Find the smallest positive integer such that when its leftmost digit is deleted, the remaining integer is $\frac{1}{19}$ of the original integer.


A $200\times 300\times 450$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


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


Find the positive real number $x$ such that $\lfloor x \rfloor + x^2 = 72$. Express $x$ as a decimal.


In parallelogram \(EFGH\), \(EF = 47\) cm, \(FG = 6y^2\) cm, \(GH = 3x + 7\) cm, and \(HE = 27\) cm. Calculate the product of \(x\) and \(y\).


Let $g(x) = \frac{x+8}{x}$. Define the sequence $(g_n)$ of functions where $g_1 = g$ and $g_n = g \circ g_{n-1}$ for all $n \geq 2$. Determine how many distinct real numbers $x$ satisfy $g_n(x) = x$ for some positive integer $n$.


Convert the point $(3, -3\sqrt{3}, 4)$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r, \theta, z)$, where $r > 0$ and $0 \leq \theta < 2\pi$.

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


How many distinct triangles can be constructed by connecting three different vertices of a regular octahedron? (Two triangles are distinct if they have different locations in space.)


John draws a regular six-pointed star in the sand, and at each of the 6 outward-pointing points and 6 inward-pointing points he places one of twelve different sea shells. How many ways can he place the shells, considering that reflections and rotations of an arrangement are deemed equivalent?


In the diagram, $O$ is the center of a circle with radii $OP = OQ = 8$. The region under consideration for the perimeter calculation is $\frac{1}{2}$ of the circle's area plus the radii. What is the perimeter of the shaded region?

[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(1,0),E);

fill(Arc((0,0),1,0,180)--cycle,mediumgray);
draw(Arc((0,0),1,0,180));
fill((0,0)--(-1,0)--(1,0)--cycle,white);
draw((-1,0)--(0,0)--(1,0));
[/asy]

Solution: \boxed{16 + 8\pi}


Alexio has 150 cards numbered 1-150, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction.


Given $f(x) = \frac{2}{3x + c}$, find the value of $c$ for which $f^{-1}(x) = \frac{2-3x}{3x}.$


Four plastic bottles can be recycled to make a new bottle. How many new bottles can eventually be made from 256 plastic bottles?


Calculate the sum of all the positive integers from 100 to 1999 inclusive.


Let $a$, $b$, $c$, $x$, $y$, and $z$ be real numbers that satisfy the three equations:
\begin{align*}
  11x +  by +  cz &= 0 \\
   ax + 19y +  cz &= 0 \\
   ax +  by + 37z &= 0.
\end{align*}
Suppose that $ a \ne 11$ and $ x \ne 0$. What is the value of
\[
\frac{a}{a - 11} + \frac{b}{b - 19} + \frac{c}{c - 37} \, ?
\]


There exist constants $c$ and $d$ such that
\[\sin^3 \theta = c \sin 3 \theta + d \sin \theta\]
for all angles $\theta.$ Find the ordered pair $(c, d)$.

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


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2f(x) y\]for all real numbers $x$ and $y.$ Additionally, suppose that $f(1) = 1$. Calculate the sum of all possible values of $f(3)$.


Choose two different numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the positive difference between the two numbers is $4$ or greater? Express your answer as a common fraction.


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$


What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$?


For the graph of a certain quadratic \( y = ax^2 + bx + c \), the vertex of the parabola is \( (2, -3) \) and one of the \( x \)-intercepts is \( (5, 0) \). What is the \( x \)-coordinate of the other \( x \)-intercept?


A rectangular picture has dimensions $x$ and $y$ (in inches), where both $x$ and $y$ are integers greater than one. It is placed in a frame such that the frame's outer dimensions are $(2x + 5)$ inches by $(y + 4)$ inches. The area of just the framed region (excluding the picture) is measured to be $84$ square inches. What is the area of the picture in square inches?


Let $\mathbf{B} = \begin{pmatrix} x & 3 \\ -4 & y \end{pmatrix}$ for some real numbers $x$ and $y$. If
\[
\mathbf{B} - \mathbf{B}^{-1} = 2 \mathbf{I},
\]
then find $\det \mathbf{B}$, where $\mathbf{I}$ is the identity matrix.


On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \).


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

pair A=(0,0), B=(0,5), C=(2.5,y), D=(5,5), E=(5,0);
pair[] dots={A,B,C,D,E};

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

axes(Arrows(4));

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


In quadrilateral $PQRS$, $\angle Q$ is a right angle, diagonal $\overline{PR}$ is perpendicular to $\overline{RS}$, $PQ=15$, $QR=20$, and $RS=9$. Find the perimeter of $PQRS$.

Solution: \boxed{44 + \sqrt{706}}


What is the greatest number of Sundays that can occur in the first $56$ days of a year?


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 9 and 16 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. Calculate the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$.


Four identical dice, each with faces numbered from 1 to 6, are stacked vertically. Five of the twenty-four faces are visible, while the other nineteen remain obscured. Calculate the total number of dots that are not visible in this view. Assume the visible faces show numbers 1, 2, 3, 5, and 6.


The polynomial $x^3 - cx^2 + dx - 2550$ has three positive integer roots. Find the smallest possible value of $c$.


Abigail, Beatrice, and Carson combine their eggs to allocate them into bigger crates at the market. Each crate can hold 18 eggs. If Abigail has 58 eggs, Beatrice has 76 eggs, and Carson has 27 eggs, and after allocating as many full crates as possible, they plan to equally share the remaining eggs among themselves. How many eggs will each get?


Determine the product of all real values of $s$ for which the equation $\frac{1}{3x}=\frac{s-3x}{8}$ has exactly one real solution.


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


If $P = 208 \div 4$, $Q = P \div 2$, and $Y = P - Q \times 10\%$, then what is the value of $Y$?


The chart below gives the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London \\ \hline
Bangkok & & 6300 & 6609 & 5944 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 \\ \hline
London & 5944 & 5989 & 7240 & \\ \hline
\end{tabular}


Christina and Alex play a game in which Christina picks an integer between 1 and 360 inclusive and Alex multiplies that integer with a factor such that the product is equal to 360. How many integers can Christina pick for which this operation is possible, and what is the sum of all such integers?


Mrs. Maple selects a random student to receive a maple leaf at the start of her Canadian Geography class. Given that there are 15 students in her class and it meets five times a week, how many different sequences of maple leaf recipients are possible during one week?


Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers \( x \).


Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]for all positive integers $x$ and $y.$ Given $g(30) = 17$ and $g(60) = 21,$ find $g(900).$


Let $O$ be the origin. Determine a scalar $k$ such that for any points $A$, $B$, $C$, and $D$ satisfying the vector equation
\[2 \overrightarrow{OA} - 3 \overrightarrow{OB} + 4 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]
the four points $A$, $B$, $C$, and $D$ must be coplanar.


Four concentric circles are drawn with radii of 3, 5, 7, and 9. The inner circle is painted white, the next ring is black, the following ring is white, and the outer ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


What is the range of the function $$s(x) = \frac{1}{|1-x|^3}~?$$ Express your answer in interval notation.

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


A $50$-gon \(Q_1\) is drawn in the Cartesian plane where the sum of the \(x\)-coordinates of the \(50\) vertices equals \(150\). A constant scaling factor \(k = 1.5\) applies only to the \(x\)-coordinates of \(Q_1\). The midpoints of the sides of \(Q_1\) form a second $50$-gon, \(Q_2\), and the midpoints of the sides of \(Q_2\) form a third $50$-gon, \(Q_3\). Find the sum of the \(x\)-coordinates of the vertices of \(Q_3\).


A wizard is brewing a rejuvenation elixir. In his formula, he can choose from four different magical herbs and six mystical crystals. However, one of the crystals does not work well with two of the herbs, and another crystal is incompatible with another herb. How many viable combinations of herbs and crystals can the wizard use for his elixir?


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


How many positive divisors does 8! have?


Thirty tiles are numbered 1 through 30 and are placed into box $A$. Another thirty tiles numbered 15 through 44 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than or equal to 20 and the tile from box $B$ is either odd or greater than 40? Express your answer as a common fraction.


Simplify $\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{7}}{\sqrt{11}} \cdot \frac{\sqrt{15}}{\sqrt{2}}$ and rationalize the denominator of the resulting fraction.


A frustum of a right circular cone is formed by slicing off a smaller cone from the top of a larger cone. The frustum has a height of 18 cm, the area of its larger base is $400\pi$ sq cm, and the area of its smaller base is $100\pi$ sq cm. Determine the height of the smaller cone that was removed.


When the decimal point of a certain positive decimal number is moved five places to the right, the new number is five times the reciprocal of the original number. What is the original number?


A relatively prime date is a date where the number of the month and the number of the day are relatively prime. Determine how many relatively prime dates are in the month of September. September has 30 days.


Calculate the sum of all the positive even factors of \(420\).


In an electrical engineering context, determine the current $I$ in a circuit where the voltage $V$ and the impedance $Z$ are given by complex numbers. Let $V = 2+3i$ and $Z = 4-2i$. Express your answer as a complex number in the form $a+bi$.

Solution: \boxed{\frac{1}{10} + \frac{4}{5}i}


Evaluate $\left|\frac{7}{4} - 3i + \sqrt{3}\right|$.

Solution: \boxed{\frac{\sqrt{241 + 56\sqrt{3}}}{4}}


How many three-digit numbers are there in which the hundreds digit is greater than both the tens digit and the units digit?


Find the remainder when \(3^{2015}\) is divided by \(13\).


Given that $x+\sin y=2010$ and $x+2010 \cos y=2009$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$.

Solution: \boxed{2009 + \frac{\pi}{2}}


The graphs of the function $f(x) = -2x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(2,a)$. If $b$ and $a$ are both integers, determine the value of $a$.


Consider the polynomial \(x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\) with integer coefficients. Determine all possible values for \(m\), which represents the exact number of integer roots of the polynomial, counting multiplicity.

