Ms. Johnson awards bonus points to students in her class whose test scores are above the median. The class consists of 81 students. What is the maximum number of students who could receive bonus points?


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


Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region between these circles. Express your answer in terms of \(\pi\).


Add $25_6 + 35_6.$ Express your answer in base $6.$


Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive integers, such that $abc = 216$. Find the smallest possible value of $b$.


What is the sum of the tens digit and the ones digit of the integer form of $(3+4)^{15}$?


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


In convex quadrilateral $ABCD, \angle A \cong \angle C, AB = CD = 200$, and $AD \neq BC$. The perimeter of $ABCD$ is $700$. Find $\lfloor 1000 \cos C \rfloor.$ (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x.$)


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


The line $y=-\frac{5}{3}x+15$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on the line segment $PQ$. If the area of $\triangle POQ$ is twice the area of $\triangle TOP$, what is the value of $r+s$?


Express the number 992 as a sum of distinct powers of 2. What is the sum of the squares of the exponents of these powers?


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


For how many values of \( c \) in the interval \([0, 500]\) does the equation \[ 9 \lfloor x \rfloor + 3 \lceil x \rceil = c \] have a solution for \( x \)?


If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 781$, what is the minimal possible value of $|x - y|$?


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


Find the sum of all integral values of \(c\) with \(c \leq 30\) for which the equation \(y = x^2 - 8x - c\) has two rational roots.

Solution: \boxed{-16 + (-15) + (-12) + (-7) + 0 + 9 + 20 = -21}


Let $r(x)$ be a monic quartic polynomial such that $r(1) = 5,$ $r(2) = 8,$ $r(3) = 13,$ and $r(4) = 20.$ Find $r(5).$


When $\sqrt[4]{2^6 \cdot 5^2}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?


Determine the number of ordered quadruples \((a, b, c, d)\) of real numbers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \begin{pmatrix} 2a & 0 \\ 0 & \frac{d}{2} \end{pmatrix}.
\]


Suppose I have 5 shirts, 4 pairs of pants, 5 ties, and 2 different blazers. If an outfit requires a shirt and pants, and can optionally include a tie and/or a blazer, how many different outfits can I create?


Thirty gremlins and twenty imps are at the Annual Mischief Convention. Due to recent disagreements, ten of these gremlins refuse to shake hands with each other, but they still shake hands with the remaining gremlins and all the imps. The imps, maintaining their usual stance, refuse to shake hands among themselves but will shake hands with all gremlins. Calculate the total number of handshakes at the convention.


Suppose $z$ and $x$ are distinct, randomly chosen roots of the equation $z^{2017} - 1 = 0.$ Find the probability that
\[
\sqrt{2+\sqrt{5}} \le |z + x|.
\]


Find the maximum value of
\[
\begin{vmatrix}
1 & 1 & 1 \\
1 & 1 + \sin(\theta + \phi) & 1 \\
1 + \cos(\theta + \phi) & 1 & 1
\end{vmatrix},
\]
as $\theta$ and $\phi$ range over all angles.


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


What is the base-10 integer 804 when expressed in base 7?


The arithmetic mean of the following six numbers is 25. What is the value of $x$?
$$x + 10, \hspace{.5cm} 18, \hspace{.5cm} 3x, \hspace{.5cm} 16, \hspace{.5cm} x + 5, \hspace{.5cm} 3x + 6$$


Suppose an integer is divisible by $6$ and the sum of its last two digits is $15$. What is the product of its last two digits?


Suppose I have a drawer containing 8 forks, 5 spoons, and 10 knives. If I reach in and randomly remove four pieces of silverware, what is the probability that I get exactly one fork, two spoons, and one knife?


Let $x$ be a real number. Consider the following five statements:
1. $0 < x^2 < 4$
2. $x^2 > 4$
3. $-2 < x < 0$
4. $0 < x < 2$
5. $0 < x - \frac{x^2}{4} < 1$

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


Let \( m \) be the smallest integer whose cube root is of the form \( n + r \), where \( n \) is a positive integer, and \( r \) is a positive real number less than \( 1/2000 \). Find \( n \).


Simplify
\[\frac{\tan^3 60^\circ + \cot^3 60^\circ}{\tan 60^\circ + \cot 60^\circ}.\]


Find the sum of the solutions to
\[\tan x + \cot x = 4\] in the interval $0 \le x \le 2\pi.$

Solution: \boxed{\frac{9\pi}{3} = 3\pi}


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


Given that $x = \frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction.


Three more than four times $C$ is equal to 25. What is the value of $C$?


The sequence begins with the number $7200$ and is formed by repeatedly dividing by $4$. How many integers are in this sequence?


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

Solution: \boxed{\begin{pmatrix} -12 \\ 22 \\ 20 \end{pmatrix}}


In trapezoid $EFGH$, sides $EF$ and $GH$ are equal. $GH$ is 12 units long, and the height of the trapezoid is 5 units. The top base $EF$ is 10 units long. Calculate the perimeter of trapezoid $EFGH$.

Solution: \boxed{22 + 2\sqrt{26}}


Three faces of a right rectangular prism have areas of 60, 75, and 100 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number.


Lucia constructs a large decorative pyramid with a height of 18 inches and a square base that measures 15 inches on each side. She needs to transport it safely to an exhibition in a cube-shaped box. What should be the volume of this cube-shaped box to fit the pyramid entirely, measured in cubic inches?


What is the sum of three consecutive multiples of 4 if the sum of the first and the third numbers is $140$?


If $\left( r + \frac{1}{r} \right)^2 = 5$, then find the value of $r^3 + \frac{1}{r^3}$.

Solution: \boxed{\pm 2\sqrt{5}}


Compute \[ \left\lfloor \dfrac {2011^3}{2009 \cdot 2010} - \dfrac {2009^3}{2010 \cdot 2011} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


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


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two unique two-digit primes `$AB$` and `$CD$`. Every digit $A$, $B$, $C$, and $D$ can be the same or different. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Jay and Sarah decide to go on walks starting from the same point but in opposite directions. Jay walks at a rate of 1 mile every 12 minutes, and Sarah jogs at a rate of 3 miles every 36 minutes. Determine the distance between Jay and Sarah after they have been walking for 2 hours.

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


Simplify $\frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{8}}{\sqrt{5}} \cdot \frac{\sqrt{9}}{\sqrt{7}}$ and rationalize the denominator of the resulting fraction.


How many ways are there to choose 4 cards from a standard deck of 52 cards, where two cards come from one suit and the other two each come from different suits?


Evaluate the infinite product \(3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \cdots.\)


Let $\mathcal{T}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$.


You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat one-third of the pizza. On each successive trip, you eat one-third of the remaining pizza. After six trips to the refrigerator, what fractional part of the pizza have you eaten?


Tiffany is constructing a fence around a new rectangular basketball court. She has 400 feet of fencing available. The length of the fence must be at least 100 feet while the width must be at least 50 feet. Tiffany aims to maximize the area enclosed by the fence to accommodate seating and storage. What is the maximum area, in square feet, she can achieve?


Convert the point \( (4, -4, 6) \) in rectangular coordinates to cylindrical coordinates. Enter your answer in the form \( (r,\theta,z) \), where \( r > 0 \) and \( 0 \leq \theta < 2\pi \).

Solution: \boxed{(4\sqrt{2}, \frac{7\pi}{4}, 6)}


Determine how many even perfect squares less than 16,000 can be represented as the difference of two squares, where the squares are $(b+2)^2 - b^2$ for some nonnegative integer $b$.


Jamie hiked the Grand Forest Trail last month. It took five days to complete the trip. The first three days she hiked a total of 42 miles. The second and third days she averaged 15 miles per day. The last two days she hiked a total of 40 miles. The total hike for the first and fourth days was 36 miles. How many miles long was the trail?


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


In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=150$, and $AD \ne BC$. The perimeter of $ABCD$ is 520. Find $\cos A$.


If two 4'' by 4'' squares are added at each successive stage, what will be the area of the rectangle at Stage 4, in square inches?


What are the rightmost three digits of \(5^{1994}\)?


What is the smallest positive integer $n$ such that $3n \equiv 8 \pmod{26}$?


Evaluate the expression $|(7 + 6i)(-5 + 3i)|$.


Let \( n \) be a positive integer, and let \( b_0, b_1, \dots, b_n \) be a sequence of real numbers such that \( b_0 = 45 \), \( b_1 = 80 \), \( b_n = 0 \), and
\[ b_{k+1} = b_{k-1} - \frac{4}{b_k} \]
for \( k = 1, 2, \dots, n-1 \). Find \( n \).


What is the sum of the last two digits of $7^{25} + 13^{25}?$


Let $\triangle ABC$ be a right triangle with $B$ being the right angle. A circle with diameter $BC$ meets side $AC$ at $D$. If the area of $\triangle ABC$ is $200$ and $AC = 40$, calculate $BD$.


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


Given a parabola $y = ax^2 + bx + c$ with vertex $(p, -p)$ and $y$-intercept $(0, p)$, where $p \neq 0$, determine the value of $b$.


The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros is twice the product of its zeros, and the sum of the coefficients equals the y-intercept of the graph, which is 12. What is $e$?


A nickel, a quarter, a half-dollar, and a dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents.


Let's define two operations $\oplus$ and $\otimes$ for any number $y$, where $y \oplus = 9 - y$ and $\oplus y = y - 9$. What is the value of $\oplus(18 \oplus)$?


Calculate the argument $\theta$ for the sum of the complex numbers represented by:
\[e^{8\pi i/40} + e^{13\pi i/40} + e^{18 \pi i/40} + e^{23\pi i /40} + e^{28 \pi i /40} + e^{33 \pi i/40}\]


Find the smallest solution to \[\lfloor x \rfloor = 10 + 150 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


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


Calculate the sum of the first five prime numbers that end with the digit 3.


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, what is the least possible value of $xy$?


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


A basketball team consists of 18 players, including a set of 3 triplets: Bob, Bill, and Ben; and a set of twins: Tim and Tom. In how many ways can we choose 7 starters if exactly two of the triplets and one of the twins must be in the starting lineup?


