Determine the integer \( n \) such that the quantity
\[
\frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i}
\]
is a twelfth root of unity, i.e., it is equal to \(\cos \frac{2n \pi}{12} + i \sin \frac{2n \pi}{12}\) for some integer \( n \) between 0 and 11 inclusive.


In the given diagram, $QRS$ is a straight line. Calculate the measure of $\angle RPS,$ in degrees.
[asy]
pair Q=(0,0);
pair R=(1.3,0);
pair SS=(2.3,0);
pair P=(.8,1);

draw(P--Q--R--SS--P--R);

label("$Q$",Q,S);
label("$R$",R,S);
label("$S$",SS,S);
label("$P$",P,N);

label("$32^\circ$",Q+(.12,.05),NE);
label("$58^\circ$",P-(.02,.15),S);
label("$45^\circ$",SS+(-.32,.05),NW);
[/asy]


Let $\mathbf{c}$ and $\mathbf{d}$ be two vectors such that
\[\|\mathbf{c} + 3\mathbf{d}\| = \|\mathbf{d}\|.\]
Find the angle between the vectors $\mathbf{c} + 3 \mathbf{d}$ and $\mathbf{c}$, in degrees.


The real number $x$ satisfies $x^2 - 7x + 12 < 0.$ Find all possible values of $x^2 + 7x + 12.$

Solution: \boxed{(42, 56)}


The lengths of the three sides of a triangle are \( 10 \), \( y+6 \), and \( 3y+2 \). The perimeter of the triangle is \( 45 \). What is the length of the longest side of the triangle?


Compute the sum:
\[\sum_{k=1}^\infty \{\zeta(2k)\}\]
where $\zeta(x)$ is the Riemann zeta function defined for $x > 1$ by
\[\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}\]
and $\{x\}$ denotes the fractional part of $x$.


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


Right triangle $ABC$ has a right angle at $C$ and $\overline{DE}$, $\overline{FG}$ are parallel to $\overline{AB}$, with $AD = DF = FB$. If $\overline{AB} = 8$ and $\overline{BC} = 6$, find the ratio of the area of trapezoid $FGAB$ to the area of triangle $ABC$.


Calculate the value of $\frac{x^2 - 5x + 4}{x - 4}$ when $x = 3$.


Find the $2023^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{7}{12}$.


At a conference with 30 executives, 15 executives drank coffee, 13 executives drank tea, and 9 executives drank soda. Seven executives drank both coffee and tea, four drank both tea and soda, and three drank both coffee and soda. Two executives drank all three beverages. How many executives drank none of these beverages?


The equations $x^3 + Kx + 20 = 0$ and $x^3 + Lx^2 + 100 = 0$ have two roots in common. Find the product of these common roots, expressed in the form $a \sqrt[b]{c}$ where $a,$ $b,$ and $c$ are positive integers, and find $a + b + c.$


An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?

Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 20 rows, use rows 0 through 19.


Find the value of $a^4 + a^5b^3 + a^3b^5 + b^4$ if $a$ and $b$ are the roots of the polynomial $x^2 - 5x + 6 = 0$.


Find the reflection of \(\begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}\) over the vector \(\begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}\).

Solution: \boxed{\begin{pmatrix} \frac{2}{3} \\ -\frac{11}{3} \\ \frac{1}{3} \end{pmatrix}}


The mean of $7, 12$ and $21$ is equal to the mean of $15$ and $z$. What is the value of $z$?


The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $kx(x+5)$, where $x$ and $k$ are positive integers. If one of the integers is 60, what is the smallest possible value of the other integer for $k=2$?


Let $g$ be a function satisfying $g(xy) = g(x)/y$ for all positive real numbers $x$ and $y$. If $g(800) = 2$, what is the value of $g(1000)$?


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers satisfying:
\begin{align*}
  17x +  b y +  c z &= 0 \\
   a x + 29 y +  c z &= 0 \\
   a x +  b y + 53 z &= 0.
\end{align*}
Assume that $ a \ne 17$ and $ x \ne 0$. Find the value of:
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53} \, ?\]


Find the maximum value of
\[
\sin (\theta_1 + \theta_2) - \sin (\theta_1 - \theta_2) + \sin (\theta_2 + \theta_3) - \sin (\theta_2 - \theta_3) + \sin (\theta_3 + \theta_4) - \sin (\theta_3 - \theta_4) + \sin (\theta_4 + \theta_5) - \sin (\theta_4 - \theta_5) + \sin (\theta_5 + \theta_1) - \sin (\theta_5 - \theta_1),
\]
over all real numbers $\theta_1, \theta_2, \theta_3, \theta_4, \theta_5.$


Rationalize the denominator of \(\frac{5}{4\sqrt{7} - 3\sqrt{2}}\) and write your answer in the form \(\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}\), where \(B < D\), the fraction is in lowest terms, and all radicals are in simplest radical form. What is \(A+B+C+D+E\)?


At breakfast, lunch, and dinner, Joe randomly chooses either an apple, an orange, a banana, or a grape to eat. The probabilities are 0.4 for an apple, 0.3 for an orange, 0.2 for a banana, and 0.1 for a grape. What is the probability that Joe will eat at least two different kinds of fruit on a given day?


Given the function $g(x) = \frac{3x - 2}{x + 4}$, find the form of its inverse $g^{-1}(x)$, and specifically compute the ratio $a/c$, where $g^{-1}(x) = \frac{ax + b}{cx + d}$.


What is the greatest 3-digit base 8 positive integer that is divisible by 7? (Express your answer in base 8.)


Jordan and his three best friends are on a relay team. In this relay race, the first runner runs a lap, then the second, then the third, and finally the fourth. If Jordan is the first person to run the second lap, how many different orders can the four team members run?


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


You plan to distribute your gold coins among 8 close friends. However, you realize that instead of dividing the gold coins into exactly 8 equal piles, 4 people will receive an extra gold coin. Given that you possess fewer than 150 gold coins, what is the maximal number of coins you could have for this scenario to occur?


What is the probability that George gets fewer heads than tails if he flips 8 coins?


My three-digit code is 0A3. Reckha can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the positions of two digits (so A30 and 30A, for example, are forbidden, but 3A0 is fine). Reckha can otherwise choose any three-digit code where each digit is in the set $\{0, 1, 2, \dots, 9, A, B, C, D, E, F\}$. How many codes are available for Reckha?


Let $f$ be defined by
\[f(x) = \left\{
\begin{array}{cl}
3-x & \text{ if } x \leq 2, \\
2x - x^2 & \text{ if } x > 2.
\end{array}
\right.\]
Calculate $f^{-1}(-2) + f^{-1}(1) + f^{-1}(4)$.

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


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


What is the sum of all integer values of $n$ such that $\frac{36}{2n - 1}$ is an integer?


I have four distinct mystery novels, four distinct fantasy novels, and four distinct biographies. I'm planning a trip and want to bring two books from different genres. How many different pairs can I select?


Find the least common multiple (LCM) and the greatest common divisor (GCD) of 36 and 48.


In a hypothetical scenario, a small country is planning an international event in 2023. Let \( A \), \( B \), and \( C \) be distinct positive integers such that their product \( A \cdot B \cdot C = 2023 \). Determine the largest possible value of the sum \( A + B + C \).


John goes to the store to buy groceries weekly with a standard list: $6$ bananas at $\$2$ each, $2$ loaves of bread at $\$3$ each, $3$ packs of butter at $\$5$ each, and $4$ boxes of cereal at $\$6$ each. This week, the store offers a 25% discount on cereal and John has a coupon for $\$10$ off if he spends $\$50$ or more. Calculate how much John pays after all discounts.


The number of degrees in the measures of the interior angles of a convex hexagon are six consecutive integers. What is the number of degrees in the largest angle?


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


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

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


How many distinct sequences of five letters can be made from the letters in EXAMPLE if each letter can be used only once and each sequence must begin with E and end with M?


What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 10 units to the sum of the numerical values of its perimeter and the length of an interior angle bisector, in units? Express your answer as a common fraction in simplest radical form.


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

Solution: \boxed{\begin{pmatrix} \frac{4}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{pmatrix}}


Let \[ q(x,y) =
\begin{cases}
x^2 + y^2 &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x^2 - 3y &\quad \text{if } x < 0 \text{ and } y < 0, \\
2x + 2y &\quad \text{otherwise}.
\end{cases}
\]
What is $q(q(2,-2), q(-3,-1))$?


Say that an integer \( B \) is delightful if there exist several consecutive integers, including \( B \), that add up to 2017. What is the smallest delightful integer?


In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.


Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 8.$  Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\]


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


If a 4'' by 4'' square is added at each successive stage, what will be the area of the rectangle at Stage 7, in square inches?


What is half of the sum of the absolute values of the differences of the squares of 20 and 15?


Jake rolls 7 fair 5-sided dice, where each die face is numbered from 1 to 5. Specifically, numbers 1, 3, and 5 are considered 'even' and 2 and 4 are 'odd' for this problem. What is the probability that exactly three of these dice show an 'even' number?


A positive five-digit integer is in the form $AB,CBA$, where $A$, $B$, and $C$ are each distinct digits. Additionally, $A$ should not equal $C$. What is the greatest possible value of $AB,CBA$ that is divisible by eleven?


If the probability that a baby born in a certain hospital will speak in the next day is 1/3, what is the probability that at least 3 babies out of a cluster of 7 babies will speak tomorrow?


Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 3,$ $\|\mathbf{b}\| = 4,$ and $\|\mathbf{a} + \mathbf{b}\|^2 = 64.$ Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.


When you simplify \(\sqrt[3]{54a^5b^9c^{14}}\), what is the sum of the exponents of the variables that are outside the radical?


A club has 40 members, and during a particular activity, they share 200 pieces of cookies such that each member gets at least one cookie. What is the maximum number of cookies one member could have taken?


