In right triangle $ABC$, we have $\angle A = 30^\circ$, $\angle B = 90^\circ$, and $BC = 12$. Find the length of side $AB$ to the nearest tenth.


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


Find the value of $p$ for which the following set of equations holds:
\[
\frac{9}{a + b} = \frac{p}{a + c} = \frac{13}{c - b}
\]


A standard die is rolled eight times. What is the probability that the product of all eight rolls is a multiple of 5?


In a right triangle $ABC$, where $\angle B = 90^\circ$ and $\angle A = 40^\circ$, side $BC$ measures 12 units. Find the length of side $AB$.


Solve the equation \( x^2 - 4x = -4 - 4i \) and determine the product of the real parts of the solutions.


Compute $\dbinom{12}{6}$ and evaluate $(x+y)^{12}$ when $x = 1$ and $y = 1$.


How many rows of Pascal's Triangle contain the number $67$?


Find the side lengths of a triangle expressed by the real roots $a$, $b$, and $c$ of the polynomial $x^3 - 6x^2 + 11x - 6 = 0$. Compute the area of this triangle.


Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time \( t = 0 \) to \( t = \infty \). Rational Man drives along the path parameterized by
\[
x = \cos t, \quad y = \sin t,
\]
and Irrational Man drives along the path parameterized by
\[
x = \cos \left(\frac{t}{2}\right), \quad y = \sin \left(\frac{t}{2}\right).
\]
If \( A \) is a point on Rational Man's racetrack, and \( B \) is a point on Irrational Man's racetrack, then find the smallest possible distance \( AB \).


If $(x+3)(x-4)=20$, find the sum of the possible values of $x$.


In a massive library storage room, there are $1500$ boxes, each holding $45$ books. Daniel's supervisor instructs him to repack all the books so that each new box contains $47$ books. After arranging as many full boxes as possible, how many books does Daniel have left over?


Three real numbers $a, b,$ and $c$ satisfy the equations $a+b+c=3$, $ab+ac+bc=-8$, and $abc=-20$. What is the largest of the three numbers? Express your answer in simplest radical form.

Solution: \boxed{\frac{1 + \sqrt{41}}{2}}


Two vertices of a right triangle are located at \((3, 3)\) and \((0,0)\). The third vertex is on the positive branch of the \(x\)-axis. What are the coordinates of the third vertex if the area of the triangle is 18 square units?


Solution: \boxed{(5, -3, 4)}


Find the positive value of $x$ which satisfies
\[\log_3(x-1) + \log_{\sqrt{3}} (x^2 - 1) + \log_{\frac{1}{3}} (x - 1) = 3.\]

Solution: \boxed{\sqrt{1 + 3 \sqrt{3}}}


Circle $\Gamma$ is the incircle of $\triangle ABC$ and is also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A=50^\circ$, $\angle B=70^\circ$, and $\angle C=60^\circ$, what is the measure of $\angle BYZ$?


The line $y = 4x + d$ is tangent to the parabola $y^2 = 16x.$  Find $d.$


For how many digits $C$ is the positive four-digit number $2C4X$ a multiple of 3, where $X$ is the smallest digit such that the number is still a multiple of 3?


Consider the polynomials \[ f(x) = 1 - 12x + 3x^2 - 4x^3 + 5x^4 \] and \[ g(x) = 3 - 2x - 6x^3 + 8x^4 + x^5. \] Find $c$ such that the polynomial $f(x) + cg(x)$ has degree 4.


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


Consider the sequence of numbers $G_n = 3^{3^n} + 1$. Determine the units digit of $G_{100}$.


On the Cartesian plane, where each unit is one meter, a goat is tied to a post at the point (5,1) with a 15-meter rope. What is the greatest distance the goat can be from the origin?

Solution: \boxed{\sqrt{26} + 15}


Given that $3 + \sqrt{5}$ is a root of the equation \[x^3 + ax^2 + bx - 40 = 0\] with $a$ and $b$ as rational numbers, compute $b$.

Solution: \boxed{b = 64}


The equation $x^2 - 6x + 15 = 27$ has two solutions, $c$ and $d$, with $c \geq d$. What is the value of $3c + 2d$?

Solution: \boxed{15 + \sqrt{21}}


If the sum of $1! + 2! + 3! + \cdots + 99! + 100!$ is divided by $30$, what is the remainder?


Define $x \Delta y = x^2 - 2y$. What is the value of $(5^{3 \Delta 4}) \Delta (2^{2 \Delta 3})$?


Suppose $x+2$ and $y-2$ are multiples of $9$.

What is the smallest positive integer, $n,$ for which $x^2 - xy + y^2 + n$ is a multiple of $9$?


Piravena is making a triangular trip starting from city $X$ to city $Y$, from $Y$ to city $Z$, and returning to $X$. The cities form a right-angled triangle with $Z$ being the right angle. The distance from $X$ to $Z$ is 4000 km, and from $X$ to $Y$ is 5000 km. The cost of taking the bus is $\$0.20$ per kilometer and to take an airplane is a $\$120$ booking fee plus $\$0.12$ per kilometer. Determine the total distance Piravena travels for her round trip.

Solution: \boxed{12000 \text{ km}}


In how many ways can five people line up in a straight line if neither of the two youngest people can be first in line?


Parallelogram $ABCD$ with $A(3,6)$, $B(5,10)$, $C(7,6)$, and $D(5,2)$ is reflected across the $x$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x+2$ to $A''B''C''D''$. This is done such that $D'$ is the image of $D$, and $D''$ is the image of $D'$. What is the ordered pair of $D''$ in the coordinate plane?

Solution: \boxed{(-4, 7)}


Given a line passing through points $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$, find the value of $b$ such that the line's direction vector can be written in the form $\begin{pmatrix} b \\ -1 \end{pmatrix}$.


The curve $y = \sin x$ intersects the horizontal line $y = \sin 60^\circ$. Determine the ordered pair of relatively prime positive integers $(p, q)$ that represent the ratio of the segments along the x-axis between successive intersection points, assuming the pattern repeats with $p < q$.

Solution: \boxed{(1, 5)}


Spinners $A$ and $B$ are spun. Spinner $A$ is divided into 6 equal sections marked 1, 1, 2, 2, 3, and 4. Spinner $B$ is divided into 4 equal sections marked 1, 3, 5, and 6. What is the probability that the product of the two spinners' numbers is even?


A wizard is preparing a magical elixir. He needs to choose one of four different herbs and one of six different gems. However, one particular gem is incompatible with three of the herbs. How many valid combinations can the wizard use to prepare his elixir?


Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $S$ is inside $\triangle PQR$, $\angle PQR = 50^\circ$, and $\angle PRS = 110^\circ$. What is the degree measure of $\angle
PQS$?


A school has between 200 and 250 students enrolled. Every afternoon, all the students come together to participate in a music assembly. The students are split into seven distinct groups. If one student is absent from school, each group can have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


What is the 15th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{8}$ and $\frac{1}{6}$?


Let $A$ and $B$ be two points on the parabola $y = 4x^2$, such that the tangent at $A$ is horizontal and the tangent at $B$ is perpendicular to that at $A$. Find the $y$-coordinate of their intersection point $P$.


Determine the maximum value of $b$ such that $0$ is in the range of the quadratic function $g(x) = x^2 - 7x + b$.


A bag contains 4 green, 3 blue and 5 red chips. If the 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 4 green chips are drawn consecutively, the 3 blue chips are drawn consecutively, and the 5 red chips are drawn consecutively, but not necessarily in the green-blue-red order? Express your answer as a common fraction.


What is the sum of the digits of the base-2 expression for $300_{10}$?


Solution: \boxed{\frac{\log_2 3}{2}}


Let $x$, $y$, and $z$ be positive real numbers. What is the smallest possible value of $(x+y+z)\left(\frac{1}{x+y-z}+\frac{1}{y+z-x}+\frac{1}{z+x-y}\right)$, considering $x+y>z$, $y+z>x$, and $z+x>y$?


Evelyn has 8 daughters and no sons. Some of her daughters have 7 daughters each, and the rest have none. Evelyn has a total of 36 daughters and granddaughters, and no great-granddaughters. How many of Evelyn's daughters and granddaughters have no daughters?


A one-inch cube of gold weighs 0.5 pounds and is worth $1000. How much is a four-inch cube of gold worth?


Find the range of the function \[h(t) = \frac{t^2+\tfrac{5}{4} t}{t^2+1},\] where $t$ can be any real number.

Solution: \boxed{\left[\frac{4 - \sqrt{41}}{8}, \frac{4 + \sqrt{41}}{8}\right]}


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


Find the largest integral value of $x$ that satisfies the inequality $\frac{2}{7}<\frac{x}{6}<\frac{7}{9}$.


Find the distance between the planes \(3x - y + 2z - 3 = 0\) and \(6x - 2y + 4z + 4 = 0\).


In Mrs. Jansen's 24-student music class, no two students have the same initials and each student's first name and last name begin with the same letter. The letter “Y” is considered a vowel. Owing to name restrictions in the class, students cannot have initials "N" and "O". What is the probability of randomly picking a student whose initials are vowels? Express your answer as a common fraction.


Add \(53_8 + 27_8\). Express your answer in base \(8\).


Find the polynomial $p(x)$ with real coefficients, such that $p(3) = 10$ and $p(x)p(y) = p(x) + p(y) + p(xy) - 2$ for all real numbers $x$ and $y$, and $p(4) = 17$.

Solution: \boxed{x^2 + 1}


Evaluate the expression $\left(8^5 \div 8^3\right) \cdot 16^4 \div 2^3$.


Monica tosses a fair 8-sided die. If the roll is a prime number, she wins that amount in dollars (for example, rolling a 7 wins her $7). If the roll is a composite number or 1, she wins nothing. Otherwise, if she rolls an 8, she loses $4. Calculate the expected value of her winnings on one die toss. Express your answer as a dollar value to the nearest cent.


What is the value of $\sqrt{49+81}-\sqrt{36-9}$?

Solution: \boxed{\sqrt{130} - 3\sqrt{3}}


In right triangle $ABC$, $\angle B = 90^\circ$, $\sin A = \frac{7}{25}$, and we are asked to find $\sin C$.


