^2 + (y - z)^2)(x + y + z) = xyz.\]Find $\frac{x^3 + y^3 + z^3}{xyz}.$


Factor the following expression: $63b^2 + 189b$.


How many integers are between $(9.2)^3$ and $(9.3)^3$?


The graph of the equation \[\sqrt{(x-2)^2 + (y-3)^2} - \sqrt{(x-8)^2 + (y-3)^2} = 5\] consists of one branch of a hyperbola. Compute the positive value for the slope of an asymptote of this hyperbola.


Suppose $\cos R = \frac{5}{13}$ in the triangle below. If $RS = 13$, find the length of $QS$.

Solution: \boxed{QS = 12}


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


Calculate the product $\left(\frac{2}{5}\right)\left(\frac{5}{8}\right)\left(\frac{8}{11}\right)\cdots\left(\frac{2001}{2004}\right)$. Express your answer as a common fraction.


In triangle $ABC$, the side lengths $a, b,$ and $c$ satisfy $2a + 3b + c = b^2$ and $2a + 3b - c = 1$. Find the largest angle of the triangle, in degrees.


What is the remainder when 371,073 is divided by 6?


Express $3.\overline{167}$ as a common fraction in lowest terms.


Find the greatest common factor of 198 and 396 and use this GCF to compute the result of adding this GCF to four times the lesser number.


Let $w$ be a complex number such that
\[w^2 + |w|^2 = 7 + 2i.\]
Find $|w|^2.$


What is the area enclosed by the graph of $\lvert x \rvert + \lvert 3y \rvert = 12$?


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


Circle $AC$ forms a semi-circle with center $B$. The shaded region $ABC$ is "rolled" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B^{\prime}$. If $BC = 1$ cm, what is the length of the path that point $B$ travels?

Solution: \boxed{2\pi \text{ cm}}


Let \(x\) and \(y\) be positive real numbers. Find the minimum value of
\[
\frac{\sqrt{(x^2 + y^2)(5x^2 + y^2)}}{xy}.
\]

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


If $a$ and $b$ are positive integers such that $\gcd(a, b) = 10$, what is the smallest possible value of $\gcd(12a, 18b)$?


In the diagram, $COB$ is a sector of a circle with $\angle COB=90^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $X.$ What is the length of $XZ$? Assume the radius $OC=CB=15.$
[asy]
draw((0,0)--(15,0),black+linewidth(1));
draw((0,0)--(15,-15),black+linewidth(1));
draw((15,0)--(15,-15),black+linewidth(1));
label("$O$",(0,0),W);
label("$C$",(15,-15),S);
label("$B$",(15,0),E);
label("$X$",(15,-7.5),E);
label("$Z$",(15,0),NE);
label("15",(0,0)--(15,-15),SW);
label("15",(0,0)--(15,0),NW);
[/asy]


What is the area enclosed by the graph of $|2x| + |3y| = 12$?


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


The side length of square $A$ is 48 cm. The side length of square $B$ is 60 cm. What is the ratio of the area of square $A$ to the area of square $B$, and what is the ratio of the perimeter of square $A$ to the perimeter of square $B$? Express your answer as a common fraction.


In a diagram, rectangle $ABCD$ is divided into five identical squares. If $ABCD$ has a perimeter of 200 cm, what is its area, in square centimeters?

Solution: \boxed{\frac{12500}{9} \text{ cm}^2}


Solution: \boxed{12x^3 + 22x^2 - 12x - 10}


The integer $m$ is between $50$ and $180$, and is a multiple of $9$. When $m$ is divided by $10$, the remainder is $7$. Similarly, when $m$ is divided by $7$, the remainder is $5$. What is the value of $m$?


What are the last two digits in the sum of the squares of the factorials of the first 5 positive integers?


Lulu has a quadratic of the form $x^2+bx+72$, where $b$ is a specific positive number. She rewrites this quadratic in the form $(x+m)^2+12$. What is $b$?


The United States Postal Service charges an extra $\$0.15$ in postage if the length of an envelope, in inches, divided by its height, in inches, is less than $1.2$ or greater than $2.8.$ How many of these four envelopes must the extra $\$0.15$ in postage be paid? \begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
A &7 &5\\
B &10 &4\\
C &8 &8\\
D &14 &5
\end{tabular}


The number $n$ is a four-digit positive integer and is the product of three distinct prime factors $x$, $y$ and $10y+x$, where $x$ and $y$ are each less than 10. What is the largest possible value of $n$?


A right cylinder has a height of 8 inches and a radius of 3 inches. Calculate both: (a) the lateral surface area of the cylinder, and (b) the total surface area of the cylinder. Express your answers in terms of $\pi$.


Find the integer $n$, where $-90 < n < 90$, such that $\tan n^\circ = \tan 1720^\circ$.


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 9 dips, and 5 dips are equivalent to 2 dups, how many daps are equivalent to 36 dups?

Solution: \boxed{37.5 \text{ daps}}


Two congruent cylinders each have a radius of 7 inches and a height of 5 inches. The radius of one cylinder is increased by 4 inches, and the height of the other cylinder is increased by a different non-zero number $y$ inches. If the resulting volumes of both cylinders are equal, what is the value of $y$?


A spiral staircase turns $180^\circ$ as it rises 15 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.


Find the value of $x$, given that $x$ is nonzero and the numbers $\{x\}$, $\lfloor x \rfloor$, $x$, and $x + \{x\}$ form an arithmetic sequence in that order.


A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere. The volume of the hemisphere is known to be $36\pi$ cm³. Find the radius of the original bubble.

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


In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 60^\circ,$ and $AC = 6.$ Find $BC.$


A container of cream cheese is 4 inches in diameter and 5 inches high, and sells for \$0.75. At the same rate, how much would a container be that is 8 inches in diameter and 10 inches high?


Simplify
\[(2 + 2\cot A - 2\csc A)(2 + 2\tan A + 2\sec A).\]


A club has 30 members and needs to select 4 officers: president, vice-president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only be officers if the other one is also an officer. In how many ways can the club choose its officers?


Determine all values of $k$ so that the domain of the function
\[c(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 3x + k}\]is all real numbers.

Solution: \boxed{\left(-\infty, -\frac{9}{20}\right)}


Find the sixth term of the geometric sequence with first term $3$ and second term $-\frac{1}{6}$.