Solution: \boxed{0, 1, 2, 3, 4, 5}


A projectile is fired with varying initial velocities $v$, but at a constant angle $\theta = 45^\circ$ from the ground. The trajectory of the projectile could again be modeled by the parametric equations:
\begin{align*}
x &= vt \cos(45^\circ), \\
y &= vt \sin(45^\circ) - \frac{1}{2} gt^2,
\end{align*}
where $t$ denotes time and $g$ denotes the acceleration due to gravity. As $v$ changes, plot the highest point of each trajectory and determine the equation of the curve that these points trace.

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


Determine the value of $x$ for which $\frac{1}{\text{log}_3x} + \frac{1}{\text{log}_5x} + \frac{1}{\text{log}_6x} = 1$.


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


The graph of the line $2x+y=c$ is the perpendicular bisector of the line segment from $(2,4)$ to $(6,10)$. What is the value of $c$?


Suppose $68 \cdot 95 \cdot 113 \equiv m \pmod{25}$, where $0 \le m < 25$.

What is the value of $m$?


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


Let $p$, $q$, and $r$ be the solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{pq}{r} + \frac{qr}{p} + \frac{rp}{q}$.


Define
\[ A' = \frac{1}{1^3} + \frac{1}{5^3} - \frac{1}{7^3} - \frac{1}{11^3} + \frac{1}{13^3} + \frac{1}{17^3} - \dotsb, \]
which omits all terms of the form \(\frac{1}{n^3}\) where \(n\) is an odd multiple of 3, and
\[ B' = \frac{1}{3^3} - \frac{1}{9^3} + \frac{1}{15^3} - \frac{1}{21^3} + \frac{1}{27^3} - \frac{1}{33^3} + \dotsb, \]
which includes only terms of the form \(\frac{1}{n^3}\) where \(n\) is an odd multiple of 3.

Determine \(\frac{A'}{B'}.\)


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's tens digit is a prime number? Express your answer as a common fraction.


How many natural numbers greater than 9 but less than 100 are relatively prime to 30?


If $a$, $b$, and $c$ are positive numbers such that $ab=30$, $ac=60$, and $bc=90$, what is the value of $a+b+c$?

Solution: \boxed{a+b+c = 11\sqrt{5}}


Calculate and simplify $\sqrt[4]{3^5 \cdot 5^3}$ to the form $c\sqrt[4]{d}$, where $c$ and $d$ are positive integers. What is $c+d$?


Let \(a\) and \(b\) be positive real numbers such that \(2a + 3b = 1.\) Find the minimum value of
\[
\frac{2}{a} + \frac{3}{b}.
\]


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


What is the smallest prime divisor of $3^{15} + 11^{21}$?


Find the sum of all angles \(x \in [0^\circ, 360^\circ]\) that satisfy
\[\cos^5 x - \sin^5 x = \frac{1}{\sin x} - \frac{1}{\cos x}.\]


If the system of equations

\begin{align*}
8x - 6y &= a, \\
9y - 12x &= b.
\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero, find the ratio $\frac{a}{b}$, assuming $b$ is nonzero.


Suppose that $\sec x + \tan x = \frac{12}{5}$ and that $\csc x + \cot x = \frac{m}{n}$, where $\frac{m}{n}$ is in lowest terms. Find $m + n$.


Let's consider California and New York license plates with different alpha-numeric formats. California license plates are formatted as two letters, followed by two digits then followed by two more letters, while New York has a format of two letters, followed by three digits and ending with one letter. Assuming all 10 digits are equally likely to appear in numeric positions and all 26 letters are equally likely to appear in alpha positions, how many more license plates can California issue than New York?


Two circles of radius 12 cm overlap such that each circle passes through the center of the other. Calculate the length of the common chord shared between these two circles.


Let $z$ be a complex number such that
\[z^2 + |z|^2 = 5 - 7i.\]Find $|z|^2.$


If \( f(x) = -3x^5 + 6x^3 - 2x^2 + 8 \), and \( g(x) \) is a polynomial such that the degree of \( f(x) + g(x) \) is 2, what is the degree of \( g(x) \)?


How many odd numbers are greater than 215 and less than 500?


If $x$ and $y$ are positive integers such that $xy - 4x + 3y = 315$, what is the minimal possible value of $|x - y|$?


Carla's age is four times Diane's age. Emma is eight years older than Diane. Carla and Emma are twins. How old is Diane?


The marble statue of a famous general in a memorial has a height of 48 feet. A scale model of the statue has a height of 3 inches. How many feet of the statue does one inch of the model represent?


How many distinct triangles can be constructed by connecting three different vertices of an octahedron? (Two triangles are distinct if they have different locations in space.)


In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$.


Expand and simplify the expression: $\frac{3}{4}\left(\frac{4}{y} - 7y^3\right)$.

Solution: \boxed{\frac{3}{y} - \frac{21y^3}{4}}


Given $k > 0$, define $J_k = 10\ldots 0128$, where there are $k$ zeros between the $1$ and the $128$. Let $M(k)$ be the number of factors of $2$ in the prime factorization of $J_k$. Determine the maximum value of $M(k)$.


A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Science. This program must contain English, Science, and at least one mathematics course. In how many ways can this program be chosen?


Martha is working on balancing a financial equation: \[fp - w = 8000\]
If $f=10$ and $w=-10 + 250i$, what is $p$?

Solution: \boxed{799 + 25i}


A car's brakes are applied, causing it to travel 12 feet less in each second than the previous second until it comes to a complete stop. The car covers 36 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time it stops?


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


A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?


A hiker moves 20 miles east, then 15 miles south, then 5 miles west, and finally 9 miles north. How far is the hiker from the starting point after completing these segments of the journey?

Solution: \boxed{\sqrt{261} \text{ miles}}


For a positive integer \(n\), let
\[
T_n = 1 + \frac{1}{2\cdot2!} + \frac{1}{3\cdot3!} + \dots + \frac{1}{n\cdot n!}.
\]
Compute
\[
\sum_{n = 1}^\infty \frac{1}{(n + 1) T_n T_{n + 1}}.
\]


Find the remainder when $x^5 - 3x^3 + x^2 + 2$ is divided by $x^2 - 4x + 6$.

Solution: \boxed{-22x - 28}


In parallelogram $EFGH$, angle $E$ measures $135^\circ$. Find the measure of angle $G$.


Two planes cut a right hexagonal pyramid parallel to the base resulting in cross sections with areas $125\sqrt{3}$ square feet and $500\sqrt{3}$ square feet. The distance between the two planes is $10$ feet. Determine the distance from the apex to the plane of the larger cross-section.


If $\mathbf{c}$ and $\mathbf{d}$ are vectors such that $\|\mathbf{c}\| = 5$ and $\|\mathbf{d}\| = 13$, and the angle $\phi$ between $\mathbf{c}$ and $\mathbf{d}$ is constrained to be within $[0,\frac{\pi}{3}]$, find all possible values of $\mathbf{c} \cdot \mathbf{d}$.


How many of the 300 smallest positive integers are congruent to 1 (mod 7)?


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


Determine all values of $a$ for which the following system has a real solution $(x,y)$:
\begin{align*}
\sqrt{x^2y^2} &= a^a, \\
\log_a (x^{\log_a y}) + 2 \cdot \log_a (y^{\log_a x}) &= 6a^3.
\end{align*}

Solution: \boxed{\left( 0, \frac{1}{8} \right]}


In the diagram below, we have $\sin \angle APB = \frac{3}{5}$. What is $\sin \angle APC$?

[asy]
pair A,P,B,C;
C = (-2,0);
P = (0,0);
B = (2,0);
A = rotate(aSin(3/5))*(1.5,0);
dot("$C$",C,S);
dot("$B$",B,S);
dot("$A$",A,N);
dot("$P$",P,S);
draw(B--C);
draw(P--A);
[/asy]


Determine how many more digits does the base-4 representation of $1729_{10}$ have than the base-6 representation of $1729_{10}$?


Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than three times any other, find the minimum value of the product $xyz$.


The graph of the line $x+y=b$ is the perpendicular bisector of the line segment from $(0, 0)$ to $(4, 8)$. What is the value of b?


In the diagram, $\triangle ABC$ is right-angled at $A$, with $AB=45$ and $AC=60.$ The point $D$ is on $BC$ such that $AD$ is perpendicular to $BC.$ Determine the length of $BD.$


In square $WXYZ$, angle $W$ is trisected by $\overline{WP}$ and $\overline{WQ}$, where $P$ is on $\overline{XY}$, $Q$ is on $\overline{WZ}$, $YP=4$, and $QZ=3$. Find the area of $WXYZ$.

[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (4*sqrt(3) - 3), width = 4*sqrt(3);
draw(origin--(length,0)--(length,width)--(0,width)--cycle);
draw((length,width)--(0,3)^^(length,width)--(length - 4,0));
dot("$W$",origin,SW); dot("$X$",(length,0),SE); dot("$Y$",(length,width),NE); dot("$Z$",(0,width),NW);
dot("$Q$",(0,3),W); dot("$P$",(length - 4,0),S);
[/asy]


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

Solution: \boxed{4 - \frac{\sqrt{15}}{3}}


What is the greatest common factor of the numbers 2550 and 7140?


Let points on a parabola be \(A = (2,0), B = (5,3),\) and \(C = (p, -p^2 + 8p - 12)\) where \(2 \le p \le 5.\) Calculate the maximum possible area of triangle ABC.


If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 17?


In a right triangle with integer-length sides, the hypotenuse has length 65 units. How many units is the length of the shorter leg?


The graph of the parabola $x = 3y^2 - 9y + 5$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


Flatville modifies its bicycle license plates which initially contain three letters: the first from $\{B, F, K, S, Z\}$, the second from $\{E, U, Y\}$, and the third from $\{G, J, Q, V\}$. To increase the number of plates, they add two new letters. These new letters can either be both added to one set or one letter can be added to one set and the other to a different set. What is the largest number of additional license plates that can be made by adding these two letters?


Evaluate the degree of the polynomial resulting from the expression $(4x^6 + 2x^5 - 3x + 6)(5x^{12} - 2x^9 + 7x^6 - 15) - (3x^3 + 4)^6$.


The denominator of a fraction is 4 less than 4 times the numerator. If the fraction is equivalent to $\frac{3}{7}$, what is the numerator of the fraction?