A ball is dropped from a height of 800 feet, and it bounces back up to three-quarters of the height it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


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


Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in the base 2 representation. What is the value of $y-x$?


In 2020, the sum of the year's digits is 4. What is the first year after 2020 in which the sum of the digits is 7?


Suppose the function $g(x)$ is defined on the domain $\{x_4, x_5, x_6\}$, and the graph of $y = g(x)$ consists of three points forming a triangle of area $50$. The graph of $y = 3g(3x)$ also consists of just three points. What is the area of the triangle formed by these points on the graph of $y = 3g(3x)$?


How many license plates can be made if they consist of 3 letters followed by 2 digits, where the first digit must be odd, the second digit must be even, and the first letter must be a vowel?


The matrix
\[\begin{pmatrix} \frac{1}{13} & -\frac{12}{13} \\ -\frac{3}{13} & \frac{10}{13} \end{pmatrix}\]
is used to project onto a certain vector $\begin{pmatrix} u \\ v \end{pmatrix}$. Find $\frac{v}{u}$.


The prime numbers are added in order starting with $3$: $3$, $3 + 5$, $3 + 5 + 7$, and so on. How many of the first 10 such sums are also prime?


In a variation of the game Mindmaster, secret codes are created by placing pegs of any of eight different colors into four slots. Colors may be repeated, any slot may remain empty, and no code can be entirely empty. How many secret codes are possible?


How many three-digit whole numbers contain at least one 7, 8, or 9 as digits?


Find the largest prime factor of $11236$.


12 people attend two separate meetings. At the first meeting, 7 people attend and shake hands with each other. At the next meeting, 5 new people attend along with 2 people from the previous meeting, and everyone shakes hands with each other without anyone from the first meeting shaking hands again. How many total handshakes occur at both meetings?


The Happy Valley Kennel now has 3 chickens, 3 dogs, and 4 cats. In how many ways can the 10 animals be placed in a row of 10 cages, such that all of the animals of each type are in adjacent cages?


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


In parallelogram $EFGH$, angle $F$ measures $130^\circ$. What is the number of degrees in the measure of angle $H$? Additionally, calculate the measure of the exterior angle at vertex $H$.

Solution: \boxed{\text{Interior angle } H = 130^\circ, \text{ Exterior angle at } H = 50^\circ}


Let $p(x)$ be a monic quartic polynomial such that $p(1) = 3,$ $p(2) = 6,$ $p(3) = 11,$ and $p(4) = 18.$  Find $p(6).$


A cone has a radius of 2 inches and a height of 8 inches. What is the radius of a sphere that has twice the volume of this cone?


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


Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[
\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 16 & 0 \\ 0 & 16 \end{pmatrix}.
\]
Find the smallest possible value of $|p| + |q| + |r| + |s|.$


Determine the ratio of \(x\) to \(y\) if \(\frac{12x - 5y}{17x - 3y} = \frac{5}{7}\). Express your answer as a common fraction.


Crestwood Elementary School has a three-square league, consisting of nine players, including Amy and Bob. Each day at recess, the nine players split into three three-square games, each with three players in no relevant order. Over the course of a semester, each possible match-up of three players occurs once. How many times did Amy play in the same game as Bob?


A rectangular box has a volume of $4608$ cubic inches and a surface area of $1824$ square inches. The sum of the lengths of its $12$ edges is $216$ inches. What is the volume of the box if its length, width, and height were each increased by two inches?

Solution: \boxed{6656 \text{ cubic inches}}


Given that 10 bahs are equal to 20 rahs, and 12 rahs are equal in value to 24 yahs, how many bahs are equivalent to 1200 yahs?


Find the number of functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) f(x - y) = (f(x) + f(y))^2 - 9x^2 f(y)\]
for all real numbers $x$ and $y$.


Calculate the value of $\frac{10! \cdot 6! \cdot 3!}{9! \cdot 7!}$.


Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined?

Solution: \boxed{8\pi - 16}


A café offers 5 types of bread, 7 types of meat (including turkey and salami), and 4 types of cheese (including mozzarella and Swiss cheese). A sandwich at this café consists of one type of bread, one type of meat, and one type of cheese. Customer Jane does not like to have a turkey/Swiss cheese combination or a salami with any type of bread. How many different sandwiches can Jane order?


Initially, an amoeba is placed in a puddle, and from that day on, each amoeba splits into two amoebas each subsequent day. However, a second amoeba is introduced into the puddle on the third day. Calculate the total number of amoebas at the end of fourteen days.


The prime factorization of 4410 is $2 \times 3^2 \times 5 \times 7^2$. How many of its positive integer factors are perfect cubes?


The digits $1, 2, 3, 4, 5,$ and $6$ can be arranged to form many different 6-digit positive integers with six distinct digits. In how many such integers are the digits $1$ to the left of the digit $2$ and the digit $3$ to the left of the digit $4$? (The digits $1$ and $2$, and $3$ and $4$ do not have to be next to each other.)


A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(3, \frac{7 \pi}{6}, \frac{\pi}{4} \right)$. Find the spherical coordinates of the point with rectangular coordinates $(x,-y,z)$.

Solution: \boxed{\left( 3, \frac{5 \pi}{6}, \frac{\pi}{4} \right)}


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.


The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many odd integer values of $x$ will the value of $12 \star x$ be a positive integer?


The repeating decimal for $\frac{4}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?


What is the 30th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{7}$, $\frac{1}{3}$, and $\frac{1}{11}$?


Calculate the number of non-zero digits to the right of the decimal point for the fraction $\frac{120}{2^4 \cdot 5^9}$.


Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$


Determine the non-zero, real value of \(x\) that satisfies \((6x)^5 = (18x)^4\).

Solution: \boxed{\frac{81}{6} = \frac{27}{2}}


What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $7(5x-3) \equiv 35 \pmod{12}$?


Solution: \boxed{t = 3.6}


Suppose that $5\triangle_3=\triangle4_{12}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?


Given that the diagonals of a rhombus are perpendicular bisectors of each other, find the area of a rhombus with side length $\sqrt{113}$ units and diagonals that differ by 8 units.


What is the least integer greater than $\sqrt{500}$?


A rectangular prism has a volume of 720 cubic units, where $a, b,$ and $c$ are integers and $1 < c < b < a$. Find the maximum possible value of $b$ if $c=3$, while ensuring $a$, $b$, and $c$ are distinct prime factors.


Given a right triangle \(DEF\) where \( \angle D = 45^\circ \) and \(DE = 8\) units, find the area of the triangle.


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

Solution: \boxed{1024 \text{ square centimeters}}


In a weekend volleyball tournament, there are six teams: A, B, C, D, E, and F. On Saturday, Team A plays Team E, Team B plays Team C, and Team D plays Team F. Then on Sunday, the three Saturday winners play round-robin matches (each team plays every other team once) for first, second, and third places, while the three losers play for fourth, fifth, and sixth places. There are no ties in any game. Given that a possible ranking of the teams from first to sixth place at the end of the tournament is AECBDF, determine the total number of possible six-team ranking sequences at the end of the tournament.


If $a * b = a^2 + ab - b^2$, find the value of $5 * (-3)$.


Suppose a right circular cylinder has a radius of $r = 5$ units and a height of $h = 12$ units. If the diagonal across the circular top and the height measures $13$ units, what is the radius of the cylinder?


What is the measure, in degrees, of the acute angle between the hour hand and the minute hand of a 12-hour clock at 8:15?


I randomly pick an integer \( p \) between 1 and 12 inclusive. What is the probability that I choose a \( p \) such that there exists an integer \( q \) for which \( p \) and \( q \) satisfy the equation \( pq - 5p - 3q = 6 \)? Express your answer as a common fraction.

Solution: \boxed{\frac{4}{12} = \frac{1}{3}}


An urn contains $5$ green balls and $5$ blue balls. A second urn contains $20$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.6$. Find $N$.

Solution: \boxed{N = 20}


Let $a$, $b$, and $c$ be the 3 roots of the polynomial $x^3 - 2x + 4 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$.


Solve for $z$ in the following equation: $3 + 2iz = 4 - 5iz$.

Express your answer in standard form.

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


An equilateral triangle is inscribed in a circle, then a circle is inscribed in this triangle, a square is then inscribed in this inner circle, and finally, a circle is inscribed in that square. What is the ratio of the area of the smallest circle to the area of the largest circle?


Given that
1. \(x\) and \(y\) are both integers between 100 and 999, inclusive.
2. \(y\) is the number formed by swapping the hundreds and tens digits of \(x\).
3. \(z = |x - y|\).
How many distinct values of \(z\) are possible?


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

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


During the first year, XYZ's stock price starts at $ \$50 $ and increases by $ 150\% $. During the second year, its stock price goes down by $ 30\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year?


Determine the height of the small cone that was removed to create a frustum from a larger cone, given that the frustum has an altitude of 16 cm, the area of its lower base is 196π sq cm, and the area of its upper base is 49π sq cm.


Billy is hiking in Colorado. He walks eastward five miles, then turns $45$ degrees northward and walks seven miles. How far is he from his starting point? Express your answer in simplest radical form.

Solution: \boxed{AD = \sqrt{74 + 35\sqrt{2}}}


What is the greatest common divisor of $1978$ and $2017$?


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

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


The second hand on a clock is 8 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.


Find the minimum value of $\tan^4 x + \sec^4 x.$


Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{7n-3}{k^n} = 2.\]
Find $k$.

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


All sides of the convex hexagon $ABCDEF$ are of equal length, and $\angle A = \angle B = \angle C = 90^\circ$.  What is the degree measure of $\angle F$?


How many positive multiples of 7 that are less than 2000 end with the digit 5?


The area of the parallelogram generated by the vectors $\mathbf{u}$ and $\mathbf{v}$ is 12. Find the area of the parallelogram generated by the vectors $3\mathbf{u} - 2\mathbf{v}$ and $4\mathbf{u} + \mathbf{v}$.


In the diagram, rectangle $ABCD$ is divided into four identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?