Add $1704_{12} + 259_{12}$. Express your answer in base $12$, using 'A' for 10 and 'B' for 11 if necessary.


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


In a game with 50 participants sitting in a circle, each holding one calculator, three calculators among them are set to 1, 0, and -1 respectively. Each participant performs specific operations on the calculators: the one showing 1 is cubed, the one showing 0 is squared, and the one showing -1 is negated. After performing these operations, each participant passes the calculators to the next person in the circle. This continues until each calculator has been passed around the entire circle. When all calculators have returned to their starting points, the numbers displayed are added together. What is the sum of the numbers on the calculators?


John is about to receive a certain amount of change less than one dollar from a cashier. If he gets the maximum number of quarters possible and the rest in nickels, he would need to receive 2 nickels to meet the amount. If he gets the maximum number of dimes possible and the rest in nickels, he would need to receive 4 nickels to meet the amount. What is the sum, in cents, of the possible amounts of change he is trying to get?


Find all values of \(x\) that satisfy
\[3x + 2 < (x - 1)^2 < 9x + 1.\]

Solution: \boxed{\left(\frac{5 + \sqrt{29}}{2}, 11\right)}


What is the greatest three-digit number that is a multiple of 17?


Compute the unique positive integer \( n \) such that
\[ 3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}. \]


The function $f$ is defined on the set of integers and satisfies
\[f(n)=
\begin{cases}
n-4 & \mbox{if } n\ge 1200 \\
f(f(n+6)) & \mbox{if } n<1200
\end{cases}\]
Find $f(125)$.


Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 750^\circ$.


What is the sum of the squares of the positive factors of 72?


In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The length of the base $BC$ is 10 units, and the height from $AB$ to $CD$ is 5 units. The base $AD$ is 22 units. Calculate the perimeter of $ABCD$.

Solution: \boxed{2\sqrt{61} + 32}


What is the arithmetic mean of all three-digit multiples of 8?


Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$?

Solution: \boxed{86.4 \text{ square units}}


Triangles $PQR$ and $PRS$ are right triangles with $PQ = 14$ units, $PR = 24$ units, and $PS = 26$ units. What is the area, in square units, of quadrilateral $PQRS$?

Solution: \boxed{288\text{ square units}}


Suppose $\sqrt{1 + \sqrt{4y-5}} = \sqrt{8}$; find $y$.


Calculate $\frac{3}{7} \cdot \frac{5}{8} \cdot \frac{9}{13} \cdot \frac{11}{17}$.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{z}$ where $z$ is an integer such that $0<z\le12$, then what is the largest possible value of $a+b+c$?


$PQRS$ is a trapezoid with the measure of base $\overline{PQ}$ three times the measure of the base $\overline{RS}$. Point $T$ is the point of intersection of the diagonals. The measure of diagonal $\overline{PR}$ is 15. Find the length of segment $\overline{RT}$. Express your answer as a common fraction.


A deck of cards has been modified to include only 36 cards, divided into 3 suits of 12 each. Two of these suits are red and one is black. If all the cards are shuffled randomly, how many ways can you pick two different cards where both must be from the red suits? Assume the order of selection matters.


If the eight solutions of $x^8 = -256$ are written in the form $a+bi$, where $a$ and $b$ are real numbers, find the product of those solutions with $b > 0$.


In triangle $ABC,$ $AB = 4,$ $AC = 7,$ $BC = 9,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$  Find $\cos \angle BAD.$


A region $S$ in the complex plane is defined by \begin{align*}
S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.
\end{align*}
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac12 + \frac12i\right)z$ is also in $S$?


A $\textit{palindrome}$ is a number which reads the same forward as backward. What is the least natural number that can be added to 134,782 to create a palindrome?


A circle has an area of \( x \) square units and a circumference of \( y \) units, with the sum \( x + y = 100\pi \). Determine the radius of this circle.

Solution: \boxed{-1 + \sqrt{101}}


Twelve people sit down at random seats around a round table. Five of them are math majors, four others are physics majors, and the three remaining people are biology majors. What is the probability that all five math majors sit in consecutive seats?


A coin is flipped ten times, and the sequence of heads and tails occurring is recorded. However, the sequence must start with at least two heads. How many distinct sequences are possible?


Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]
for all positive integers $x$ and $y.$ Given $f(10) = 16$, $f(40) = 24$, and $f(3) = 5$, find $f(800)$.


In triangle $ABC$, where $\angle A = 90^\circ$, $BC = 30$, and $\tan C = 3\sin C$. Find the length of $AB$.


In the diagram shown, $\overrightarrow{OA} \perp \overrightarrow{OC}$ and $\overrightarrow{OB} \perp \overrightarrow{OD}$. If $\angle{AOD}$ is 4 times $\angle{BOC}$, what is $\angle{AOD}$?


What is the smallest positive integer $n$ such that $19n \equiv 789 \pmod{11}?$


Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine?


The equation $y = -16t^2 + 100t - 1.5t$ describes the height (in feet) of a projectile launched from the ground at 100 feet per second with a linear air resistance term of $1.5$ feet per second per second slowing its ascent. At what $t$ will the projectile first reach a height of 45 feet? Express your answer as a decimal rounded to the nearest tenth.


In the expansion of $(x + 1)^{36}$, what is the coefficient of the $x^3$ term?


Pentagon ABCDE has a vertical line of symmetry through CD. What is the $y$-coordinate of vertex C so that the area of the pentagon is 90 square units? [asy]
unitsize(2mm);
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[/asy]


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$, meaning that if the length of the object was originally $x$, then it is now $x + \frac{1}{3} x$. On the second day he increased the object's length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on, following the pattern where on the $n^{\text{th}}$ day the increase is $\frac{1}{n+2}$. At the end of performing his trick for $n$ consecutive days, the object’s length becomes exactly 50 times its original length. What is the value of $n$?


If \(\sec x + \tan x = 5,\) then find \(\sec x - \tan x.\)


In a sequence, the first term is \(a_1 = 1007\) and the second term is \(a_2 = 1008.\) The values of the remaining terms satisfy the condition:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\] for all \(n \geq 1.\) Determine \(a_{1000}.\)


How many terms of the arithmetic sequence 100, 96, 92, $\dots$ appear before the number 4 appears?


A ball is thrown off a cliff and its height can be represented by the equation $h=60-9t-8t^2$, where $t$ is time in seconds. Simultaneously, a bird starts flying upwards from the base of the cliff with a height equation represented by $h=3t^2+4t$. Find the time at which both will be at the same height.


Let $a$ and $b$ be nonzero real numbers. Let $m'$ and $M'$ be the minimum value and maximum value of
\[\frac{|a - b|}{|a| + |b|},\] respectively. Find $M' - m'$.


\(ABC\) is a triangle with \(A = (0,0)\), \(B = (42,18)\), and \(C\) having integer coordinates. What is the minimum area \(\triangle ABC\) can have?


In the diagram, $F$ and $G$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively in $\triangle ABC$, where $A(0,5)$, $B(0,0)$, and $C(7,0)$. Determine the area of $\triangle FGC$.


What is the area, in square units, of a triangle that has sides of $5, 4$ and $4$ units? Express your answer in simplest radical form.


An enormous bronze statue of a mythical Griffin stands 90 feet tall. A miniature scale model of this statue, displayed in a local museum, measures 6 inches in height. Determine how many feet of the actual statue does one inch of the scale model represent?

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


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


How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?


Let $z = x + yi$ be a non-real complex number such that $|z| = 1.$ Find the real part of $\frac{1}{2 - z}.$

Solution: \boxed{\frac{2 - x}{5 - 4x}}


How many real numbers \( y \) are solutions to the following equation?
\[
|y - 2| = |y - 1| + |y - 4|
\]


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 24 units, respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. What is the area of quadrilateral $GHCD$ in square units?


Find the sum of all possible values of \( a \) for which the zeroes of the function \( f(x) = x^2 - ax + a - 3 \) are integers.


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


After a hurricane in Miami, the estimated damage was $\$$45 million in local (US) currency. Later, it was discovered that some assets were valued in Canadian dollars. The total value of these assets was $\$$15 million. At that time, 1 US dollar was worth 1.25 Canadian dollars. Determine the damage in Canadian dollars including the assets valued in Canadian dollars. Additionally, there is a recovery tax of 10% on the total damage calculated in Canadian dollars that needs to be considered.

Solution: \boxed{78,375,000 \text{ Canadian dollars}}


Determine the value of the expression
\[
\log_3 (64 + \log_3 (64 + \log_3 (64 + \cdots))),
\]
assuming it is positive.


How many positive factors of 180 are perfect squares?


Two distinct integers, \(x\) and \(y\), are randomly chosen from the set \(\{1, 2, 3, \ldots, 15\}\). What is the probability that \(xy - x - y\) is odd?


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

Solution: \boxed{\frac{9\pi - 18}{2}}


A rectangle has its length increased by $30\%$ and its width increased by $15\%$. By what percent is the area of the rectangle increased?


In heptagon $GEOMETRY$, $\angle G \cong \angle E \cong \angle T \cong \angle R$ and $\angle M$ is supplementary to $\angle Y$, $\angle J$ is supplementary to $\angle O$. Find the measure of $\angle T$.


Bob now has an eight-sided die with sides numbered from 1 to 8. Every morning, he rolls the die to decide his breakfast:
- If he rolls a composite number, he chooses sweetened cereal.
- If he rolls a prime number, he goes for unsweetened cereal.
- If he rolls a 1, 7, or 8, he needs to roll the die again.
What is the expected number of times Bob will roll his die in a non-leap year?


Find the quadratic polynomial with real coefficients, which has $-2 - 5i$ as a root, and where the coefficient of $x$ is $-6$.

Solution: \boxed{-\frac{3}{2}x^2 - 6x - \frac{87}{2}}


What is the minimum number of equilateral triangles needed to cover an equilateral triangle of side length 12 units, using triangles of side lengths 1 unit and 2 units?