The points $P, Q,$ and $R$ are represented by the complex numbers $z, 2z,$ and $3\overline{z},$ respectively. Suppose $|z|= 1.$ When $P,$ $Q$, and $R$ are not collinear, let $T$ be the fourth vertex of the parallelogram $PQRT.$ What is the maximum distance between $T$ and the origin of the complex plane?


Determine the value of $x$ that satisfies $\sqrt[3]{2x\sqrt{x^3}} = 6$.


The number $0.478125$ can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are positive integers. When this fraction is in its simplest form, what is the sum of $a + b$?


The graph of the function \( g(x) \) is defined as \( g(x) = x^2 - 4x + 3 \) for \( -2 \le x \le 5 \). How many values of \( x \) satisfy \( g(g(x)) = 3 \)?


In how many distinct ways can I arrange my six keys on a keychain, if I want to put my house key next to my car key and my office key next to my bike key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation).


If $k$ and $\ell$ are positive 4-digit integers such that $\gcd(k,\ell)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[k,\ell]$?


Define $a * b$ as $2a - b^2$. If $a * 4 = 9$, what is the value of $a$?


Marguerite drove 100 miles in 2.4 hours. If Sam drove for 4 hours at the same average rate as Marguerite but rested for 1 hour during this period, how many miles did he drive in total?

Solution: \boxed{125 \text{ miles}}


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


The expression $\dfrac{\sqrt[5]{5}}{\sqrt[4]{5}}$ equals 5 raised to what power?


Suppose
\[\frac{1}{x^3-7x^2+11x+15} = \frac{A}{x-5} + \frac{B}{x+3} + \frac{C}{(x+3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?


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


How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?


Given six positive integers from a list of ten positive integers as $6, 7, 2, 4, 8, 5$, what is the largest possible value of the median of this list of ten positive integers?


$200_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$?


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

Solution: \boxed{(8, 9)}


What is the greatest integer less than 200 for which the greatest common factor of that integer and 45 is 9?


In a circle with center $O$, the measure of $\angle SIQ$ is $45^\circ$ and $OS=12$ cm. Find the number of centimeters in the length of arc $SQ$. Express your answer in terms of $\pi$.


A plane cuts through a right circular cylinder with a radius of $2$. The resulting ellipse has a major axis that is $40\%$ longer than its minor axis. What is the length of the major axis?


If \(x+y=\frac{10}{12}\) (which simplifies to \(\frac{5}{6}\)) and \(xy = \frac{7}{36}\), what is the value of \(x^2 + y^2\)?


Triangle $ABC$ is an isosceles triangle with $AB=BC$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, while point $F$ is the midpoint of $\overline{AC}$. If $\overline{CE}$ is 15 units long, determine the length of $\overline{BD}$. Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(3,112^.5)--(6,0)--cycle);
draw((6,0)--(9,112^.5)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (3,112^.5), N);
label("$C$", (6,0), SE);
label("$D$", (4.75,6), N);
label("$E$", (9,112^.5), N);
label("$F$", (1.5,112^.5/2), NW);
[/asy]


When the base-10 integers 300 and 1500 are expressed in base 2, and then subtract the smaller number from the larger number, how many digits does the result have in base 2?


In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 6. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region formed by the areas of $\triangle POS$, $\triangle ROQ$, and the sectors $POR$ and $SOQ$?

[asy]
size(120);
import graph;
fill((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray);
fill(Arc((0,0),sqrt(3),45,135)--cycle,mediumgray);fill(Arc((0,0),sqrt(3),225,315)--cycle,mediumgray);
draw(Circle((0,0),sqrt(3)));
draw((-1,-1)--(1,1)--(1,-1)--(-1,1)--cycle);
label("$P$",(-1,1),NW); label("$R$",(1,1),NE); label("$S$",(-1,-1),SW); label("$Q$",(1,-1),SE);
[/asy]


What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form.


Let $a \clubsuit b = a + \frac{b}{a}$. Calculate $(4 \clubsuit (2 \clubsuit 3)) \clubsuit 5$.


Suppose you have 8 shirts, 5 pairs of pants, 6 ties, and 2 jackets. An outfit requires a shirt and a pair of pants and can optionally include a tie and/or a jacket. How many different outfits can you create?


Compute the product $\begin{pmatrix} 3 & -2 \\ -4 & 5 \end{pmatrix} \begin{pmatrix} 4 \\ -2 \end{pmatrix}$, then multiply the resulting matrix by $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 16 \\ 26 \end{pmatrix}}


Determine the sum of all values of $x$ for which the function $$g(x) = \frac{1}{2+\frac{1}{2+\frac{1}{x}}}$$ is undefined.


How many four-character license plates can be constructed where the plate consists of a consonant, followed by two vowels, each independently selected, followed by a digit? (For this problem, consider Y a vowel.)


Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[ f(x - y) = \sqrt{f(xy) + 2} \]
for all $x > y > 0.$ Determine $f(2021)$.


Determine the point of intersection for the lines $8x - 5y = 40$ and $6x + 2y = 14$.

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


Amanda, Chloe, and Ethan are siblings. Their names consist of 6, 5, and 4 letters respectively, placed on identical cards so that each card contains one letter. Without replacement, two cards are selected at random from the 15 cards. What is the probability that one letter is from Amanda's name and the other from either Chloe's or Ethan's name? Express your answer as a common fraction.


How many values of $x$, $-10 < x < 50$, satisfy $\cos^2 x + 3\sin^2 x = 1?$ (Note: $x$ is measured in radians.)


Twelve people sit down at random seats around a round table. Five of them are computer science majors, four are engineering majors, and the three remaining individuals are art majors. What is the probability that all five computer science majors sit in consecutive seats?


It takes five painters working at the same rate 1.5 work-days to finish a job. If only four painters are available, how many work-days will it take them to finish the job, working at the same rate? Express your answer as a mixed number.

Solution: \boxed{1\frac{7}{8} \text{ work-days}}


A circle with center \(O\) has a radius of \(10\) units and circle \(P\) has a radius of \(5\) units. The circles are externally tangent to each other at point \(Q\). Segment \(TS\) is the common external tangent to circle \(O\) and circle \(P\) at points \(T\) and \(S\), respectively. What is the length of segment \(OS\)? Express your answer in simplest radical form.

Solution: \boxed{OS = 10\sqrt{3}}


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units.