Solution: \boxed{1600 \text{ square centimeters}}


When three standard dice are tossed, the numbers \( a, b, c \) are obtained. Find the probability that
\[ (a-1)(a-6)(b-1)(b-6)(c-1)(c-6) \neq 0. \]


Jason's safe now uses a six-digit combination, with each digit ranging from 1 to 6. Digits from 1 to 3 are odd, and digits from 4 to 6 are even. He remembers that each odd digit in his combination is followed by an even digit, and each even digit is followed by an odd digit. How many possible combinations does Jason need to try?


Consider the modulo \(m\) graph paper for \(m=9\), and the graphs of the congruences $$y \equiv 4x + 3 \pmod{9}$$ and $$y \equiv 7x + 6 \pmod{9}.$$ Determine the sum of the \(x\)-coordinates of the points where these graphs intersect.


Find the sum of the $50^{\mbox{th}}$ and $51^{\mbox{st}}$ terms of the sequence which consists of all those positive integers that are powers of 3 or sums of distinct powers of 3.


Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number, and the tens digit is 2.


Calculate $\frac{4 \cdot 7! + 28 \cdot 6!}{8!}$


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


A box contains 8 white balls and 7 black balls. Seven balls are drawn out of the box at random. What is the probability that all the balls drawn are white?


The United States Postal Service charges an extra postage based on the length-to-height ratio of envelopes. If the length of an envelope, in inches, divided by its height, in inches, is less than $1.5$ or greater than $3.0$, the extra charge is $\$0.15$. For the following envelopes, determine the total extra postage that must be paid:
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
A &7 &6\\
B &8 &2\\
C &7 &7\\
D &13 &4
\end{tabular}


A right cone has a base with a circumference of $24\pi$ inches and an original height of 40 inches. The height of this cone is reduced while the base circumference remains unchanged. The volume of the shorter cone is $432\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fraction.


Let $D$ be the circle with the equation $x^2 + 8x + 20y + 89 = -y^2 - 6x$. Find the value of $c + d + s$ where $(c, d)$ is the center of $D$ and $s$ is its radius.

Solution: \boxed{-17 + 2\sqrt{15}}


For how many integer values of $a$ does the equation $$x^2 + ax + 9a = 0$$ have integer solutions for $x$?


We know the following to be true:

$\bullet$ 1.  \( X \) and \( M \) are integers with \( 1000 < X < 8000 \) and \( M > 1 \);

$\bullet$ 2.  \( X = M \times M^2 \).

What is the value of \( M \) for which \( X \) is a perfect square?


Elina defined a $\emph{smart integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.

Solution: \boxed{\frac{0}{4} = 0}


How many zeros are in the expansion of $(10^{11} - 2)^2$?


The dilation, centered at $1 + 2i$, with a scale factor of 2, takes $3 + 4i$ to which complex number? Also find the image of $-2 + i$ under the same dilation.


Evaluate the following expression:
\[ \frac{x^1 \cdot x^2 \cdot x^3 \cdots x^{20}}{x^2 \cdot x^4 \cdot x^6 \cdots x^{30}} \]
given \( x = 3 \).


The surface area of a sphere is $256\pi\text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of $\pi$.


The function \( g : \mathbb{R} \to \mathbb{R} \) satisfies
\[ g(x) + 2g(1 - x) = 5x^2 \]
for all \( x \). Find \( g(5) \).


If ten apples cost the same as five oranges, and three oranges cost the same as four pears, how many pears can Tyler buy for the price of 20 apples?


Two parabolas are the graphs of the equations \(y = 3x^2 - 4x + 2\) and \(y = 9x^2 + 6x + 2\). Determine all points where they intersect. List the points in order of increasing \(x\)-coordinate, separated by semicolons.

Solution: \boxed{\left(-\frac{5}{3}, 17\right); (0, 2)}


Given rows 2015, 2016, and 2017 of Pascal's triangle, let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 2015th, 2016th, and 2017th rows, respectively, with the leftmost element occurring at $i = 0.$ Compute
\[
\sum_{i = 1}^{2016} \frac{b_i}{c_i} - \sum_{i = 1}^{2015} \frac{a_i}{b_i}.
\]


Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\)


Calculate the value of $$\frac{1}{4}\times8\times\frac{1}{16}\times32\times\frac{1}{64}\times128\times\frac{1}{256}\times512\times\frac{1}{1024}\times2048.$$


If I evaluate $30 \cdot 29 \cdot 28 \cdots 3 \cdot 2 \cdot 1$ and then double the result, how many zeros are there at the end of the number I get?


If two cards from a standard deck, now with 5 Aces, are selected randomly, what is the probability that either two aces or at least 1 king occurs? (The deck still contains 52 cards in total, including 4 kings.)


If $\left|\frac{18}{x} - 6\right| = 3$, find the product of all possible values of $x$. Express your answer as an improper fraction.


How many integers from 0 through 9999, inclusive, do not contain any of the digits 2, 3, 4, 5, 8, or 9?


What is the value of $\frac{10! \cdot 7! \cdot 3!}{9! \cdot 8!}$?


What is the remainder when 2,567,139 is divided by 6?


A new edition of a popular music album is priced at $\$30$. The store offers two discounts: $\$5$ off and $15\%$ off. Calculate the difference in final price between applying the $\$5$ deduction first and then the $15\%$, versus applying the $15\%$ discount first and then subtracting $\$5$. Express the result in cents.

Solution: \boxed{75\text{ cents}}


Consider the ellipse \[4(x+2)^2 + 16y^2 = 64.\]
Let \(C\) be one of the endpoints of its major axis, and let \(D\) be one of the endpoints of its minor axis. Find the distance \(CD.\)


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 24, determine the area of $\triangle ABC$.


In a regular tetrahedron, the midpoints of the edges are connected to form a smaller tetrahedron. Find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron, expressed as a sum of integers $m+n$ where $m/n$ is the simplest form of the ratio.


Two numbers $126$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?


Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert.


A point $(x,y)$ is a distance of 8 units from the $x$-axis and 12 units away from the point $(1,6)$. Furthermore, it is a distance $n$ from the origin. Given $x > 1$, what is $n$?

Solution: \boxed{\sqrt{205 + 2\sqrt{140}}}


If a number $n$ has $4$ divisors, how many divisors does $n^2$ have?


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

Solution: \boxed{[-4, -\frac{3}{2})}


The line \( y = -\frac{5}{3}x + 10 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r, s) \) is on line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?


Michael's school offers 10 different math classes. He must take at least one specific compulsory course but he wants to take a total of 4 math classes this semester. In how many ways can he choose his additional classes?


Suppose functions $g$ and $f$ have the properties that $g(x)=4f^{-1}(x)$ and $f(x)=\frac{36}{x+4}$. For what value of $x$ does $g(x)=16$?


Triangle $PQR$ has side-lengths $PQ = 15, QR = 30,$ and $PR = 22.5$. The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. What is the perimeter of $\triangle PXY?$


Consider a "Modulo $m$ graph paper" where $m=11$. Graph the congruence $$2x \equiv 3y + 1 \pmod{11}$$ and determine the sum of the coordinates $x_0 + y_0$ where $(x_0, 0)$ is the $x$-intercept and $(0, y_0)$ is the $y$-intercept.


A triangular array of numbers has a first row consisting of the even integers $2, 4, 6, \ldots, 100$ in increasing order. Each row below the first has one fewer entry than the row above, and the bottom row has a single entry. Each entry in any subsequent row equals the sum of the two entries diagonally above it in the row immediately above it. Determine the number of entries in the array that are multiples of $53$.


How many positive multiples of 12 are two-digit numbers?


Let \(\omega\) be a nonreal root of \(x^4 = 1.\) Compute
\[(1 - \omega + \omega^3)^4 + (1 + \omega - \omega^3)^4.\]


What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 58\cdot 3x + 58^2$ is a multiple of 53?


Let $x(x+y) = 40$ and $y(x+y) = 90$. Additionally, $x - y = 5$. What is the value of $(x+y)^2$?


How many perfect cubes, divisible by 4, are there between 50 and 1500?


Calculate the value of the infinite product $3^{\frac{1}{2}} \cdot 9^{\frac{1}{4}} \cdot 27^{\frac{1}{8}} \cdot 81^{\frac{1}{16}} \dotsm.$


In a 4 by 4 grid, each of the 16 small squares measures 2 cm by 2 cm and is shaded. Four unshaded shapes, one ellipse, and three circles, are placed on top of the grid. The ellipse has a major axis equal to 4 cm and a minor axis equal to 2 cm, positioned horizontally. The three circles each have radii of 1 cm and are distributed within the grid. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$?


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$ such that the graph of $y=g(x)$ consists of just three points. Suppose those three points form a triangle of area $45$.

The graph of $y = 4g(3x)$ also consists of just three points. What is the area of the triangle formed by those three points?


Suppose $A$, $B$, and $C$ are non-zero distinct digits less than $5$, and suppose we have ${AB_5}+{C_5}={C}0_5$ and ${AB_5}+{BA_5}={CC_5}$. Find the three-digit number ${ABC}$.


In a new configuration, $\overrightarrow{OA}\perp\overrightarrow{OC}$ and $\overrightarrow{OB}\perp\overrightarrow{OD}$. Assume $\angle{AOD}$ is twice $\angle{BOC}$. Determine the measure of $\angle{AOD}$.


A prism has 18 edges. How many faces does the prism have?


Convert $256_{10}$ to base 12. Represent $10$ as $A$, and $11$ as $B$, if necessary.


Evaluate the expression \[(5^{500} + 6^{501})^2 - (5^{500} - 6^{501})^2\] and express it in the form $k \cdot 30^{500}$, where $k$ is a positive integer. What is $k$?


What is the smallest positive integer that can be written in the form $5013m + 111111n$, where $m$ and $n$ are integers?


Consider the diagram below where each grid square is 1 unit by 1 unit:
[asy]
size(100,0);
add(shift(-5,-5)*grid(10,10));
draw((-5,0)--(5,0),linewidth(2));
draw((0,-5)--(0,5),linewidth(2));
label("",(5,0),E);
label("",(0,5),N);
draw((0,-3) -- (3,5), blue, Arrows);
[/asy]
The equation of the line shown can be written as $y=mx+b$. Find the product $mb$.