What is the smallest positive integer $n$ such that $725n \equiv 1025n \pmod{40}?$


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le12$, then what is the largest possible value of $a+b+c$?


Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15.


Solve the inequality:
\[\frac{x}{x + 2} \ge 0,\ \text{where}\ x < 5.\]
Provide your answer using interval notation.

Solution: \boxed{(-\infty, -2)\cup[0, 5)}


What is the least five-digit positive integer which is congruent to 6 (mod 17)?


Let \(x^2 + bx + c = 0\) be a quadratic whose roots are each three more than the roots of \(4x^2 - 6x - 8\). What is \(c\)?

Solution: \boxed{c = 23/2}


For $j > 0$, let $J_j = 10\ldots 032$, where there are $j$ zeros between the $1$ and the $3$. Let $M(j)$ be the number of factors of $2$ in the prime factorization of $J_j$. What is the maximum value of $M(j)$?


A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Find the number of values of $z$ that satisfy both $|z| = 8$ and $f(z) = z$.


Many states are considering a new license-plate pattern that consists of a sequence of four letters followed by a sequence of four digits. Assuming that each four-letter four-digit arrangement is equally likely, what is the probability that such a license plate will contain at least one palindrome (a four-letter arrangement or a four-digit arrangement that reads the same left-to-right as it does right-to-left)? Express your answer as a fraction $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n.$


In triangle $DEF$, $DE = 5$, $EF = 12$, and $DF = 13$. Point $G$ is the midpoint of $EF$. If $DG = m \sqrt{2}$, find $m$.


What is the volume of the region in three-dimensional space defined by the inequalities $\max(|x|, |y|, |z|) \leq 1$ and $\max(|x-1|, |y-1|, |z-1|) \leq 1$?

Solution: \boxed{1^3 = 1}


Let \[ g(x) =
\begin{cases}
x/3 &\quad \text{if } x \text{ is divisible by 3}, \\
4x-1 &\quad \text{if } x \text{ is not divisible by 3}.
\end{cases}
\]
What is $g(g(g(g(3))))$?


Consider a grid where each block is 1 unit by 1 unit. You need to move from point $A$ to point $B$ which now exists on a grid sized 6 units wide by 5 units high. Starting at point $A$, the first step must be to the right. Following that, calculate how many different paths can take you to point $B$ without cutting across blocks and under the condition that no diagonal moves are allowed and each path must consist of a total of 11 unit moves.


Simplify $\frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{8}}{\sqrt{9}} \cdot \frac{\sqrt{10}}{\sqrt{11}}$ and rationalize the denominator of the resulting fraction.


The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards such that one is black and the other is red? Express your answer as a common fraction.


Find the number of four-digit numbers where the digits form an increasing arithmetic sequence and each digit is distinct.


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


Express $2.\overline{17}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


Suppose $3 - \sqrt{8}$ and $5 + \sqrt{13}$ are two roots of a polynomial with rational coefficients. What is the smallest possible degree of this polynomial, if we also know it includes a permutation of signs for $\sqrt{8}$ and $\sqrt{13}$ in other roots?


Find the smallest positive solution to the equation:
\[\tan 4x + \tan 5x = \sec 5x.\]


Determine the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ within the interval $(0.001, 0.002)$.

Solution: \boxed{318 - 159 = 159}


The polynomial \[x^4 + bx^3 + cx^2 + 7dx + e = 0\] has integer coefficients, where $d$ is nonzero. Let $m$ be the exact number of integer roots of the polynomial, counting multiplicities. Determine all possible values for $m$.

Solution: \boxed{0, 1, 2, 4}


Find the positive real number $x$ such that
\[ 3\sqrt{x^2 + x} + 3\sqrt{x^2 - x} = 6\sqrt{2}. \]


Find all solutions to the equation $\sqrt{3 + z} = 12$. Express your answer as a common fraction.


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


Consider a sequence of $64$ terms, each with the form $Q+m$ where $Q$ is the product $2 \times 3 \times 5 \times \ldots \times 67$ of all prime numbers less than or equal to $67$, and $m$ successively takes the values $2, 3, 4, \ldots, 65$. Let $M$ be the number of primes appearing in this sequence. What is the value of $M$?


Suppose $\sin N = \frac{3}{5}$ in a right triangle LMN, where LM is 15 units. What is the length of LN?


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


What is the sum of all two-digit positive integers whose squares end with the digits 25?


How many digits are located to the right of the decimal point when $\frac{5^8}{10^5 \cdot 16}$ is expressed as a decimal?


Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be vectors such that \(\|\mathbf{a}\| = 5,\) \(\|\mathbf{b}\| = 3,\) and \(\|\mathbf{c}\| = 4\) and
\[
\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{k},
\]
where \(\mathbf{k}\) is a known vector with \(\|\mathbf{k}\| = 2.\) Compute \(\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.\)


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


Three dice, each numbered from 1 to 6, are tossed. What is the probability that the sum of the numbers on the top faces is exactly 10?

Solution: \boxed{\frac{27}{216} = \frac{1}{8}}


In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 3\angle D$, $\angle C = 2\angle B$, and the lengths of sides $\overline{AB} = 2$ units and $\overline{CD} = 1$ unit. Find $\angle A$.


Monica tosses a fair 8-sided die. If the roll is a prime number, she wins that amount in dollars. If the roll is composite, she wins nothing. If the roll is a multiple of 3, she loses that amount in dollars. Calculate the expected value of her winnings on one die toss.


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


Determine the residue of $-935 \pmod{24}$. Your answer should be an integer in the range $0, 1, 2, \ldots, 22, 23$.


Let $x$ be a positive real number and $\theta$ a real number such that $0 \leq \theta \leq \pi$. Find the maximum possible value of $$\frac{x^2 + 2 - \sqrt{x^4 + 4\sin^2\theta}}{x}.$$

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


Consider sequences of positive real numbers of the form $x, 3000, y, \dots$ where every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 3001 appear somewhere in the sequence?


How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?


Express $1.\overline{07}$ as a reduced fraction, given that $0.\overline{01}$ is $\frac{1}{99}$.


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


Determine the maximum value of $\sin a$ given that \[\sin (a + b) = \sin a + \sin b.\]


Determine the number of ways to arrange the letters of the word "OCCUR".


Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.


Let $b_n = 6^n + 8^n$. Determine the remainder when $b_{41}$ is divided by $49$.


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


Henry walked on a flat field 10 meters due north from a tree. He then turned due east and walked 30 feet. He then turned due south and walked 10 meters plus 40 feet. How many feet away from his original starting point is Henry?

Solution: \boxed{50\text{ feet}}


If the ratio of $b$ to $a$ is 4, then what is the value of $a$ when $b = 15 - 6a$?


Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be nonzero vectors, no two of which are parallel, such that:
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]

Let \(\theta\) be the angle between \(\mathbf{b}\) and \(\mathbf{c}\). Find \(\sin \theta\).

Solution: \boxed{\sin \theta = \frac{\sqrt{3}}{2}}


Let $AB$ be a diameter of a circle centered at $O$.  Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 30^\circ$, find $\angle CED$, in degrees.


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


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


If $0.9 = \frac{y}{151+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.


Berengere and her exchange student Emily are at a French café that accepts both euros and American dollars. They want to buy a cake which now costs 8 euros, but individually, they don't have enough money. Emily has a ten-dollar bill, and the exchange rate is 1 euro = 1.10 USD. How many euros does Berengere need to contribute so they can buy the cake?

Solution: \boxed{0 \text{ euros}}


How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8?


For how many integer values of $n$ between 1 and 719 inclusive does the decimal representation of $\frac{n}{720}$ terminate?


Consider finding the smallest possible value of the sum $\lvert x + 3\rvert + \lvert x + 5\rvert + \lvert x + 6\rvert + \lvert x + 7\rvert$.


The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = x, \quad g(x,y) = g(y,x), \quad (x + y) g(x,y) = y g(x, x + y).
\]
Calculate $g(16,40)$.


If $e^{i\theta} = \frac{4 + i\sqrt{7}}{5}$, then find $\sin 4\theta$.


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


Find constants $A$, $B$, and $C$ such that the equation
\[
\frac{(x+B)(Ax+40)}{(x+C)(x+8)} = 3
\]
has infinitely many solutions for $x$. For these values of $A$, $B$, and $C$, find the sum of those $x$ values which do not satisfy the equation.


What is the digit in the tens place when $8^{1998}$ is expressed in decimal notation?


Circle $D$ has a radius of 9 cm. Calculate the area of the largest possible inscribed equilateral triangle in circle $D$.

Solution: \boxed{\frac{243 \sqrt{3}}{4} \text{ square centimeters}}


Find the integer $n$, $0 \le n \le 180$, such that $\cos n^\circ = \cos 812^\circ$.


Contractor Maria agreed to complete a job in 40 days. After 10 days, she found that the 10 people assigned to the work had already completed $\frac{2}{5}$ of the job. Assuming each person works at the same rate, what is the minimum number of people she must keep on the job to ensure that the job will be completed on time?


Let $O$ be the origin. Determine the scalar $k$ such that for any points $A,$ $B,$ $C,$ and $D$ satisfying the equation
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]
the four points $A,$ $B,$ $C,$ and $D$ are coplanar.


In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 15$, and $\tan C = 3\sin C$. What is $AC$?


I have 6 shirts and 6 pairs of pants, each available in 6 colors. There are 12 hats in total; 6 of these hats have patterns and 6 hats are the solid colors that match the shirts and pants. I refuse to wear an outfit where all three items are the same color or where a patterned hat is worn with two solid items of different colors. How many outfit choices do I have?


Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or red) and the eighth-graders will pick the color of the jersey (black, white, gold, or blue), and each group will not confer with the other. Additionally, the ninth-graders will choose whether to include a cap (yes or no). If for all items, each possible choice is equally likely to be chosen, what is the probability that the shorts will be a different color than the jersey, given that the team decides to include a cap?