Two concentric circles have the same center $O$. Point $X$ is the midpoint of segment $OP$, and point $Y$ on segment $OP$ such that the length of $XY$ is half the length of $OX$. Calculate the ratio of the area of the circle with radius $OY$ to the area of the circle with radius $OP$.


What is the base ten equivalent of $43210_7$?


The line $y = 5x - 7$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ 2 \end{pmatrix} + t \begin{pmatrix} 3 \\ h \end{pmatrix}.\]
Determine the ordered pair $(s,h)$.

Solution: \boxed{\left(\frac{9}{5}, 15\right)}


Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. What is the probability that the positive difference between the two numbers is $3$ or greater?


For how many integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 6?


What is the modular inverse of $17$ modulo $1001$?

Express your answer as an integer from $0$ to $1000$, inclusive.


What is the smallest positive integer \( n \) such that the fraction \( \frac{n}{n+150} \) is a terminating decimal?


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

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


Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $300\sqrt{3}$ square feet and $675\sqrt{3}$ square feet. The two planes are $12$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?

Solution: \boxed{36 \text{ feet}}


What is the largest integer less than $\log_3 \frac{3}{1} + \log_3 \frac{7}{3} + \cdots + \log_3 \frac{4039}{4037} + \log_3 \frac{4041}{4039}$?


The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$.


Evaluate the expression $(a^b)^a - (b^a)^b$ multiplied by $c$ for $a = 3$, $b = 2$, and $c = 4$.


All the numbers 0, 1, 2, 3, 4, 5 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?


How many pairs of positive integers $(x,y)$ satisfy $x^2-y^2=77$?


Consider two lines: line $l$ parametrized as
\begin{align*}
x &= 2 + 3t,\\
y &=  3 + 2t,
\end{align*}
and line $m$ parametrized as
\begin{align*}
x &= 4  + 2s,\\
y &=  5 + 3s.
\end{align*}
Let $C$ be a point on line $l$, $D$ be a point on line $m$, and let $Q$ be the foot of the perpendicular from $C$ to line $m$.

Find a vector $\begin{pmatrix} v_1\\v_2\end{pmatrix}$ onto which $\overrightarrow{QC}$ (the projection of $\overrightarrow{DC}$) is projected, such that $v_1 + v_2 = 3$.

Solution: \boxed{\begin{pmatrix} 9 \\ -6 \end{pmatrix}}


Determine the number of zeros located directly after the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^7 \cdot 5^6} \cdot \frac{3}{5^2}$.


In a square layout, the vertices of the square are the centers of four circles. Each side of the square is 10 cm, and the radius of each circle is 3 cm. Calculate the area in square centimeters of the shaded region formed by the exclusion of the parts of the circles that overlap with the square.

Solution: \boxed{100 - 9\pi \text{ cm}^2}


A circular cake with diameter $10 \text{ cm}$ is cut into four equal-sized sector-shaped pieces. Determine $l^2$, where $l$ is the length of the longest line segment that can be drawn in one of these pieces.


There are $30$ different complex numbers $z$ such that $z^{30}=1$. For how many of these is $z^{10}$ a real number?


Each triangle is a 45-45-90 triangle, and the hypotenuse of one triangle is the leg of an adjacent triangle. The hypotenuse of the largest triangle is 10 centimeters. What is the length of the leg of the smallest triangle? Express your answer as a common fraction.


Suppose I have 7 shirts, 5 pairs of pants, 4 ties, and 2 different jackets. If an outfit requires a shirt and pants, can have a tie or not, and can be worn with one jacket or without a jacket, how many outfits can I make?


How many odd whole numbers are factors of 252 that are not divisible by 3?


Suppose a function $f(x)$ is defined on a domain $\{x_1, x_2, x_3\}$ so that the graph of $y=f(x)$ consists of three distinct points which form a triangle of area $50$. Consider another function defined as $y = 3f(3x)$. The graph of this function also consists of just three points. What is the area of the triangle formed by these points?


Let $B$ be the set of all numbers that can be represented as the sum of three consecutive even integers. What is the greatest common divisor of all numbers in $B$?


Calculate the product of the least common multiple (LCM) of 15, 10, and 6, and the greatest common divisor (GCD) of 15, 10, and 6.


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 20 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. Calculate the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$. Express your answer as a common fraction.


If $(5, 20)$ and $(11, -8)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $y$-coordinates of the other two vertices after the rectangle is rotated 90 degrees around its center.


According to the table below, how many dollars are in the median value of the 75 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$140,000$\\\hline
Vice-President&10&$\$100,000$\\\hline
Director&15&$\$80,000$\\\hline
Manager&5&$\$55,000$\\\hline
Associate Director&9&$\$52,000$\\\hline
Administrative Specialist&35&$\$25,000$\\\hline
\end{tabular}


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


Joe now chooses randomly with equal probabilities from four types of fruits (apple, orange, banana, grape) for his four meals (breakfast, lunch, tea, and dinner). What is the probability that Joe will eat at least two different kinds of fruit in one day?


In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 21 respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction.


Calculate the sum of all positive even factors of $360$.


In triangle \(ABC,\) points \(D\) and \(E\) are on \(\overline{AB}\) and \(\overline{AC},\) respectively, and angle bisector \(\overline{AT}\) intersects \(\overline{DE}\) at \(F.\)  If \(AD = 2,\) \(DB = 2,\) \(AE = 3,\) and \(EC = 3,\) compute \(\frac{AF}{AT}.\)


In triangle $XYZ$, we have $\angle Z = 90^\circ$, $XY = 20$, and $YZ = \sqrt{51}$. What is $\tan X$?


Rationalize the denominator of $\frac{5}{3\sqrt[4]{7}}$. The answer can be written in the form $\frac{P\sqrt[4]{Q}}{R}$, where $P$, $Q$, and $R$ are integers, $R$ is positive, and $Q$ is not divisible by the fourth power of any prime. Find $P+Q+R$.


Convert $147_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in the base 2 representation. What is the value of $y-x$?


Three years ago, there were 30 trailer homes on Maple Drive with an average age of 15 years. Later, a group of brand new trailer homes was added. Today, the average age of all the trailers on Maple Drive is 12 years. If each of the new trailer homes added is now 3 years old (since they were new), how many new trailer homes were added three years ago?


Paul's test scores so far are 78, 84, 76, 82, and 88. What score must he achieve on his sixth test for the arithmetic mean of his six scores to equal 85?