We have a triangle $\triangle PQR$ such that $PQ = PR = 13$ and $QR = 24$. What is the length of the median $PM$?


Write the expression $\frac{6+4d}{9} + 3$ as a single fraction.

Solution: \boxed{\frac{33 + 4d}{9}}


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


From the following infinite list of numbers, how many are integers? $$\sqrt{7776},\sqrt[3]{7776},\sqrt[4]{7776},\sqrt[5]{7776},\sqrt[6]{7776},\ldots$$


Suppose $z$ and $w$ are complex numbers such that
\[|z| = |w| = 2, \quad \text{and} \quad z \overline{w} + \overline{z} w = 2.\]
Find the largest possible value of the real part of $z + w.$


Liam earned scores of 92, 85, 89, and 93 on his first four mathematics examinations. If Liam needs an average of 90 after his fifth exam, what is the minimum score he must achieve on the fifth exam?


Given the six digits 1, 3, 5, 7, 8, and 9, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer, and the sum of the two digits must be even?


Let $x^2+bx+c = 0$ be a quadratic whose roots are each three less than the roots of $2x^2-4x-5$. What is $c$?


A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.


Find the remainder when $x^4 + 4x^2 + 20x + 1$ is divided by $x^2 - 2x + 7.$

Solution: \boxed{8x - 6}


How many of the divisors of \( 10! \) are larger than \( 9! \)?


Suppose that $2n, 2n+2, 2n+4, 2n+6, 2n+8$ are five consecutive even integers.

Determine a simplified expression for the sum of these five consecutive even integers.

Solution: \boxed{10n + 20}


Determine the first term in the geometric sequence $a, b, c, d, e, 32, 64$.


For what base is the representation of $346_{10}$ a four digit number whose final digit is even?


What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 780 = 0$ has integral solutions?


If $\frac{8}{5} (p + q + r + s) = 20$, what is the average of $p$, $q$, $r$, and $s$?


In writing the integers from 10 through 199 inclusive, how many times is the digit 7 written?


What is the smallest positive integer $n$ such that $23n \equiv 5678 \pmod{11}$?


How many different counting numbers will each leave a remainder of 11 when divided into 101?


How many ways are there to place 6 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?


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


Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor + \lceil \sqrt{26} \rceil.\]
Here $\lceil x \rceil$ denotes the smallest integer not less than $x$.


How many integers are common solutions to these three inequalities?
\[
\begin{array}{cccc}
(1) & -3y & \geq & y+9 \\
(2) & -2y & \leq & 18 \\
(3) & -4y & \geq & 2y+20
\end{array}
\]


A number $x$ is equal to $3 \cdot 40 \cdot 75$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


Lucas, Mia, and Grace are planning a hike. Lucas hikes at a speed of 5 miles per hour. If Mia hikes $\frac{3}{4}$ as fast as Lucas, and Grace hikes $\frac{6}{7}$ as fast as Mia, how fast does Grace hike? Additionally, if Liam hikes $\frac{4}{3}$ times faster than Grace, determine Liam’s hiking speed.

Solution: \boxed{\frac{30}{7} \text{ mph}}


A regular hexagon $ABCDEF$ has sides of length 3. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form.


For which positive integer values of $k$ does $kx^2 + 24x + 3k = 0$ have rational solutions? Express your answers separated by commas and in increasing order.


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


Determine all $x$ such that $\lfloor \lfloor 3x \rfloor + 1/3 \rfloor = \lfloor x + 3 \rfloor.$

Solution: \boxed{\left[ \frac{4}{3}, \frac{5}{3} \right)}


When $500^{150}$ is expanded out, the result is $1$ followed by how many zeros?


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


In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are then placed on top of the grid. Five of the circles have a diameter equal to a side of a small square (1 cm), and one large circle has a diameter equal to the length of four sides of a small square (4 cm). The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?


When the base-16 number $ABCD_{16}$ is converted to base 2, how many base-2 digits (bits) does it have?


Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional vectors such that the angle between them is $45^\circ$, and both have length 2. Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 3\mathbf{q}$ and $3\mathbf{p} + \mathbf{q}$.


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


Let $r_1, r_2,$ and $r_3$ be the roots of the polynomial equation
\[x^3 - 5x^2 + 16 = 0.\]
Find the monic polynomial in terms of $x$, whose roots are $3r_1, 3r_2, 3r_3$.

Solution: \boxed{x^3 - 15x^2 + 432}


I have four distinct mystery novels, four distinct fantasy novels, and four distinct biographies. I'm planning a trip and I want to take two books of different genres. How many different pairs of books can I choose?


Charlie received a bag of coins from his uncle. On his way to the shop, he accidentally dropped one-third of them. Later, he realized his loss and managed to find two-thirds of the coins he had dropped. How many of Charlie's original coins are still missing? Express your answer as a common fraction.


The square quilt block shown is made from sixteen unit squares, some of which have been divided into either half diagonally or into quarters to form smaller triangles. Determine the fraction of the square quilt that is shaded. The shaded parts are arranged as follows:
- In the first row, two squares are each divided into two triangles, shading one triangle per square.
- In the second row, one square is divided into quarters, shading two of the quarters.
- In the third row, there are no divisions or shadings.
- In the fourth row, one square is divided into half diagonally, with one half shaded.


Jenny baked 48 pies for a large family gathering. Here's how her pies were divided on the basis of their ingredients:
  - $5/8$ of the pies contained strawberries.
  - $3/4$ contained bananas.
  - $2/3$ contained kiwifruit.
  - $1/4$ contained coconut flakes.
What is the largest possible number of pies that had none of these ingredients?


Given the observed pattern where \( 9^2 = 81 \) contains no zeros, \( 999^2 = 998001 \) contains two zeros, and \( 9999^2 = 99980001 \) contains three zeros, determine how many zeros are in the decimal expansion of \( 99999999^2 \).


Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 1230^\circ.$


Let $x_1, x_2, \ldots, x_n$ be real numbers that satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and
\[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\]
What is the smallest possible value of $n$?


What is the median number of moons per planet? (Include Pluto, and consider a hypothetical scenario where Mars has 3 moons and Jupiter has 18 moons) \begin{tabular}{c|c}
Planet & $\#$ of Moons\\
\hline
Mercury&0\\
Venus &0\\
Earth &1\\
Mars &3\\
Jupiter&18\\
Saturn&23\\
Uranus&15\\
Neptune&2\\
Pluto&5\\
\end{tabular}


**How many positive factors does 72 have, and what is their sum?**


If the consecutive integers from $1$ to $50$ were written as $$123456789101112...4950,$$ what would be the $75^{\text{th}}$ digit to be written?


Liam took a computer science examination which consisted of 50 problems: 15 on programming, 20 on data structures, and 15 on algorithms. He answered $80\%$ of the programming, $50\%$ of the data structures, and $70\%$ of the algorithms questions correctly. The passing grade for the test is $65\%$. How many more questions would Liam have needed to answer correctly to pass?


The surface area of a sphere is $400\pi\text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of $\pi$.


In triangle $ABC$, $AB = 8$, $AC = 15$, and $BC = 17$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $BC$. Find $GP$.


A cylinder has a height of 15 and a radius of 5. Determine the total surface area, including the two ends, and the volume of the cylinder.


Calculate the sum of the infinite series $1 + 3\left(\frac{1}{2000}\right) + 5\left(\frac{1}{2000}\right)^2 + 7\left(\frac{1}{2000}\right)^3 + \cdots$.


Let \( m = 2^{35}5^{21} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)?


In 1960, there were 600,000 cases of a disease reported in a country. By 2000, the number of cases had dropped to 600. Assuming a linear decrease, how many cases would have been reported in 1990?


The lattice is modified so that it continues for $9$ rows with each row containing $7$ numbers instead of $5$. What will be the fourth number in the $9$th row?
\[
\begin{tabular}{rccccccc}
Row 1: & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
Row 2: & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\
Row 3: & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\
Row 4: & 22 & 23 & 24 & 25 & 26 & 27 & 28
\end{tabular}
\]


Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where  \[g(n) = \left\{
\begin{array}{cl} 12 & \text{ if }n\text{ is a multiple of 3}, \\
3 & \text{ if }n\text{ is only a multiple of 2}, \\
0 & \text{ if }n\text{ is neither a multiple of 2 nor 3}.
\end{array}
\right.\] Allie rolls the die five times and gets a 5, 4, 1, 2, and 6. Betty rolls and gets 6, 3, 3, 2, and 1. What is the product of Allie's total points and Betty's total points?


The points $(2, 8), (14, 17)$ and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is minimum?


Find the greatest common divisor of $45123$ and $32768$.


Eight fair coins are to be flipped. What is the probability that at least seven of them will be heads or at least seven of them will be tails?


How many ways are there to distribute 7 indistinguishable balls into 4 indistinguishable boxes such that each box has at least one ball?


A box of 200 personalized pencils costs $\$50$. If a company wants to buy 3000 pencils and receives a 10% discount on every additional set of 1000 pencils beyond the first 1000, how much does it cost in total?


When \( n \) is divided by 5, the remainder is 2. What is the remainder when \( n + 2023 \) is divided by 5?


Eight students participate in a banana eating contest. The graph shows the number of bananas eaten by each participating student. Alice ate the most bananas and Tom ate the fewest. How many more bananas than Tom did Alice eat?

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
for( int i = 0 ; i < 10 ; ++i ){

yaxis[i] = (0,i);

draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,9));
draw((17,9)--(17,0));

fill((1,0)--(1,2)--(2,2)--(2,0)--cycle,grey);
fill((3,0)--(3,3)--(4,3)--(4,0)--cycle,grey);
fill((5,0)--(5,7)--(6,7)--(6,0)--cycle,grey);
fill((7,0)--(7,4)--(8,4)--(8,0)--cycle,grey);
fill((9,0)--(9,5)--(10,5)--(10,0)--cycle,grey);
fill((11,0)--(11,3)--(12,3)--(12,0)--cycle,grey);
fill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);
fill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students/Participants",(8.5,0),S);
label("Results of a Banana Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Bananas Eaten",(-1,4.5),W);
[/asy]


Consider the parabolas $y = (x - 2)^2$ and $x + 3 = (y + 2)^2$. Find the sum of the $x$-coordinates and the $y$-coordinates where these parabolas intersect, i.e., calculate
\[
x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.
\]


Express \( 3.\overline{56} \) as a common fraction in lowest terms.