A quadrilateral is intersected by two diagonals, dividing it into four smaller triangles labeled from 1 to 4. Assume that the intersection of the diagonals is a point such that no side of the quadrilateral passes through this point. How many degrees are in the sum of the measures of the angles at the vertices of the smaller triangles numbered 1 through 4, where each vertex's angle is part of the quadrilateral or a diagonal intersection?


Let \(S\) be the sum of all positive integers \(n\) such that \(n^2 + 12n - 2006\) is a perfect square. Find the remainder when \(S\) is divided by \(1000\).


For how many bases between two and nine inclusive does the representation of $725_{10}$ have a final digit of 1?


On a 24-hour clock, an elapsed time of six hours looks the same as an elapsed time of $30$ hours. Generally, a time $n$ is "clock equivalent" to its square number of hours, $n^2$ on a 24-hour clock. What is the least whole number of hours that is greater than $6$ hours and is "clock equivalent" to its square number of hours on a 24-hour clock?


What is the smallest possible area, in square units, of a right triangle with two sides measuring $4$ units and $6$ units?


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


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


Find the value of $b$ for the smallest positive real number $c$ such that there exists a positive real number $b$, for which all the roots of the polynomial $x^4 - cx^3 + bx^2 - cx + 1 = 0$ are real.


Find the domain of the function
\[g(x) = \sqrt{4 - \sqrt{5 - \sqrt{6 - x}}}.\]

Solution: \boxed{[-19, 6]}


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, what is the least possible value of $xy$?


A spherical balloon collapses into a wet horizontal surface and settles into a shape of a hemisphere while keeping the same volume. The minor radius of the original balloon, when viewed as an ellipsoid due to unequal pressure distribution, was $4\sqrt[3]{3}$ cm. Find the major radius of the original balloon, assuming the major and minor axes were proportional and the proportionality constant is 2 before it became a hemisphere.


How many integers $m \neq 10$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{10}$?


Six students participated in a math contest, and their scores were 59, 67, 97, 103, 109, and 113. Four of these students have an average score of 94. What is the average score of the remaining two students?


A point is selected at random from the portion of the number line from 0 to 8. What is the probability that the point is closer to 6 than to 2? Express your answer as a decimal to the nearest tenth.


For each positive integer \( n \), let \( T(n) \) denote the sum of the digits of \( n \). For how many values of \( n \) is \( n + T(n) + T(T(n)) = 2187 \)?


Determine the range of the function
\[ g(x) = \left( \arctan \frac{x}{2} \right)^2 + \pi \arccot \frac{x}{2} - \left( \arccot \frac{x}{2} \right)^2 + \frac{\pi^2}{8} (x^2 - 2x + 2). \]

Solution: \boxed{\left[ \frac{3\pi^2}{8}, \infty \right)}


What is the units digit of the sum of the squares of the first 2053 odd positive integers?


Let $T_n$ denote the sum of the first $n$ terms of an arithmetic sequence with a common difference of 5. If $\frac{T_{2n}}{T_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term.


The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square nor a perfect cube? Express your answer as a common fraction.


Let $a$, $b$, $c$, $d$, $e$, and $f$ be positive integers with $a+b+c+d+e+f = 2512$. Let $M$ be the largest of the sums $a+b$, $b+c$, $c+d$, $d+e$, and $e+f$. What is the smallest possible value of $M$?


Find the greatest common divisor of $8!$ and $(6!)^2$.


Marty wants to paint a box and has the option to use red, blue, green, yellow, or black paint. He can choose to style the paint by using a brush, a roller, or a sponge. However, he is not allowed to use a roller if he chooses black paint. How many different combinations of color and painting method can Marty choose?


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

- The only integer in the list that occurs more than once is $10$,
- Its median is $11$,
- Its average (mean) is $12$.

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


James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins?


A person has two eight-sided dice. One die is fair, and the other die is biased such that it rolls an eight with a probability of $\frac{3}{4}$, and each of the other seven faces has a probability of $\frac{1}{28}$. The person selects a die at random and rolls it four times. Given that the first three rolls are all eights, calculate the probability that the fourth roll is also an eight.


In a shooting match, nine clay targets are arranged in three hanging columns with four targets in the first column, three in the second column, and two in the third. A marksman must break all the targets following these rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
Determine the number of different orders in which the nine targets can be broken.


In triangle $ABC$, $AB = 10$, $AC = 14$, and $BC = 16$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $BC$. Determine the length of $GP$.


In triangle $XYZ,$ the midpoint of $\overline{YZ}$ is $(0,2,1),$ the midpoint of $\overline{XZ}$ is $(1,1,-1),$ and the midpoint of $\overline{XY}$ is $(-1,3,2)$. Find the coordinates of vertex $X$.

Solution: \boxed{(0, 2, 0)}


Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \[
n^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx + 3x + 3y + 3z - 8
\]


What is the greatest number of points of intersection that can occur when $3$ different circles and $1$ straight line are drawn on the same piece of paper?


Square $ABCD$ has area $144$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $50$, find the area of triangle $GCD$.


Let \[f(x) = \left\{
\begin{array}{cl}
\frac{x}{14} & \text{ if }x\text{ is a multiple of 2 and 7}, \\
2x & \text{ if }x\text{ is only a multiple of 7}, \\
7x & \text{ if }x\text{ is only a multiple of 2}, \\
x+2 & \text{ if }x\text{ is not a multiple of 2 or 7}.
\end{array}
\right.\]If $f^a(x)$ means the function is nested $a$ times, what is the smallest value of $a$ greater than 1 that satisfies $f(3)=f^a(3)$?


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

Solution: \boxed{\begin{pmatrix} \frac{16}{5} \\ -\frac{12}{5} \end{pmatrix}}


Calculate the cross product of the vectors \(\begin{pmatrix} 3 \\ -4 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\).

Solution: \boxed{\begin{pmatrix} -20 \\ -13 \\ 8 \end{pmatrix}}


Find all values of $x > 9$ which satisfy
\[\sqrt{x - 6 \sqrt{x - 9}} + 3 = \sqrt{x + 6 \sqrt{x - 9}} - 3.\]

Solution: \boxed{[18, \infty)}


Consider the line equation \(y = 4x + 2\). This line has a parameterization given by:
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -3 \end{pmatrix} + u \begin{pmatrix} 3 \\ h \end{pmatrix}.
\]
Find the ordered pair \((s, h)\).

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


A pet store has 20 puppies, 4 kittens, 6 hamsters, and 10 rabbits. Alice, Bob, Charlie, and David each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, Charlie, and David buy pets and leave the store satisfied?


Solve the double inequality for \( x \):
\[
-2 < \frac{x^2 - 16x + 15}{x^2 - 2x + 5} < 1.
\]

Solution: \boxed{\left(\frac{5}{7}, \frac{5}{3}\right) \cup (5,\infty)}


Solution: \boxed{-1 - i}


A list of five positive integers has the following properties:

$\bullet$ The only integer in the list that occurs more than once is $7,$

$\bullet$ its median is $12,$ and

$\bullet$ its average (mean) is $13.$

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


How many positive integer multiples of $77$ (product of $7$ and $11$) can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$?


Sixty ping-pong balls are numbered from 1 to 60. However, balls numbered 58 and 60 are damaged and cannot be chosen. A ball is selected at random from the remaining balls. What is the probability that the number on the selected ball is a multiple of 4 or 6? Express your answer as a common fraction.


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

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


Bonnie constructs the frame of a larger cube using 12 pieces of wire, each 8 inches long. Simultaneously, Roark uses pieces of wire, each 2 inches long, to construct several smaller non-interconnected cube frames. The total volume of Roark's collection of cubes is equal to the volume of Bonnie's cube. Calculate the ratio of the total length of Bonnie's wire to the total length of Roark's wire.


Find the units digit of the following within the indicated number base: $45_8 + 37_8$


How many different isosceles triangles have integer side lengths and a perimeter of 27?


The faces of a decahedral die are labeled with digits $1$ through $10$. What is the probability, expressed as a common fraction, of rolling a sum of $17$ with a pair of such decahedral dice?


A sports league has 12 teams divided into three divisions of 4 teams each. How many games are in a complete season for the league if each team must play every other team in its own division three times and every team in the other divisions once?


Mika has 3 gallons of purified water and uses 11/8 gallons for cooking. How much water does she have left?

Solution: \boxed{1 \frac{5}{8}}


Three different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these three numbers is one? Express your answer as a common fraction.


You are looking to evenly distribute your collection of gold coins among 13 friends. After distributing them, you find that 3 of your friends would receive one extra coin if you gave away all your coins. You own fewer than 150 gold coins. Determine the maximum number of gold coins you could have for this scenario to occur.


Solution: \boxed{x^6 - x^5 + x^4 + x^3 + x - 5}


How many car license plates can be made if they consist of three letters followed by 4 digits, where the first letter can be any letter, the second letter must be a consonant, the third letter must be a vowel, the first two digits are odd, and the last two digits are even?


In the diagram below, we have $\cos \angle RPQ = \frac{8}{17}$. What is $\cos \angle RPS$?

[asy]
pair R,P,Q,SS;
SS = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aCos(8/17))*(1.5,0);
dot("$S$",SS,S);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--SS);
draw(P--R);
[/asy]


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


Find the sum of the first five prime numbers that have a units digit of 3.


When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(8, 6)$, and $H(8, 0)$. Calculate the area of trapezoid $EFGH$.


What is the twelfth positive integer that is both odd and a multiple of 5?


200 people are surveyed inquiring if they think teal is a shade of blue or green. 130 people believe teal is a shade of blue, and 45 of these also consider it a shade of green. Additionally, 35 people think teal is neither blue nor green.

How many people believe that teal is a shade of green?


If Heidi can paint a wall in 60 minutes, and her friend Zoe helps her, such that together they can do it in 40 minutes, what fractional part of the wall can they paint together in 8 minutes?


If Ella rolls a standard eight-sided die until she rolls the same number on consecutive rolls, what is the probability that her 15th roll is her last roll? Express your answer as a decimal to the nearest thousandth.


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


How many non-congruent triangles with a perimeter of 12 and integer side lengths can be formed?