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


For certain real numbers \(p\), \(q\), and \(r\), the polynomial \[g(x) = x^3 + px^2 + 2x + 20\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + 2x^3 + qx^2 + 200x + r.\] What is \(f(-1)\)?


At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First, #6 bowls #5. The loser receives 6th prize and the winner proceeds to bowl #4. The loser of this game receives 5th prize and the winner plays #3. The loser of this game receives 4th prize and the winner bowls #2. The loser of this game receives 3rd prize and the winner bowls #1. The winner of the final game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes?


A $\textit{composite number}$ is a number that has more than one prime factor, while a $\textit{prime number}$ is a number that has exactly two distinct positive divisors: 1 and itself. The number 95 can be expressed as the sum of a composite number and a prime number. What is the minimum positive difference between the prime number and the composite number in such pairings?


An ellipse has its foci at $(2, 0)$ and $(2, 4)$. Given that it passes through the point $(7, 2)$, 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 $a+k$.


The DeBose family has two boys with ages $8$ each, a girl aged $16$, and a boy aged $18$. What is the mean (average) age of the children?


How many odd numbers between $1000$ and $9999$ have distinct digits?


A circle is circumscribed around $ABCD$ with given angles $\angle ACB = 50^\circ$ and $\angle CAD = 20^\circ$. How many degrees are in $\angle CAB + \angle ACD$?


If the product $(2x^2 - 3x + 5)(8 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $9a + 3b + 6c + d$.


A projection takes $\begin{pmatrix} 2 \\ -4 \end{pmatrix}$ to $\begin{pmatrix} 3 \\ -3 \end{pmatrix}.$  Which vector does the projection take $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$ to?

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


A bus arrives randomly sometime between 1:00 and 2:30, waits for 20 minutes, and then leaves. If Laura also arrives randomly between 1:00 and 2:30, what is the probability that the bus will be there when Laura arrives?


Suppose $\sin N = \frac{4}{5}$ in the diagram below. What is $LN$?

[asy]
pair L,M,N;
M = (0,0);
N = (20,0);
L = (0,25);
draw(L--M--N--L);
draw(rightanglemark(L,M,N,20));
label("$M$",M,SW);
label("$N$",N,SE);
label("$L$",L,NE);
label("$25$",L/2,W);
[/asy]


The image of a point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(8,6)$. Find $m+b$.


Let $s$ be the positive real solution to $x^3 + \frac{1}{4} x - 1 = 0.$ Find the exact numerical value of
\[s^2 + 2s^5 + 3s^8 + 4s^{11} + \dotsb.\]


Let $p(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$  Among these roots, there are exactly 1009 distinct values among the absolute values
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] Determine the minimum number of real roots that $p(x)$ can have.


Find the $3 \times 3$ matrix $\mathbf{M}$ such that for any $3 \times 3$ matrix $\mathbf{N}$, $\mathbf{M} \mathbf{N}$ results in the transformation where the first row and third row of $\mathbf{N}$ are swapped and the second row of $\mathbf{N}$ is tripled. In mathematical terms,
\[\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} g & h & i \\ 3d & 3e & 3f \\ a & b & c \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} 0 & 0 & 1 \\ 0 & 3 & 0 \\ 1 & 0 & 0 \end{pmatrix}}


An environmental agency now decides to enhance its reach and needs to hire new employees such that 120 of the new employees will be able to monitor water pollution, 95 of the new employees will be able to monitor air pollution, and exactly 40 of the new employees will be able to monitor both water and air pollution (these 40 are included in both the 120 and 95). What is the minimum number of employees that must be hired?


Construct the matrix $\mathbf{P}$ such that for any vector $\mathbf{v}$, $\mathbf{P} \mathbf{v}$ represents the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 3 \\ 1 \\ -4 \end{pmatrix}.$

Solution: \boxed{\mathbf{P} = \begin{pmatrix} \frac{9}{26} & \frac{3}{26} & -\frac{12}{26} \\ \frac{3}{26} & \frac{1}{26} & -\frac{4}{26} \\ -\frac{12}{26} & -\frac{4}{26} & \frac{16}{26} \end{pmatrix}}


Let $a,$ $b,$ and $c$ be real numbers, and let $P,$ $Q,$ $R$ be points such that the midpoint of $\overline{QR}$ is $(a,0,0),$ the midpoint of $\overline{PR}$ is $(0,b,0),$ and the midpoint of $\overline{PQ}$ is $(0,0,c).$  Find
\[
\frac{PQ^2 + PR^2 + QR^2}{a^2 + b^2 + c^2}.
\]


What is the equation of the line parallel to $3x + 6y = 12$ and passing through the point $(2, -1)$? Write the equation in slope-intercept form.

Solution: \boxed{y = -\frac{1}{2}x}


Compute $\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}^4.$

Solution: \boxed{\begin{pmatrix} -8 & 8\sqrt{3} \\ -8\sqrt{3} & -8 \end{pmatrix}}


A math conference is presenting a lecture series with eight different lecturers. Dr. Smith must lecture after Dr. Jones, and Dr. Green must lecture after Dr. Brown. In how many orders can the eight lecturers be scheduled?


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


The lengths of the sides of a triangle with positive area are $\log_2 8$, $\log_2 48$, and $\log_2 n$, where $n$ is a positive integer. Find the number of possible values for $n$.


If $999,999^2$ has a certain number of zeros following the observed pattern, how many zeros are in the expansion of $999,999,999^2$?


What is the sum of the digits of the base $7$ representation of $2023_{10}$?


Solve the quadratic equation $2x^2 - 7x + 1 = x + 31$. Find the positive difference between the solutions.


If $n$ is a positive integer, then let $g(n)$ be the sum of the digits of $\frac{1}{3^n}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 15$?


Compute without using a calculator: $\dfrac{10!}{7!3!}$.


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


A rectangle with integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area?


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


Factor the expression \(4x(x+2) + 10(x+2) + 2(x+2)\).


The 5 a.m. temperatures for seven consecutive days were \( -6^\circ \), \( -2^\circ \), \( -2^\circ \), \( -3^\circ \), \( 2^\circ \), \( 4^\circ \), and \( 3^\circ \) Celsius. Additionally, the relative humidity percentages for these days were 70%, 65%, 65%, 72%, 80%, 75%, and 77% respectively. Calculate both the mean 5 a.m. temperature for the week and the mean relative humidity.


If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$.