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


Let \(x,\) \(y,\) and \(z\) be positive real numbers such that \(x + y + z = 2.\) Find the minimum value of
\[
\frac{x + y}{xyz}.
\]


At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department must be represented by two of its members. Every committee must include at least one woman from the biology department. Find the number of possible committees that can be formed subject to these requirements.


The first and fifteenth terms of an arithmetic sequence are \(\frac{10}{11}\) and \(\frac{8}{9}\), respectively. What is the eighth term?


I have a bag containing 7 marbles, each uniquely numbered from 1 to 7. I draw three different marbles at random. What is the expected value of the sum of the numbers on these marbles?


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


The number $n$ is a prime number between 30 and 40. If you divide $n$ by 6, the remainder is 1. What is the value of $n$?


The solutions to $7x^2 - 2x + 45 = 0$ can be written in the form $x = p + qi$, where $p$ and $q$ are real numbers. What is $p + q^2$? Express your answer as a fraction.


If \(x\) is real, compute the maximum integer value of
\[
\frac{4x^2 + 8x + 19}{4x^2 + 8x + 5}.
\]


Let $x$, $y$, and $z$ be real numbers such that
\[\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 1.\]Find the sum of all possible values of $\cos 2x + \cos 2y + \cos 2z.$


What is the digit in the tens place when $13^{3007}$ is expressed in decimal notation?


What is the least common multiple of the numbers 1584 and 1188?


Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3?


Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction.


What is the average of all the integer values of \( N \) such that \( \frac{N}{84} \) is strictly between \( \frac{2}{7} \) and \( \frac{1}{6} \)?


Eight 6-sided dice are rolled. What is the probability that four of the dice show even numbers and the remaining four dice show odd numbers?


Suppose
\[\frac{1}{x^3-7x^2+10x+24}=\frac{A}{x-2}+\frac{B}{x-6} + \frac{C}{(x-6)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?


Alice can buy pins, pons, and puns for $3$, $4$, and $9$ each, respectively. What is the largest number of puns she can purchase if she must buy at least one of each item and will spend exactly $108$?


Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$.


The fourth and sixth terms of a geometric sequence of real numbers are $6!$ and $7!$ respectively. What is the first term?

Solution: \boxed{a = \frac{720\sqrt{7}}{49}}


If $(x+3)(x-4)=12$, find the sum of the possible values of $x$.


Sarah has 6 children. What is the probability that at least half of them are girls, given that the probability of a child being a girl is $\frac{3}{5}$?


For how many values of \(c\) in the interval \([0, 2000]\) does the equation \[5 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for \(x\)?


When plotted in the standard rectangular coordinate system, trapezoid $ABCD$ has vertices $A(1, -2)$, $B(1, 1)$, $C(7, 7)$, and $D(7, -1)$. What is the area of trapezoid $ABCD$?


Let $\mathbf{v} = \begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}.$  The columns of a matrix are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w},$ where $\mathbf{u}$ is a unit vector. Find the largest possible determinant of this matrix.


A drawer in a darkened room contains $120$ red socks, $90$ green socks, $70$ blue socks, $50$ black socks, and $30$ yellow socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks being drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $12$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)


Three tour guides are leading eight tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that no guide can take all the tourists. How many different groupings of guides and tourists are possible?


A convex polyhedron \(Q\) has \(30\) vertices, \(70\) edges, and \(40\) faces, of which \(30\) are triangular and \(10\) are pentagonal. Compute the number of space diagonals in polyhedron \(Q\).


The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ remains $C = \frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, then converted back to Fahrenheit using the formula $F' = \frac{9}{5}C' + 32$ where $C'$ is the rounded Celsius temperature, and $F'$ is finally rounded to the nearest integer. For how many integer Fahrenheit temperatures between 32 and 2000 inclusive does the original temperature equal the final temperature?


How many factors of 51200 are perfect squares?


A penny, nickel, dime, quarter, and fifty-cent piece are simultaneously flipped. The fifty-cent piece is biased and has a $\dfrac{3}{4}$ probability of coming up heads. What is the expected value of the amount of these coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent.


Consider two functions, \( p(x) = |x| + 1 \) and \( q(x) = -x^2 \). Evaluate \( q(p(x)) \) at \( x = -3, -2, -1, 0, 1, 2, 3 \) and find the sum of these seven values.


Given vectors $\mathbf{v} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 6 \\ -4 \\ 8 \end{pmatrix}$, compute $\text{proj}_{\mathbf{w}} \mathbf{v}$.

Solution: \boxed{\begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix}}


In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 200$, and $AD \neq BC$. The perimeter of $ABCD$ is $780$. Find $\lfloor 1000 \cos A \rfloor.$


In the given right-angled triangle $ABC$, with the angle at $B$ being a right angle (90 degrees), $D$ is a point on $BC$. The ratio of $BD$ to $DC$ is $5$ to $2$. The area of $\triangle ABD$ is given as $35$ square centimeters. Calculate the area of $\triangle ADC$.
[asy] size(85); defaultpen(linewidth(1)+fontsize(10));

pair A = (0,5.5), B=(0,0), D = (3.5,0), C = (5,0);

draw(A--B--C--A--D); label("A",A,N); label("B",B,S); label("C",C,S); label("D",D,S);
draw(rightanglemark(A,B,C,8),linewidth(0.7));
[/asy]


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


Find the modular inverse of \( 5 \), modulo \( 23 \).

Express your answer as an integer from \( 0 \) to \( 22 \), inclusive.


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


Suppose that the roots of $x^3 + 4x^2 + 5x - 14 = 0$ are $p$, $q$, and $r$, and the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find the value of $w$.


Teressa needs to travel from city $X$ to city $Y$, then from $Y$ to city $Z$, and then from $Z$ to $X$. Each leg of her journey can be done entirely by bus or entirely by airplane. The cities form a right-angled triangle, with $Z$ being a distance of $4000\text{ km}$ from $X$ and $Y$ is $4800\text{ km}$ from $X$. The bus fare costs Teressa $\$0.15$ per kilometer. For flying, she pays a $\$150$ booking fee plus $\$0.12$ per kilometer. If Teressa decides to fly from $X$ to $Y$, determine the cost of her flight.


Let $g$ be a non-constant polynomial such that
\[g(x - 1) + g(x) + g(x + 1) = \frac{[g(x)]^2}{x^2}\]
for all nonzero real numbers $x$. Find the sum of all possible values of $g(0)$.


What is the ratio of the volume of cone $C$ to the volume of cone $D$? Cone $C$ has a radius of 15.6 and a height of 29.5. Cone $D$ has a radius of 29.5 and a height of 15.6.


A curious archaeologist holds a game where participants guess the age of a recently discovered fossil based on clues. The age of the fossil consists of the digits 1, 1, 1, 4, 8, and 9 with the condition that the age must end with an even digit. How many possible ages are there for the fossil?


Let $x, y, z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^2+y^2&=&z^2+w^2&=&2010,\\ xz&=&yw&=&1008.\end{array}\]
If $T=x+y+z+w$, compute the value of $\lfloor T \rfloor$.


Let $d$ be a complex number. Suppose there exist distinct complex numbers $x$, $y$, and $z$ such that for every complex number $w$, we have
\[
  (w - x)(w - y)(w - z) = (w - dx)(w - dy)(w - dz).
\]
Compute the number of distinct possible values of $d$.


A rectangle has a perimeter of 60 feet, and each side has an integer length. How many non-congruent rectangles meet these criteria?


If the least common multiple of $A$ and $B$ is $180$, and the ratio of $A$ to $B$ is $2:3$, then what is their greatest common divisor?


In triangle $ABC$, $\angle C=90^\circ$, $AC=9$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=6$, then what is the length of $BD$?


How many unordered pairs of prime numbers have a sum of 52?


Suppose $\triangle ABC$ and $\triangle ACD$ are two congruent triangles with $AB = AC = AD$ and $\angle BAC = 30^\circ$. Determine the measure of $\angle BDC$.


How many subsets of the set $\{1, 2, 3, 4, 5, 6\}$ must contain the number 6 and at least one of the numbers 1 or 2?


How many four-digit numbers are multiples of neither 5 nor 7?


A weather forecast predicts that for each day from Monday to Friday this week, there is a $30\%$ chance of sun, a $20\%$ chance of getting 5 inches of rain, and a $50\%$ chance of getting 8 inches of rain. Over the weekend, there is a $50\%$ chance of sun, a $25\%$ chance of getting 2 inches of rain, and a $25\%$ chance of getting 6 inches of rain. What is the expected value of the total number of inches of rain which will fall from Monday to Sunday this week?

Solution: \boxed{29\text{ inches}}


For each positive integer $n$, let $h(n) = n^4 - 380n^2 + 600$. What is the sum of all values of $h(n)$ that are prime numbers?


Let $b_n = \frac{7^n-1}{6}$. 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?


A matrix represents reflection over a line $\ell$ that passes through the origin, given by:
\[\begin{pmatrix} \frac{9}{41} & \frac{40}{41} \\ \frac{40}{41} & -\frac{9}{41} \end{pmatrix}.\]
Find the direction vector of line $\ell$, in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a, b$ are integers, $a > 0$, and $\gcd(|a|, |b|) = 1$.

Solution: \boxed{\begin{pmatrix} 5 \\ 4 \end{pmatrix}}


Let $x = (3 + \sqrt{8})^{12},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


There is an angle $\theta$ in the range $45^\circ < \theta < 90^\circ$ which satisfies
\[\tan \theta + \tan 3\theta + \tan 5\theta = 0.\]
Calculate $\tan \theta$ for this angle.


Each of the first twelve prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number?


What is the sum of the greatest common divisor of $75$ and $4500$ and the least common multiple of $75$ and $4500$?


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


In right $\Delta XYZ$, $\angle XYZ$ is a right angle. Point $N$ is the midpoint of $\overline{XZ}$. If $\overline{XY} = 5$ cm and $\overline{YZ} = 12$ cm, calculate the following:
1. The length of median $\overline{YN}$.
2. The area of $\Delta XYZ$.