An isosceles trapezoid has sides labeled as follows: \(AB = 25\) units, \(BC = 12\) units, \(CD = 11\) units, and \(DA = 12\) units. Compute the length of the diagonal \( AC \).


What are the last two digits in the sum of factorials of the first 50 positive integers?


In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=2\sqrt{11}$ and $AD=2\sqrt{1001}$, find $BC^2$.


Express $3.\overline{145}$ as a common fraction in lowest terms.


I have 6 books, two of which are identical copies of a math book, and another two that are identical copies of a physics book. The remaining two books are different from each other and from the others. In how many ways can I arrange them on a shelf?


What is the remainder when $5^{96} - 8$ is divided by $14$?


The average of four different positive whole numbers is $5$. If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?


An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees capable of monitoring air and water pollution. On top of this, there should be another team where 40 of the new employees will also monitor soil pollution (including taking roles in air and water tasks). Determine the minimum number of new employees the agency must hire.


Two circles intersect at points $C$ and $D$. The first circle is centered at $(3, -2)$ with a radius $5$, and the second circle is centered at $(3, 6)$ with a radius $3$. Find $(CD)^2$.

Solution: \boxed{(CD)^2 = 0}


There is a unique two-digit positive integer \( u \) such that when multiplied by 13, the last two digits of the result are 52.


In the diagram, three circles are arranged such that each has an area divided into two equal parts, and $O$ is the center of the largest circle. The area of the largest circle is $100\pi$. A second smaller circle is centered at a point on the diameter of the largest circle and touches the circumference of the largest circle. A third circle of radius equal to that of the second circle is centered at a point such that its circumference passes through the center $O$ of the largest circle and it intersects the second circle. Calculate the total area of the shaded regions in all three circles. [asy]
size(120);
import graph;
fill(Arc((0,0),3.16,180,360)--cycle,mediumgray);
fill(Arc((0,1.58),1.58,0,180)--cycle,mediumgray);
fill(Arc((1.58,0),1.58,90,270)--cycle,mediumgray);
draw(Circle((0,0),3.16));
draw(Circle((0,1.58),1.58));
draw(Circle((1.58,0),1.58));

dot((0,0)); label("$O$",(0,0),N);
draw((-3.16,0)--(3.16,0)); draw((-1.58,1.58)--(1.58,1.58));
[/asy]


How many perfect squares are factors of 450?


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$.

Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.


Tom wishes to organize his coin collection into $X$ stacks, where each stack contains the same number of coins, $Y$. Each stack must have more than one coin, and no single stack can contain all his coins. If there are 11 possible values for $Y$ under these conditions, what is the smallest number of coins Tom could have?


Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$


A club consists of three board members and a certain number of regular members. Every year, the board members retire and are not replaced. Each regular member recruits one new person to join as a regular member. Initially, there are nine people in the club total. How many people total will be in the club after four years?


Find the least common multiple of 45 and 200.


Express $\frac{0.\overline{81}}{2.\overline{72}}$ as a common fraction.


Consider the polynomial $g(x) = 3x^5 + 2x^4 - x^3 + 2x^2 - 5x + s$. For what value of $s$ is $g(3) = 0$?


If the sum $1! + 2! + 3! + \cdots + 19! + 20!$ is divided by $21$, what is the remainder?


In a press conference before the All-Star Game, nine All-Stars are taking questions. Four are from the Cubs, three are from the Red Sox, and two are from the Yankees. Additionally, the coach of the Cubs insists on sitting with his team. If teammates and their coach insist on sitting together, how many ways can the nine All-Stars and the Cubs' coach be seated in a row?


Calculate how many positive $3$-digit numbers are multiples of $15$, but not of $45$.


Simplify the expression $(-8) \cdot \frac{18}{14} \cdot \frac{49}{27} + \frac{4}{3}$.


The sum of the areas of two squares is 130, while the difference in their areas is 42. Find the sum of their perimeters.

Solution: \boxed{4\sqrt{86} + 8\sqrt{11}}


Convex hexagon $ABCDEF$ has exactly three distinct side lengths. Side $AB$ measures 5 units, side $BC$ measures 6 units, and side $CD$ measures 7 units. The perimeter of hexagon $ABCDEF$ is 38 units. Determine the number of sides of the hexagon that measure 7 units.


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


Participation in the local basketball league is $15\%$ higher this year than it was last year. From last year to this year, the number of males increased by $10\%$ and the number of females increased by $25\%$. There were $30$ males participating in the league last year. What fraction of the league's participants is female this year? Express your answer as a common fraction.


A modified deck of 54 cards includes 13 ranks of each suit ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), plus two jokers. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\heartsuit$ and $\diamondsuit$) are red. The deck is randomly arranged. What is the probability that the top card is a Queen?


Sequence \(A\) is a geometric sequence. Sequence \(B\) is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than \(450\). What is the least positive difference between a number selected from sequence \(A\) and a number selected from sequence \(B\)?

\(\bullet\) Sequence \(A:\) \(5,\) \(15,\) \(45,\) \(135,\) \(\ldots\)

\(\bullet\) Sequence \(B:\) \(30,\) \(70,\) \(110,\) \(150,\) \(\ldots\)


In writing the integers from 20 through 99 inclusive, how many times is the digit 7 written?


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 315 hours, 58 minutes, and 36 seconds, the time will be $A:B:C$. What is the value of $A + B + C$?


A bowl contains 15 jellybeans (five red, three blue, two green, and five white). If you pick four jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red and one is green? Express your answer as a common fraction.


Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal.


Triangle $ABC$ is a right triangle with $AC$ as the hypotenuse and $AB=BC$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, and $\overline{CE}$ is 10 units long. What is the length of $\overline{BD}$? Express your answer as a decimal to the nearest tenth.


The stem-and-leaf plot shows the duration (in minutes and seconds) of songs played during a concert by a band. There are 19 songs in total. In the stem-and-leaf plot, $3 \ 15$ represents 3 minutes, 15 seconds, which is the same as 195 seconds. Determine the median duration of these songs. Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&45&50&55&&\\
1&10&12&15&18&20\\
2&05&10&15&25&30\\
3&00&05&10&15&20\\
4&10&15&&&\\
\end{tabular}


What is the area, in square units, of a regular octagon inscribed in a circle whose area is $64\pi$ square units? Express your answer in simplest radical form.

Solution: \boxed{128 \sqrt{2}}


Emily's quiz scores so far are: 85, 88, 90, 94, 96, and 92. What score does Emily need to get on her seventh quiz to make the arithmetic mean of the seven scores equal 91?


Evaluate $\lfloor (12.7)^2 \rfloor - \lfloor 12.7 \rfloor \cdot \lfloor 12.7 \rfloor.$


Simplify $\frac{4^5 + 4^3}{4^4 - 4^2}$ and express your answer as a common fraction.


Selena wants to buy tacos that are sold in packages of 4 and taco shells that are sold in packages of 6. What is the smallest number of taco packages she can buy so that she has an equal number of tacos and taco shells, but wants at least 60 of each?


Let $M = 123456789101112\ldots4950$ be the number formed by writing integers from 1 to 50 in sequence. What is the remainder when $M$ is divided by 45?


Each of six standard six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three-of-a-kind), with the remaining die showing a different number?


Given a cylinder with volume $V = 500 \pi$ cm³, the total surface area including the top and bottom circles is minimized for a radius $R$ and height $H$. Determine the ratio $\frac{H}{R}$.


Our club has expanded and now has 30 members, 15 boys and 15 girls. In how many ways can we elect a President, a Vice-President, and a Secretary if the President and Vice-President must be of different genders and the Secretary must be the same gender as the President?


How many zeros are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^7 \cdot 5^9}$?


When $\sqrt[3]{2^{11} \cdot 3^8}$ is fully simplified, the result is $a \sqrt[3]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?


For the function $\lfloor x\rfloor$, which represents the greatest integer less than or equal to $x$, find the sum of the three smallest positive solutions to $x-\lfloor x\rfloor=\frac{1}{\lfloor x\rfloor + 0.5}$. Express your answer as a mixed number.


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


In right triangle $ABC$, we have $\sin B = \frac{3}{5}$ and $\sin C = \frac{4}{5}$. Find $\sin A$.


What is the smallest abundant number that is not a multiple of 5?


Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram. If $AB = 35$ units and $CD = 65$ units, what is the area of square $BCFE$?


A parallelogram is defined by the vertices at coordinates $(1, 2)$, $(5, 6)$, $(12, 6)$, and $(8, 2)$. Calculate the sum of the perimeter and the area of this parallelogram.

Solution: \boxed{42 + 8\sqrt{2}}


A rectangular box has a volume of 5000 cubic inches and a surface area of 1800 square inches. The sum of the lengths of its 12 edges is 240 inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by 2 inches?

Solution: \boxed{7048 \text{ cubic inches}}


How many triangles can be formed using the vertices of a regular hexacontagon (a 60-sided polygon), avoiding the use of any three consecutive vertices in forming these triangles?


Evaluate \[ \left( (c^c - c(c-1)^c + k) \right)^c \] when \( c=4 \) and \( k=2 \).


If $\tan x = 3,$ then find $\tan \left( x + \frac{\pi}{3} \right).$

Solution: \boxed{-\frac{6 + 5\sqrt{3}}{13}}


Compute $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}^{100}.$

Solution: \boxed{\begin{pmatrix} 1 & 0 \\ 200 & 1 \end{pmatrix}}


Let $m = \underbrace{333333333}_{\text{9 digits}}$ and $n = \underbrace{9999999999}_{\text{10 digits}}$.

Determine $\gcd(m, n)$.


If $x$ is doubled, increased by $7$, and then divided by $5$, the result is $22$. What is the value of $x$?


In Cyclonia, bicycle license plates also contain three letters. The first letter is chosen from the set $\{B, G, K, Q, W\}$, the second from $\{E, U\}$, and the third from $\{F, J, S, X, Z\}$.

Cyclonia decided to make more license plates by adding three new letters. These new letters can all be added to one set, or they can be distributed among the sets. Determine the largest possible number of ADDITIONAL license plates that can be made by adding these three letters.