Circle $\omega$ has radius 3 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=10$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=9$.

Solution: \boxed{2\sqrt{91} + 9}


The Gnollish language now consists of 4 words, "splargh," "glumph," "amr," and "kreet." In a sentence, "splargh" cannot come directly before "glumph," and "kreet" cannot come directly before "amr." How many valid 3-word sentences are there in Gnollish?


What is the arithmetic mean of all positive three-digit multiples of 7?


Circle \(D\) has a radius of 8 cm. What is the area of the largest possible inscribed triangle having one side as a diameter of circle \(D\)?

Solution: \boxed{64\text{ square centimeters}}


Find the unique positive real number $b$ for which there exists a smallest positive real number $a$ such that all roots of the polynomial $x^3 - 2ax^2 + bx - 2a$ are real.


Calculate the sum of the angles of the solutions in degrees to the equation $w^4 = 81i$. Express the solutions in the form:
\[
w_1 = r_1 (\cos \theta_1 + i \sin \theta_1),
\]
\[
w_2 = r_2 (\cos \theta_2 + i \sin \theta_2),
\]
\[
w_3 = r_3 (\cos \theta_3 + i \sin \theta_3),
\]
\[
w_4 = r_4 (\cos \theta_4 + i \sin \theta_4),
\]
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$


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


Determine the value of $k$ if the roots of the quadratic equation $2x^2 + 8x + k = 0$ are $x = -2 \pm \sqrt{6}$.


Solution: \boxed{45x^6(1 + 2x - 6x^6)}


In triangle $ABC$, $AB = 8$, $BC = 13$, and $AC = 15$. If $D$ and $E$ are selected on $\overline{AB}$ and $\overline{AC}$ respectively, such that $AD = 3$ and $AE = 10$, find the area of triangle $ADE$.


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


What is the last digit of the decimal expansion of $\frac{1}{2^{15}}$?


Six balls are numbered 1 through 6 and placed in a bowl. Josh will randomly choose a ball from the bowl, note its number, and then put it back into the bowl. He will then randomly choose another ball from the bowl and note its number. What is the probability that the product of the two numbers will be odd and greater than 14? Express your answer as a common fraction.


A square pyramid has a base edge of 24 inches and an altitude of 18 inches. A smaller square pyramid whose altitude is one-third of the original pyramid's altitude is cut away from the apex. What fractional part of the original pyramid's volume is the resultant frustum's volume?


Nilesh wants to distribute 480 beads into $r$ total containers such that each container contains an equal number of beads. There is more than one container, and each container contains more than one bead. For how many values of $r$ can this be achieved?


Find the phase shift of the graph of \( y = 3 \sin \left( 3x + \frac{\pi}{4} \right) \).


Tim has $\$4.50$ in U.S. coins. He has the same number of quarters, nickels, and dimes. What is the greatest number of dimes he could have?


Find the focus of the parabola $x = -\frac{1}{4} y^2.$


Find the matrix $\mathbf{M}$ that doubles the first row and triples the second row of any given 2x2 matrix $\mathbf{A}= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. That is, find $\mathbf{M}$ such that
\[
\mathbf{M} \mathbf{A} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.
\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


A circle is tangent to the lines $3x + 4y = 24$ and $3x + 4y = -16.$ The center of the circle lies on the line $x - 3y = 0.$ Find the center of the circle.

Solution: \boxed{\left(\frac{12}{13}, \frac{4}{13}\right)}


The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Alice is on the surface of Earth, 6,400 kilometers from the center, the gravitational force is 800 Newtons. What is the gravitational force (in Newtons) that the Earth exerts on her when she's standing on a satellite orbiting at 384,000 kilometers from the center of the earth? Express your answer as a fraction.


At a national conference, 31 participants are assigned to individual rooms in a hotel where the rooms are numbered consecutively from 1 to 31. Due to late arrivals, the participants staying in rooms 15 and 20 are absent. What is the median room number of the remaining 29 participants?


In the diagram, $\triangle ABC$ is right-angled at $A,$ with $AB=40$ and $AC=30.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $BD.$


On the game show $\text{\emph{Wheel of Fraction New Spin}}$, the spinner is now divided into the following sections: "$0", "$2000", "$300", "$7000", "$400". Assuming each section is of equal area, calculate the probability of earning exactly $\$2400$ in your first three spins.


In the diagram, each of the four identical circles touch the other two adjacent circles directly. The circumference of each circle is 72. What is the perimeter of the shaded region formed at the center where all four circles meet? Assume each pair of adjacent circles' centers form an angle of 90 degrees at the center of the shaded region.


In $\triangle ABC,$ $AB=AC=20$ and $BC=18.$  Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. Suppose $AD = 12$. What is the perimeter of parallelogram $ADEF$?


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


Fully factor the following expression: $3x^2 - 75y^2$

Solution: \boxed{3(x + 5y)(x - 5y)}


The function \( g(x) \) satisfies
\[ g(xz) = g(x)g(z) \]
for all real numbers \( x \) and \( z \), and \( g(1) \neq 0 \). Find \( g(5) \).


A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. If the radius of the hemisphere is $5$ cm, find the radius of the original bubble.


Let \(ABCDEFGH\) be a cube with side length 2. Find the volume of the pyramid with base \(ACG\) and vertex \(B\).


What is the first year after 2010 in which the sum of the year's digits is 7?


Alicia has \( n \) candies, where \( n \) is a positive integer with three digits. If she buys \( 6 \) more, she will have a multiple of \( 9 \). If she loses \( 9 \), she will have a multiple of \( 6 \). What is the smallest possible value of \( n \)?


Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form.


An ellipse is centered at the coordinates (3, -5). It has a semi-major axis of length 6 and a semi-minor axis of length 2. Evaluate the sum of the center coordinates and the lengths of the axes, \( h + k + a + b \).


The first $30$ numbers of a similar arrangement are shown below. What would be the value of the $50^{\mathrm{th}}$ number if the arrangement were continued?

$\bullet$ Row 1: $3,$ $3$

$\bullet$ Row 2: $6,$ $6,$ $6,$ $6$

$\bullet$ Row 3: $9,$ $9,$ $9,$ $9,$ $9,$ $9$

$\bullet$ Row 4: $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12$


Determine the sum of the common positive divisors of the numbers $$18, 54, 36,$$ and $$90.$$


What integer $n$ satisfies $0 \leq n < 31$ and $$-527 \equiv n \pmod{31}~?$$


Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin.


Seventeen is the seventh prime number. What is the 12th prime number, and what is the sum of the prime numbers from the seventh to the 12th?


In a certain hyperbola, the center is at \((3,1)\), one focus is at \((3,9)\), and one vertex is at \((3,-2)\). The equation of this hyperbola can be written as
\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.
\]
Find \(h + k + a + b.\)

Solution: \boxed{7 + \sqrt{55}}


What is the greatest common divisor of 1729 and 1309?


How many different 8-digit positive integers exist that end with an even number? (Note: the first digit cannot be 0, and an even number ends in 0, 2, 4, 6, or 8.)


Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $12$ digits. Express your answer in base $10$.


A fair 8-sided die is rolled. If the roll is even, then you win that amount of dollars, and if the roll is a 3 or 5, you win $\$2$. If the roll results in any other odd number, you win nothing. What is the expected value of your winnings?


A box contains seven cards. Two of the cards are black on both sides, three 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? Express your answer as a common fraction.


A rectangular wooden block is 3 inches long, 4 inches wide, and 2 inches high. The block is painted blue on all six faces and then cut into 1-inch cubes. How many of the cubes each have a total number of blue faces that is an even number?


Determine the slope \(m\) of the asymptotes for the hyperbola given by the equation \[\frac{x^2}{144} - \frac{y^2}{81} = 1.\]


In the diagram, regular hexagon $ABCDGH$ has sides of length $10$. Triangle $ABE$ is equilateral. Line segments $BE$ and $AD$ intersect at $P$. Point $Q$ is on $BD$ such that $PQ$ is perpendicular to $BD$ and $PQ = x$. Determine the measure of angle $BPC$.


The numbers \(a, b, c, d\) are equal to \(2, 3, 5, 6\), in some order. Find the largest possible value of
\[ab + bc + cd + da.\]


How many ways are there to choose 4 cards from a standard deck of 52 cards, if exactly three suits are represented such that two cards come from one suit and one card comes from each of the remaining two suits?


A basketball team has 16 players, including a set of triplets: Alice, Betty, and Cindy, as well as a set of twins: Donna and Elly. In how many ways can we choose 7 starters if the only restriction is that not all three triplets or both twins can be in the starting lineup together?


Let \( m \) be the smallest integer whose cube root is in the form \( n+r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \( \frac{1}{500} \). Find \( n \).


What is the largest number, all of whose digits are 4 or 5, and whose digits add up to $17$?


Given that $b$ is an even multiple of $7768$, find the greatest common divisor of $4b^2 + 37b + 72$ and $3b + 8$.


Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?


For how many values of $c$ in the interval $[0, 2000]$ does the equation \[9 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for $x$?


A pirate is searching for buried treasure on 5 islands. On each island, there is a $\frac{1}{3}$ chance that the island has buried treasure and no traps, a $\frac{1}{6}$ chance that the island has traps but no treasure, and a $\frac{1}{2}$ chance that the island has neither traps nor treasure. What is the probability that while searching all 5 islands, the pirate will encounter exactly 4 islands with treasure, and none with traps?