Solution: \boxed{30 \text{ square cm}}


The digits 1, 3, 5, and 9 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Express your answer as a common fraction.


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


Let $a$ and $b$ be the roots of the equation $x^2-mx+3=0$. Suppose $a + \frac{2}{b}$ and $b + \frac{2}{a}$ are the roots of another equation $x^2-px+q=0$. Find $q$.


What is the smallest positive integer that has a remainder of 1 when divided by 3, a remainder of 2 when divided by 4, and a remainder of 3 when divided by 5?


What is the greatest number of Sundays that can occur in the first $51$ days of a year?


A deck of eighty cards includes four cards each of numbers $1$ to $20$. If three pairs—each pair consisting of two cards with the same number—are removed from distinct number sets (e.g., a pair of `7`s, a pair of `15`s, and a pair of `20`s), find the probability that two randomly selected cards from the remaining deck also form a pair. Present your answer as a sum of relatively prime integers $m + n$ where the probability is $m/n$.


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

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


The Soccer club has 24 members: 12 boys and 12 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?


A square and a right triangle are adjacent to each other, with each having one side on the $x$-axis. The lower right vertex of the square and the lower left vertex of the triangle are at $(12, 0)$. They share one vertex, which is the top right corner of the square and the hypotenuse-end of the triangle. The side of the square and the base of the triangle on the x-axis each equal $12$ units. A segment is drawn from the top left vertex of the square to the farthest vertex of the triangle, creating a shaded area. Calculate the area of this shaded region.

Solution: \boxed{36 \text{ square units}}


Let $a, b, c, d, e$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 + e^2 = 9.\] Find the maximum value of $a^3 + b^3 + c^3 + d^3 + e^3.$


The average of five different positive whole numbers is $6.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the middle three numbers?


What is the value of $y$ if $y = \left(\frac{2023^2 - 2023}{2023}\right)^2$?


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 circle, and the piece of length \(b\) is bent to form a square. The circle and the square have equal perimeter. What is \(\frac{a}{b}\)?


Monica is now playing a different game involving two fair 6-sided dice. She wins $4 times the sum of the two dice if the total is odd. If the total is even, she wins nothing unless both numbers rolled are identical (doubles), in which case she wins $2 times the sum of the two dice. Otherwise, she loses $6. What is the expected value of her winnings on one roll of two dice? Express your answer as a dollar value to the nearest cent.


Let $r$ be a real number such that $|r| < 4$, and let $z$ be a complex number such that
\[ z + \frac{1}{z} + 2 = r. \]
Find $|z|$.


In triangle $ABC$, $AB = 6$, $AC = 10$, and $BC = 8$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find the sum $GP + GQ + GR$.


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 317 hours, 58 minutes, and 33 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$?


In rectangle $ABCD$, $AB=7$ and $BC=4$. Points $J$ and $K$ are on $\overline{CD}$ such that $DJ = 2$ and $KC=3$. Lines $AJ$ and $BK$ intersect at point $L$. Find the area of $\triangle ABL$. Visualize as follows:
[asy]
pair A,B,C,D,L,J,K;
A=(0,0);
B=(7,0);
C=(7,4);
D=(0,4);
J=(2,4);
K=(4,4);
L=(3.5,6);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--J--L--B--K--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$J$",J,S);
label("$K$",K,S);
label("$L$",L,N);
label("2",(1,4),N);
label("1",(3,4),N);
label("3",(5.5,4),N);
label("4",(0,2),W);
label("4",(7,2),E);
label("7",(3.5,0),S);
[/asy]


Find the mean of all solutions for $x$ when $x^3 + 3x^2 - 14x = 0$.


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


Andrew's grandfather's age is nine times Andrew's age. If Andrew's grandfather was 63 years old when Andrew was born, how many years old is Andrew now?


Find $a + b$ if the range of the function $h(x) = \frac{3}{1+3x^3}$ can be written as an interval $(a, b]$.


Let $x$ and $y$ be positive real numbers such that $5x + 3y < 75$. Find the maximum value of
\[ xy(75 - 5x - 3y). \]


What is the sum of the last two digits of $9^{25} + 11^{25}?$


How many whole numbers are between $\sqrt[4]{17}$ and $\sqrt[4]{340}$?


Find the range of the function \( f(x) = \arcsin x + \arccos x + \ln(1 + x) \) for \( x \in [-1, 1] \).

Solution: \boxed{\left( -\infty, \frac{\pi}{2} + \ln 2 \right]}


Find the number of integers $j$ such that $1 \le j \le 3000$ and $f(j) = 1 + j + \sqrt{j}$, where $f(j)$ denotes the sum of all positive divisors of $j$.


Determine how many subsets of $\{1, 2, 3, \dots, 8\}$, including the empty set, are "spacy". A set is called "spacy" if it contains no more than one out of any three consecutive integers.


When Emma divides her oranges into groups of 6, she has 5 remaining, and when she divides her oranges into groups of 8, she has 7 remaining. If Emma has fewer than 100 oranges, what is the sum of all possible quantities of oranges that she could have?


Determine the base seven product of the numbers $321_7$ and $13_7$.


What integer $n$ satisfies $0 \leq n < 23$ and $$-215 \equiv n \pmod{23}~?$$


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


The bacteria in a petri dish double every 30 seconds. After two minutes and thirty seconds, there are 20,480 bacteria in the dish. How many were in the dish at the beginning of the experiment?


Mrs. Vale's mathematics class has expanded to include 52 students, with each pair of students sharing the same initials, and no other students having those initials. Each student's first name and last name begin with the same letter. If the letter "Y" is not considered a vowel, what is the probability of randomly picking a student whose initials are vowels? Express your answer as a common fraction.


In how many ways can 100 be written as the sum of three distinct positive perfect squares if the order of the squares does not matter?


Let $a_1, a_2, \ldots$ be a sequence of positive real numbers such that
\[ a_n = 9a_{n-1} - n \] for all $n > 1$. Find the smallest possible value of $a_1$ that ensures all terms in the sequence remain positive.


Determine the value of $k$ if the two roots of the quadratic equation $5x^2 + 7x + k$ are $\frac{-7 \pm i\sqrt{399}}{10}$.


A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y = x^2$ and $y = 4-x^2$?


There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 8 \\ 0 \\ -4 \end{pmatrix},\]
where $\mathbf{a}$ is parallel to $\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}.$  Find $\mathbf{b}.$

Solution: \boxed{\begin{pmatrix} \frac{36}{7} \\ -\frac{30}{7} \\ -\frac{18}{7} \end{pmatrix}}


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


If the probability that a baby born in a certain hospital will speak in the next day is 2/5, what is the probability that at least 3 babies out of a group of 6 babies will speak tomorrow?


Sasha has \(\$4.80\) in U.S. coins. She has twice as many nickels as quarters, and the number of dimes is equal to the number of quarters. What is the greatest number of quarters she could have?


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


Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?


The infinite sequence $T=\{t_1, t_2, t_3, \ldots\}$ is defined by $t_1=3$ and $t_n=3^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{100}$ is divided by 7?


In the diagram, $\angle FAB$ and $\angle ABC$ are right angles, $AB=5,$ $BC=3,$ $AF=7,$ and $\overline{AC}$ and $\overline{BF}$ intersect at $D.$ What is the difference between the areas of $\triangle ADF$ and $\triangle BDC?$


In triangle $DEF$, the side lengths are $DE = 15$, $EF = 20$, and $FD = 25$. A rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. Letting $WX = \lambda$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \lambda - \delta \lambda^2.\]
Then the coefficient $\gamma = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Find the number of solutions to:
\[\sin x = \left(\frac{1}{3}\right)^x\]
on the interval $(0, 50\pi)$.


In the coordinate plane, the hyperbola $xy = 2$ intersects a circle at four points, of which three are $\left( 4, \frac{1}{2} \right),$ $\left( -2, -1 \right),$ and $\left( \frac{2}{3}, 3 \right).$ Find the fourth point of intersection.

Solution: \boxed{\left( -\frac{3}{4}, -\frac{8}{3} \right)}


Find the largest integral value of \( y \) which solves: \( \frac{1}{4} < \frac{y}{7} < \frac{7}{11} \).


Let $x_1, x_2, x_3$ be positive real numbers such that $x_1 + 3x_2 + 4x_3 = 100.$ Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2.\]


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


The sequence $(a_n)$ satisfies
\[a_1 + a_2 + a_3 + \dots + a_n = n^2 a_n\]for all $n \ge 2.$ If $a_{50} = 1,$ find $a_1.$


Our football team has 12 members, of which only 4 are strong enough to play as offensive or defensive linemen, while all other positions can be played by anyone. In how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, a defensive lineman, and a wide receiver?


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


Subtract $222.22$ from $555.55$ and then multiply the result by $1.5$. Express the final result as a decimal to the nearest hundredth.


A torus (donut shape) with an inner radius of $3$ and an outer radius of $5$ is placed on a flat table. Determine the radius of the largest spherical ball that can be positioned on top of the center of the torus so that the ball still touches the horizontal plane, given that the torus is formed by revolving a circle in the $xz$-plane centered at $(4,0,1)$ with a radius of $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis.


Find the next smallest abundant number after 12 by checking its immediate successors among composite numbers.


Find the quotient when $x^6 + 5x^3 - 8$ is divided by $x - 2.$

Solution: \boxed{x^5 + 2x^4 + 4x^3 + 13x^2 + 26x + 52}


How many nonnegative integers can be represented in the form \[ b_6 \cdot 4^6 + b_5 \cdot 4^5 + b_4 \cdot 4^4 + b_3 \cdot 4^3 + b_2 \cdot 4^2 + b_1 \cdot 4^1 + b_0 \cdot 4^0, \] where $b_i \in \{-2, -1, 0, 1, 2\}$ for $0 \leq i \leq 6$?


Find the domain of the function
\[f(x) = \sqrt{2 - \sqrt{4 - \sqrt{5 - x}}}.\]

Solution: \boxed{[-11, 5]}


Let $\alpha$ and $\beta$ be the roots of $x^2 + px - 1 = 0$, and let $\gamma$ and $\delta$ be the roots of $x^2 + qx - 1 = 0$. Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha - \delta)(\beta - \delta)\]
in terms of $p$ and $q$.