How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 11?


A box contains 7 white balls and 8 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white or they all are black?


The three positive integers $x,$ $y,$ $z$ satisfy
\[x \times y \times z = 8!.\]
Find the smallest possible value of $x + y + z.$


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, a perfect cube, nor a sixth power? Express your answer as a common fraction.


Compute the sum of the squares of the sines and cosines for every integer degree from \(0^\circ\) to \(45^\circ\):
\[
\sin^2 0^\circ + \sin^2 1^\circ + \dots + \sin^2 45^\circ + \cos^2 0^\circ + \cos^2 1^\circ + \dots + \cos^2 45^\circ.
\]


Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1, 2, 3, 4, 5, 6)$ such that
\[\left(\frac{a_1 + 1}{2}\right)^3 \cdot \left(\frac{a_2 + 2}{2}\right)^3 \cdot \left(\frac{a_3 + 3}{2}\right)^3 \cdot \left(\frac{a_4 + 4}{2}\right)^3 \cdot \left(\frac{a_5 + 5}{2}\right)^3 \cdot \left(\frac{a_6 + 6}{2}\right)^3 > 6!. \]


When \(x\) is divided by each of \(5\), \(7\), and \(9\), remainders of \(4\), \(6\), and \(8\) (respectively) are obtained. What is the smallest possible positive integer value of \(x\)?


What is the coefficient of $a^3b^2$ in $(a+b)^5 \left(c + \frac{1}{c}\right)^8$?


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


John purchased a car for \$8000 to use for his pizza delivery job. He earns \$12 for each pizza delivered, and spends \$4 on gas per delivery. Additionally, John pays a fixed monthly cost of \$200 for the maintenance of the car. How many pizzas does John need to deliver to cover his car purchase and maintenance costs for the first month?


Expand and simplify $-(2x - 4)(2x + 3(2x - 4))$. After expanding, what is the sum of the coefficients in the final expression?


A $3$-digit number is termed geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence with a non-integer rational common ratio. Find the difference between the largest and smallest geometric numbers that start with $8$.


What is the arithmetic mean of the integers from -6 through 7, inclusive? Express your answer as a decimal to the nearest tenth.


In the diagram, triangles $ABC$ and $BDE$ are isosceles. The perimeter of $\triangle BDE$ is $26,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $BE$ is $10.$ What is the length of $AB?$

[asy]
size(7cm);
defaultpen(fontsize(11));
pair b = (0, 0);
pair e = 10 * dir(-30);
pair a = 8 * dir(-140);
pair d = 6 * dir(-90);
draw(a--b--e--d--cycle);
draw(b--d);
label("$y^\circ$", a, 2 * (E + NE));
label("$y^\circ$", b, 2 * (S + SW));
label("$x^\circ$", b, 2 * (S + SE));
label("$x^\circ$", e, 2 * (2 * W));
label("$A$", a, W);
label("$B$", b, N);
label("$E$", e, E);
label("$D$", d, S);
[/asy]


For how many positive integral values of $b$ is it true that $x = 3$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
3x > 4x - 4 \\
2x - b > -3
\end{cases}
$$


The ratio of the areas of two squares is $\frac{75}{98}$. After rationalizing the denominator, what is the simplified form of the ratio of their side lengths, and calculate the sum of all integers involved.


Let $b_n = \frac{5^n - 1}{4}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


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


If the two roots of the quadratic $9x^2 + 5x + k$ are $\frac{-5\pm i\sqrt{371}}{18}$, what is $k$?


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


For $k > 0$, let $J_k = 10\ldots 025$, where there are $k$ zeros between the $1$ and the $2$. Let $M(k)$ be the number of factors of $5$ in the prime factorization of $J_k$. Determine the maximum value of $M(k)$.


Solution: \boxed{\frac{1}{8} \log_2 5}


If a school bus leaves school with 60 students on board, and one-third of the students get off the bus at each of the first four stops, how many students remain on the bus after the fourth stop?


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


On a $12$-hour clock, an elapsed time of five hours looks the same as an elapsed time of $25$ hours. Because of this, we can say that five hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $5$ hours and is "clock equivalent'' to its square number of hours?


Calculate how many distinct four-digit positive integers have a product of their digits equal to 18.


Alice's favorite number is between 50 and 100. It is a multiple of 11, but not a multiple of 2. The sum of its digits is a multiple of 5. What is Alice's favorite number?


Find the ordered pair $(a, b)$ of positive integers, with $a < b,$ for which
\[\sqrt{1 + \sqrt{45 + 20 \sqrt{5}}} = \sqrt{a} + \sqrt{b}.\]

Solution: \boxed{(1, 5)}


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


If $a$ and $b$ are nonzero real numbers such that $a^2 + b^2 = 9ab$, find the value of $\left|\frac{a+b}{a-b}\right|$.


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


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

Solution: \boxed{1440 \text{ cubic inches}}


A cube with edges of length 2 cm has a dot marked in the center of one of its faces. The cube is sitting on a flat surface and rolls over its edges to make exactly four quarter-turns (90 degrees each), without slipping, until the dot is back on the same face, initially facing up. Calculate the length, in centimeters, of the path followed by the dot. The path's length is expressed as $k\pi$, where $k$ is a constant. Determine the value of $k$.

Solution: \boxed{k = 2}


**Container I** holds 5 red balls and 5 green balls. **Containers II** holds 3 red balls and 3 green balls, and **Container III** holds 4 red balls and 2 green balls. **Container IV** holds 6 red balls and 6 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction.


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

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


When the base-10 integers 400, 1600, and 3200 are expressed in base 2, how many more digits does the number with the most digits have compared to the one with the least digits?


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

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


Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in the resulting binary representation. Calculate the value of $y - x$.


Consider the following equations of lines:
\begin{align*}
y &= 2x + 4 \\
y &= -x + 1 \\
2y &= 6x + 10 \\
y &= -4x + 7 \\
3y &= 3x - 9 \\
\end{align*}
Determine how many pairs of these lines are either parallel or perpendicular to each other.


When we say Ray climbs the stairs $m$ at a time, it means he starts on the ground (step $0$), then jumps to step $m$, then to step $2m$, and so on until the number of steps to the top is less than $m$. Ray climbs a flight of stairs of $n$ steps in two ways. When he does it $6$ steps at a time, there are $5$ steps left at the top. When he does it $7$ steps at a time, there are $4$ steps left at the top. What is the smallest possible value of $n$ that is greater than $20$?


A department store offers two promotions. Promotion A says, "Buy one pair of shoes at full price, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $15 off the second pair." Michael wants to buy two pairs of shoes where the first pair costs $50 and the second pair costs $40. He can only use one of the promotions, A or B. Michael decides to use the promotion that will save him the most money. How many dollars does Michael save by picking one promotion over the other?

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


Solution: \boxed{1280 \cdot 16^7}


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


A cruise ship has three types of cabins: Deluxe (30 cabins), Premium (20% of the total number of cabins), and Standard ($\frac{3}{4}$ of the total number of cabins). How many cabins does the cruise ship have in total?


Sam has 4 red lava lamps and 4 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 4 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on?


Determine the coefficient $b$ in the polynomial $Q(x) = x^3 + dx^2 + bx + e$, knowing that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph $y= Q(x)$ is 6, what is the value of $b$?


Suppose $a$ and $b$ are positive integers such that $\gcd(a, b)$ is divisible by exactly $10$ distinct primes and $\mathop{\text{lcm}}[a, b]$ is divisible by exactly $25$ distinct primes.

If $b$ has fewer distinct prime factors than $a$, then $b$ has at most how many distinct prime factors?


James is preparing for a duathlon and decides to skip the swimming segment. He bikes 30 miles at a speed of $3x + 1$ mph, takes a 10-minute break to switch gear, and then runs 10 miles at a speed of $x + 2$ mph. The entire session, including the break, lasts 180 minutes. Determine James's running speed, to the nearest hundredth of a mile per hour.

Solution: \boxed{7.99 \text{ mph}}


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


Anna's seven assignment scores are 86, 90, 92, 98, 87, 95, and 93. What are the arithmetic mean and median of these seven scores?


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, follows the rule: at each vertex, it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. We define $p = \frac{n}{6561}$ as the probability that the bug is at vertex $A$ after it has crawled exactly 8 meters. Find the value of $n$.


A square pyramid with a base edge of 40 inches and an altitude of 20 inches has a smaller pyramid cut away from its apex. This smaller pyramid has an altitude which is one-third of the altitude of the original pyramid. What is the volume of the remaining frustum as a fractional part of the volume of the original pyramid?


Find the radius of the circle with equation $4x^2 - 8x + 4y^2 + 16y + 20 = 0.$


Compute \[\lfloor 1 \rfloor + \lfloor 1.5 \rfloor + \lfloor 2.0 \rfloor + \lfloor 2.5 \rfloor + \dots + \lfloor 104.5 \rfloor + \lfloor 105 \rfloor,\] where the arguments of the floor functions are in an arithmetic progression with a common difference of $0.5$ and the first term is $1$.


An ellipse has foci at \( F_1 = (0, 3) \) and \( F_2 = (4, 0) \). It intersects the \(x\)-axis at two points, one of which is the origin. Find the other point of intersection.

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


What is the remainder when $2431 \cdot 1587$ is divided by $800$?


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)}


Let $a, b, c, d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 \leq 4.\]
Find the maximum value of $e^a + e^b + e^c + e^d$.


Call a positive integer $n$ $15$-pretty if $n$ has exactly $15$ positive divisors and $n$ is divisible by $15$. For example, $45$ is $15$-pretty. Let $T$ be the sum of all positive integers less than $3000$ that are $15$-pretty. Find $\tfrac{T}{15}$.


In how many ways can 8 distinct beads be placed on a necklace, considering that the necklace can be flipped and also has a pendant attached at a fixed position (which breaks the rotational symmetry of the arrangement)?