Divide the product of the first four odd composite numbers by the product of the next four odd composite numbers. Express your answer as a common fraction.


A cube with an edge length of 5 units has the same volume as a square-based pyramid with base edge lengths of 10 units. What is the height \( h \) of the pyramid?


Suppose lines $\overline{AD} \parallel \overline{FG}$. How many degrees are in angle $EFG$?

[asy]
import olympiad;

pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,20);
pair F = (0,0);
pair G = (40,0);

draw(F--G);
draw(F--C);
draw(A--D);
draw(B--E);

label("F", F, W);
label("G", G, ENE);
label("C", C, N);
label("A", A, NW);
label("D", D, E);
label("E", E, S);
label("B", B, NW);

draw(scale(20)*anglemark(G, F, C));
draw(shift(E)*scale(35)*shift(-E)*anglemark(B, E, A));
draw(shift(E)*scale(20)*shift(-E)*anglemark(C, E, B));

label("$x$", (6,20), NW);
label("$3x$", (13,25), N);
label("$2x$", (5,0), NE);
[/asy]


Find the largest three-digit number divisible by each of its distinct, non-zero digits with its tens digit smaller than 5.


Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 35 square units? [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,3), C=(2,14), D=(4,3), E=(4,0);
pair[] dots={A,B,C,D,E};

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

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(4,0)",E,SE);
label("D(4,3)",D,NE);
label("C",C,NE);
label("B(0,3)",B,NW);
[/asy]


You have $6$ blue shirts, $4$ yellow shirts, $7$ pairs of pants, $9$ blue hats, and $6$ yellow hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?

Solution: \boxed{504 \text{ outfits}}


What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity?


If $q(x) = x^5 - 2x^3 + 3$, find the coefficient of the $x^{12}$ term in the polynomial $(q(x))^2$.


The stem and leaf plot represents the heights, in inches, of players on the Westwood High School boys' basketball team. Compute the mean height of the players. (Note: $6|4$ represents 64 inches.)

Height of the Players on the Basketball Team (inches)

$5|7\;9$

$6|2\;4\;4\;5\;5\;8\;9$

$7|0\;1\;3\;5\;5\;7\;8$

Solution: \boxed{68.25 \text{ inches}}


A book with 63 pages numbered from 1 to 63 has its pages renumbered in reverse, from 63 to 1. For how many pages do the new page number and old page number share the same units digit?


Find the units digit of the sum $53_8 + 64_8$ in base 8.


What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6?


A box contains 8 cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?


Find the smallest two-digit prime number such that reversing the digits results in an odd composite number.


Determine how many numbers less than 30 are abundant numbers.


Container I holds 10 red balls and 2 green balls; container II holds 3 red balls and 5 green balls, container III holds 2 red balls and 6 green balls, and newly added container IV holds 5 red balls and 3 green balls. A container is selected at random, and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction.


Find the largest integer $n$ for which $18^n$ evenly divides $25!$.


A standard deck of 52 cards is divided into 4 suits, each containing 13 cards. Two of the suits are red and the other two are black. Each suit includes one unique card called the Ace. In how many ways can you pick two cards in sequence such that the two cards are from different suits?


Find the non-zero values of \( c \) for which there are exactly three positive values of \( b \) for which the quadratic equation \( x^2 + a\left(b + \frac{1}{b}\right)x + c = 0 \) has exactly one solution, with \( a = 2 \).


Given two triangles, where the sides of the first triangle are 8 cm, 18 cm, and 12 cm, and the sides of the second smaller triangle are 4 cm, 9 cm, and 6 cm. The angle between the first two sides of both triangles is $120^\circ$. Find the length of the segment corresponding to the 12 cm side in the smaller triangle.

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


Let $a$ and $b$ be integers. Consider the following six statements:
1. $\frac{1}{a} < \frac{1}{b}$
2. $a^2 > b^2$
3. $a < b$
4. $a < 0$
5. $b < 0$
6. $|a| > |b|$

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


If 31,254,389 candies are packaged in sets of 6, how many candies will be left over?


Consider a hexagonal-shaped center in an array of hexagons, each side of the hexagon being one unit in length. The first ring of hexagons around this center hexagon contains 6 hexagons. The second ring contains 12 hexagons. If this pattern continues, how many hexagons are in the $100^{th}$ ring?


Here are two functions:
$$
\begin{array}{ccc}
f(x) & = & 4x^2 + 3x + 5 \\
g(x) & = & x^2 - mx - 9
\end{array}
$$
If $f(5) - g(5) = 20,$ what is the value of $m$?

Solution: \boxed{m = -16.8}


The endpoints of a diameter of circle $C$ are $(2, 1)$ and $(8, 7)$. What are the coordinates of the center of circle $C$? Additionally, calculate the length of the diameter.


A rectangular box $Q$ is inscribed in a sphere of radius $r$. The surface area of $Q$ is 432, and the sum of the lengths of its 12 edges is 120. What is $r$?


Let $a$ and $b$ be the roots of the equation $x^2 - nx + 6 = 0.$ Suppose that $a + (2/b)$ and $b + (2/a)$ are the roots of the equation $x^2 - rx + s = 0.$ What is $s?$


The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:3:5$ that can be written in the form \[{n\choose k} : {n\choose k+1} : {n\choose k+2}\]. Find the sum of all possible values of $n+k$.


The function \( g(x) \) satisfies
\[ g(x + y) = g(x) + g(y) \]
for all real numbers \( x \) and \( y \), and \( g(5) = 6 \). Find \( g(6) \).


If 2 cards from a deck of 54 cards (which includes 4 kings, 6 aces, and 44 other cards) are selected randomly, what is the probability that either two aces or at least one king occurs?


What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 7:30 a.m. on a 12-hour analog clock?


If 2 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 1 king occurs?


Maria has some 5 cent stamps and some 7 cent stamps. What is the least number of stamps she can combine so the value of the stamps is 50 cents?


Rectangle $ABCD$ has sides $\overline {AB}$ of length 8 and $\overline {CB}$ of length 6. Divide $\overline {AB}$ into 200 congruent segments with points $A=P_0, P_1, \ldots, P_{200}=B$, and divide $\overline {CB}$ into 200 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{200}=B$. For $1 \le k \le 199$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 399 parallel segments drawn.


When the fourth root of $x$ is squared, the result is 16. What is the value of $x$?