Suppose that $x^3$ varies inversely with $y^2$. Given $x = 5$ when $y = 4$, determine the value of $x^3$ when $y = 8$.


Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two different letters are each repeated once, and only odd digits are used without repeating any digit?


Three positive integers have an arithmetic mean of 30 and a median of 28. If the median is 6 less than the largest number, what is the smallest number?


The complete graph of $y=g(x)$, which consists of five line segments, is provided. The graph includes the following segments: $(-5, -4) \rightarrow (-3, 0)$, $(-3, 0) \rightarrow (-1, -1)$, $(-1, -1) \rightarrow (1, 3)$, $(1, 3) \rightarrow (3, 2)$, $(3, 2) \rightarrow (5, 6)$. Calculate the sum of the $x$-coordinates of all points where $g(x) = 2.4$.


What is the largest $5$-digit integer congruent to $19 \pmod{26}$?


Find the distance between the points (0,12) and (9,0).


In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 30$, and $\tan Z = 3\sin Z$. What is the length of $XZ$?


What is the value of $\sqrt{49+121}-\sqrt{36-9}$?

Solution: \boxed{\sqrt{170} - 3\sqrt{3}}


What is the remainder when 845,307 is divided by 6?


Let $f(x)= \frac{ax+b}{cx+d}$ where $a$, $b$, $c$, and $d$ are nonzero real numbers. The function has the properties $f(5)=5$, $f(50)=50$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$. Determine the unique number that is not in the range of $f$.


A survey conducted by a local TV station WXLT results in the following partially filled table. Determine how many of the females surveyed watch this TV station?

\begin{tabular}{| l | c | r |}
\hline
& watch & don't watch \\
\hline
males & 85 & \\ \hline
females & & 120 \\ \hline
total & 160 & 180 \\
\hline
\end{tabular}


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.


Find the smallest positive integer $y$ which satisfies the congruence $56y + 8 \equiv 6 \pmod {26}$.


A club has 18 members and needs to select 2 members to be co-presidents and a third member to be secretary. In how many ways can the club select these positions?


There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that \[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 8 \\ 4 \\ -3 \end{pmatrix},\] where $\mathbf{a}$ is parallel to $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}.$  Find $\mathbf{b}.$

Solution: \boxed{\begin{pmatrix} 5 \\ \frac{11}{2} \\ -\frac{9}{2} \end{pmatrix}}


A rectangular array of tables is arranged such that each row contains the same number of tables as every other row, and each column contains the same number of tables as every other column. With at least two tables in every row and column, how many such arrays are possible in a meting room containing $48$ tables? Note that $12$ rows of $4$ tables is considered different from $4$ rows of $12$ tables.


When $\frac{7}{8000}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


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


Solution: \boxed{(-\infty, 2) \cup (3, 5) \cup (6, 7) \cup (8, \infty)}


The sum of two positive integers $c$ and $d$ is 1023. What is the largest possible value of $\gcd(c,d)$?


Four primes \(p_1\), \(p_2\), \(p_3\), and \(p_4\) form an increasing arithmetic sequence with \(p_1 > 3\) and common difference 4. What is the ones digit of \(p_1\)?


If the area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is 12, find the area of the parallelogram generated by the vectors $3 \mathbf{a} + 4 \mathbf{b}$ and $2 \mathbf{a} - 6 \mathbf{b}$.


Solution: \boxed{\frac{33 - \sqrt{129}}{32}}


Suppose \( z \) and \( w \) are complex numbers such that
\[
|z| = \sqrt{2}, |w| = 2, \text{ and } z \overline{w} + \overline{z} w = 1.
\]
Find the largest possible value of the real part of \( z + w \).


Find the middle integer of three consecutive even integers where the sum is one-fifth of the product of these integers.


A rectangle, with integer length and width, has a perimeter of 60 units. What is the area of the rectangle when the shorter dimension is minimized?


A modified deck of 78 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 6 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, $\clubsuit$, $\star$, and $\moon$), each suit consisting of one card of each rank. The deck is randomly arranged. What is the probability that the top card is a $\star$?


For what value of \(n\) is \(5 \times 8 \times 2 \times 6 \times n = 9!\)?


In a circle with radius $6$, two chords $PQ$ and $RS$ intersect at a point $T$. Chord $RS$ bisects chord $PQ$ such that $PT = TQ = 8$ and $PQ$ is the only chord from $P$ that $RS$ bisects. It is known that the sine of the central angle subtended by minor arc $PR$ is a rational number. If expressed as a fraction $\frac{p}{q}$ in lowest terms, what is the product $pq$?


Consider the following scenario where two stores offer discounts on the same laptop with a list price of $\$59.99$:
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Laptop Y} \\ \hline
Tech Bargains & $\$15$~off the list price \\ \hline
Budget Bytes & $30\%$~off the list price \\ \hline
\end{tabular}
How much cheaper, in cents, is the cheaper store's price for Laptop $Y$ compared to the more expensive store?


The number $r$ can be expressed as a four-place decimal $0.abcd$, where $a, b, c, d$ represent digits. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac{2}{5}$. What is the number of possible values for $r$?


The noon temperatures for eight consecutive days were $82^\circ$, $80^\circ$, $84^\circ$, $86^\circ$, $88^\circ$, $90^\circ$, $88^\circ$, and an unknown temperature on the last day. If the mean noon temperature for the week was $86^\circ$ Fahrenheit, what was the temperature on the last day?


In the cube $ABCDEFGH$, find $\sin \angle BAC$ ensuring that the angle is uniquely determined and forms a right angle.


Liam received a bill for $600, but if he fails to pay it on time, a late charge of 2% is added every 30 days. After failing to pay the bill for 90 days, what is the amount due?


How many cubic feet are in one cubic meter? Assume 1 meter is approximately equal to 3.28084 feet.


What is the smallest positive value of $n$ so that the equation $15x^2 - nx + 630 = 0$ has integral solutions?


Consider two friends, Anna and Bella, each thinking of a monic polynomial of degree 3. Both polynomials have the same positive constant term and the same coefficient for $z^2$. The product of their polynomials is \[z^6 + 2z^5 + 5z^4 + 10z^3 + 10z^2 + 8z + 9.\] What is the constant term of Bella's polynomial?


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


A new proposal suggests that years ending in double zeroes will be considered leap years only if they leave a remainder of 300 or 700 when divided by 1000. Determine how many double-zero ending leap years occur between 1996 and 4096 under this new rule.


The expression $512x^3 - 27$ can be rewritten as $(ax+b)(cx^2+dx+e)$. Find $a+b+c+d+e$.


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

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


Simplify $\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{5}}{\sqrt{11}} \cdot \frac{\sqrt{6}}{\sqrt{13}}$ and rationalize the denominator of the resulting fraction.


Let $M$ be the largest integer multiple of 9, no two of whose digits are the same. What is the remainder when $M$ is divided by 1000?


For some positive integers $c$ and $d$, the product \[\log_c(c+1) \cdot \log_{c+1} (c+2) \dotsm \log_{d-2} (d-1)  \cdot\log_{d-1} d\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$


The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the Earth's center and the object. When Alice is on the surface of the Earth, 4,000 miles from the center, the gravitational force is 500 Newtons. How much gravitational force (in Newtons) will Earth exert on her when she is on a space station located 40,000 miles from the Earth's center?

Solution: \boxed{5 \text{ Newtons}}


Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$?


Solution: \boxed{-\frac{1}{2} - \frac{4}{5} i}


The measures of the interior angles of a convex pentagon form an increasing arithmetic sequence. Determine how many such sequences are possible if the pentagon is not equiangular, all of the angle degree measures are positive integers less than 140 degrees, and the sum of the angles in the pentagon is exactly 540 degrees.


How many two-digit numbers can be written in the form \( 3^n \) where \( n \) is an integer?


Calculate the product $\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{2004}{2008}\right)$. Express your answer as a common fraction.


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


Find the sum of the first eight terms in the geometric sequence starting at $\frac{1}{5}$ where each term is $\frac{2}{5}$ of the previous term.


How many five-digit positive integers have at least one digit that is a 4 or a 7?


The set of points $(x,y)$ as $t$ takes on all real values is given by:
\[
x = t^2 - 3,
\]
\[
y = t^3 - 6t + 2.
\]
Find the ordered pair $(x, y)$ where the curve crosses itself.

Solution: \boxed{(3, 2)}


Define an integer $N$ to be $4$-nice or $6$-nice if there exists a positive integer $a$ such that $a^4$ or $a^6$ respectively, has exactly $N$ positive divisors. Determine the number of positive integers less than or equal to $500$ that are neither $4$-nice nor $6$-nice.


In the geometric sequence with a first term of $8$ and a second term of $-16$, what is the $302^{nd}$ term?


Factor $w^4 - 81$ as far as possible, where the factors are monic polynomials with real or complex coefficients.


Jessica needs to convert her height from inches to centimeters for a medical form. Given that there are 10 inches in a foot and 25 centimeters in a foot, what should Jessica specify as her height in centimeters if her height in inches is 65 inches?


In triangle $ABC$, $\angle C=90^\circ$, $AC=9$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=5$, then what is the length of $BD$?


A rectangular band formation is a formation with $x$ band members in each of $r$ rows, where $x$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has four members left over. If he increases the number of members in each row by 2 and reduces the number of rows by 3, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have?


The base of a triangular piece of paper $ABC$ is $18\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $9\%$ that of the area of the triangle $ABC.$ What is the length of $DE,$ in cm?

Solution: \boxed{5.4\text{ cm}}


A truncated cone has horizontal bases with radii 10 and 3. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


What is the smallest positive integer $n$ such that $13n \equiv 456 \pmod{5}?$


If $x^5 - 2x^4 + 3x^3 - px^2 + qx - 6$ is divisible by $(x + 3)(x - 2),$ find the ordered pair $(p,q).$

Solution: \boxed{(-31, -71)}


What is the value of $x$ in the equation $\frac{3}{x} - \frac{1}{x} \times \frac{6}{x} = -2.5$?