How many positive three-digit integers are multiples of 4 and of 9?


Simplify the product \[\frac{10}{5}\cdot\frac{15}{10}\cdot\frac{20}{15}\dotsm \frac{5n+5}{5n}\dotsm \frac{2010}{2005}.\]


What is the 25th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{8}$ and $\frac{1}{5}$?


The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$.


John borrows $15$ dollars from Clara with a simple interest rate of $10\%$ per day. However, the interest starts accumulating only after the first 3 days. What is the least integer number of days after which John will have to pay her back at least twice as much as he borrowed?

Solution: \boxed{13 \text{ days}}


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


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}}


If 3 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 2 aces are selected? (Assuming there are 4 Aces, 4 Queens, and 52 total cards in the deck.)


The arithmetic mean of five numbers is 20. Three of the numbers are 12, 25, and 13. The other two numbers are equal. What is the sum of the squares of these two equal numbers?


The graph of the line $x-y=c$ is a perpendicular bisector of the line segment from $(2,4)$ to $(6,12)$. What is the value of $c$?


A triangle has sides of lengths 7 cm and 24 cm, and it is a right triangle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth.

Solution: \boxed{22.96 \text{ cm}}


The graphs of $y=|x|$ and $y=-x^2-5x-4$ are drawn.  For every $x$, a vertical segment connecting these two graphs can be drawn as well.  Find the smallest possible length of one of these vertical segments.


For what real value of $v$ is $\frac{-31-\sqrt{481}}{12}$ a root of $6x^2 + 31x + v$?


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


In parallelogram $EFGH$, the measure of angle $EFG$ is twice the measure of angle $FGH$. Determine the measure of angle $EHG$.


Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$.


We define two sequences of vectors $(\mathbf{u}_n)$ and $(\mathbf{z}_n)$ as follows: First, $\mathbf{u}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix},$ $\mathbf{z}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.$ 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} 10 \\ 10 \end{pmatrix}}


Two lines are defined by the equations \( y = mx + 7 \) and \( y = 4x + b \). The lines intersect at the point \( (8, 11) \). Find the value of \( b + m \).


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - x^2y^2\] for all $x, y \in \mathbb{R}$. Find the sum of all possible values of $g(1)$.


For real numbers $s,$ consider the intersection of the lines $2x + 3y = 6s + 4$ and $x + 2y = 3s - 1$. All the points of intersection are plotted on a plane. Determine the slope of the line on which all these points lie.


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

[asy]
import graph; size(8cm); real lsf=0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen ds = black; real xmin = -3.25, xmax = 3.25, ymin = -6.25, ymax = 7.25;
pen cqcqcq = rgb(0.75, 0.75, 0.75);

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

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

xaxis("", xmin, xmax, Ticks(laxis, Step = 1.0, Size = 2, NoZero), Arrows(6), above=true); yaxis("", ymin, ymax, Ticks(laxis, Step = 1.0, Size = 2, NoZero), Arrows(6), above=true);
real f1(real x) { return (x - 2)*(x)*(x + 1)/6 + x + 2; }
draw(graph(f1, -3.25, 3.25), linewidth(1));
clip((xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle);
label("$y=f(x)$", (3.5, 0.6), E);
[/asy]

If \( f(a) = b \) and \( f(b) = 3 \), then what is the value of \( a - b \)?


Let $O$ be the origin, and $(2a, 2b, 2c)$ be a fixed point. A plane passes through $(2a, 2b, 2c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A$, $B$, and $C$, respectively. Assume none of these points coincides with $O$ and let $(p, q, r)$ be the center of the sphere passing through $A$, $B$, $C$, $O$. Find the value of $\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.$


In polar coordinates, the point $\left( -3, \frac{\pi}{6} \right)$ corresponds to which point in the standard polar coordinate representation? Enter your answer as $(r,\theta)$, where $r > 0$ and $0 \leq \theta < 2\pi$.

Solution: \boxed{(3, \frac{7\pi}{6})}


Thirty gremlins and twelve pixies are at the Forest Gathering. Due to a recent dispute, half of the gremlins (15 gremlins) refuse to shake hands with each other but will shake hands with the pixies and the other half of the gremlins. The rest of the gremlins are friendly and shake hands with all gremlins and all pixies. The pixies, neutral in the disputes, shake hands with all gremlins. Each pair of creatures shakes hands at most once. How many handshakes occur at this gathering?


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


Susan went to a nearby cafe, which recently updated its menu. Not knowing that Susan prefers vegan food and is allergic to nuts, her friend recommended this cafe. The menu features six dishes that are vegan, which represents one-third of the total dishes offered. Out of these vegan options, however, four include nuts. What fraction of the items on the menu can Susan eat?


Calculate the angle between the hour-hand and minute-hand of a clock at 3:15.


On a whiteboard, "CALCULATIONS" is spelled out with $12$ magnets, one letter per magnet. Three vowels and three consonants fall off and are put away in a bag. If the C's, A's, and L's are indistinguishable, how many distinct possible collections of letters could be put in the bag?


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


What is the modular inverse of $13$, modulo $2000$?

Express your answer as an integer from $0$ to $1999$, inclusive.


Determine how many terms are in the simplified form of the expression \[(x+y+z)^{2010}+(x-y-z)^{2010}.\]


In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?


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

Solution: \boxed{9(\frac{\pi}{2} - 1)}


Calculate the value of \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{12}}{x^2\cdot x^4 \cdot x^6 \cdots x^{24}}\] if \( x = 3 \).


In $\triangle ABC$, $AC = BC = 8$ and $AB = 4$. Point $D$ is on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 10$. Determine the length of $BD$.

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


Simplify $\sqrt{11 \cdot 3} \cdot \sqrt{3^3 \cdot 11^3}$.


What is the least natural number that can be added to 123,456 to create a palindrome?


What is the sum of all of the positive even factors of $720$?


Simplify the expression $(2x + 20) + (150x + 30) + y$.

Solution: \boxed{152x + 50 + y}


For what values of $x$ is $\frac{\log{(x-2)}}{\sqrt{5-x}}$ defined?


Determine the matrix $\mathbf{M}$ such that:
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 3b \\ c & d \end{pmatrix}.\]If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


I planned to work for 25 hours a week for 15 weeks this summer to earn $\$4000$ to buy a used car. However, due to an illness, I couldn't work for the first three weeks. If the hourly rate for the first 5 weeks (including sick weeks) was planned to be $\$20$ per hour, and the rate for the remaining weeks is $\$16$ per hour, how many hours per week must I work for the rest of the summer to meet my financial goal?

Solution: \boxed{18.75 \text{ hours per week}}


Determine the tens digit of $13^{1987}$.


Let $p, q, r, s, t, u, v$, and $w$ be distinct elements in the set \[
\{-6, -4, -3, -1, 1, 3, 5, 8\}.
\]
What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


Simplify $\sqrt{45x} \cdot \sqrt{20x} \cdot \sqrt{28x} \cdot \sqrt{5x}$.


The product of two positive integers plus their sum is 159. The integers are relatively prime, and each is less than 30. What is the sum of the two integers?


A set of seven positive integers has a median of 5 and a mean of 14. What is the maximum possible value of the list's largest element?


The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than $450$ people. When the freshmen are told to line up in columns of $19$, $18$ people are in the last column. When the freshmen are told to line up in columns of $17$, $10$ people are in the last column. How many people are in the incoming freshman class?


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


Triangle $PQR$ is an obtuse, isosceles triangle. Angle $P$ measures 30 degrees. What is the number of degrees in the measure of the largest interior angle of triangle $PQR$?


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


Consider the graph of the function $y=g(x)$, characterized by the function $g(x) = \frac{x^2}{Px^2 + Qx + R}$, where $P$, $Q$, and $R$ are integers. For $x > 5$, it is true that $g(x) > 0.5$. From the graph, it is observed that there are vertical asymptotes at $x = -3$ and $x = 4$, and the horizontal asymptote occurs at a line $y = 1/P$. Determine the value of $P+Q+R$.


Let \(a, b, c, z\) be complex numbers such that \(|a| = |c| = r > 0\) and \(|b| = 2r\), where \(r\) is a positive real number, and they satisfy:
\[az^2 + bz + c = 0.\]
Find the largest possible value of \(|z|\).

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


Let $b_n = \frac{15^n - 1}{14}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


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


One of the roots of the equation \( z^2 = -104 + 63i \) is \( 5 + 8i \). What is the other root?

Solution: \boxed{-5 - 8i}


Let \( p, q, r \) be real numbers and let \( A, B, C \) be points such that the midpoint of \( \overline{BC} \) is \( (p,0,0) \), the midpoint of \( \overline{AC} \) is \( (0,q,0) \), and the midpoint of \( \overline{AB} \) is \( (0,0,r) \). Furthermore, let \( p = l+1, q = m+1, r = n+1 \) where \( l, m, n \) are initially given constants. Find
\[
\frac{AB^2 + AC^2 + BC^2}{(l+1)^2 + (m+1)^2 + (n+1)^2}.
\]


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $256\sqrt{2}$ square feet and $576\sqrt{2}$ square feet. The two planes are $12$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


We roll a fair 6-sided die 8 times.  What is the probability that we get an odd number in exactly 6 of the 8 rolls, and among those odd numbers, exactly 2 are the number 3?


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


Triangle $PQR$ has side-lengths $PQ = 10$, $QR = 20$, and $PR = 16$. The line through the incenter of $\triangle PQR$, parallel to $\overline{QR}$, intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. What is the perimeter of $\triangle PXY$?


The sum of three numbers \(a\), \(b\), and \(c\) is 150. If we increase \(a\) by 10, decrease \(b\) by 3, and multiply \(c\) by 4, the three resulting numbers are equal. What is the value of \(b\)?


Suppose \( x \) is an integer and \( y = 0.5 \). Evaluate \(\lceil x + y \rceil - \lfloor x + y \rfloor\). What is \(\lceil x + y \rceil - (x + y)\)?