During the first part of a marathon run, John maintains a speed of 8 miles per hour. As he becomes fatigued, his speed decreases to 6 miles per hour, and during the last stretch, he pushes through at a speed of 5 miles per hour. If John completes a total of 95 miles in 15 hours, and the time spent at 8 mph and 6 mph are the same, how many hours did John run at each speed?

Solution: \boxed{x = 5}


Find all solutions to the equation:
\[\sqrt[3]{18x - 2} + \sqrt[3]{16x + 2} = 6 \sqrt[3]{x}.\]


A triangular region is enclosed by the lines with equations $y = \frac{2}{3} x + 4$, $y = -3x + 9$, and $y = 2$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth.


Let \(x, y, z\) be positive real numbers. Find the maximum value of
\[
\frac{(x+y+z)^2}{x^2+y^2+z^2}.
\]


Ella needs to buy 20 identical books and has a budget of $\$200$. There is a $\$5$ entry fee to access the book fair. Each book has the same cost in whole dollars. If a $8\%$ sales tax is applied to the price of each book, what is the highest possible price per book that Ella can afford?


Parallelogram $PQRS$ has vertices $P(5,5)$, $Q(-1, -5)$, $R(-11, -5)$, and $S(-5,5)$. If a point is selected at random within the parallelogram, what is the probability that this point is not below the $x$-axis?


Let \(B\) be a point on the x-axis at \(B(x, 0)\) and \(Q\) at \(Q(0, 15)\) on the y-axis, where \(x > 0\). Point \(C\) is on the y-axis at \(C(0, p)\) such that \(0 < p < 15\). Find an expression for the area of triangle \(COB\) in terms of \(p\) and \(x\). Illustrate the problem as shown:

[asy]
size(6cm);
defaultpen(fontsize(10));
pair o = (0, 0);
pair q = (0, 15);
pair b = (18, 0);
pair c = (0, 9);

draw((-2, 0)--(20, 0), Arrow);
draw((0, -2)--(0, 20), Arrow);
draw(q--b--o--cycle);
draw(c--b);

label("$Q(0, 15)$", q, W);
label("$B(x, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (20, 0), E);
label("$y$", (0, 20), N);
label("$C(0, p)$", c, W);
[/asy]


In a press conference before a championship game, ten players from four teams will be taking questions. The teams are as follows: three Celtics, three Lakers, two Warriors, and two Nuggets. If teammates insist on sitting together and one specific Warrior must sit at the end of the row on the left, how many ways can the ten players be seated in a row?


Let $r(x)$ have a domain $\{-2, -1, 0, 1, 3\}$ and range $\{-1, 0, 1, 3, 5\}$. Define $s(x)$ for $x \geq 1$ by $s(x) = 2x + 1$. What is the sum of all possible values of $s(r(x))$?


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


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


When the graph of $y = 3x^2 + 2x + 1$ is shifted five units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.


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

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


Triangle $GHI$ is similar to triangle $JKL$. If $GH=8$, $HI=16$, and $KL=24$ units, what is the length of segment $JK$?


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


Consider two arithmetic sequences, one starting from $5$ with a common difference of $8$, and another starting from $3$ with a common difference of $9$. Determine the largest number common to both sequences from $1$ to $150$.


Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction.


Determine how many integers satisfy all of the following inequalities:
\[
\begin{array}{cccc}
(1) & -5x & \geq & 3x + 12 \\
(2) & -3x & \leq & 15 \\
(3) & -6x & \geq & x + 26
\end{array}
\]


A cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\%$ instead of 0.1000.)


Let $x$, $y$, and $z$ be positive real numbers. Determine the smallest possible value of $(x+y+z)\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{y+z}\right)$.


In the diagram, the larger circle is divided into four equal sectors and the smaller circle into two equal halves. $O$ is the center of the larger circle. The area of the larger circle is $100\pi.$ What is the total area of the shaded regions? [asy]
size(100);
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fill(Arc((0,0),3,0,90)--cycle,mediumgray);
fill(Arc((0,0),3,180,270)--cycle,mediumgray);
fill(Arc((0,1.5),1.5,90,270)--cycle,mediumgray);
draw(Circle((0,0),3));
draw(Circle((0,1.5),1.5));

dot((0,0)); label("$O$",(0,0),N);
[/asy]


What is the arithmetic mean of the integers from -3 through 6, inclusive? Express your answer as a decimal to the nearest tenth.


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


What is the quotient when \(9x^4 + 27x^3 - 8x^2 + 8x + 5\) is divided by \(3x + 4\)?

Solution: \boxed{3x^3 + 5x^2 - \frac{28}{3}x + \frac{136}{9}}


There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4$ such that
\[\sin^4 \theta = b_1 \sin \theta + b_2 \sin 2 \theta + b_3 \sin 3 \theta + b_4 \sin 4 \theta\]for all angles $\theta.$  Find $b_1^2 + b_2^2 + b_3^2 + b_4^2.$


The letters of the alphabet are each assigned a random integer value, and $G=15$. The value of a word comes from the sum of its letters' values. If $GAME$ is 50 points, $MEGA$ is 55 points, and $AGE$ is 40 points, what is the value of $M$?


Ms. Hamilton's eighth-grade class is organizing a large school basketball tournament with 32 teams divided into 8 groups of 4 teams each. Each group will play a round-robin, where every team plays every other team in their group. The top two teams from each group will advance to the knockout rounds, where the losing team in each match is eliminated. How many total games will be played to determine the winner?


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

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


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

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


For all composite integers $n$, what is the largest integer that always divides into the sum of $n$ and the square of $n$?


Suppose we flip six coins: a penny, a nickel, a dime, a quarter, a half-dollar, and a dollar coin. What is the probability that the penny and dime come up differently, but the nickel and quarter come up the same?


Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number.

Solution: \boxed{3 \frac{1}{5}}


Express $\frac{0.\overline{888}}{1.\overline{222}}$ as a common fraction.


Suppose in a right triangle $DEF$ with the right angle at $E$ and $DF = \sqrt{85}$, $DE = 7$. A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF$. Let $Q$ be the point where the circle and side $DF$ meet. Determine the length of $FQ$.


A right triangle with legs $15$ and $12$ is similar to another right triangle with corresponding legs $y$ and $10$, respectively. What is the value of $y$? Express your answer as a decimal to the nearest hundredth.


Margaret began collecting postcards. She collected 10 postcards on the first day and each subsequent day, she collected 12 more postcards than the previous day. If she collected postcards for 7 consecutive days, what was the average number of postcards collected per day?


What is the smallest prime whose digits sum to $23$?


Find the maximum number of real roots for the polynomial
\[x^n + x^{n-1} + \dots + x^2 + x + c = 0,\] where $n$ is a positive integer and $c$ is a real constant different from 1.


Let $c$ and $d$ be real numbers. One of the roots of $x^3 + cx + d = 0$ is $2 + i \sqrt{2}$. Find $c + d$.


Determine the value of \(a\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = k \] is a non-degenerate ellipse if and only if \( k > a \).


A pyramid with a volume of 120 cubic inches has a rectangular base. If the length of the base is quadrupled, the width quadrupled, and the height doubled, what is the new volume of the pyramid, in cubic inches?


Mary wants to purchase two large pizzas, one medium pizza, and four drinks. The drinks cost $p$ dollars each, the medium pizza costs two and a half times as much as one drink, and each large pizza costs four times as much as one drink. If Mary started with $50$ dollars, how much money would she have left after making her purchases? Give your answer as an expression of $p$.

Solution: \boxed{50 - 14.5p}


During the first year, XYZ's stock price starts at $ \$50 $ and increases by $ 200\% $. During the second year, its stock price decreases by $ 50\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year?


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

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


If the six solutions of $x^6 = -729$ are written in the form $a+bi$, where $a$ and $b$ are real, find the product of those solutions with $a > 0$.


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


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


The quadratic $-8x^2 + 16x + 320$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?


If $x^2 + y^2 = 2$, what is the largest possible value of $|x| + |y|$?


Solve for $x$: $x = \displaystyle\frac{\sqrt{7^2+24^2}}{\sqrt{49+16}}$. Express as a common fraction.


Suppose $c$ and $d$ are the roots of the equation \(x^2 - 5x + 3 = 0\). Consider \(c + \frac{2}{d}\) and \(d + \frac{2}{c}\) as roots of another quadratic equation \(x^2 - rx + s = 0\). Find the value of \(s\).

Solution: \boxed{s = \frac{25}{3}}


Solution: \boxed{\left(\frac{29}{32}\right)^{\frac{1}{7}}}


I now have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?


Two animals, a koala and a kangaroo, absorb different percentages of the fiber they eat. The koala absorbs 30% of the fiber it eats and absorbs 12 ounces of fiber in one day. The kangaroo absorbs 40% of the fiber it eats and absorbs 16 ounces of fiber in the same day. How many ounces of fiber did each animal eat that day?


Compute: $12 \cdot \frac{1}{26} \cdot 52 \cdot 4.$


A cone is formed by aligning the two straight sides of a 240-degree sector of a circle with radius 20. What is the result when the volume of the cone is divided by $\pi$?


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


In parallelogram $ABCD$, the area is $50$ square units. Diagonal $AC$ bisects $\angle BAD$ and $\angle BCD$. Point $M$ is the midpoint of diagonal $AC$. Calculate the area of triangle $ADM$.


If $x$, $y$ are nonzero real numbers such that $x^2 + y^2 = 18xy$, find the value of $\left|\frac{x+y}{x-y}\right|$.


Consider all polynomials of the form
\[x^{10} + b_9 x^9 + b_8 x^8 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \(b_i \in \{0,1\}\) for all \(0 \le i \le 9\). Find the number of such polynomials that have exactly three different integer roots: -1, 0, and 1.


For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?


What is the base ten equivalent of $76543_7$?


Find the sum of the distinct prime factors of $7^7 - 7^3$.


What is the area, in square units, of the region enclosed by the lines \(y = 2x - 5\), \(y = -3x + 18\), and the \(y\)-axis?


The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ -4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{9}{5} \\ -\frac{12}{5} \end{pmatrix}\] lie on a line. Find the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = \frac{3}{4} x - \frac{15}{4}}


In the diagram, rectangle $ABCD$ is divided into four identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?