All numbers $2452$, $2030$ and $2372$ have something in common: each is a 4-digit number beginning with $2$ that has exactly two identical digits. How many such numbers are there?


Determine all values of $a$ such that the points $(0,0,0),$ $(1,a,0),$ $(0,2,a),$ and $(a,0,1)$ are coplanar.


Find the sum of all real solutions to the equation
\[\frac{x-3}{x^2+3x+2} = \frac{x-6}{x^2-12x+32}.\]


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


Tamika selects two different numbers at random from the set $\{7, 9, 11\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 4, 5\}$ and 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{6}{9} = \frac{2}{3}}


Point $(x, y)$ is randomly picked from the rectangular region with vertices at $(0, 0), (3000, 0), (3000, 4000),$ and $(0, 4000)$. What is the probability that $x > 3y$? Express your answer as a common fraction.


In trapezoid $EFGH$, sides $\overline{EF}$ and $\overline{GH}$ are parallel, $\angle E = 3\angle H$, and $\angle G = 4\angle F$. Find $\angle E$.


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


Fifteen points are marked on a circle. How many distinct convex polygons of four or more sides can be drawn using some (or all) of the fifteen points as vertices?


Given that \( q(x) \) is a quadratic polynomial and \( p(x) \) is a linear polynomial, with the graph of \( \frac{p(x)}{q(x)} \) having a vertical asymptote at \( x=1 \), a hole at \( x=0 \), and \( p(1)=1 \), \( q(3) = 3 \), find \( p(x) + q(x) \).

Solution: \boxed{\frac{1}{2}x^2 + \frac{1}{2}x}


Find the remainder when $7^{2023}$ is divided by $17$.


Let \( t \) be a parameter that varies over all real numbers. Any parabola of the form
\[ y = 4x^2 + tx - 3t \]
passes through a fixed point. Find this fixed point.

Solution: \boxed{(3, 36)}


Solution: \boxed{84x^4 + 2x^5}


Write $\sqrt{\frac{25}{49}+\frac{16}{81}}$ as a common fraction.


A bowling ball is redesigned with a spherical surface and a diameter of 24 cm. For custom fitting, three holes are drilled for a new bowler, each 6 cm deep. The diameters of the holes are 2 cm, 2.5 cm, and 4 cm. Assuming the holes are right circular cylinders, calculate the volume of the fitted bowling ball in cubic centimeters, expressed in terms of \(\pi\).

Solution: \boxed{2264.625\pi \text{ cubic cm}}


How many positive integers less than $151$ are multiples of either $6$ or $8$, but not both at once?


If I roll 6 standard 8-sided dice (each die has sides numbered from 1 to 8) and multiply the number on the face of each die, what is the probability that the result is a composite number?


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


Calculate the sum of the first 6 terms of the geometric series: \( \frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4 + \left(\frac{1}{4}\right)^5 + \left(\frac{1}{4}\right)^6 \).


Determine the radius of the circle described by the equation \( x^2 + y^2 + 36 = 6x + 24y \).


The coefficients of the polynomial
\[b_6 x^6 + b_5 x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + b_0 = 0\] are all integers, and its roots $s_1, s_2, \dots, s_6$ are all integers. Furthermore, the roots of the polynomial
\[b_0 x^6 + b_1 x^5 + b_2 x^4 + b_3 x^3 + b_4 x^2 + b_5 x + b_6 = 0\] are also $s_1, s_2, \dots, s_6.$ Find the number of possible multisets $T = \{s_1, s_2, \dots, s_6\}.$


The line $y = \frac{2}{3}x - 5$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -6 \\ p \end{pmatrix} + t \begin{pmatrix} m \\ 7 \end{pmatrix}.\]
Find the ordered pair $(p, m).$

Solution: \boxed{\left( -9, \frac{21}{2} \right)}


Let
\[ g(x) = x^3 + 5x^2 + 15x + 35. \]
The graphs of \( y = g(x) \) and \( y = g^{-1}(x) \) intersect at exactly one point \((c,d)\). Enter the ordered pair \((c,d)\).

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


Ray climbs up a flight of stairs with $n$ steps in two ways. Climbing $6$ steps at a time leaves $4$ steps at the top, and climbing $7$ steps at a time leaves $3$ steps at the top. What is the smallest possible value of $n$ that is greater than $15$?


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


Find the remainder when the polynomial \(x^{1010}\) is divided by the polynomial \((x^2 + 1)(x+1)(x-1)\).


Let set $\mathcal{C}$ be an 80-element subset of $\{10,11,12,\ldots,120\}$, and let $U$ be the sum of the elements of $\mathcal{C}$. Find the number of possible values of $U$.


In $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and $CD \parallel AB$. Point $E$ is on extension of $CD$ such that $DE \parallel BC$. Determine $m\angle ECD$.


For a winter coat, Maria must specify her chest size in centimeters. Given that:
- $1$ inch equals $0.0254$ meters,
- $1$ meter equals $100$ centimeters,
- and due to material shrinkage in cold temperatures, the chest size increases by $1\%$ in cold weather,
what size should Maria specify, in centimeters, if her normal chest size in inches is $38$ inches? (You may use a calculator; answer to the nearest tenth.)

Solution: \boxed{97.5 \text{ cm}}


Determine an expression for the area of $\triangle RDS$ in terms of $k$. Your answer should be simplified as much as possible. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair r = (0, 15); pair b = (15, 0);
pair d = (3, 15); pair s = (0, 10);

draw((-2, 0)--(18, 0), Arrow);
draw((0, -2)--(0, 18), Arrow);
draw(r--d--b);
draw(d--s--b);

label("$R(0, 15)$", r, W);
label("$D(3, 15)$", d, NE);
label("$B(15, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (18, 0), E);
label("$y$", (0, 18), N);
label("$S(0, k)$", s, W);
[/asy]

Solution: \boxed{\frac{45 - 3k}{2}}


Determine the degree of $d(x)$ if a polynomial of degree $17$ is divided by $d(x)$ to yield a quotient of degree $10$ and a remainder of $2x^4 + 3x^3 - 5x + 7$.


If the least common multiple of two positive integers divided by their greatest common divisor equals 20, and one of the integers is 36, what is the smallest possible value of the other integer?


Consider the system of equations:
\[
4x - 3y = c, \\
2y - 8x = d.
\]
Determine \(\frac{c}{d}\), assuming \(d \neq 0.\)


Convert $2e^{15\pi i/4}$ to rectangular form.