Solution: \boxed{\frac{-3 - \sqrt{69}}{5}}


Suppose that \( a \) varies inversely with \( c \), where \( c = \sqrt{b} \). If \( a = 15 \) when \( b = 9 \), find the value of \( a \) when \( b = 16 \).


Let $A,$ $B,$ and $C$ be points on a circle of radius $15.$ If $\angle ACB = 45^\circ,$ what is the circumference of the minor arc ${AB}$? Express your answer in terms of $\pi.$


How many positive three-digit integers with a $0$ in the units place are divisible by 20?


Convert the point $(\rho,\theta,\phi) = \left( 15, \frac{5 \pi}{4}, \frac{\pi}{6} \right)$ in spherical coordinates to rectangular coordinates.

Solution: \boxed{\left(-\frac{15\sqrt{2}}{4}, -\frac{15\sqrt{2}}{4}, \frac{15\sqrt{3}}{2}\right)}


Let \( m \) be the smallest positive three-digit integer congruent to 4 (mod 7). Let \( n \) be the smallest positive four-digit integer congruent to 4 (mod 7). What is \( n-m \)?


If you roll 6 standard 8-sided dice, what is the probability that the product of the numbers on the faces is a prime number?


Find the minimum value of
\[(15 - x)(9 - x)(15 + x)(9 + x).\]


In how many ways is it possible to arrange the digits of 12250 to get a five-digit multiple of 5?


How many perfect squares are factors of 450?


Suppose that $c$ is a positive integer greater than or equal to $2.$ When $250$ is converted to base $c$, the resulting representation has $4$ digits. What is the number of possible values for $c$?


What is the least positive integer which, when divided by 4, gives a remainder of 3, when divided by 5, gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, and when divided by 11 gives a remainder of 10?


The current time is 5:00 p.m. What time will it be in 1567 minutes?

Solution: \boxed{7:07 \text{ p.m.}}


Find \(453_{8} - 326_{8}\). Express your answer in base \(8\).


Find the product of all real values of $s$ for which $\frac{3}{4x} = \frac{s-2x}{5}$ has exactly one real solution.


A $\emph{convex}$ polygon is defined as one in which each interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a segment that connects two non-adjacent vertices. How many diagonals can be drawn from one vertex in a convex polygon that has 25 sides?


Find the range of the function $f(x) = \arcsin x + \arccos x + \arctan(2x).$ Assume all functions are in radians and defined for $-1 \leq x \leq 1$.

Solution: \boxed{\left[\frac{\pi}{2} - \arctan(2), \frac{\pi}{2} + \arctan(2)\right]}


A triangle in a Cartesian coordinate plane has vertices at (3, -3), (3, 4), and (12, 4). Determine the area of the triangle in square units.

Solution: \boxed{31.5 \text{ square units}}


Ramon now sells three enchiladas and four tacos for $\$3.50$ and he sells four enchiladas and three tacos for $\$3.90$. Assuming a fixed price per item, what is the cost, in dollars, of four enchiladas and five tacos? Express your answer as a decimal to the nearest hundredth.


Solution: \boxed{16 + 6i}


Mr. and Mrs. Lopez have three children. When they get into their family car, which has two seats in the front (including driver’s seat) and three in the back, either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?


Solution: \boxed{(-\infty, 1) \cup (2, 3) \cup (4, \infty)}


Stuart has drawn a pair of concentric circles. He starts at point $A$ on the larger circle and draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the smaller one. If $m\angle ABC = 60^\circ$, how many segments will he draw before returning to his starting point at $A$?


Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?


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

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


Let \( p, q, r, \) and \( s \) be the roots of the polynomial \[ x^4 + 6x^3 + 13x^2 + 7x + 3 = 0. \] Find the value of \[ \frac{1}{pqr} + \frac{1}{pqs} + \frac{1}{prs} + \frac{1}{qrs}. \]


What is the value of $n$ such that $10^n = 10^{4} \times \sqrt{\frac{10^{155}}{0.0001}}$?


How many perfect squares less than 10,000 have a ones digit of 4, 5, or 6?


The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years in the ages of any two consecutive children. How old is the oldest child?


Let \(x, y, z\) be real numbers such that \(9x^2 + 4y^2 + 25z^2 = 1\). Find the maximum value of \(8x + 3y + 10z\).


Bob took six quizzes during the semester with scores of 85, 90, 88, 92, 86, and 94. The quizzes had weights of 1, 2, 1, 3, 1, and 2 respectively. What was his weighted mean score?


Robert decides to visit the milk bottling plant for 7 days. Each day, the plant has a 3/4 chance of bottling chocolate milk. What is the probability that the bottling plant bottles chocolate milk exactly 5 out of those 7 days?


Let $O$ be the origin and let $(a, b, c)$ be the midpoint of the line segment connecting $O$ and a point $(d, e, f)$. A plane passes through $(d, e, f)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A$, $B$, and $C$ respectively, all distinct from $O$. Let $(p, q, r)$ be the center of the sphere passing through $A$, $B$, $C$, and $O$. Find
\[
\frac{d}{p} + \frac{e}{q} + \frac{f}{r}.
\]


What is the remainder when 425,421 is divided by 12?


In a similar press conference setting, nine All-Stars are taking questions. Three are Cubs, three are Red Sox, two are Yankees, and one is from the Dodgers. If teammates insist on sitting together, how many ways can the nine All-Stars be seated in a row?


In triangle $ABC,$ $\angle B = 45^\circ$ and $\angle C = 60^\circ.$ The point $D$ divides $\overline{BC}$ in the ratio $1:1$. Find
\[\frac{\sin \angle BAD}{\sin \angle CAD}.\]


Find the arithmetic mean of the prime numbers in this list: 31, 33, 35, 37, 39, 41.


Find the $2 \times 2$ matrix $\mathbf{N}$ such that $\mathbf{N} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -6 \end{pmatrix}$ and $\mathbf{N} \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 19 \\ -2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{30}{7} & -\frac{13}{7} \\ -\frac{6}{7} & -\frac{10}{7} \end{pmatrix}}


If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 9$, what is the value of $\cos{2\theta}$?


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


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King, the second card is an Ace, and the third card is a 10?


Find the range of the function
\[ g(x) = \frac{\sin^3 x + 7 \sin^2 x + 2 \sin x + 3 \cos^2 x - 10}{\sin x - 1}, \]
as \( x \) ranges over all real numbers such that \( \sin x \neq 1 \).

Solution: \boxed{[3, 13)}


How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 4, 4, 7, 8, 9\}$ if no digit may be used more times than it appears in the given set?


Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 19?


Find the numerical value of
\[
\frac{\sin 20^\circ \cos 10^\circ + \cos 160^\circ \cos 110^\circ}{\sin 24^\circ \cos 6^\circ + \cos 156^\circ \cos 106^\circ}.
\]


The diagram shows the miles traveled by cyclists Carla and Daniel. After five hours, by how many more miles has Carla cycled than Daniel? The graph indicates that Carla travels at a consistent rate reaching 100 miles in 5 hours, while Daniel travels at a consistent rate reaching 75 miles in 5 hours.


Find the $2 \times 2$ matrix $\mathbf{N}$ such that $\mathbf{N} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$ and $\mathbf{N} \begin{pmatrix} 4 \\ 0 \end{pmatrix} = \begin{pmatrix} 20 \\ -12 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 5 & 5 \\ -3 & -3 \end{pmatrix}}


What is the greatest common factor of the numbers 2750 and 9450?


What is the smallest positive integer value of $x$ such that $(3x + 28)^2$ is a multiple of 53?


Let $a, b, c, d$ be nonzero real numbers. Find the number of real roots of the equation
\[
\begin{vmatrix} x & -c & b \\ c & x & -d \\ -b & d & x \end{vmatrix} = 0.
\]


In triangle ABC, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, what is ${AC}$?


The number $1027$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$?


A hypersphere is inscribed in a 4-dimensional hypercube (tesseract). What is the ratio of the volume of the inscribed hypersphere to the volume of the tesseract? Express your answer as a common fraction in terms of $\pi$.


Let $a,$ $b,$ $c,$ and $d$ be nonzero real numbers. Find the number of real roots of the equation
\[\begin{vmatrix} x & d & -b \\ -d & x & a \\ b & -a & x \end{vmatrix} = 0.\]


Jack rewrites the quadratic \(4x^2 - 16x + 2\) in the form of \((dx + e)^2 + f,\) where \(d,\) \(e,\) and \(f\) are all integers. What is \(de\)?


How many of the first 30 rows of Pascal's triangle consist only of even numbers, except for the $1$ at each end? (Exclude row $0$ and row $1$ in your count).


According to the table below, how many dollars are in the median value of the 70 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$140{,}000$\\\hline
Vice-President&8&$\$95{,}000$\\\hline
Director&10&$\$78{,}000$\\\hline
Manager&4&$\$60{,}000$\\\hline
Associate Director&7&$\$55{,}000$\\\hline
Administrative Specialist&40&$\$25{,}000$\\\hline
\end{tabular}


Find the sum of all numbers $y$ satisfying $y + \frac{49}{y} = 14$.


Let $x$ be a real number. Consider the following five statements:
1. $0 < x^3 < 1$
2. $x^3 > 1$
3. $-1 < x < 0$
4. $0 < x < 1$
5. $0 < x - x^3 < 1$

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


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{16384}}}}$$


In a new game, I walk on a line, starting at point 0. On move 1, I stay still, but on move $n$ where $2 \le n \le 30$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After all 30 moves, how many steps away am I from the starting point?


Find the solutions to $z^5 = -32i$ and determine the sum of their angles, $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5$, where the solutions can be expressed as:
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 &= r_5 (\cos \theta_5 + i \sin \theta_5),
\end{align*}
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ$.


How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $7$, $24$, and $25$?


How many three-digit numbers exist if we exclude all numbers where the hundreds and units digits are the same but the tens digit is different from them?