Solution: \boxed{-(p - q)^2}


Define 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}


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


When Ray attempts to climb stairs $m$ steps at a time, starting from the floor (step 0), he jumps to step $m$ then to $2m$ and so forth until the remaining steps to the top are less than $m$. Ray climbs a flight of stairs $n$ steps in two ways. Firstly, when he climbs $6$ steps at a time, there are $4$ steps left at the top. Secondly, when he climbs $7$ steps at a time, there are $3$ steps left at the top. Determine the smallest possible value of $n$ that is greater than $10$.


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


Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$.


For $\mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}$, \[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -8 \\ -4 \end{pmatrix}.\] Find $y$.


Simplify \(\cot 20^\circ + \tan 10^\circ.\)

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


It is currently 3:00:00 PM, as read on a 12-hour digital clock. In 85 hours, 58 minutes, and 30 seconds, the time will be $A:B:C$. What is the value of $A + B + C$?


Lines $l$ and $k$ are parallel, and a third line $m$, a transversal, intersects them at points $P$ and $Q$ respectively. $m\angle A = 110^\circ$ at point $P$ on line $l$. At point $Q$ on line $k$, there is an angle $m\angle C = 70^\circ$. Determine the measure of $m\angle B$, which is the supplement of the angle formed between line $m$ and line $k$ at point $Q$.


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

[asy]
draw((0,0)--(6,6)--(13,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(13,2),dashed);
label("9",(3,3),NW);
label("12",(9.5,4),NE);
label("15",(10.5, -2.5),SE);
label("14",(4,-3.5),SW);
[/asy]


The quadratic equation $x^2 + kx + n = 0$ has roots that are three times those of $x^2 + px + k = 0$, where none of $k$, $n$, and $p$ is zero. Determine the value of $n/p$.


The real number $x$ satisfies $x^2 - 7x + 12 < 0.$ Find all possible values of $x^2 - 5x + 6.$


If $p$ is a constant such that $9x^2 + 27x + p$ is the square of a binomial, then what is $p$?


The marble statue of George Washington in a museum has a height of 120 feet. A scale model of the statue has a height of 6 inches. How many feet of the statue does one inch of the model represent?


Consider a truncated pyramid formed by slicing the top off a regular pyramid. The larger base now has 6 vertices, and the smaller top base is a regular hexagon. The ant starts at the top vertex of the truncated pyramid, walks randomly to one of the six adjacent vertices on the larger base, and then from there, it walks to one of the six adjacent vertices on the smaller top base. What is the probability that this final vertex is the initially started top vertex? Express your answer as a common fraction.


Find the total number of digits in the value of $2^{15} \times 5^{10}$.


Six identical rectangles are arranged to form a larger rectangle $ABCD$, where $AB = 3y$ and $BC = 2x$. If the area of $ABCD$ is $2400$ square units, find the length $y$, rounding off to the nearest integer.


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations:
\begin{align*}
  5x + by + cz &= 0 \\
  ax + 7y + cz &= 0 \\
  ax + by + 9z &= 0.
\end{align*}
Suppose that $a \ne 5$ and $x \ne 0$. What is the value of
\[ \frac{a}{a - 5} + \frac{b}{b - 7} + \frac{c}{c - 9} \, ?\]


A line is parameterized by a parameter \( t \), such that the vector at \( t = 1 \) is \( \begin{pmatrix} 2 \\ 5 \end{pmatrix} \), and the vector at \( t = 4 \) is \( \begin{pmatrix} 11 \\ -7 \end{pmatrix} \). Find the vector on the line at \( t = 0 \).

Solution: \boxed{\begin{pmatrix} -1 \\ 9 \end{pmatrix}}


A stock investment went up by \( 30\% \) in 2006. Starting at this increased value, what percent would it need to decrease in 2007 to return to its original price at the beginning of 2006?


Let \( f(x) = x^2 + ax + b \) and \( g(x) = x^2 + cx + d \) be two distinct quadratic polynomials with real coefficients. Suppose the \( x \)-coordinate of the vertex of \( f \) is a root of \( g \), the \( x \)-coordinate of the vertex of \( g \) is a root of \( f \), and both \( f \) and \( g \) yield the same minimum value. Additionally, the graphs of the two polynomials intersect at the point \( (150, -200) \). Determine the value of \( a + c \).

Solution: \boxed{a+c = -600}


The median of the set $\{m, m + 2, m + 7, m + 11, m + 18\}$ is 16. What is the mean?


A sports league has 20 teams split into four divisions of 5 teams each. How many games are there in a complete season if each team plays every other team in its own division three times and every team in the other divisions once?


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


Find the sum of all \(x\) that satisfy the equation \(\frac{-12x}{x^2-1} = \frac{3x}{x+1} - \frac{7}{x-1}\).


In a new sequence, the first term is \(a_1 = 5000\) and the second term is \(a_2 = 5001\). Furthermore, the values of the remaining terms are designed so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \( n \geq 1 \). Determine \(a_{1000}\).


Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $500$. What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$?

$\bullet$ Sequence $A$: $3$, $9$, $27$, $81$, $\ldots$

$\bullet$ Sequence $B$: $10$, $25$, $40$, $55$, $\ldots$


The graph of the parabola $x = 3y^2 - 9y + 5$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


A ball is dropped from 2000 feet high and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


Given that $$(x+y+z)(xy+xz+yz)=36$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=12$$ for real numbers $x$, $y$, and $z$, and also given that $$(x+y+z)^2=x^2+y^2+z^2+12,$$ what is the value of $xyz$?


If the operation $\Diamond$ is redefined as $a \Diamond b = a + ab^2 - b + 1$, what is the value of $(-2) \Diamond 3$?


John draws a regular six-pointed star in the sand, placing a unique sea shell at each of the 6 outward-pointing and 6 inward-pointing vertices, making a total of 12 different points with 12 different sea shells. How many ways can he arrange the shells if reflections and rotations of the arrangement are considered equivalent?


For how many integers $n$ between 1 and 19 (inclusive) is $\frac{n}{18}$ a repeating decimal?


An arithmetic sequence with a first term of $3$ has a common difference of $8$. A second sequence begins with $5$ and has a common difference of $9$. In the range of $1$ to $150$, what is the largest number common to both sequences?


Find the inverse of the matrix
\[\begin{pmatrix} 4 & x \\ -2 & 6 \end{pmatrix},\]
where \(x\) is a parameter such that the matrix is invertible. If the inverse does not exist, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} \frac{6}{24 + 2x} & \frac{-x}{24 + 2x} \\ \frac{2}{24 + 2x} & \frac{4}{24 + 2x} \end{pmatrix}}


Define a set of integers as "chilly" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are chilly?


Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 12$, what is $a_2a_7$?


Find the sum of all integers $n$ such that $n^2 - 21n + 110$ is a perfect square and $n$ is a multiple of $4$.


If $a$, $b$, and $c$ are integers such that $x^3 - x^2 - x - 1$ is a factor of $ax^4 + bx^3 + cx^2 + 1,$ find $c.$


In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written?


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

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


Let $a,$ $b,$ $c$ be positive real numbers. Find the set of all possible values of
\[ g(a,b,c) = \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}. \]


When a polynomial is divided by $5x^3 - 7x^2 + 3x - 8,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Find the radius of the circle with the equation $x^2 + 8x + y^2 - 10y + 41 = 0$.


A right triangle has sides of length 6, 8, and 10. What is the distance between the centers of the inscribed and circumscribed circles of the triangle?


A four-inch wooden cube is fully painted blue on all six faces. This cube is then uniformly cut into one-inch cubes. Determine how many of the one-inch cubes have blue paint on at least two faces.


Each of the numbers $a_1, a_2, \dots, a_{100}$ is $\pm 1.$ Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 100} a_i a_j.\]


In Morse code, each symbol is represented by a sequence of dashes and dots. How many distinct symbols can be represented using sequences of 1, 2, 3, 4, or 5 total dots and/or dashes, given that each sequence must contain at least one dot?


Convert the binary number $11011001_2$ to base 4.


Consider the set of all triangles $ABC$ where $A$ is the origin and $B$ and $C$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $37x + y = 1853$. Find the number of such distinct triangles whose area is a positive integer.


What is the least integer value of $x$ such that $\lvert 3x + 4 \rvert \le 18$?


How many positive divisors does 8! have?


The planet Zephyr moves in an elliptical orbit with its star at one of the foci. At its nearest point (perihelion), it is 3 astronomical units (AU) from the star, while at its furthest point (aphelion) it is 15 AU away. Determine the distance of Zephyr from its star when it is exactly halfway through its orbit.

Solution: \boxed{9 \text{ AU}}


At the end of the year, the Science Club decided to hold an election for which 6 equal officer positions were available. However, 20 candidates were nominated, of whom 8 were past officers. Of all possible elections of the officers, how many will have at least 2 of the past officers?


The volume of a certain rectangular solid is $512 \text{ cm}^3$, its total surface area is $384 \text{ cm}^2$, and its dimensions are in geometric progression. Find the shortest side length of this solid.

Solution: \boxed{8 \text{ cm}}


Compute the distance between the parallel lines given by
\[
\begin{pmatrix} 3 \\ -4 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}
\]
and
\[
\begin{pmatrix} 0 \\ 0 \end{pmatrix} + s \begin{pmatrix} 2 \\ -3 \end{pmatrix}.
\]


Mike sells one enchilada and four tacos for $3.50, and he sells four enchiladas and one taco for $4.20. Assuming a fixed price per item, what is the cost, in dollars, of two enchiladas and five tacos? Express your answer as a decimal to the nearest hundredth.


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 370^\circ.$


A fair 8-sided die is rolled once. If I roll $n$, then I win $8-2n$ dollars. What is the expected value of my win, in dollars?


The product of two consecutive negative integers is 2720. What is the sum of these two integers?


For how many integer values of \( n \) between 1 and 449 inclusive does the decimal representation of \( \frac{n}{560} \) terminate?