Two real numbers $x$ and $y$ satisfy $x - y = 5$ and $x^3 - y^3 = 35$. Compute $xy$.


If two numbers are randomly chosen without replacement from $\{2, 3, 5, 6, 9\}$, what is the probability that their product will be a multiple of 12? Express your answer as a common fraction.


How many more digits does the base-4 representation of $1600_{10}$ have than the base-7 representation of $1600_{10}$?


Let $Q$ represent the product of all prime numbers less than or equal to $31$. Consider the sequence of terms given by $Q + m$ for each integer $m$ from $2$ to $32$. Count the number of prime numbers, $M$, that appear in this sequence.


Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \frac{9}{2} = 0$ is true for all values of $z$.


Calculate the greatest common divisor (GCD) of $7!$ and $\frac{11!}{4!}$. Express your answer as an integer.


Find the largest constant $m,$ such that for any positive real numbers $a,$ $b,$ $c,$ $d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > m.\]

Solution: \boxed{m = 2}


Let \( a \) and \( b \) be real numbers such that \( 4a + 3b \leq 10 \) and \( 3a + 6b \leq 12 \). Find the largest possible value of \( a + b \).


Points \(A\), \(B\), \(C\), and \(D\) are located on line segment \(\overline{AB}\) such that \(AB = 4AD = 3BC\). If a point is selected at random on \(\overline{AB}\), what is the probability that it is located between \(C\) and \(D\)?


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ f(x)f(y) - f(xy) = x^2 + y^2 \]
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 \( n \times s \).


The equation \(x^2 + 16x = 100\) has two solutions. The positive solution has the form \(\sqrt{c} - d\) for positive natural numbers \(c\) and \(d\). What is \(c+d\)?


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


Let  \[ g(x) =
\begin{cases}
-2x^2 & \text{if } x \geq 0,\\
x+5 & \text{if } x <0.
\end{cases}
\]Compute $g(g(g(g(g(2)))))$.


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


Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-15,225),$ $(-1,1),$ and $(14,196),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the intersection points.


The function \( g(x) \) satisfies
\[
g(2^x) + xg(2^{-x}) + x = 1
\]
for all real numbers \( x \). Find \( g(2) \).


What is the units digit of the sum of the nine terms of the sequence $1! + 2, \, 2! + 3, \, 3! + 4, \, ..., \, 8! + 9, \, 9! + 10$?


A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6, 2)$, and $(0,2)$. What is the probability that $x < y$ and $x + y < 2$?


A teacher wants to arrange 5 copies of different editions of Introduction to Geometry and 2 copies of Introduction to Number Theory on a bookshelf. How many different ways can he arrange these books?


Compute the product $1213_4 \times 21_4$ in base 4.


Suppose in a right-angled triangle PQR, point Q is the right angle, $\sin R = \frac{3}{5}$. PQ is 9 units long. What is the length of PR?


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


For how many integers $n$ between 1 and 200 is the greatest common divisor of 77 and $n$ equal to 7?


Find the largest value of $n$ such that $6x^2 + nx + 144$ can be factored as the product of two linear factors with integer coefficients.


How many integers (positive and negative) are divisors of 8?


The median of a set of consecutive odd integers is 157. If the greatest integer in the set is 169, what is the smallest integer in the set?


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


What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{3}{7} < \frac{n}{n + k} < \frac{8}{19}$?


If Ella rolls a standard twenty-sided die until she rolls the same number on consecutive rolls, what is the probability that her 15th roll is her last roll?

Solution: \boxed{P = \frac{19^{13}}{20^{14}}}


At the end of a game, each of the six members of a basketball team shakes hands with each of the six members of the opposing team, and all of the players shake hands with the three referees. Additionally, each team also has one coach who shakes hands with all players from both teams and the three referees. How many handshakes occur?


In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 48. What is the perimeter of the shaded region, if each arc forming the shaded region now subtends an angle of 90° at the center of its respective circle?


Both $a$ and $b$ are positive integers, and $b > 1$. When $a^b$ is the greatest possible value less than 500, and additionally, $a + b$ is even, what is the sum of $a$ and $b$?


What is the sum of all the odd integers between $200$ and $400$ that are multiples of $5$?


If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $70^\circ$, and the angle between $\mathbf{b}$ and $\mathbf{c}$ is $50^\circ$, what is the angle between the vectors $-\mathbf{a}$ and $\mathbf{c}$?


How many integers between 400 and 600 have the sum of their digits equal to 17?


Sam adds all the positive integers from 1 to 100. Alex does a comparable task, but he rounds each integer to its nearest multiple of 20 before summing. Calculate the positive difference between Sam’s sum and Alex’s sum.


Determine the greatest common divisor of 15225, 20335, and 35475.


Calculate the sum of the coefficients of the terms which contain a nonzero power of $y$ when the product $(5x+3y+2)(2x+5y+3)$ is expanded.


A certain integer has $4$ digits when written in base $9$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?


Let \(0 \le x, y, z, w \le 1.\) Find the possible values of the expression
\[
\sqrt{x^2 + (1 - y)^2} + \sqrt{y^2 + (1 - z)^2} + \sqrt{z^2 + (1 - w)^2} + \sqrt{w^2 + (1 - x)^2}.
\]

Solution: \boxed{[2\sqrt{2}, 4]}


In convex quadrilateral $ABCD$, $\angle A = \angle B = \angle C$, $AB=CD=150$, and $AD \ne BC$.  The perimeter of $ABCD$ is 580. Find $\cos A$.


Points $P$ and $Q$ are midpoints of two adjacent sides of a rectangle. What fraction of the interior of the rectangle is shaded? Express your answer as a common fraction.

[asy]
filldraw((0,0)--(4,0)--(4,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(2,2)--(4,2)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
label("Q",(2,2),N);
[/asy]


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


Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*}
x + yz &= 9, \\
y + xz &= 13, \\
z + xy &= 13.
\end{align*}
be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$ Find $x_1 + x_2 + \dots + x_n.$


The function \( f(x) \) satisfies
\[ f(2^x) + (x+1)f(2^{-x}) = x \]
for all real numbers \( x \). Find \( f(2) \).


Let $r_1, r_2,$ and $r_3$ be the roots of the polynomial
\[ x^3 - 4x^2 + 5 = 0. \]
Find the monic polynomial, in $x,$ whose roots are $3r_1, 3r_2,$ and $3r_3.$

Solution: \boxed{x^3 - 12x^2 + 135}


A circle is tangent to the lines $3x - 4y = 40$ and $3x - 4y = 0$. The center of the circle lies on the line $x - 2y = 0$. Find the center of the circle.

Solution: \boxed{(20, 10)}


Suppose we have two equations: $2020a + 2024b = 2040$ and $2022a + 2026b = 2044$. What is the value of $a - b$?


Let \( f(x) = \frac{x + 8}{x} \). The sequence \( (f_n) \) of functions is defined by \( f_1 = f \) and
\[ f_n = f \circ f_{n - 1} \]
for all \( n \geq 2 \). For example,
\[ f_2(x) = f(f(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
Let \( S \) be the set of all real numbers \( x \) such that
\[ f_n(x) = x \]
for some positive integer \( n \). Find the number of elements in \( S \).


Let
\[
\mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ -\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix} \renewcommand{\arraystretch}{1}
.\]
Compute $\mathbf{B}^{2018}$.

Solution: \boxed{\begin{pmatrix} -\frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ -\frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \end{pmatrix}}


Let $\mathbf{B} =\begin{pmatrix} 0 & 1 \\ 4 & 5 \end{pmatrix}.$ Determine if there exist scalars $r$ and $s$ such that
\[
\mathbf{B}^4 = r \mathbf{B} + s \mathbf{I}.
\]
If so, find the ordered pair $(r,s)$.

Solution: \boxed{(165, 116)}


A prism has 36 edges. How many faces does the prism have?


If \( n = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7 \), how many even positive factors does \( n \) have?


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


A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


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.


A line passes through the vectors \(\mathbf{a}\) and \(\mathbf{b}\). For some value of \(k\), the vector
\[k \mathbf{a} + \frac{2}{5} \mathbf{b}\]
also lies on the same line. Find \(k\).


Calculate the value of $(a^3+b^3+c^3)\div(a^2-ab+b^2+c^2)$ when $a = 7$, $b = 5$, and $c = -2$.


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


Elizabeth has two numbers, $c$ and $d$. When she divides $c$ by 60, she gets a remainder of 47. When she divides $d$ by 42, she gets a remainder of 17. What remainder does she get when she divides $c+d$ by 21?


Find the residue of $198 \cdot 6 - 16 \cdot 8^2 + 5 \pmod{16}$.


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


Calculate the value of $2^4 \times 3^2 \times 5^2 \times 11$.


Simplify
\[
\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.
\]

Solution: \boxed{2 \csc x}


Determine the value of $b$ such that the remainder is constant when the polynomial $12x^3 - 9x^2 + bx + 8$ is divided by $3x^2 - 4x + 2$.


What is the greatest common divisor of $8164$ and $2937$?


Place each of the digits 5, 6, 7, and 8 in exactly one square to make the smallest possible product.


Let $n$ be a positive integer. If $b \equiv (5^{2n} + 6)^{-1} \pmod{11}$, what is the remainder when $b$ is divided by 11?

Solution: \boxed{b = 8}


Determine the value of $s$ for which the vector
\[\bold{v} = \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix} + s \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} 5 \\ 5 \\ 6 \end{pmatrix}.\]


Consider a square $EFGH$ with sides of length 8, and $N$ is the midpoint of $\overline{GH}$. A circle with radius 4 and center $N$ intersects a circle with radius 8 and center $E$ at points $Q$ and $H$. What is the distance from $Q$ to $\overline{EF}$?


On a balance scale, $4$ green balls balance $8$ blue balls, $3$ yellow balls balance $6$ blue balls, and $2$ blue balls balance $3$ white balls. How many blue balls are needed to balance $3$ green, $4$ yellow and $3$ white balls?


Two numbers are randomly selected from the set $T = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. What number $n$ should be removed to increase the probability that their sum is 10?