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


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


Let $f(x) = x^3 - x^2 + 2x + 1007$. What is the greatest common divisor of $f(10)$ and $f(11)$?


What is the base ten equivalent of $23456_{7}$?


In right triangle $PQR$, angle $P$ measures 45 degrees and angle $Q$ measures 45 degrees. When drawn, the angle bisectors of angles $P$ and $Q$ intersect at a point $S$. What is the measure of obtuse angle $PSQ$?


In a right prism with triangular bases, given that the sum of the areas of three mutually adjacent faces (two lateral faces and one base) is 30, find the maximum volume of the prism.


Consider the complex roots of the equation:
\[ z^6 - z^4 + z^2 - 1 = 0. \]
Among these roots, find the maximum imaginary part that can be expressed as $\sin \phi$, where $-90^\circ \leq \phi \leq 90^\circ$. Determine $\phi$.


Find $XY$ in the triangle below.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (2,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$12$", (Q+R)/2,NE);
label("$60^\circ$", (0.7,0),N);
[/asy]


What is the smallest positive integer $n$ for which $n^2$ is divisible by 24 and $n^3$ is divisible by 720?


Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$.

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


Simplify the expression $\sqrt[3]{54a^5b^9d^{14}}$ and determine the sum of the exponents of the variables outside the radical.


Jamie paid for a $2.40 meal using exactly 50 coins consisting of pennies, nickels, and dimes. How many dimes did Jamie use if no change was received?


How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?


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

Solution: \boxed{x^3 - 12x^2 + 45x + 54}


Given that $m$ is an integer such that $1 < 3m \leq 45$, what is the sum of all possible integer values of $m$?


How many even numbers between $1000$ and $9999$ have all distinct digits?


Starting with 100 million, Jordan alternates steps between dividing by 2 and multiplying by 5. The sequence begins with 100,000,000, 50,000,000, 250,000,000, and so on. After 14 steps, what is the last integer in Jordan’s sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers, and $a$ is as small as possible.

Solution: \boxed{2 \cdot 5^{15}}


What integer $n$ satisfies $0 \le n < 17$ and $$-150 \equiv n \pmod{17}~?$$ Additionally, find $n$ such that it also divides 102 exactly.


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


A list of five positive integers has the following properties:
- The number $6$ occurs twice in the list.
- The median of the list is $12$.
- The average (mean) of the list is $14$.
What is the smallest possible integer that could appear in the list?


Triangle $DEF$ has side lengths $DE=8$, $EF=10$, and $FD=12$. Two bugs start simultaneously from vertex $D$ and crawl along the perimeter of the triangle at different speeds. The first bug crawls at 1 unit per minute, and the second bug at 2 units per minute in opposite directions. They meet at point $G$. What is $EG$?


Calculate $45 \cdot 68 \cdot 99 \equiv m \pmod{25}$, where $0 \leq m < 25$.


Suppose $p$ is a prime number and $2017_p + 305_p + 211_p + 145_p + 7_p = 153_p + 280_p + 367_p$. Determine the number of possible values of $p$.


Let a square have a side length of \(10^{0.2}\) meters. Compute the area of the square and then multiply it by \(10^{0.1}\), \(10^{-0.3}\), and \(10^{0.4}\).


Compute the sum of the roots of the equation
\[ x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0, \]
given that all of the roots are real and nonnegative.


How many integers from 1 through 9999, inclusive, do not contain any of the digits 1, 3, 5, 7, or 9?


Josef and Timothy play a game where Josef picks an integer between 1 and 500 inclusive, and Timothy divides 360 by that integer and states whether the quotient is an integer. How many integers could Josef pick such that Timothy’s quotient is an integer?


My friend reads four times as fast as I do. If it takes me 90 minutes to read a book, how many minutes will it take my friend to read the same book?

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


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


Twelve points are marked on a circle. How many distinct convex polygons of four or more sides can be drawn using some (or all) of these twelve points as vertices?


Given that $a$, $b$ are nonzero real numbers such that $a^2 + b^2 = 10ab$, find the value of $\left|\frac{a+b}{a-b}\right|$.


Find the projection **p** of the vectors $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 5 \\ -1 \end{pmatrix}$ onto a vector if the resultant projections are the same.

Solution: \boxed{\begin{pmatrix} \frac{21}{73} \\ \frac{56}{73} \end{pmatrix}}


What is the sum of all positive integer solutions less than or equal to $25$ to the congruence $7(5x-3) \equiv 35 \pmod{9}$?


Find the reflection of the vector $\begin{pmatrix} 2 \\ 6 \end{pmatrix}$ over the vector $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$, and then scale the reflection by a factor of 3.

Solution: \boxed{\begin{pmatrix} \frac{78}{5} \\ -\frac{54}{5} \end{pmatrix}}


Crestwood Elementary School has a popular dodgeball league, consisting of twelve players, including Sarah and Mia. Each day at recess, the twelve players split into two dodgeball games, each with six players in no particular order. Over the course of a semester, each possible matchup of six players occurs once. How many times did Sarah play in the same game as Mia?


What is the smallest four-digit number that is divisible by $35$?


There are 4 math clubs in the school district, with 6, 8, 9, and 10 students respectively. Each club has three co-presidents. If I randomly select a club, and then randomly select four members of that club to give a copy of *Introduction to Counting and Probability*, what is the probability that exactly two of the people who receive books are co-presidents?


A spinner is divided into 8 equal sectors numbered from 1 to 8. Jane and her sister each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her sister wins. What is the probability that Jane wins?


A new game is underway with 50 participants sitting in a circle, including the organizer, Sam. Sam starts with 3 calculators: one displaying 1, another displaying 0, and the last with -1. He performs the following actions: he squares the number on the 1-display calculator, presses the square button on the 0-display calculator, and on the -1-display calculator, he toggles between squaring and negating it on every alternate pass starting with squaring. Each participant performs the same operations in sequence, passing the calculators to the next person. After a full round, when all calculators return to Sam, he computes the sum of the numbers on the calculators. What is the final sum?


Calculate $\sqrt{48x} \cdot \sqrt{3x} \cdot \sqrt[3]{81x^2}$ . Express your answer in simplest radical form in terms of $x$.

Solution: \boxed{36x \sqrt[3]{3x^2}}


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

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


Calculate the number of base-2 digits (bits) required to represent the hexadecimal number $ABC12_{16}$.