If $n$ is a positive integer, determine $f(n)$ as the sum of the digits of $\frac{1}{25^n}$. Find the smallest positive integer $n$ such that $f(n) > 20$.


Simplify $\frac{2x-3}{4} + \frac{3x+5}{5} - \frac{x-1}{2}$. Express your answer as a single fraction.


Find the remainder when $2^{2^{2^2}}$ is divided by 500.


The center of a circle has coordinates $(3, -7)$. The circle is reflected about the line $y = -x$. What are the $x, y$ coordinates of the center of the image circle?

Solution: \boxed{(7, -3)}


If $\cos x,$ $\sin x,$ $\cot x$ form a geometric sequence, in this order, then find $\tan^6 x - \tan^2 x.$


Let \[ f(x) =
\begin{cases}
-x^2 - 3x + 2 & \text{if } x \geq 0,\\
x + 10 & \text{if } x < 0.
\end{cases}
\] Compute $f(f(f(f(f(2)))))$.


Determine the value of $x$ for which $(2023+x)^2 = x^2$.


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

What is the remainder when \( 10! \) is divided by \( 13 \)?


Let \(x\) and \(y\) be real numbers such that:
\[
xy + \frac{x}{y} + \frac{y}{x} = 3.
\]
Find the sum of all possible values of \((x + 1)(y + 1).\)


How many of the positive divisors of 8820 are multiples of 7?


Expand and find the sum of the coefficients of the expanded expression for $-(4-d)(2d+3(4-d))$.


How many positive divisors does 8! have?


If $(x+2)^2(x-3)=40$, find the sum of the possible values of $x$.


Solution: \boxed{1 + i}


A survey was done by a local community center to find out how many people were interested in participating in a new yoga class. The table provided below shows partially completed results of the survey. Determine how many of the males surveyed were not interested in joining the yoga class.

\begin{tabular}{| l | c | r |}
\hline
& interested & not interested \\
\hline
males & & 78 \\ \hline
females & 70 & 90 \\ \hline
total & 180 & 200 \\
\hline
\end{tabular}


Determine the radius of the round cake whose boundary is given by the equation \( x^2 + y^2 + 1 = 2x + 5y \).


Let the line $q$ be the perpendicular bisector of $D = (30, 10)$ and $E = (6, 8).$ Given that $DE$ meets $q$ at $F = (x, y),$ what is $3x - 5y$?


Find the phase shift of the graph of $y = 5 \sin \left( 2(x - \frac{\pi}{4}) \right)$.


Angela has $a$ marbles. Brian has $1.5$ times as many marbles as Angela. Caden has $2.5$ times as many marbles as Brian, and Daryl has $4$ times the number of marbles Caden has. If in total Angela, Brian, Caden, and Daryl have $90$ marbles, what is the value of $a$?


For how many integer values of $n$ between 1 and 499 inclusive does the decimal representation of $\frac{n}{500}$ terminate?


In a debate tournament, $60\%$ of the participants were eliminated in the preliminary round. Only $\frac{1}{4}$ of the remaining participants managed to advance to the semi-finals after the intermediate round. If 30 participants were still in the tournament after the intermediate round, how many participants started the tournament?


An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $18$, and one of the base angles is $\arcsin(0.6)$. Find the area of the trapezoid.


How many of the integers between 1 and 500, inclusive, can be expressed as the difference of the squares of two nonnegative integers?


How many integer values of $x$ make the inequality $4x^2 + 16x + 15 > 23$ not true?


Find the smallest positive integer $b$ for which $x^2+bx+1200$ factors into a product of two binomials, each having integer coefficients.


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


Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number.


The graphs of a function $f(x) = 4x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-4, a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?


If \( g(x) = \dfrac{3x + 2}{x - 2} \), find the value of \( g(8) \).


How many pairs of positive integers $(x, y)$ satisfy the equation $x^2 - y^2 = 72$?


Container I holds 10 red balls, 5 green balls, and 3 blue balls; containers II and III each hold 3 red balls, 5 green balls, and 2 blue balls. A container is randomly selected, and then a ball is randomly selected from that container. What is the probability that the selected ball is green? Express your answer as a common fraction.


What is the remainder when 482,157 is divided by 6?


An equilateral triangle has two vertices at $(0,6)$ and $(10,6)$. If the third vertex is in the first quadrant, what is the y-coordinate?

Solution: \boxed{6 + 5\sqrt{3}}


Mr. and Mrs. Lopez have three children. When they get into their family car, two people sit in the front, and two people sit in the back. Either Mr. or Mrs. Lopez must sit in the driver's seat. How many different seating arrangements are possible?


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1800$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$.


Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 200$ such that $i^x+i^y$ is a real number.


In triangle $ABC,$ $E$ lies on $\overline{BC}$ and $G$ lies on $\overline{AB}.$ Let $\overline{AE}$ and $\overline{CG}$ intersect at $Q.$

If $AQ:QE = 3:2$ and $GQ:QC = 2:3,$ find $\frac{AG}{GB}.$


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


Calculate how many whole numbers are between \(\sqrt[3]{50}\) and \(\sqrt[3]{500}\).


Three cards are drawn with replacement from a standard deck of 52 cards. What is the probability that at least one of the cards is a heart or a king?


Simplify the expression:

$$8y + 15 - 3y + 20 + 2x$$

Solution: \boxed{5y + 2x + 35}


A pyramid with volume 80 cubic inches has a rectangular base. If the length of the base is tripled, the width remains the same, and the height is increased by $80\%$, what is the volume of the new pyramid, in cubic inches?


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


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

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


In the expansion of \((x+2)^{50}\), what is the coefficient of the \(x^3\) term?

Solution: \boxed{19600 \cdot 2^{47}}


Suppose the point $(2,5)$ is on the graph of $y = 2f(x)$. Then there is one point which must be on the graph of $y = \frac{f^{-1}(x)}{3}$. What is the sum of that point's coordinates?


Ray climbs a staircase by taking different numbers of steps at a time. When he climbs 5 steps at a time, there are 4 steps left at the top. When he climbs 6 steps at a time, there are 3 steps left. When he climbs 7 steps at a time, there are 5 steps left. What is the smallest number of steps in the staircase that is greater than 20?

Solution: \boxed{n = 159}


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

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


Given vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$, find the scalar $r$ such that
\[\begin{pmatrix} 5 \\ -2 \\ 3 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).\]


What is the product of all possible values for $b$, if the length of the segment between the points $(3b, b+2)$ and $(6, 3)$ is $3\sqrt{5}$ units?


What is the total number of digits used when the first 2500 positive odd integers are written?


Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \geq 1$. If $g(16) = 8$, compute $[g(4)]^4$.


Two concentric circles are centered at point Q. The sides of a 60-degree angle at Q form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 48-degree angle at Q. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.


For how many integer values of $d$ in the interval $[0,2000]$ does the equation \[8 \lfloor x \rfloor + 3 \lceil x \rceil = d\] have a solution for $x$?


Add $5.623$ and $4.76$ and write the result as a decimal.


Find the distance between the planes \(3x - 2y + 6z = 9\) and \(6x - 4y + 12z = 18.\)


Let $\mathbf{c}$ and $\mathbf{d}$ be orthogonal vectors in $\mathbb{R}^2$. Given $\operatorname{proj}_{\mathbf{c}} \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ find $\operatorname{proj}_{\mathbf{d}} \begin{pmatrix} 4 \\ -1 \end{pmatrix}.$

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


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


A truncated cone has horizontal bases with radii 20 and 5. The slant height of the truncated cone is 25. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


Find the point in the plane $4x - 3y + 6z = 42$ that is closest to the point $(2,1,4)$.

Solution: \boxed{\left(\frac{174}{61}, \frac{22}{61}, \frac{322}{61}\right)}


A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 600, and the sum of the lengths of its 12 edges is 160. What is $s$?


The American Mathematics College welcomes its incoming freshman class, which totals fewer than $500$ people. When the freshmen are requested to line up in columns of $25$, $24$ are left in the last column. When the freshmen are asked to form columns of $19$, $11$ remain in the last column. Determine the total number of people in the incoming freshman class.


Let
\[
\mathbf{A} = \begin{pmatrix} 2 & 4 \\ 3 & 2 \end{pmatrix}
\]
compute $\det(\mathbf{A}^2 - 3\mathbf{A})$.


A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\triangle DEF$.


The numbers $a, b, c, d$ are equal to $2, 3, 5, 7$ in some order. Find the largest possible value of
\[(a+b)(c+d).\]


A printer prints 25 pages per minute. How many minutes will it take to print 350 pages, considering that the printer requires a 5-minute break after every 150 pages printed?


Let \( r_1, r_2, \) and \( r_3 \) be the roots of the polynomial
\[
x^3 - 3x^2 + 5 = 0.
\]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3 \).

Solution: \boxed{x^3 - 9x^2 + 135}


Let $u, v, w$ be the roots of the equation $x^3 - 15x^2 + 13x - 6 = 0$. Find the value of $(1+u)(1+v)(1+w)$.


A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\pi$.

Solution: \boxed{\frac{500}{3} \pi}


If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\).


Let \( m \) be the smallest integer whose fourth root is of the form \( n+r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \( 1/10000 \). Find \( n \).


In triangle $DEF,$ $d = 8,$ $e = 10,$ and $f = 6.$ Let $J$ be the incenter. Then
\[\overrightarrow{J} = p \overrightarrow{D} + q \overrightarrow{E} + r \overrightarrow{F},\]
where $p,$ $q,$ and $r$ are constants such that $p + q + r = 1.$ Find the ordered triple $(p,q,r).$

Solution: \boxed{\left(\frac{1}{3}, \frac{5}{12}, \frac{1}{4}\right)}


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(12, -5)$?


Define $a * b$ as $2a - b^2$. If $a * 7 = -20$, what is the value of $a$?