Let $x = (2 + \sqrt{2})^6,$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[
x(1 - f).
\]


Determine the number of roots of unity, where a root of unity is defined as a solution to $z^4 = 1$, that are also roots of the quadratic equation $z^2 + pz + q = 0$, for some integers $p$ and $q$.


Given that
\[
\frac{\sin^6 \theta}{\sin^3 \phi} + \frac{\cos^6 \theta}{\cos^3 \phi} = 1,
\]
find the sum of all possible values of
\[
\frac{\sin^6 \phi}{\sin^3 \theta} + \frac{\cos^6 \phi}{\cos^3 \theta}.
\]


Ed and Sue bike at equal and constant rates. Additionally, they jog and swim at equal and constant rates respectively. Ed covers 90 kilometers after biking for 1 hour, jogging for 5 hours, and swimming for 4 hours, while Sue covers 115 kilometers after jogging for 1 hour, swimming for 5 hours, and biking for 4 hours. Their biking, jogging, and swimming rates are whole numbers of kilometers per hour. What is the sum of the squares of Ed's biking, jogging, and swimming rates?


Suppose
\[\frac{1}{x^3 + 2x^2 - 17x - 30} = \frac{A}{x-5} + \frac{B}{x+2} + \frac{C}{(x+2)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?


Determine how many "double prime dates" occurred in 2007, where a "double prime date" is a date where both the day and the month numbers are prime, and additionally, the sum of the day and the month is also a prime number.


How many four-digit positive integers exist with the hundreds digit as $3$?


How many diagonals within a regular nine-sided polygon span an odd number of vertices between their endpoints?


Find the next smallest abundant number after 12.


What is the sum of all four-digit integers from 1000 to 5000, where each number increases by 4 from the previous one?


If $F\,825\,G72$ is divisible by $9$, where $F$ and $G$ each represent a single digit, what is the sum of all possible values of the sum $F+G$?


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.


Express as a common fraction: $$\frac{\frac{3}{7} + \frac{5}{8}}{\frac{5}{12} + \frac{2}{9}}.$$


A triangle has vertices at $(-2, 3), (7, -1), (4, 6)$. Calculate the area of this triangle.


Compute the product of $0.\overline{456}$ and $7$, and write your result as a fraction in simplified form.


A traffic light runs repeatedly through the cycle: green for 40 seconds, yellow for 5 seconds, and red for 45 seconds. Mary selects a random four-second interval to watch the light. What is the probability that the color changes while she watches?


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


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}


A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form a square, and the piece of length $b$ is bent to form a circle. The perimeter of the square is equal to the circumference of the circle. What is $\frac{a}{b}$?


If $3x - 2y = 7$ and $x + 3y = 8$, what is the value of $x$?


Given the polynomial:
\[ax^4 + bx^3 + cx^2 + dx + e = 0,\]
if the coefficients are all integers and the roots are \(1\), \(-3\), \(7\), and \(-2/5\), and \(e\) is a positive integer, find its smallest possible value.


Jane's quiz scores in her math class were 82, 90, 88, 95, and 91. After reviewing her performance, her teacher decided to give an additional 2 points on each quiz as extra credit. Calculate her mean quiz score after the extra credit is added.


What is the base-10 integer 803 when expressed in base 7?


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $15$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


What is the remainder when 987,654 is divided by 6?


What is the coefficient of $x^7$ in the expansion of $(3x + 2)^{10}$?


The sum of five positive integers that form an arithmetic sequence is 95. Of all such possible sequences, what is the smallest possible second term?


Given $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4$, find the value of the determinant
\[\begin{vmatrix} a & 5a + 3b \\ c & 5c + 3d \end{vmatrix}.\]


A $180\times 270\times 360$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. Determine how many $1\times 1\times 1$ cubes the internal diagonal of this solid passes through.


Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=5$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form.

Solution: \boxed{6 + 6 = 12}


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

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


What fraction of $\frac{3}{4}$ is $\frac{1}{5}$?


Given that $b$ is an odd multiple of $1187$, find the greatest common divisor of $3b^2 + 34b + 76$ and $b + 16$.


The remainders when three positive integers are divided by 6, 7, and 8 are 2, 3, and 4 respectively. Find the remainder when their product is divided by 9.


Find the quotient and remainder when $x^5 - 25x^3 + 9x^2 - 17x + 12$ is divided by $x - 3$.

Solution: \boxed{x^4 + 3x^3 - 16x^2 - 39x - 134}


What is the largest three-digit multiple of 8 whose digits' sum is 24?


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


A circular pie with a diameter of $20\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Determine the square of the length of the longest line segment that can be drawn in one of these pieces.


Define $a \Delta b = a^3 - b^2$. What is the value of $(3^{5 \Delta 14}) \Delta (4^{4 \Delta 6})$?


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


Evaluate
\[
\begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \end{vmatrix}.
\]

Solution: \boxed{x^3 + y^3 + z^3 - 3xyz}


The denominator of a fraction is 4 more than 4 times the numerator. If the fraction is equivalent to $3/7$, what is the numerator of the fraction?


Suppose \( g(x) \) is a rational function such that \( 4g\left(\frac{1}{x}\right) + \frac{3g(x)}{x} = x^3 \) for \( x \neq 0 \). Find \( g(-3) \).


Compute the smallest base-10 positive integer greater than 6 that is a palindrome when written in both base 3 and base 5.


Consider the prime numbers starting from $3$: $3$, $3 + 5$, $3 + 5 + 7$, and so on. Determine how many of the first 15 such sums are prime and greater than 50.


The Grunters play the Screamers 6 times. The Grunters are a better team and have a 60% likelihood of winning any given game. What is the probability that the Grunters win exactly 4 out of the 6 games? Express your answer as a common fraction.


Find the integers $x$ for which $x^2 < 3x$, where $x$ is between 1 and 10 inclusive.

Solution: \boxed{1 \text{ and } 2}


In triangle \(XYZ\), sides are given by \(x = 8\), \(y = 11\), and \(z = 5\). Let \(J\) be the incenter of triangle \(XYZ\).

Determine the position vector \(\overrightarrow{J}\) in terms of \(\overrightarrow{X}\), \(\overrightarrow{Y}\), and \(\overrightarrow{Z}\), such that \(\overrightarrow{J} = p \overrightarrow{X} + q \overrightarrow{Y} + r \overrightarrow{Z}\), where \(p\), \(q\), and \(r\) are constants with \(p + q + r = 1\). Calculate the ordered triple \((p, q, r)\).

Solution: \boxed{\left(\frac{1}{3}, \frac{11}{24}, \frac{5}{24}\right)}


In right triangle $GHI$, we have $\angle G = 30^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.


Simplify the expression \( i^0 + i^1 + \cdots + i^{2012} \).


Suppose that $a$ and $b$ are positive integers such that $a-b=8$ and $\text{gcd}\left(\frac{a^4+b^4}{a+b}, ab\right) = 16$. Find the smallest possible value of $b$.


How many integers between 400 and 600 have the sum of their digits equal to 17?


Let $x, y, z$ be real numbers such that
\[
x + y + z = 5,
\]
\[
x^2 + y^2 + z^2 = 11.
\]
Find the smallest and largest possible values of $x$, and compute their sum.


Calculate the area of the region defined by the equation $x^2+y^2 + 8x - 6y + 17 = 0$.


Let $x$ and $y$ be positive real numbers such that $10x + 15y = 150$. Find the maximum value of $xy$ under the condition that $x^2 + y^2 \leq 100$.


How many positive four-digit integers of the form $\_\_90$ are divisible by 90?


Fifteen students are taking both Geometry and History. There are 30 students taking Geometry. There are 15 students who are taking History only. Additionally, 8 students are taking both Geometry and Science, with 10 students taking Science only. How many students are taking Geometry or History or Science but not a combination of these subjects?


What is the arithmetic mean of the integers from -6 through 7, inclusive? Express your answer as a decimal to the nearest tenth.


Determine the least positive number $t$ such that the sequence:
\[\arccos (\cos \alpha), \ \arccos (\cos 3 \alpha), \ \arccos (\cos 5 \alpha), \ \arccos (\cos t \alpha)\]
is an arithmetic progression for some $\alpha$ where $0 < \alpha < \frac{\pi}{2}$.


Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, Daryl has five times as many marbles as Caden, and Emma has twice the number of marbles Daryl has. Compute $a$ if together they all have 212 marbles.


How many factors of \(108000 = 2^5 \cdot 3^3 \cdot 5^3\) are perfect squares?


How many positive three-digit integers less than 700 have at least two digits that are the same and none of the digits can be zero?


Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$?


Find a positive integer that is divisible by 20 and whose cube root is a number between 9 and 9.1.


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


Two circles are drawn in an 18-inch by 20-inch rectangle. Each circle has a diameter of 8 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{2\sqrt{61} \text{ inches}}


A line contains the points $(2, 9)$, $(10, m)$, and $(25, 4)$. What is the value of $m$?


A new chart shows the air distances in miles between a different set of world cities. If two different cities from this chart are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Also, what is the probability that one of the cities in the pair is Tokyo? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
& Tokyo & Cairo & Sydney & Toronto & Paris \\ \hline
Tokyo & & 5900 & 4800 & 6450 & 6040 \\ \hline
Cairo & 5900 & & 8650 & 5260 & 2140 \\ \hline
Sydney & 4800 & 8650 & & 9440 & 10600 \\ \hline
Toronto & 6450 & 5260 & 9440 & & 3700 \\ \hline
Paris & 6040 & 2140 & 10600 & 3700 & \\ \hline
\end{tabular}


Two cylindrical tanks have the same volume. The height of one tank is four times the height of the other. If the radius of the taller tank is 10 units, find the radius of the shorter tank. Express your answer in simplest radical form.


If \(0 \le p \le 1\) and \(0 \le q \le 1\), define \(H(p, q)\) by
\[
  H(p, q) = -3pq + 4p(1-q) + 5(1-p)q - 6(1-p)(1-q) + 2p.
\]
Define \(J(p)\) to be the maximum of \(H(p, q)\) over all \(q\) (in the interval \(0 \le q \le 1\)). What is the value of \(p\) (in the interval \(0 \le p \le 1\)) that minimizes \(J(p)\)?