A dealer sold 250 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were BMWs, how many BMWs did she sell?

\begin{tabular}{ |c | c|}
\hline \textbf{Type of Car} & \textbf{$\%$ of Total Cars Sold} \\ \hline
Mercedes & $18\%$ \\ \hline
Toyota & $25\%$ \\ \hline
Acura & $15\%$ \\ \hline
\end{tabular}


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

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


Let $u$, $v$, and $w$ be the roots of the equation $x^3 - 6x^2 + 11x - 6 = 0$. Compute $\frac{uv}{w} + \frac{vw}{u} + \frac{wu}{v}$.


How many distinct numbers can you get by multiplying two or more distinct members of the set $\{2, 3, 5, 7, 13\}$ together?


Find the smallest positive integer that is both an integer power of 12 and is not a palindrome.


In a new scenario, city $D$, $E$, and $F$ form a right-angled triangle with $D$ being the right-angle. The distance from $D$ to $E$ is $4000\text{ km}$, from $E$ to $F$ is $4500\text{ km}$, and from $F$ to $D$ is $5000\text{ km}$. For buses, the fare is $\$0.20$ per kilometer. Airplanes charge a $\$120$ booking fee when flying from $D$ or $F$ but a $\$150$ booking fee from $E$, plus $\$0.12$ per kilometer regardless of the departure. Piravena decides to take a bus from $D$ to $E$ and fly from $E$ to $F$. Calculate the total travel cost for Piravena from $D$ to $F$ via $E$.


Express $\frac{0.\overline{444}}{1.\overline{666}}$ as a common fraction.


What is the value of the expression \(\frac{y^2 - y - 12}{y - 4}\) for \(y = 5\)? Express your answer in simplest form.


A $6 \times 9$ rectangle can be rolled to form two different cylinders. Determine the ratio of the volume of the cylinder with the larger volume to the cylinder with the smaller volume when one cylinder is formed by using 6 as the height and the other using 9 as the height.


The graph of the equation \[\sqrt{x^2+y^2} + |y-1| = 5\] consists of portions of two different parabolas. Compute the distance between the vertices of the parabolas.


Find all 6-digit multiples of 66 that have the form $4d6,\!45e$ where $d$ and $e$ are digits. What is the maximum value of $d$?


Five standard, six-sided dice are to be rolled. If the product of their values is an even number, what is the probability that their sum is divisible by 3?


In a trapezoid $ABCD$ where $AB$ is parallel to $CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Given that the area of triangle $ABE$ is 60 square units and the area of triangle $ADE$ is 30 square units, what is the area of trapezoid $ABCD$?


Let $p$ be the largest prime number with 2023 digits. What is the smallest positive integer $k$ such that $p^2 - k$ is divisible by 30?

Solution: \boxed{k = 1}


Find the smallest positive five-digit number that is divisible by each of the three smallest odd prime numbers.


Define the operation $\spadesuit$ as $a \spadesuit b = |a - b|$. What is the value of $(2 \spadesuit 3) \spadesuit (6 \spadesuit (9 \spadesuit 4))$?


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

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


Given that $-2 - 3\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 54 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Calculate the value of $0.\overline{246} - 0.\overline{135} + 0.\overline{579}$. Express your answer as a fraction in its lowest terms.


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

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


The average of four different positive whole numbers is $5.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?


The circles whose equations are \(x^2 + y^2 - 6x + 4y - 8 = 0\) and \(x^2 + y^2 - 8x - 2y + 10 = 0\) intersect in points \(C\) and \(D.\) Compute the slope of \(\overline{CD}.\)


Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$.


A right triangle has sides with lengths 6, 8, and 10. There are both an inscribed circle and a circumscribed circle around the triangle. What is the distance between the centers of these two circles?


Convert the point \((4, -4, 6)\) in rectangular coordinates to cylindrical coordinates. Enter your answer in the form \((r, \theta, z)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\).

Solution: \boxed{(4\sqrt{2}, \frac{7\pi}{4}, 6)}


A $7 \times 10$ rectangle is used to form two different cylindrical shapes by rolling it along its length and width. Calculate the ratio of the larger volume to the smaller volume when the rectangle is formed into cylinders in these two ways. Express your answer as a common fraction.


A club has 24 members, 12 boys and 12 girls. There are 6 senior boys, 6 junior boys, 6 senior girls, and 6 junior girls. In how many ways can we choose a president and a vice-president if they must be of the same gender, and one must be a senior while the other must be a junior?


The functions given are $p(x) = x^2 - 4$ and $q(x) = -|x| + 1$. Evaluate $q(p(x))$ at $x = -3, -2, -1, 0, 1, 2, 3$ and find the sum of the values.


The first and the seventeenth terms of an arithmetic sequence are $\frac{4}{7}$ and $\frac{5}{6}$, respectively. What is the ninth term?


A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure 1 unit. What is the perimeter of this region?


Tracy initially had a bag of candies. She first ate $\frac{1}{4}$ of them, then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. After that, Tracy and her mom each ate 12 candies from what was left. Lastly, her brother took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the beginning?


In a special tournament series, the Miami Heat are facing off against the New York Knicks. To win this tournament, a team must win 3 games out of a possible 5 games. If the Heat wins each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the Knicks will win the tournament while requiring all 5 games to be played?


What is the coefficient of $x^6$ in the expansion of $(3x+2)^8$?


For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$?


Define $A \star B$ as $A \star B = \frac{A \cdot B + A}{5}$. What is the value of $(3 \star 6) \star 4$?


Find the sum of the first 10 terms of the geometric sequence $12, 4, \frac{4}{3}, \ldots$.


Let $G_n$ be the Fibonacci sequence defined by $G_0 = 0$, $G_1 = 1$, and $G_{n+2} = G_{n+1} + G_n$. Compute
\[\sum_{n=0}^\infty \frac{G_n}{2^n}.\]


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

Solution: \boxed{\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}}


In hexagon $PROBLEMS$, $\angle P \cong \angle R \cong \angle B$, $\angle O$ and $\angle S$ are supplementary, and $\angle L = 90^\circ$. Calculate the measure of $\angle B$.


What is the sum of all the prime numbers between 10 and 30?


Six standard, eight-sided dice are rolled. If the product of their values is an even number, what is the probability that their sum is odd? Express your answer as a common fraction.


Let $x = (3 + \sqrt{8})^{100},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


In the diagram, $ABCD$ is a trapezoid with an area of $20$. $CD$ is three times the length of $AB$. A line segment $CE$ is drawn from point $C$ to meet $AB$ extended at point $E$, such that $AE = AD$. What is the area of $\triangle ACD?$

[asy]
draw((0,0)--(-3,4)--(9,4)--(12,0)--cycle);
draw((0,0)--(9,4));
label("$D$",(0,0),W);
label("$A$",(-3,4),NW);
label("$B$",(9,4),NE);
label("$C$",(12,0),E);
label("$E$",(-6,4),NW);
[/asy]


Find the number of ordered 19-tuples $(a_1, a_2, a_3, \dots, a_{19})$ of integers, such that the cube of any number in the 19-tuple is equal to three times the sum of the other 18 numbers.


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


Find the distance between the planes $2x - 4y + 4z = 10$ and $4x - 8y + 8z = 20.$


**Find the largest prime divisor of $17^2 + 144^2$.**


In right triangle $\triangle ABC$ with $\angle BAC = 90^\circ$, medians $\overline{AD}$ and $\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\overline{AD}$ is the altitude from $A$, find the area of $\triangle ABC$.


In the shown diagram, segment $CD$ is parallel to segment $WX$. If $CX = 30$ units, $DP = 15$ units, and $PW = 45$ units, find the length of segment $PX$. Diagram:
[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair W = (0,0), X = (24,0), C = (0,12), D = (9,12);
draw(C--D--W--X--cycle);
label("$C$",C,W); label("$D$",D,E); label("$W$",W,W); label("$X$",X,E);
pair P = intersectionpoint(C--X,D--W);
label("$P$",P,E);
[/asy]

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


In the diagram, $PQRS$ is a trapezoid with an area of $18.$ $RS$ is three times the length of $PQ.$ What is the area of $\triangle PQS?$

[asy]
draw((0,0)--(1,6)--(10,6)--(15,0)--cycle);
draw((10,6)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,6),NW);
label("$Q$",(10,6),NE);
label("$R$",(15,0),E);
[/asy]


Find the smallest, positive five-digit multiple of $18$.


Let $N$ be the product of all odd primes less than $2^5$. What remainder does $N$ leave when divided by $2^5$?


Let $x$ be a real number. Consider the following five statements:

1. $0 < x^2 < 1$
2. $x^2 > 1$
3. $-1 < x < 0$
4. $0 < x < 1$
5. $0 < x^2 - x^3 < 1$

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


Let $Q$ be a point outside of circle $C.$ A line from $Q$ is tangent to circle $C$ at point $R.$ A secant from $Q$ intersects $C$ at $X$ and $Y,$ such that $QX < QY.$ If $QX = 5$ and $QR = XY - QX,$ what is $QY$?


Fido's leash is tied to a stake at the center of his yard, which is in the shape of an equilateral triangle. His leash is exactly long enough to reach the midpoint of each side of the triangle. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest radical form as \(\frac{\sqrt{a}}{b}\pi\), what is the value of the product \(ab\)?


What is the units digit of the product of all odd positive integers between 20 and 200 that are not divisible by 3?


The function \( g(x) \) satisfies
\[ g(xy) = xg(y) + y \]
for all real numbers \( x \) and \( y \). If \( g(1) = 3 \), find \( g(5) \).


Let $P$ be the centroid of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. The area of triangle $ABC$ is 24. Find the area of triangle $G_1 G_2 G_3$.


Solution: \boxed{-1 + i}


Simplify
\[\frac{\tan^3 60^\circ + \cot^3 60^\circ}{\tan 60^\circ + \cot 60^\circ}.\]


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

Solution: \boxed{\begin{pmatrix} \frac{3}{5} & \frac{3}{10} \\ \frac{3}{5} & \frac{3}{10} \end{pmatrix}}


Given \( ab + ac + bd + cd = 42 \) and \( b + c = 6 \), find \( a + d \).