What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot $5|8$ represents $58.$

\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
1 & $0 \hspace{2mm} 0 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
2 & $1 \hspace{2mm} 3 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 6$ \\ \hline
3 & $0 \hspace{2mm} 0 \hspace{2mm} 2 \hspace{2mm} 7 \hspace{2mm} 7 \hspace{2mm} 8$ \\ \hline
4 & $1 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 6 \hspace{2mm} 9$ \\ \hline
5 & $1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 8 \hspace{2mm} 9$ \\ \hline
\end{tabular}


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


Point $Q$ lies on the line $x=4$ and is 12 units from the point $(0,-3)$. Find the product of all possible $y$-coordinates that satisfy the given condition.


In the right triangle $ABC$, the hypotenuse $AC = 5$, and side $AB = 3$. Find the length of side $BC$ and calculate $\tan A$.


Compute $\sqrt{75} \cdot \sqrt{48} \cdot \sqrt{12}$.


Find the sum of the rational roots of the polynomial $h(x)=x^3-7x^2+10x-2$.


In a chess tournament, each player played exactly one game against each of the other players. A win awarded the winner 1 point, a loss 0 points, and a tie awarded each player 0.5 points. After the tournament, it was found that exactly half of the points earned by each player were earned against the fifteen players with the least number of points. Out of these fifteen lowest-scoring players, each player earned half of their total points against the other fourteen of the lowest-scoring group. Determine the total number of players in the tournament.


For some complex number $\omega$ with $|\omega| = 3,$ there is some real $\lambda > 0$ such that $\omega,$ $\omega^2,$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Find $\lambda.$

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


Two new spinners are being used in a game. The first spinner is divided into thirds, with options 3, 4, and 7. The second spinner is divided into fifths, with options 1, 4, 5, 6, and 8. If each spinner is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.


Lisa has 12 friends and 44 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least 2 marbles and no two friends receive the same number of marbles?


How many ways are there to distribute 6 indistinguishable balls into 4 indistinguishable boxes?


Solution: \boxed{5 + 3^{2/3}}


What is the sum of all integer values of $n$ such that $\frac{36}{2n - 1}$ is an integer?


Given that
\[
\cos^2 \alpha \sin^2 \beta + \sin^2 \alpha \cos^2 \beta = \cos \alpha \sin \alpha \cos \beta \sin \beta,
\]
find the value of
\[
\frac{\sin^2 \beta \cos^2 \alpha}{\sin^2 \alpha} + \frac{\cos^2 \beta \sin^2 \alpha}{\cos^2 \alpha}.
\]


Given the cubic polynomial $x^3 - 7x^2 + px - q$ has three positive integer roots, find the average of all distinct possible values of $q$.


Compute the square of 1085 without using a calculator.


The positive integers $E, F$ and $G$ form an arithmetic sequence, while the integers $F, G$ and $H$ form a geometric sequence. If $\frac{G}{F} = \frac{7}{4},$ what is the smallest possible value of $E + F + G + H$?


Determine the smallest non-zero value of $y$ that satisfies the equation $\sqrt{6y+3}=3y+1$. Provide your answer in simplest fractional form.


Given that $x$ is a multiple of $11739$, what is the greatest common divisor of $g(x) = (3x + 4)(5x + 3)(11x + 5)(x + 11)$ and $x$?


Let $(a_i), (b_i), (c_i)$ represent the sequence, from left to right, of elements in the 3001st, 3002nd, and 3003rd rows, respectively, of Pascal's triangle. The leftmost elements occur at $i = 0.$ Compute
\[\sum_{i = 0}^{3002} \frac{b_i}{c_i} - \sum_{i = 0}^{3001} \frac{a_i}{b_i}.\]


The longer leg of a right triangle is $3$ feet shorter than three times the length of the shorter leg. The area of the triangle is $108$ square feet. What is the length of the hypotenuse, in feet?


Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 73 \times 83 \times 93$ is divided by $5$.


Evaluate the expression $\frac{10^{2003} + 10^{2005}}{10^{2004} + 10^{2004}}$ and determine the whole number closest to this value.


The matrix for reflecting over a certain line $\ell$, which passes through the origin, is given by
\[
\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.
\]
Find the direction vector of line $\ell$. Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a,$ and $b$ are integers, $a > 0,$ and $\gcd(|a|,|b|) = 1.$

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


Find $3_6 + 4_6 + 5_6 + \cdots + 100_6$. Express your answer in base $6$.


In how many ways can five people line up in a straight line if the two youngest persons cannot be first in line?


What is the sum of all integer solutions to $4 < (x-3)^2 < 36$?


How many solutions of the equation $\sin x = \sin (\sin x)$ are on the interval $0 \le x \le \sin^{-1} 0.99$?


Given that $b$ is a multiple of $2142$, find the greatest common divisor of $b^2 + 11b + 28$ and $b + 6$.


An infinite geometric series has a first term of $172$ and a sum of $400$. What is its common ratio?

Solution: \boxed{r = \frac{57}{100}}


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and
\[ f(x+y) = f(x) + f(y) - xy \]
for all real numbers $x$ and $y.$

Let $m$ be the number of possible values of $f(2),$ and let $t$ be the sum of all possible values of $f(2).$ Find $m \times t.$


What is the area, in square units, of a triangle whose vertices are at $(2, 2)$, $(2, -3)$, and $(7, 2)$?


Four years ago, the number of years in Anna's age was a perfect square, and three years from now, her age will be a perfect cube. How many years old is Anna?


A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 300 or 700 when divided by 1100. How many leap years will there be that end in double zeroes between 2100 and 4500?


Let $p$, $q$, and $r$ be integers such that
\[
\mathbf{B} = \frac{1}{3} \begin{pmatrix} 2 & p \\ q & r \end{pmatrix}
\]
and \(\mathbf{B}^2 = 2\mathbf{I}\). Find the largest possible value of \(p + q + r\).


At the mall's food court, Crystal wants to buy a meal consisting of one entree and one drink, and she may also choose to add one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals, including those with and without dessert, can she purchase from these options?

\begin{tabular}{ |c|c|c| }
\hline
\textbf{Entrees} & \textbf{Drinks} & \textbf{Desserts} \\
\hline
Pizza & Lemonade & Ice Cream \\
\hline
Chicken Teriyaki & Cola & Apple Pie \\
\hline
Corn Dog & Iced Tea & \\
\hline
Fish and Chips & Root Beer & \\
\hline
\end{tabular}


In triangle $DEF$, the measure of $\angle D$ is $90$ degrees. The measure of $\angle E$ is $10$ degrees less than four times the measure of $\angle F$. What is the measure, in degrees, of $\angle F$?

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


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

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


Eight balls, numbered 1, 3, 5, 7, 9, 10, 11, 12, are placed in a hat. Each ball is equally likely to be chosen. If one ball is chosen, what is the probability that the number on the selected ball is a prime number or divisible by 4?


Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values?


Determine the smallest value of the expression
\[
\frac{n}{3} + \frac{27}{n}
\]
for which positive integer \( n \).


The sum of four positive integers that form an arithmetic sequence is 58. Of all such possible sequences, what is the greatest possible third term?


Let $a$ and $b$ be real numbers. Determine the ordered pair $(a,b)$ such that one of the roots of the polynomial
\[x^3 + ax^2 + 6x + b = 0\]
is $1 - 3i$.

Solution: \boxed{(0, 20)}


Suppose positive integers $x$ and $y$ satisfy the equation $\frac{1}{x} + \frac{1}{3y} = \frac{1}{9}$. Find the least possible value of $xy$.


A rectangle, whose sides are parallel to the coordinate axes, is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1.\]
Find the area of the rectangle if the length of the rectangle is twice its width.


In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 26$, and $BC = 10$. What is $\sin A$?


Cory has $4$ apples, $3$ oranges, $2$ bananas, and $1$ grape. If Cory eats one piece of his fruit per day for ten days and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit if he must start with an apple?


Samantha is planning to build a rectangular garden and needs to enclose it with a fence. She has 400 feet of fencing material available. The regulations for the garden require that the length of the garden be at least 100 feet. She wants the area of the garden to be as large as possible to fit various plants and decorations. What dimensions should Samantha choose for her garden to maximize the enclosed area, and what is the optimal area?


If $\cos x,$ $\sin x,$ $\cot x$ form a geometric sequence, in this order, then find $\tan^6 x - \tan^2 x.$


Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 50 square units? [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,5), C=(3,20), D=(6,5), E=(6,0);
pair[] dots={A,B,C,D,E};

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

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(6,0)",E,SE);
label("D(6,5)",D,NE);
label("C",C,NE);
label("B(0,5)",B,NW);
[/asy]


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


Two rays with common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=2$. What is the maximum possible length of $\overline{OB}$?


Find the difference between $5321_7$ and $1234_7$ in base $7$.


Given that $x + \cos y = 2023$ and $x + 2023 \sin y = 2022$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x + y$.


Jenny eats 30% of the jellybeans in her jar at the beginning of each day. At the end of the third day, 24 jellybeans remained. How many jellybeans were there originally in the jar?


Find all real values of $x$ that satisfy $\frac{x^2+2x^3-3x^4}{x+2x^2-3x^3} \le 2.$ (Give your answer in interval notation.)

Solution: \boxed{(-\infty, -\frac{1}{3}) \cup (-\frac{1}{3}, 0) \cup (0, 1) \cup (1, 2]}


Find the sum of the base 6 values $5_6 + 7_6 + 9_6 + \cdots + 31_6$. Express your answer in base 6.


In a rectangular configuration $ABCD$, there are three squares with non-overlapping interiors. One of them, which is shaded, has an area of 4 square inches. The side length of the larger square is twice the side length of the shaded square. What is the area of rectangle $ABCD$, given that all squares fit exactly within $ABCD$ with no other spaces?


A line is given by the equation $y = 2x + 5$. This line is parameterized as
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -3 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \end{pmatrix}.\]
Determine the ordered pair $(s, m)$.