Points \(P\), \(Q\), \(R\), and \(S\) are midpoints of the sides of a large rectangle with length \(10\) and width \(6\). Calculate the area of the smaller quadrilateral formed by connecting these midpoints.


Two cylindrical silos are used to store grain, and both have the same volume. The height of one silo is four times the height of the other. If the radius of the taller silo is 15 units, determine the radius of the shorter silo in simplest radical form.


When a polynomial is divided by $3x^3 - 4x^2 + 9x - 5,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


A sphere is inscribed in a right cone with base radius $16$ cm and height $32$ cm. Determine the radius of the sphere, which can be expressed in the form $a\sqrt{c} - a$ cm. What is the value of $a + c$?


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2cf(x)y\]
for all real numbers $x$ and $y$, where $c$ is a constant. Determine all possible values of $f(2)$, find the sum of these values, and calculate the product of the number of possible values and their sum.


Let $\mathbf{B}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{B}^4 = \mathbf{0}.$  Determine the number of different possible matrices that $\mathbf{B}^2$ can be.  If you think the answer is infinite, then enter "infinite".


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


For my French exam, there are 800 vocabulary words. The exam grade is the percentage of these words that I recall correctly. If I learn a word, I will recall it correctly. Surprisingly, due to similar sounding words, for every 10 words I guess, I recall 1 word correctly by chance. What is the minimum number of words I should learn to get at least $90\%$ on my exam?


A school has between 150 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into seven distinct sections of students. 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?


What is the value of $a$ for which $\frac{1}{\log_2a} + \frac{1}{\log_3a} + \frac{1}{\log_4a} + \frac{1}{\log_5a} = 1$?


The measures of angles \( X \) and \( Y \) are both positive, integer numbers of degrees. The measure of angle \( X \) is a multiple of the measure of angle \( Y \), and angles \( X \) and \( Y \) are supplementary angles. How many measures are possible for angle \( X \)?


$PQRS$ is a trapezoid where base $\overline{PQ}$ is three times the length of base $\overline{RS}$. Point $T$ is the intersection of the diagonals. The length of diagonal $\overline{PR}$ is 15. Find the length of segment $\overline{RT}$.


Parallelogram $ABCD$ has vertices $A(2,5)$, $B(4,9)$, $C(6,5)$, and $D(4,1)$. Parallelogram $ABCD$ is first reflected across the $x$-axis to form $A'B'C'D'$, and then $A'B'C'D'$ is reflected across the line $y=-x+1$. Find the coordinates of $D''$, the image of $D'$ after the second reflection.

Solution: \boxed{(2, -3)}


In a club of 25 members, 10 are left-handed, and 18 members enjoy jazz music. 2 of the members are right-handed and dislike jazz music. How many club members are left-handed and enjoy jazz music? Assume people are either left-handed or right-handed, but not both.


Compute
\[\frac{1}{\sin 70^\circ} - \frac{\sqrt{3}}{\cos 70^\circ}.\]


Coach Randall is preparing a 6-person starting lineup for her soccer team, the Rangers, which has 15 players. Among the players, three are league All-Stars (Tom, Jerry, and Spike), and they are guaranteed to be in the starting lineup. Additionally, the lineup must include at least one goalkeeper, and there is only one goalkeeper available among the remaining players. How many different starting lineups are possible?


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=10$. Find the volume of the tetrahedron.


Let $S$ be a positive integer whose only digits are 0s and 1s. If $Y = S \div 15$ and $Y$ is an integer, what is the smallest possible value of $Y$?


The line \(y = -\frac{2}{3}x + 10\) crosses the \(x\)-axis at point \(P\) and the \(y\)-axis at point \(Q\). Point \(T(r, s)\) is on the segment \(PQ\). If the area of \(\triangle POQ\) is four times the area of \(\triangle TOP\), find the value of \(r + s\).


The United States Postal Service charges an extra $\$0.11$ in postage if the ratio of the length of an envelope, in inches, to its height, in inches, is either less than $1.2$ or greater than $2.6.$ Determine for how many of the following envelopes the extra charge applies:
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
A &5 &4\\
B &10 &4\\
C &8 &8\\
D &12 &5
\end{tabular}


Find the number of square units in the area of the shaded region. The region is bounded by a square and two triangles within the square. The vertices of the square are at (0,0), (40,0), (40,40), and (0,40). The vertices of the shaded region are (0,0), (10,0), (40,30), (40,40), (30,40), and (0,20).


Let \( x, y, \) and \( z \) be positive real numbers such that \( x + y + z = 9 \). Find the minimum value of
\[
\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.
\]


Let $n$ be a positive integer and $a,b,c,d$ be integers such that $a\equiv c^{-1} \pmod{n}$ and $b\equiv d^{-1} \pmod{n}$. What is the remainder when $(ab + cd)$ is divided by $n$?


How many prime numbers between 50 and 100 have a prime remainder when divided by 7?


The arithmetic mean, geometric mean, and harmonic mean of numbers $x$, $y$, $z$ are $10$, $5$, and $4$, respectively. What is the value of $x^2+y^2+z^2$?

Solution: \boxed{x^2+y^2+z^2 = 712.5}


Alice and Charlie each arrive at a café at a random time between 3:00 and 4:00 PM. Each stays for 20 minutes. What is the probability that Alice and Charlie see each other at the café?


Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she gives away $7$ candies, she will have a multiple of $9$. If she finds $9$ candies, she will have a multiple of $7$. What is the smallest possible value of $n$?


Erich has now evolved his tastes and likes only numbers that are divisible by both 5 and 3. How many different last digits are possible in numbers that Erich now likes?


A bus arrives randomly some time between 1:00 PM and 2:30 PM at a station and waits for 30 minutes before leaving. If Joe arrives randomly at the station any time between 1:00 PM and 2:30 PM, what is the probability that he will find the bus still at the station?


The Lakers are playing the Celtics in the NBA finals. To win the NBA finals, a team must win 4 games. If the probability of the Celtics winning each game is $\frac{3}{4}$ and there are no ties, what is the probability that the Lakers win the NBA finals in exactly seven games? Express your answer as a fraction.


How many ways are there to put 6 balls in 3 boxes if the balls are indistinguishable and the boxes are also indistinguishable?


Find the least integer value of $x$ for which $3|2x - 1| + 6 < 24$.


Write $x^{12} + x^6 + 1$ as the product of two polynomials with integer coefficients, where $x^3 \neq 1$.

Solution: \boxed{(x^6 + x^3 + 1)(x^6 - x^3 + 1)}


A $\frac{1}{5}$-array is structured with similar rules:
\[
1 \qquad \frac{1}{5}\quad \frac{1}{25} \quad \frac{1}{125} \quad \cdots\\
\frac{1}{10} \quad \frac{1}{50} \quad \frac{1}{250} \quad \cdots\\
\frac{1}{100} \quad \frac{1}{500} \quad \cdots\\
\frac{1}{1000} \quad \cdots\\
\ddots
\]
Each entry in the $r$th row and the $c$th column is derived from multiplying the first value of the row by $\frac{1}{5}$ repeatedly. The first entry of each row is $\frac{1}{10}$ times the first entry of the previous row. If the sum of all terms in a $\frac{1}{5}$ -array can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, compute $m+n$ modulo $100$.


Hannah drives her car to visit her friend. The odometer reads 32,150 when she leaves her house, and it reads 32,178 when she arrives at her friend's house. Hannah's car consumes fuel at a rate of 25 miles per gallon, and the price of one gallon of gas is $\$3.75$. Calculate the cost of the gas used for Hannah's trip. (Express your answer in dollars and round to the nearest cent.)


Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x + 5.
\]


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.


Simplify $\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{5}}{\sqrt{11}} \cdot \frac{\sqrt{6}}{\sqrt{8}}$ and rationalize the denominator of the resulting fraction.


Given that $$(x+y+z)(xy+xz+yz)=36$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=10$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


On a standard die, one additional dot is added to a randomly chosen face with each face equally likely to receive the dot. The die is then rolled. What is the probability that the top face has an even number of dots?


What is the last digit of the decimal expansion of $\frac{1}{3^{15}}$?


For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[6+6r+6r^2+6r^3+\cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=504$. Find $T(b)+T(-b)$.


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


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


Calculate the value of $\left(\frac{\frac{1}{2} + \frac{1}{3}}{\frac{2}{7} + \frac{1}{4}}\right) \times \frac{3}{5}$ and express your answer as a simplified fraction.


A rectangle is divided by line segments. Initially, one line segment, $PQ$, divides the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be divided when $5$ line segments (including $PQ$) are drawn through the rectangle?


For what values of the constant \( k \) does the graph of \( g(x) = \frac{x^2 + 3x + k}{x^2 - 3x - 18} \) have exactly one vertical asymptote?

Solution: \boxed{-54 \text{ or } 0}


A revised proposal suggests making years that end in double zeroes a leap year only if the year leaves a remainder of 200, 600, 300, or 700 when divided by 1000. How many leap years will there be that end in double zeroes between 2000 and 4000 under this new proposal?


Let \(a,\) \(b,\) and \(c\) be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 1)(b^2 + 4b + 1)(c^2 + 4c + 1)}{abc}.
\]


Let $r(\theta) = \frac{1}{2-\theta}$. What is $r(r(r(r(r(r(30))))))$ (where $r$ is applied $6$ times)?


Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC=12$. Find the radius of the incircle of $\triangle ABC$.

Solution: \boxed{12 - 6\sqrt{2}}


A segment with endpoints at $A(3, 3)$ and $B(9, 9)$ is extended beyond point $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates for point $C$?

Solution: \boxed{(12, 12)}


Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be vectors, and let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$. Then the determinant of the matrix whose column vectors are $\mathbf{b} \times \mathbf{c},$ $\mathbf{c} \times \mathbf{a},$ and $\mathbf{a} \times \mathbf{b}$ is equal to
\[k \cdot D^n.\]
Determine the ordered pair $(k,n)$.


If Alex gives Bob a penny, Bob will have four times as many pennies as Alex has. If Bob gives Alex two pennies, Bob will have twice as many pennies as Alex has. How many pennies does Bob currently have?