Let $p$ and $q$ be any two integers such that $p > q$ and $p$ is odd and $q$ is even. Determine the largest integer that divides all possible numbers of the form $p^2 - q^2$.


Determine the sum of the coefficients in the polynomial $5(2x^8 + 5x^3 - 9) + 3(2x^6 - 3x^3 + 4)$ when fully simplified.


Rachel is trying to remember the four-digit code to her bike lock. She knows she used the digits from 1 to 4 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible codes does she need to try?


Cara is sitting at a circular table with her seven friends. One of her friends, named Emma, must always sit next to her. How many different possible pairs of people could Cara be sitting between, given that one of them must always be Emma?


The matrix
\[\begin{pmatrix} a & \frac{20}{49} \\ c & \frac{29}{49} \end{pmatrix}\]
is a projection matrix. Find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{20}{49}, \frac{29}{49}\right)}


What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 720 = 0$ has integral solutions?


Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?


We define two sequences of vectors $(\mathbf{u}_n)$ and $(\mathbf{z}_n)$ with initial vectors $\mathbf{u}_0 = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\mathbf{z}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$. For $n \geq 1$, $\mathbf{u}_n$ is the projection of $\mathbf{z}_{n-1}$ onto $\mathbf{u}_0$, and $\mathbf{z}_n$ is the projection of $\mathbf{u}_n$ onto $\mathbf{z}_0$. Find the sum:
\[\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dotsb.\]

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


In an equilateral triangle $\triangle ABC$ with each side measuring $6$ units, $I$ is the incenter of the triangle. Determine the distance $BI$, where $B$ is one of the vertices of $\triangle ABC$.


Determine the area of trapezoid EFGH with vertices E(0,0), F(0,-3), G(7,0), and H(7,9).


Let \( r \), \( s \), and \( t \) be the three roots of the polynomial equation \( 9x^3 + 2023x + 4047 = 0 \). Find \(( r+s )^3 + ( s+t )^3 + ( t+r )^3\).


Three different integers from 1 through 30 inclusive are chosen at random. What is the probability that all three numbers are prime? Express your answer as a common fraction.


How many positive integers, not exceeding 200, are multiples of 2 or 5 but not 6?


What is the number of degrees in the acute angle formed by the hands of a clock at 9:57?


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


The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both red? Express your answer as a common fraction.


What is the probability that the square root of a randomly selected two-digit whole number is less than 7? Express your answer as a common fraction.


Simplify $\sqrt[4]{48a^5b^8c^{14}}$ and find the sum of the exponents of the variables that are outside the radical.


John has 15 marbles of different colors, including one red, one green, one blue, and one yellow marble. In how many ways can he choose 5 marbles, if exactly one of the chosen marbles is red, green, blue, or yellow?


What is the smallest positive value of $x$ such that $x + 8321$ results in a palindrome?


Alice and Bob play a similar game with a basketball. On each turn, if Alice has the basketball, there is a 5/8 chance that she will toss it to Bob and a 3/8 chance that she will keep the basketball. If Bob has the basketball, there is a 1/4 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with the basketball. What is the probability that Alice has the basketball again after two turns?


Find the solution to $x |x| = 3x + 4$ that has the smallest value.


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


How many integers, $x$, satisfy $|7x - 5| \le 9$?


The sequence begins with 800,000 and each subsequent term is obtained by dividing the previous term by 2. What is the last integer in this sequence?


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a $\diamondsuit$?


How many total days were there in the years from 2000 through 2003?


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

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


Consider a scenario where you have two regular polygons, one above the other, as shown in the diagram. One polygon is a regular pentagon and the other is a rectangle. Points $A$, $B$, $C$, and $D$ are as described in the diagram with $A$ connected to the vertices of both. Calculate the sum of the interior angles at $ABC$ and $ABD$. Assume both polygons are regular and sides of the rectangle are vertical or horizontal.


What is the remainder when 245,245 is divided by 35?


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


In the graph of $\frac{x^2 + 4x + 3}{x^3 - 2x^2 - x + 2}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asymptotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Determine $a+2b+3c+4d$.


Find the least positive integer $x$ such that $x + 127 \equiv 53 \pmod{15}$ and simultaneously $x+104 \equiv 76 \pmod{7}$.


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $.4\overline{37}$?


$\triangle ABC$ is inscribed in a circle of radius $r$ such that its base $AB$ coincides with the diameter $AB$. Point $C$ lies somewhere on the circle different from points $A$ and $B$. Let $s = AC + BC$. Determine the maximum possible value of $s^2$ given all permissible positions of $C$ on the circle.

Solution: \boxed{s^2 = 8r^2}


In triangle $ABC,$ $AB = 7,$ $BC = 8,$ and $AC = 9.$ If $D$ and $E$ are chosen on $\overline{AB}$ and $\overline{AC}$ so that $AD = 3$ and $AE = 6,$ then find the area of triangle $ADE.$


Find the polynomial $p(x),$ with real coefficients, such that $p(3) = 10$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\] for all real numbers $x$ and $y.$

Solution: \boxed{x^2 + 1}


Triangle $PQR$ has a perimeter of 2021 units. The sides have lengths that are all integer values. Define $PQ < PR \leq QR$. What is the smallest possible value of $PR - PQ$?


Find the quotient when $x^5 - 10x^4 + 31x^3 - 28x^2 + 12x - 18$ is divided by $x - 3$.

Solution: \boxed{x^4 - 7x^3 + 10x^2 + 2x + 18}


Let \( w \) be a complex number such that \( |w - 3 + 2i| = 4 \). Find the minimum value of
\[
|w + 1 + 2i|^2 + |w - 7 - 2i|^2.
\]


How many three-digit numbers satisfy the property that the middle digit is twice the average of the first and the last digits?


I have 7 books, three of which are identical copies of the same novel, and the others are distinct. If a particular book among these must always be placed at the start of the shelf, in how many ways can I arrange the rest of the books?


Our club has 30 members, 15 boys and 15 girls. In how many ways can we choose a president, a vice-president, and a secretary, if the president and vice-president must be of different genders and the secretary must be of the same gender as the president?


Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{d} & \frac{1}{c} \\ \frac{1}{b} & \frac{1}{a} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]


A bowling ball is a solid sphere with a diameter of 24 cm. To custom fit the ball for a bowler, three holes are drilled into the ball. The depths of the holes drilled are 6 cm, 6 cm, and 4 cm, with diameters of 1 cm, 2 cm, and 4 cm respectively. Each of the three holes is a right circular cylinder. Determine the volume of the custom fitted bowling ball in cubic centimeters, expressing your answer in terms of $\pi$.


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


Cylinder $C$'s height is equal to the radius of cylinder $D$ and cylinder $C$'s radius is equal to the height $k$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \pi k^3$ cubic units. What is the value of $M$?


A girl has the following eight coins in her purse: $3$ pennies, $1$ nickel, $2$ dimes, $1$ quarter, and $1$ half-dollar. She takes out two coins, records the sum of their values, and then puts them back with the other coins. She continues to take out two coins, record the sum of their values, and put them back. How many different sums can she record at most?


How many numbers are in the list $-58, -51, -44, \ldots, 71, 78$?


Calculate the product $\frac{2}{3} \cdot \frac{4}{7} \cdot \frac{9}{13}$.


A Mersenne number is defined to be a number of the form $2^n - 1$, where $n$ is an integer. Determine the largest Mersenne number less than 300 that is also a composite number.


Let \(a\), \(b\), \(c\) be nonnegative real numbers such that \(a + b + c = 2.\) Find the maximum value of
\[
\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.
\]


Among all triangles $ABC$, with angle $A$ fixed at $\frac{\pi}{3}$, find the maximum value of $\sin A + \cos B \cos C.$

Solution: \boxed{\frac{2\sqrt{3} + 1}{4}}


Define $g(x) = 3x + 2$. Suppose $g(x) = 2f^{-1}(x) + 4$, and $f^{-1}(x)$ is the inverse of the function $f(x) = bx + c$. Find the value of $3b + 4c$.


Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 2) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$


Let $b\equiv (2^{-1}+3^{-1}+5^{-1})^{-1}\pmod{13}$. What is the remainder when $b$ is divided by 13?


What is the units digit of the product of all the odd positive integers between 10 and 50 that are not multiples of 5?


In the provided geometric diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$ if the length of segment $AC$ is 8 units, and segment $CD$ is 30 units?

[asy]

draw((0,0)--(38,0)--(11,18)--(0,0));

dot((0,0));
label("$A$",(0,0),SW);
label("8",(4,0),S);
dot((8,0));
label("$C$",(8,0),S);
label("30",(23,0),S);
dot((38,0));
label("$D$",(38,0),SE);
dot((11,18));
label("$B$",(11,18),N);

draw((8,0)--(11,18));

[/asy]


Let \( y = \frac{\sum\limits_{n=1}^{30} \sin n^\circ}{\sum\limits_{n=1}^{30} \cos n^\circ} \). Find the smallest integer that exceeds \( 50y \).


Find the remainder when $123456789012$ is divided by $252$.


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


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


If $4x + 9 \equiv 3 \pmod{25}$, then $3x + 14$ is congruent $\pmod{25}$ to what integer between $0$ and $24$, inclusive?


The four zeros of the polynomial $x^4 + mx^2 + nx + 144$ are distinct real numbers in arithmetic progression. Compute the value of $m.$


June and Julia live 2 miles apart, and it takes June 6 minutes to ride her bike from her house directly to Julia's house. If June decides to visit Lee's house first, which is halfway between her and Julia's house before proceeding to Julia's house, how many minutes will it take June to ride from her house to Julia's house via Lee's house if the total distance becomes 3 miles?


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

1. $0 < x^3 < 1$
2. $x^2 > 1$
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 $x$?


Find the quotient when \( x^6 - 2x^5 + 3x^4 - 4x^3 + 5x^2 - 6x + 12 \) is divided by \( x - 1 \).

Solution: \boxed{x^5 - x^4 + 2x^3 - 2x^2 + 3x - 3}