Solution: \boxed{(-4, 6)}


The graphs of a function $g(x) = 2x + c$ and its inverse function $g^{-1}(x)$ intersect at the point $(-4, d)$. Given that $c$ and $d$ are both integers, what is the value of $d$?


In the diagram, $BP$ bisects $\angle ABQ$ and $BQ$ is an arbitrary ray within $\angle ABC$. $BM$ bisects $\angle PBQ$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$.


One digit of the decimal representation of $\frac{6}{7}$ is randomly selected. What is the probability that the digit is a 3? Express your answer as a common fraction.


I have four apples and twelve oranges. If a fruit basket must contain at least one piece of fruit and must include at least two oranges, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)


Point $P$ is located inside triangle $ABC$ so that angles $PAB$, $PBC$, and $PCA$ are all congruent. The sides of the triangle have lengths $AB=8$, $BC=15$, and $CA=17$. Find $\tan \angle PAB$.


What is the sum and the average of the prime numbers between 20 and 40?


A triangle in a Cartesian coordinate plane has vertices at (5, -2), (5, 5), and (12, 5). Determine the area and the perimeter of the triangle. Express your answer in exact form for the area and as a decimal for the perimeter.


Ten office employees attend a conference. If each employee shakes hands exactly once with everyone else, except for their direct manager and someone they directly manage (assume each person manages exactly one other person and each has exactly one direct manager, and there is a closed managerial loop that includes all ten people), how many handshakes occur?


In a different company, the below table shows the salary distribution among employees:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$180,000$\\\hline
Senior Manager&8&$\$95,000$\\\hline
Manager&12&$\$70,000$\\\hline
Assistant Manager&10&$\$55,000$\\\hline
Clerk&40&$\$28,000$\\\hline
\end{tabular}

With a total of 71 employees in this company, what is the median salary?


In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$.


Suppose that $a$, $b$, and $c$ are digits, where $a$, $b$, and $c$ are not all nine and not all zero, and the repeating decimal $0.\overline{abc}$ is expressed as a fraction in lowest terms. How many different denominators are possible?


Solution: \boxed{3 + 5z}


Consider a diagram showing the miles traveled by bikers Alice and Bob. After six hours, how many more miles has Alice biked than Bob? The diagram depicts Alice traveling at a constant rate forming a line that is steeper than Bob's, indicating a faster rate of biking. Assume Alice's line corresponds to a rate of about 15 miles per hour and Bob's line, about 10 miles per hour.


In a right-angled triangle, the square of the hypotenuse plus the squares of the other two sides is 500. Additionally, one leg is 3 times longer than the other. What is the length of the hypotenuse?


Completely factor the following expression: \[(8b^3 + 104b^2 - 9) - (-9b^3 + b^2 - 9)\]

Solution: \boxed{b^2(17b + 103)}


Two sides of an isosceles triangle are 15 inches and 30 inches. If the shortest side of a similar triangle is 75 inches, what is the perimeter of this larger triangle?

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


Jane and her brother each spin a spinner once. The spinner has six congruent sectors, numbered from 1 to 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.


The line $y = \frac{5x - 7}{6}$ is parameterized in the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{v} + t \mathbf{d},\]
so that for $x \ge 4,$ the distance between $\begin{pmatrix} x \\ y \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$ is $t$. Find $\mathbf{d}$.

Solution: \boxed{\begin{pmatrix} \frac{6}{\sqrt{61}} \\ \frac{5}{\sqrt{61}} \end{pmatrix}}


How many integers $-12 \leq n \leq 12$ satisfy $(n-3)(n+3)(n+7)<0$?


How many distinct three-digit numbers can be written with the digits $5$, $6$, $7$, $8$ if no digit may be used more than once in a three-digit number, and the number must include the digit $8$?


Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$  If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{4}{\omega^2},
\]
then find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.
\]


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $256\sqrt{2}$ square feet and $576\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross-section, in feet?


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1200$ and $AD = 2400$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.


What is the positive difference between the $2500^{\mathrm{th}}$ term and the $2510^{\mathrm{th}}$ term of the arithmetic sequence $-12,$ $-5,$ $2,$ $9,$ $\ldots$? Additionally, calculate the sum of the first $10$ terms of this sequence.


$36$ oranges are required to make $48$ quarts of orange juice. How many oranges are needed to make $6$ quarts of orange juice?


Mr. Collector has eight different Marvel comic books, six different DC comic books, and five different Image comic books. When stacked, all Marvel comic books are grouped together, all DC comic books are grouped together, and all Image comic books are grouped together. In how many different orders can these 19 comic books be stacked in a pile with the covers facing up and all of them facing the same direction?


In a bag, there are 7 blue chips, 5 red chips, 4 yellow chips, and 3 green chips. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Twelve 6-sided dice are rolled. What is the probability that exactly four of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.


How many different positive values of \( x \) will make this statement true: there are exactly \( 3 \) positive two-digit multiples of \( x \)?


Determine the remainder when $(x^6-1)(x^3-1)$ is divided by $1+x+x^2$.


You've recently purchased more than 130 eggs. The eggs are organized in containers that can each store precisely 15 eggs. However, upon checking, you discover that three containers hold only 14 eggs each, while the rest are full. What is the smallest number of eggs you could have right now?


In a Cartesian plane in which each unit is meters, a goat is tied to a stake at the point $(3, -4)$ with an $8$ meter rope. What is the maximum distance the goat can be from the point $(1, 2)$?

Solution: \boxed{8 + 2\sqrt{10} \text{ meters}}


Solve the quadratic $x^2 - 5x + 15 = x + 35$ and find the positive difference between its solutions.


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


Solve for the ordered pair $(x,y)$ that satisfies the simultaneous equations:
\begin{align*}
3x - 4y &= -2, \\
4x + 5y &= 23.
\end{align*}

Solution: \boxed{\left(\frac{82}{31}, \frac{77}{31}\right)}


Find the remainder when \( x^6 - x^5 - x^4 + x^3 + x^2 \) is divided by \( (x^2 - 1)(x - 2) \).

Solution: \boxed{9x^2 - 8}


Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 1230^\circ.$


Let
\[\mathbf{B} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.\]
Compute $\mathbf{B}^{87}$.

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


A student is asked to calculate the value of $\left(7^2 - 5^2\right)^4$. What is the final value of this expression?


How many non-similar regular 12-pointed stars are there, considering the same conditions as originally described for the 1000-pointed stars?


In triangle $XYZ$, angle bisectors $\overline{XF}$ and $\overline{YG}$ intersect at $Q$. Given $XY = 8$, $XZ = 6$, and $YZ = 4$, find $\frac{YQ}{QG}$.


Find the least common multiple (LCM) and the greatest common divisor (GCD) of 48 and 180.


A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 15.$ Find $m$.


Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and Germany, $7$ cents each, Peru $5$ cents each, and Italy $6$ cents each. (Brazil and Peru are South American countries and Germany and Italy are in Europe.) In his collection, before the '80s, Juan has 5 Brazil stamps from the '50s, 9 from the '60s, 12 Peru stamps from the '50s, and 8 from the '60s. In dollars and cents, how much did his South American stamps issued before the '80s cost him?


In a mathematics class at Riverside School, Mr. Thompson selects a random student to solve a problem on the board at the beginning of each class session. There are 12 students in his class, and the class meets five times a week. How many different sequences of students solving problems can occur in one week?


Find the integer \( n \) such that \( -150 < n < 150 \), and \( \tan n^\circ = \tan 1600^\circ \).


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z), \\
y &= 2023 - 2024 \operatorname{sign}(x + z), \\
z &= 2023 - 2024 \operatorname{sign}(x + y).
\end{align*}


Determine the focus of the parabola given by the equation $y = 4(x-1)^2 - 3.$

Solution: \boxed{(1, -\frac{47}{16})}


Solve for $x$ in the equation $25 - (3 \times 5) = (2 \times x) + 1$.

Solution: \boxed{x = \frac{9}{2}}


Convert the point $(0, 3\sqrt{3}, -3)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2\pi,$ and $0 \le \phi \le \pi.$

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


Ray climbs a flight of stairs in three different ways. When he takes $3$ steps at a time, there are $2$ steps left at the top. When he takes $7$ steps at a time, there are $1$ step left at the top. When he takes $4$ steps at a time, there are $3$ steps left at the top. What is the smallest possible value of the total number of steps ($n$) that is greater than $15$?


Find the minimum value of
\[
\frac{\sin^8 x + \cos^8 x + 1}{\sin^6 x + \cos^6 x + 1}
\]
over all real values \( x \).


Find the matrix $\mathbf{N}$ that swaps the rows of a given $2 \times 2$ matrix. In other words, calculate the matrix $\mathbf{N}$ 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}}


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}$, and $\mathbf{r}$ be the projection of $\mathbf{q}$ onto $\mathbf{w}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{5},$ then find $\frac{\|\mathbf{r}\|}{\|\mathbf{v}\|}.$

Solution: \boxed{\frac{\|\mathbf{r}\|}{\|\mathbf{v}\|} = \frac{27}{125}}


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


A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/3 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 14.$ Find $m.$


Sector $OAB$ is a quarter of a circle with a radius of 4 cm. Inside this sector, a circle is inscribed and is tangent at three points. Determine the radius of the inscribed circle. Express your answer in simplest radical form.

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


A solid sphere used in a shot put competition has a diameter of 12 cm. To customize it, two cylindrical holes are drilled into the sphere. Each hole has a depth of 5 cm and diameters of 1 cm and 1.5 cm respectively. Calculate the remaining volume of the sphere in cubic centimeters, expressed in terms of \(\pi\).


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


What is the remainder when $2537 \cdot 1985$ is divided by $600$?