The sequence $5184, 1296, 324, \ldots$, is created by repeatedly dividing by 4. How many integers are in this sequence?


The mean of the set of numbers $\{92, 88, 94, 90, x\}$ is 91. What is the median of these five numbers when another number, $y=93$, is added to the set?


Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$, and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(150, -150),$ what is the value of $a + c$?


Compute the value of:
\[
\left( 1 - \frac{1}{\cos 40^\circ} \right) \left( 1 + \frac{1}{\sin 50^\circ} \right) \left( 1 - \frac{1}{\sin 40^\circ} \right) \left( 1 + \frac{1}{\cos 50^\circ} \right).
\]


A cube has side length 10 units. Its vertices are alternately colored blue and red similar to a standard chessboard arrangement. What is the volume of the tetrahedron whose vertices are the blue vertices of the cube?


In a circle with center $R$, there are two radii $RS$ and $RT$ each measuring 7 units. Calculate the perimeter of the shaded region formed by these two radii and a $\frac{5}{6}$ arc connecting points $S$ and $T$.

Solution: \boxed{14 + \frac{35\pi}{3}}


There are approximately 0.4545 kilograms in a pound. To the nearest whole pound, how many pounds does a bear that weighs 300 kg weigh?

Solution: \boxed{660\text{ pounds}}


Given that $x$ is a positive integer less than 70, how many solutions does the congruence $x + 20 \equiv 45 \pmod{26}$ have?


Suppose $x$ is directly proportional to $y^3$ and inversely proportional to $w^2$, and $y$ is inversely proportional to $\sqrt{z}$. If $x = 5$ when $z = 16$ and $w = 2$, find the value of $x$ when $z = 64$ and $w = 4$.


Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote $\sin x$, $\tan x$, and $\sec x$ on three different cards. He then distributed these cards to three students: Isabelle, Edgar, and Sam, one card to each. After sharing the values on their cards with each other, only Isabelle could accurately identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Isabelle's card.


In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$.


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


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


A circle is tangent to the lines \(3x + 4y = 40\) and \(3x + 4y = -20.\) The center of the circle lies on the line \(x - 2y = 0.\) Find the center of the circle.


How many three-digit whole numbers have no 5's and no 8's as digits?


Jordan was issued an invoice of $\$600$ which was overlooked and incurred a $2\%$ late charge on the $30^{\text{th}}$ day past its due date. Thirty days after this, the total sum had another $2\%$ late fee added. Furthermore, Jordan requested a service which added an additional $30$ charge before the first late fee was applied. Calculate Jordan's total bill.


The sides of a triangle have lengths $13, 17,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse and scalene?


Given that $\sin A+\sin B=0.75$ and $\cos A + \cos B = 1$, determine the value of $\cos(A-B)$.


The planet Zytron orbits in an elliptical path around its star with the star located at one of the foci of the ellipse. Zytron’s closest approach to its star (perihelion) is 3 astronomical units (AU), and its furthest point (aphelion) is 8 AU away. How far is Zytron from its star when it is exactly halfway in its orbit along the major axis?

Solution: \boxed{5.5 \text{ AU}}


Let $C$ be a point not on line $AF$ and $D$ a point on line $AF$ such that $CD \perp AF$. Meanwhile, $B$ is a point on line $CF$ such that $AB \perp CF$. If $AB = 6$, $CD = 3$, and $AF = 12$, then what is the length of $CF$?


What is the sum of the tens digit and the units digit in the decimal representation of $8^{1502}$?


Allison, Brian, and Noah each have a die. All of the faces on Allison's die have a 5. The faces on Brian's die are numbered 1, 2, 3, 4, 4, 5, 5, and 6. Noah's die has an 8-sided die with the numbers 2, 2, 6, 6, 3, 3, 7, and 7. All three dice are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction.


How many integers, \( x \), satisfy \( |4x - 5| < 9 \)?


Let \( p \) be the smallest three-digit integer congruent to 7 (mod 13). Let \( q \) be the smallest four-digit integer congruent to 7 (mod 13). What is \( q-p \)?


At what points do the line $2y + 5x = 10$ intersect the x-axis and y-axis? Express your answers as ordered pairs.


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


How many integers \( m \neq -4, 4 \) satisfy the inequality \( \frac{1}{|m|} \geq \frac{1}{5} \)?


What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{1176}$ is defined?


A flagpole is supported by a wire which extends from the top of the pole to a point on the ground 4 meters from its base. When Ana walks 3 meters from the base of the pole toward the point where the wire is attached to the ground, her head just touches the wire. Ana is 1.6 meters tall. How tall is the flagpole?

Solution: \boxed{6.4\text{ meters}}


Suppose the graph of a function, $y=f(x)$, has the property that if it is shifted $12$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.

What is the smallest positive $a$ such that if the graph of $y=f\left(\frac x3\right)$ is shifted $a$ units to the right, then the resulting graph is identical to the original graph of $y=f\left(\frac x3\right)$?


Evaluate the determinant:
\[
\begin{vmatrix}
1 & \sin \alpha & -\cos \alpha \\
-\sin \alpha & 0 & \sin(\alpha + \beta) \\
\cos \alpha & -\sin(\alpha + \beta) & 1
\end{vmatrix}.
\]

Solution: \boxed{\sin^2(\alpha + \beta) + \sin^2 \alpha}


In rectangle $ABCD$, $AB=150$. Let $E$ be the midpoint of $\overline{AD}$. Given that line $AC$ and line $BE$ are perpendicular, find the length of $AD$.

Solution: \boxed{150 \sqrt{2}}


A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood small blue hexagon' in exactly 2 ways?


Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be two matrices such that $\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}.$ Assuming $2b \neq 3c,$ find $\frac{a - d}{c - 2b}.$


Find all real numbers $x$ such that \[4 \le \frac{2x}{3x-7} < 9.\] (Give your answer in interval notation.)

Solution: \boxed{(\frac{63}{25}, 2.8]}


Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $k$ such that $Q$ is divisible by $3^k.$


Consider the expansion of \((x + 2)^n\) where three consecutive terms have coefficients in the ratio \(1:3:5\). These can be expressed as \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Determine the sum of all possible values of \(n+k\).


How many different three-letter sets of initials are possible using the letters $A$ through $J$ if one and only one of the letters must be a vowel?


Determine how many zeros are at the end of the product $35 \times 4900$.


Seven couples are at a social gathering. If each person shakes hands exactly once with everyone else except their spouse and one other person they choose not to shake hands with, how many handshakes were exchanged?


Jane is constructing a rectangular cardboard box with square bases for a school project. She wants the height of the box to be 5 units greater than the side of the square bases. What should the height be if she wants the surface area of the box to be at least 150 square units while using the least amount of cardboard? Additionally, the perimeter of one square base plus the height should be at least 25 units.


A quadrilateral with one pair of parallel sides (trapezoid) has its consecutive angles in an arithmetic sequence. The sum of the smallest and largest angles equals $200^{\circ}$. If the second-smallest angle is $70^{\circ}$, find the measure of the third-largest angle.


How many different positive three-digit integers can be formed using only the digits in the set $\{2, 2, 3, 4, 4, 4, 7\}$ if no digit may be used more times than it appears in the given set of available digits?


Let \(a, b, c, d, e\) be real numbers such that
\[a^2 + b^2 + c^2 + d^2 + e^2 = 5.\]
Find the maximum value of \(a^3 + b^3 + c^3 + d^3 + e^3.\)


If $x$, $y$, and $z$ are positive with $xy=40$, $xz=80$, and $yz=160$, what is the value of $x+y+z$?


Determine if there exists a matrix $\mathbf{N}$ such that
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & c \\ b & a \end{pmatrix}.\]
If no such matrix $\mathbf{N}$ exists, then provide the zero matrix.

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


A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 5 meters from its base. Additionally, a second cable extends from the midpoint of the pole to a point on the ground 2 meters from its base. When Leah walks 4 meters from the base of the pole toward the point where the first cable is anchored to the ground, her head just touches the first cable. Leah is 1.7 meters tall. How many meters tall is the pole?

Solution: \boxed{8.5\text{ meters}}


Suppose that $b$ is a nonzero constant for which the equation $2bx^2 + 16x + 5 = 0$ has only one solution. Find this solution.


**Find the remainder when \( x^4 + 1 \) is divided by \( x^2 - 4x + 7 \).**

Solution: \boxed{8x - 62}


What is the largest four-digit number that is divisible by 8?


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


Calculate the product \(\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{3004}{3008}\right)\). Express your answer as a common fraction and then add \(\frac{1}{100}\) to the result.


Determine the rightmost three digits of $7^{1987}$.


15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and does NOT place the ball back in the bin. How many different lists are possible?


Solution: \boxed{36x^3(2 - 7x^4)}


Find the matrix that corresponds to rotating about the origin by an angle of $45^\circ$ counter-clockwise followed by a reflection across the x-axis.

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


In the circle with center $O$, the measure of $\angle SIP$ is $45^\circ$ and $OS=12$ cm. Find the number of centimeters in the length of arc $SP$.


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


A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet?


In how many ways is it possible to arrange the digits of 11250 to get a five-digit multiple of 5?


Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[Area(WXYZ) = \alpha \omega - \beta \omega^2.\]
Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


A large circular pizza with a diameter of $16\text{ cm}$ is to be equally divided into four quarter-circle sectors by two perpendicular diameter cuts. What is the length of the longest line segment that can be drawn within one of these quarter-circle sectors, and find $l^2$?


Determine how many prime dates occurred in 2008, a leap year. A "prime date" is when both the month and the day are prime numbers.


Find the largest solution to \[\lfloor x \rfloor = 7 + 50 \{ x \},\]where $\{x\} = x - \lfloor x \rfloor.$


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


Find $x$ such that $\log_x 16 = \log_{64} 4$.


Calculate $\sqrt{48x} \cdot \sqrt{3x} \cdot \sqrt{50x}$. Express your answer in simplest radical form in terms of $x$.

Solution: \boxed{60x \sqrt{2x}}


Determine the ratio of the area of a square inscribed in a semicircle with radius $2r$ to the area of a square inscribed in a circle with radius $r$. Express your answer as a common fraction.