Simplify \(\tan \frac{\pi}{12} + \tan \frac{5\pi}{12}\).


Bill is tasked with purchasing exactly eight donuts from a shop offering six different kinds. He must buy at least two of each of three specific kinds but has no restrictions on the others. How many combinations will satisfy Bill's order requirements?


Larry initially writes the number 4 on his thumb. He then applies function $f$ to 4 and writes the output on his next finger. Larry repeats this process, applying $f$ and writing the output on a new finger each time. If the new function $f$ maps numbers as follows seen in a graph: \(f(4) = 7\), \(f(7) = 8\), \(f(8) = 3\), \(f(3) = 5\), \(f(5) = 4\) (forming a cycle every five applications), what number will Larry write on his twelfth finger?


Given a triangle $ABC$ with a perimeter of 2023 units where the side lengths are all integers and $AB < BC \leq AC$, identify the smallest possible value for the difference $BC - AB$.

Solution: \boxed{BC - AB = 1}


The stem and leaf plot shows the heights, in inches, of the players on the Westvale High School boys' basketball team. Calculate the mean height of the players on the team. (Note: $6|2$ represents 62 inches.)

Height of the Players on the Basketball Team (inches)

$5|7$

$6|2\;4\;4\;5\;7\;8$

$7|0\;1\;2\;2\;3\;4\;5\;5$


How many total days were there in the years 2000 through 2005?


Find the value of $x$ such that $\sqrt{5x + 9} = 11$.


Consider the sequence defined by \( G_n = 2^{3^n} + 1 \). Determine the units digit of \( G_{500} \).


If $a \equiv 25 \pmod{48}$ and $b \equiv 80 \pmod{48}$, then for what integer $n$ in the set $\{150, 151, 152, \ldots, 190, 191\}$ is it true that $$a-b \equiv n \pmod{48}~?$$


For $-1<s<1$, let $T(s)$ denote the sum of the geometric series \[9+9s+9s^2+9s^3+\cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=1458$. Find $T(b)+T(-b)$.


Let the graph of \(y = f(x)\) for \(-5 \leq x \leq 5\) be defined as follows:
- \(f(x) = -\frac{1}{2}x\) when \(-5 \leq x < 0\)
- \(f(x) = \frac{1}{3}x\) when \(0 \leq x \leq 5\)

On this interval, what is the range of \(f(x) + x\)?

Solution: \boxed{\left[-2.5, \frac{20}{3}\right]}


Crestview school has updated its school colors to include purple, gold, and white. The students are redesigning a flag with three solid-colored horizontal stripes. How many different flags are possible using one, two, or all three of the school colors, if adjacent stripes must not have the same color?


James and Lindsay have a jar containing a collective sum of 100 quarters and 185 dimes. The capacity of a roll for quarters is now 45, and for dimes, it is 55. Determine the total dollar value of the quarters and dimes that cannot be rolled.


Three different integers are randomly chosen from the set $$\{-9, -5, -8, 7, 4, -2, 6\}.$$
What is the probability that their product is negative? Express your answer as a common fraction.


In a triangle, the area is numerically equal to twice the perimeter. Find the radius of the inscribed circle.


Consider the quadratic equation $k(x^2 - x) + 2x + 7 = 0$. Let $a$ and $b$ be the roots of this equation for which $\frac{a}{b} + \frac{b}{a} = 3$. Let $k_1$ and $k_2$ be the values of $k$ fulfilling the previous condition. Determine the value of $\frac{k_1}{k_2} + \frac{k_2}{k_1}$.


Let $n$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $n$ is divided by $5!$.


The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}.$ There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} 5 \\ 2 \\ -3 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).\] Find $r.$


A plane's passengers consist of 50% women and 50% men. Twenty percent of the women and fifteen percent of the men are in first class. What is the total number of passengers in first class if the plane is carrying 300 passengers?


The numbers $a,$ $b,$ $c,$ $d$ are equal to 2, 3, 4, 5, in some order. Find the largest possible value of
\[
(a+b)(c+d).
\]


In a circle $P$, angle $MBC$ measures 60 degrees. What is the measure of minor arc $MB$ in degrees? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
draw(unitcircle);
draw(dir(120)--(-1,0)--(1,0));
dot("$B$",(-1,0),W); dot("$M$",dir(120),N); dot("$C$",(1,0),E); dot("$P$",(0,0),S);
[/asy]


Calculate the value of $\sqrt{31 - 12\sqrt{3}} + \sqrt{31 + 12\sqrt{3}}$.


Let \(x, y, z\) be nonzero complex numbers such that \(x + y + z = 18\) and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz.\]
Find \(\frac{x^3 + y^3 + z^3}{xyz}\).


In a quiz competition, scoring is set such that $7$ points are awarded for each correct answer, $0$ points for each incorrect answer, and $2$ points for each question left unanswered. John decides to answer the first $18$ questions and leave the last $7$ unanswered in a $25$-question quiz. Determine the minimum number of questions John must answer correctly to score at least $120$ points.


What is the value of $\dfrac{5 \times 7}{10}$?


Evaluate the expression $4^4 - 4 \times 4^3 + 6 \times 4^2 - 4$.


What is the sum of five consecutive even integers if the sum of the first and last integers is 204? Also, find the smallest integer in this sequence.


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

Solution: \boxed{-4x - 7}


Given that $x + \cos y = 3005$ and $x + 3005 \sin y = 3004$, where $0 \leq y \leq \pi$, find the value of $x + y$.


Suppose two distinct integers are chosen from between 1 and 25, inclusive. What is the probability that their sum is divisible by 4?


"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x < m$. To graph a congruence, we mark every point $(x, y)$ that satisfies the congruence. Consider the graphs of the following equations on modulo $9$ graph paper: $$y \equiv 2x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$
Determine the sum of the $x$-coordinates of the points where the graphs of the two equations intersect.


Find the product of all $x$ such that the expression $\frac{x^2 - 4x + 4}{x^2 - 5x + 6}$ is undefined.


Determine the value of $n$ such that $10^n = 10^{-3} \times \sqrt{\frac{10^{53}}{0.0001}}$.


A pentagonal prism has 7 faces, 10 vertices, and 15 edges. A new pyramid is to be attached to one of its pentagonal faces. Calculate the sum of the number of exterior faces, vertices, and edges of the combined solid when the pyramid is attached to the pentagonal face.