Determine the smallest value of $x$ satisfying the equation \[\lfloor x \rfloor = 3 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


What is the modulo $7$ remainder of the sum $1+2+3+4+ \ldots + 199+200+201+202+203?$


Let \(x\) and \(y\) be positive real numbers, and let \(k\) be a positive constant. Find the maximum value of
\[
\frac{(kx + y)^2}{x^2 + y^2}.
\]

Solution: \boxed{k^2 + 1}


Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[2ab \sqrt{3} + 2bc.\]


Let $G$ be the centroid of triangle $DEF$. If $GD^2 + GE^2 + GF^2 = 84$, then find $DE^2 + DF^2 + EF^2$.


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

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


A regular triangle and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$?

[asy]
for(int i=0; i <=9; ++i) {
draw(dir(360*i/9+90)--dir(360*(i+1)/9+90));
}
pair A = dir(360*5/9+90);
pair D = dir(360*6/9+90);
pair C = A+dir(-30)*(D-A);
pair B = dir(360*4/9+90);

draw(A--C--D);

label("$A$",A,NE);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,NE);

[/asy]


What is the base ten equivalent of $65432_7$?


Solution: \boxed{x = 67.5}


Solution: \boxed{p = 12.75}


Given the function $f(x) = 3x^4 + 6$, find the value of $f^{-1}(150)$.


A flagpole is supported by a wire which extends from the top of the pole to a point on the ground 4 meters from its base. When Tom walks 3 meters from the base of the pole toward the point where the wire is attached to the ground, his head just touches the wire. Tom is 1.8 meters tall. How tall is the flagpole?


The initial fare for a taxi ride is $\$2.00$, with an added rate of $\$0.30$ per mile. If the ride distance exceeds 10 miles, a 10% discount is applied to the total fare before the 10-mile mark is reached. How much does a 12-mile taxi ride cost?


Let \( z \) and \( w \) be complex numbers such that
\[ |z| = 2, |w| = 2, \text{ and } z \overline{w} + \overline{z} w = 4. \]
Find the largest possible value of the real part of \( z + w \).


What is the product of all real numbers that are tripled when added to their reciprocals?


In trapezoid $EFGH$, sides $\overline{EF}$ and $\overline{GH}$ are parallel, $\angle E = 3\angle H$, and $\angle G = 2\angle F$. Find $\angle F$.


Determine the product of the digits $X$ and $Y$ such that the number $590\,X34\,52Y$ is divisible by $45$. What is the sum of all distinct possible products $XY$, with each product counted only once regardless of the pair ($X$, $Y$) variations that produce it?


A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\%$ longer than the minor axis, the length of the major axis is:


Define \( a \Delta b = a^3 - b \). What is the value of \( (2^{5 \Delta 6}) \Delta (4^{2 \Delta 3}) \)?

Solution: \boxed{2^{357} - 1024}


How many integers $n$ satisfy the condition $200 < n < 400$ and the condition $n$ has the same remainder whether it is divided by $7$ or by $9$?


**Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $45^{\circ}$, what is the length, in inches, of segment $XZ$?**


In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 20$ and $EF = 15$. Find $\cos D$.


Write the following expression as a polynomial: $$(3x^2 + 4x + 8)(x - 2) - (x - 2)(x^2 + 5x - 72) + (4x - 15)(x - 2)(x + 3).$$

Solution: \boxed{6x^3 - 16x^2 + 43x - 70}


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $15$. Let the vertices have $x$-coordinates $a, b$, and $a+5$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


What is the value of \[\frac{x^2 \cdot x^4 \cdot x^6 \cdots x^{30}}{x^3 \cdot x^6 \cdot x^9 \cdots x^{27}}\] if \( x=3 \)?


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


Given the polynomial \( ax^4 + bx^3 + 27x^2 + dx + 10 \) has a factor of \( 2x^2 - 3x + 2 \). Find the ordered pair \( (a, b) \).

Solution: \boxed{(2, -13)}


The measures of angles $A$ and $B$ are both positive, integer numbers of degrees. The measure of angle $A$ is a multiple of the measure of angle $B$, and angles $A$ and $B$ are supplementary angles. Determine the number of possible measures for angle $A$.


A rectangular box has a volume of $3456$ cubic inches and a surface area of $1368$ square inches. The sum of the lengths of its $12$ edges is $192$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{5024 \text{ cubic inches}}


In a classroom, 15 out of 22 students voted in favor of extending the class duration. What is the ratio of votes in favor to the total number of students, rounded to the nearest tenth?


Find the value of $a$ for which the equation $2a + E(a, 3, 2) = 4 + E(a, 5, 3)$ holds, where $E(a, b, c) = a \cdot b^2 + c$.

Solution: \boxed{a = -\frac{5}{14}}


The digits 8, 5, 9, and 7 are arranged randomly to form a four-digit number. What is the probability that the number is odd and divisible by 5?


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


There are 6 blue chips and 3 yellow chips in a bag. One chip is drawn from the bag and is not placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


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


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


Solve for \( z \) in the following equation: \( 5 - 2iz = 1 + 5iz \).

Solution: \boxed{z = -\frac{4i}{7}}


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

Solution: \boxed{x = 3^{\frac{6}{5}}}


Square each integer $n$ in the range $1 \leq n \leq 7$ and find the remainders when the squares are divided by $13$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $13$?


What is the smallest integer $x$ such that $\frac{x}{4} + \frac{3}{7} > \frac{2}{3}$?


What is the smallest whole number $b$ such that 82 can be expressed in base $b$ using only three digits?


Find the reflection of \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\) over the vector \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\).

Solution: \boxed{\begin{pmatrix} -6 \\ 2 \end{pmatrix}}


The square shown is divided into 16 equal smaller squares, and each small square is further divided into two rectangles along one of its diagonals. If the perimeter of the larger square is 256 units, what is the perimeter of one of these smaller rectangles?

Solution: \boxed{32 + 16\sqrt{2} \text{ units}}


Express $\cos 3x + \cos 5x + \tan 2x$ as a combination of trigonometric products.

Solution: \boxed{2 \cos 4x \cos x + \frac{\sin 2x}{\cos 2x}}


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


A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to twice the number of units in its perimeter. What is the number of units in the perimeter of this rectangle?


Find the coordinates of the center of the hyperbola given by \[\frac{(4y-6)^2}{6^2} - \frac{(5x-3)^2}{7^2} = -1.\]

Solution: \boxed{\left(\frac{3}{5}, \frac{3}{2}\right)}


Pete thinks of a number. He triples it, adds 15, multiplies the result by 5, and ends up with 245. What was his original number?


Find the largest prime divisor of $28^2 + 45^2$.


Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=4$, and $|r-s|=5$. What is the sum of all possible values of $|p-s|$?


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


Compute $45 \times 28 + 45 \times 72 - 10 \times 45$.


Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form.

Solution: \boxed{\sqrt{113 + 56\sqrt{2}}}


The mean of the set of numbers $\{92, 90, 86, 89, 91, y\}$ is 89.5. What is the median of the set of six numbers? Express your answer as a decimal to the nearest tenth.


At the CleverCat Training Center, cats can learn to do three tricks: jump, fetch, and meow. Of the cats at the center:
- 45 cats can jump
- 25 cats can fetch
- 40 cats can meow
- 15 cats can jump and fetch
- 20 cats can fetch and meow
- 23 cats can jump and meow
- 10 cats can do all three
- 5 cats can do none

How many cats are in the training center?


Two students go to a study session each day. They arrive at the study room independently, at random times between 9 a.m. and 9:30 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the study room is $25\%$, and $m = a - b\sqrt{c}$, where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$


Determine if the numbers $\sqrt{4u-2}$, $\sqrt{4u+2}$, and $2\sqrt{2u}$ can be the side lengths of a triangle. If so, what is the measure of the largest angle?


Determine the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \times \mathbf{v}\] for all vectors $\mathbf{v}.$

Solution: \boxed{\begin{pmatrix} 0 & -4 & -1 \\ 4 & 0 & -3 \\ 1 & 3 & 0 \end{pmatrix}}


It can be shown that for any positive integer \( n \),
\[
\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}^n = \begin{pmatrix} G_{n + 1} & G_n \\ G_n & G_{n - 1} \end{pmatrix},
\]
where \( G_n \) denotes the \( n \)th term of a sequence similar to Fibonacci but starts with \( G_0 = 1, G_1 = 2 \) and \( G_{n} = 2G_{n-1} + G_{n-2} \) for \( n \geq 2 \).

Compute \( G_{784} G_{786} - G_{785}^2 \).


Mr. Chang has 12 grandchildren. Assuming that the gender of each child is determined independently and with an equal likelihood of male and female, what is the probability that Mr. Chang has more granddaughters than grandsons?


After having coffee together, Jay, Paul, and Susan start walking away from the same point. Jay walks northeast at 1 mile every 20 minutes, Paul walks southeast at 3 miles every 40 minutes, and Susan walks east at 2 miles every 30 minutes. In miles, how far apart are Jay and Paul after 2 hours?


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


Let $x = \frac{1}{3}$ and $y = \frac{2}{3}$. Given $z = xy$, find the value of $3x^2y^5z^3$.


A triangle is made from wooden sticks of lengths 9, 18, and 21 inches joined end-to-end. Identical pieces of integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the minimum length of the piece that can be cut from each stick to achieve this?


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


If the roots of the quadratic equation $2x^2 + 16x + c = 0$ are $x = \frac{-16 \pm \sqrt{24}}{4}$, what is the value of $c$?


Subtract $6A3_{12}$ from $9B5_{12}$ and express your answer in base $12$, using $A$ for $10$ and $B$ for $11$.


A relatively prime date is defined as a date where the month number and day number are relatively prime. Determine how many relatively prime dates are there in July, assuming July is chosen as potentially having one of the fewest relatively prime dates.


Compute the domain of the real-valued function
$$g(x) = \sqrt{7 - \sqrt{9 - \sqrt{2x}}}.$$

Solution: \boxed{[0, 40.5]}