Each vertex of a new parallelogram, also having integer coordinates, is located at $(1,3)$, $(6,8)$, $(13,8)$, and $(8,3)$. Calculate the perimeter $p$ and the area $a$ of this parallelogram, and find the value of the sum $p + a$.

Solution: \boxed{p + a = 10\sqrt{2} + 49}


Let $x = (3 + \sqrt{8})^{1000}$, let $n = \lfloor x \rfloor$, and let $f = x - n.$  Find
\[
x(1 - f).
\]


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 5$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right|$?


The function \( g(x) \) satisfies \( g(xy) = g(x) g(y) \) for all real numbers \( x \) and \( y \), and \( g(1) \neq 0 \). Find \( g(10) \).


Find the value of $m$ such that the parabola $y = x^2 + 4$ and the hyperbola $y^2 - mx^2 = 4$ are tangent.

Solution: \boxed{m = 8 + 4\sqrt{3}}


We have a triangle $\triangle DEF$ such that $DE = DF = 13$ and $EF = 24.$ What is the length of the median $DM$?


Six horizontal lines and five vertical lines are drawn on a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed with an area of at least 1 square unit?


In triangle $ABC$, $AB=130$, $AC=110$ and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.


Alex runs 4 miles in twice the time it took Jordan to run 3 miles. Jordan runs 3 miles in one-third the time it takes Steve to run 9 miles. If it takes Steve 36 minutes to run 9 miles, using similar rates, how long would it take Alex to run 6 miles?


A square has sides of length 20, and a circle centered at the center of the square has a radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.


What is the sum of the first twelve positive multiples of $17$, and what is the sum of the squares of these multiples?


Find $y$ such that $\log_y 64=\log_4 256 + 1$.


Simplify the following expression:

$$7y + 8 - 3y + 15 + 2x$$

Solution: \boxed{4y + 2x + 23}


If 10 bahs are equal to 16 rahs, and 6 rahs are equal in value to 10 yahs, how many bahs are equal in value to 500 yahs?


Point \( G \) is the midpoint of \( \overline{XY} \), point \( H \) is the midpoint of \( \overline{XG} \), point \( I \) is the midpoint of \( \overline{XH} \), and point \( J \) is the midpoint of \( \overline{XI} \). If \( XJ = 4 \), what is the number of units in the length of \( \overline{XY} \)?


What integer \( n \) satisfies \( 0 \leq n < 11 \) and
$$
1234 \equiv n \pmod{11}~?
$$


Determine the number of real solutions for $x$ in the equation: $$(-4x + 12)^2 + 1 = (x-1)^2.$$


The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 15 cents. If she had one more quarter, the average value would be 16 cents. How many nickels does she have in her purse?


Initially, Jenny has a certain number of jellybeans in her jar. Each day, she eats 25% of the jellybeans that were in her jar at the beginning of that day. After consuming jellybeans on the first day, an additional 20 jellybeans were put into the jar. At the end of the third day, 45 jellybeans remained. How many jellybeans were in the jar originally?


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 $\angle F$?


Find the number of odd digits in the base-4 representation of $157_{10}$.


Express the number twenty-five in base 3.


Circle $C$ has its center at $C(5, 5)$ and has a radius of 3 units. Circle $D$ has its center at $D(14, 5)$ and has a radius of 3 units. What is the area of the gray region bound by the circles and the $x$-axis?
```asy
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,18,Ticks("%",1.0));
yaxis(0,9,Ticks("%",1.0));
fill((5,5)--(14,5)--(14,0)--(5,0)--cycle,gray(0.7));
filldraw(circle((5,5),3),fillpen=white);
filldraw(circle((14,5),3),fillpen=white);
dot("$C$",(5,5),S); dot("$D$",(14,5),S);
```

Solution: \boxed{45 - \frac{9\pi}{2}}


In triangle $ABC$, angle $ACB$ is 60 degrees, and angle $CBA$ is 70 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.


Let
\[x^6 - 2x^4 - x^2 + 2 = q_1(x) q_2(x) \dotsm q_m(x),\] where each \(q_i(x)\) is a monic polynomial with integer coefficients, and cannot be factored further over the integers. Compute \(q_1(3) + q_2(3) + \dots + q_m(3).\)


Given that eleven is the fifth prime number, what is the fifteenth prime number?


The arithmetic mean of four numbers is 20. Three of the numbers are 12, 24, and 36, and the fourth number is the square of another number. What is the fourth number?


What integer $n$ satisfies $0 \leq n < 23$ and $$45689 \equiv n \pmod{23}~?$$


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


Determine the real values of $b$ where the expression $\frac{b+3}{b^2 - 9}$ is undefined.

Solution: \boxed{-3, 3}


How many even integers between 3000 and 8000 have all different digits?


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product $abc = 72$.


For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $3n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $2n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there is at least one value of $n$ such that $C_n - B_n = A_n^2$?


Determine the equation of the line on which all vectors $\mathbf{v}$ lie such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{9}{2} \\ 6 \end{pmatrix}.\]
The equation should be in the form "$y = mx + b$".

Solution: \boxed{y = -\frac{3}{4}x + \frac{75}{8}}


Consider a center square surrounded by concentric rings of unit squares, where each ring represents a step further from the center. The center square is of size 1x1 unit. The first ring around the center square comprises 8 unit squares, the second ring 16 unit squares, continuing in this pattern. Calculate the total number of unit squares present up to and including the $50^{th}$ ring.


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


Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?


A traffic light operates in a cycle as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 40 seconds. Sam decides to watch the light for a four-second interval. What is the probability that the light changes color during his observation?


In Pascal's Triangle, each entry is the sum of the two entries directly above it. Determine the row number where three consecutive entries have the ratios 2:3:4.


The matrix for reflecting over a certain line $\ell$, which passes through the origin, is given by
\[
\begin{pmatrix} \frac{5}{13} & \frac{12}{13} \\ \frac{12}{13} & -\frac{5}{13} \end{pmatrix}.
\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix},$ where $a,$ and $b$ are integers, $a > 0,$ and $\gcd(|a|,|b|) = 1.$

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


In how many ways can 8 people be seated in a row of chairs if two specific pairs of people, namely Wilma and Paul, and Harry and Sally, refuse to sit next to each other?


Evaluate the polynomial \[ x^3 - 3x^2 - 10x + 5, \] where \(x\) is the positive number such that \(x^2 - 3x - 10 = 0\).


Dr. Math's four-digit house number $ABCD$ is such that it can be split into $AB$ and $CD$, both different two-digit primes. Additionally, each prime is less than 50, and $AB > CD$. How many such house numbers are possible?


Determine the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{14} & \frac{3}{14} & -\frac{1}{7} \\ \frac{3}{14} & \frac{9}{14} & -\frac{3}{7} \\ -\frac{1}{7} & -\frac{3}{7} & \frac{2}{7} \end{pmatrix}}


Express $\frac{0.\overline{714}}{2.\overline{857}}$ as a common fraction.


What is the smallest whole number \(b\) such that 100 can be expressed in base \(b\) using only three digits?


In $\triangle PQR$, point $M$ is the midpoint of side $PQ$. Point $N$ is on $PR$ such that $PN:NR = 2:1$. Point $S$ is on $PM$ such that $PS:SM = 1:2$. If the area of $\triangle MNS$ is 10, determine the area of $\triangle PQR$.


John tracked his walking distance over several weeks, and his pedometer showed a total of $39457_9$ miles. How many miles (in base ten) did John walk?


The tail of a 1-mile long train exits a tunnel 3 minutes after the front of the train entered the tunnel. If the train is moving at 80 miles per hour, how many miles long is the tunnel?

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


Let $O$ be the origin. A variable plane has a distance of 2 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the centroid of triangle $ABC.$ Find
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.\]


A fair 8-sided die is rolled. If I roll $n$, then I win $n^3$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


For what real value of $u$ is $\frac{-15-\sqrt{145}}{6}$ a root of $3x^2 + 15x + u$?


In a right triangle $DEF$ where $\angle D = 90^\circ$, sides $DE = 5$ and $EF = 13$. Find $\cos F$.


Olivia, Sophie, and Max share a sum of money. Their portions are in the ratio of 2:3:8, respectively. If Olivia's portion is $\$$30, what is the total amount of money shared?


A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to twice the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$.


What is the sum of the different prime factors of $172944?$


A region $S$ in the complex plane is defined by
\[
S = \{x + iy: -2 \le x \le 2, -2 \le y \le 2\}.
\]
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac12 + \frac12i\right)z$ is also in $S$?


A club has $4$ members from each of $4$ different schools, for a total of $16$ members. How many possible ways are there to arrange a presidency meeting under the following conditions:

i. The club must choose one of the $4$ schools at which to host the meeting, and

ii. The host school sends $3$ representatives to the meeting, and each of the other three schools sends $1$ representative.


Solve for $y$ in the equation by completing the square: $$y^2 - 6y + 5 = 0.$$

Solution: \boxed{1 \text{ and } 5}


Below is the graph of \(y = a \sec bx\) for some positive constants \(a\) and \(b\). The maximum value of \(y\) observed in the first positive section of the graph is 3. Find \(a\).

[No actual graph provided, but imagine a similar setup as the original, with possible compression or expansion in the x-axis and shifted vertical asymptotes]

Solution: \boxed{a = 3}


Evaluate $\lfloor -6 -0.5 \rfloor \cdot \lceil 6 + 0.5 \rceil \cdot \lfloor -5 -0.5 \rfloor \cdot \lceil 5 + 0.5 \rceil \cdot \dots \cdot \lfloor -0.5 \rfloor \cdot \lceil 0.5 \rceil$.


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 21$, what is $b_3b_6$?


What is the largest five-digit integer whose digits have a product equal to the product $(6)(5)(4)(3)(2)(1)$?


Matilda places a total of 30 purple Easter eggs in several yellow baskets and a total of 45 pink Easter eggs in red baskets. Each basket contains the same number of eggs and there must be at least 3 eggs in each basket. How many eggs did Matilda put in each basket?


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

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


Solution: \boxed{\log_2 \left(\frac{44}{3}\right)}