The base-10 numbers 359 and 72 are multiplied. The product is then written in base-7. What is the units digit of the base-7 representation?


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


Rectangle A has a perimeter of 40 cm, and its length is twice its width. Rectangle B has an area equal to one-half the area of rectangle A. What is the perimeter of rectangle B?

Solution: \boxed{20\sqrt{2} \text{ cm}}


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


You have a rectangular prism box with length $x+3$ units, width $x-3$ units, and height $x^2 + 9$ units. For how many positive integer values of $x$ is the volume of the box less than 500 units?


Lisa has 12 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles? Additionally, How many marbles did the 3rd, 7th, and 11th friends receive in total?


A hemisphere with radius 10 cm is attached to a cylindrical base of the same radius and 1 cm thickness. What is the total surface area of this structure? Express your answer in terms of $\pi$.


A triangle is inscribed in a circle. The triangle's vertices divide the circle into three arcs. The central angles opposite these arcs measure $60^\circ$, $120^\circ$, and $180^\circ$. Find the area of the triangle if the circle's radius is $\frac{18}{\pi}$.


A survey done by a radio station FM100 shows the following data. Determine how many of the males surveyed do not listen to this station.

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males & & 105 \\ \hline
females & 75 & \\ \hline
total & 160 & 200 \\
\hline
\end{tabular}


The stem-and-leaf plot shows the duration in minutes and seconds for one ride on each of the $21$ top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents $1$ minute, $45$ seconds, which is equivalent to $105$ seconds. What is the median of this data set? Express your answer in seconds.

\begin{tabular}{c|cccccc}
0 & 50 & 55 & 58 & 58 & 59 & \\
1 & 10 & 20 & 30 & 45 & 55 & 58\\
2 & 05 & 15 & 23 & 30 & 48 & 50\\
3 & 10 & 20 & 30 & 40 & \\
\end{tabular}


What is the minimum value of $w$ if $w=3x^2 + 4y^2 - 12x + 8y + 15$?


Solution: \boxed{25(3x + 2)}


Consider the polynomial \( Q(x) = x^3 + px^2 + qx + d \) where the mean of its zeros, the product of its zeros taken two at a time, and the sum of its coefficients are all equal. If the \( y \)-intercept of the graph of \( y = Q(x) \) is 7, determine the value of \( q \).


Suppose the graph of a function, $y = g(x)$, has the property that if it is shifted $30$ units to the right, then the resulting graph is identical to the original graph of $y = g(x)$.

What is the smallest positive $b$ such that if the graph of $y = g\left(\frac{x}{3}\right)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y = g\left(\frac{x}{3}\right)$?


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


How many digits are there in the base-7 representation of $2000$?


Determine the ordered pair $(x, y)$ if

\begin{align*}
x + y &= (7 - x) + (7 - y),\\
x - y &= (x - 2) + (y - 2).
\end{align*}

Solution: \boxed{(5, 2)}


Both \(a\) and \(b\) are positive integers and \(b > 1\). When \(a^b\) is the greatest possible value less than 500, what is the sum of \(a\) and \(b\)?


Determine the sum of the first eight terms of the geometric sequence starting with $\frac{1}{3}$ and having a common ratio of $\frac{1}{3}$. Express your result as a simplified fraction.


How many positive multiples of 7 that are less than 200 have a units digit of 7?


Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\]


The isosceles triangle and the rectangle shown here have the same area in square units. If the length of the rectangle is $l$ and the breadth is $b$, with $l > b$, what is the height of the triangle, $h$, in terms of $l$ and $b$?

[asy]
draw((0,0)--(0,5)--(18,5)--(18,0)--cycle);
fill((0,0)--(18,7)--(0,10)--cycle,white);
draw((0,0)--(18,7)--(0,10)--cycle);
label("$l$",(9,5),N);
label("$h$",(9,7),N);
label("$b$",(18,2.5),E);
draw((0,5)--(18,5),dashed);
draw((0,7.5)--(0.5,7.5)--(0.5,7));
[/asy]


The sum of the terms in an infinite geometric series is 12, and the sum of their squares is 54. Find the first term.


What is the area of the region defined by the equation \(x^2 + y^2 + 5 = 6y - 10x + 7\)?


Determine the number of solutions for the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-200^2)} = 0.\]


Let \(\bold{v} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}\) and \(\bold{w} = \begin{pmatrix} 2 \\ -9 \end{pmatrix}\). Find the area of the parallelogram with vertices \(\bold{0}\), \(\bold{v}\), \(2\bold{w}\), and \(\bold{v} + 2\bold{w}\).


Bert receives a box that can accommodate exactly $150$ jellybeans. A month later, Lisa receives a box, which is thrice as long, twice as high, and four times wider than Bert's box. Assuming the jellybean capacity is proportional to the volume, how many jellybeans did Lisa receive in her box?


The sum of three different numbers is 100. The largest number is twice the smallest number, and the difference between the two larger numbers is 10. What is the value of the smallest number?


Find an ordered pair $(x, y)$ that solves the system:
\[
\begin{align*}
7x + 3y &= 5,\\
4x^2 + 5y &= 9
\end{align*}
\]

Solution: \boxed{\left(\frac{35 - \sqrt{1321}}{24}, \frac{-125 + 7\sqrt{1321}}{72}\right)}


Suppose $-5 \leq x \leq -3$ and $3 \leq y \leq 5$, and $y$ must be an odd integer. What is the largest possible value of $\frac{x+y}{x}$?


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that
\[
\mathbf{a} + \mathbf{b} + 2 \mathbf{c} = \mathbf{0}.
\]
Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.


Solution: \boxed{\frac{7 - x}{8}}


In parallelogram $EFGH$, $EF = 42$ cm, $FG = 4y^2$ cm, $GH = 3x + 6$ cm, and $HE = 32$ cm. What is the product of $x$ and $y$?


Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible, meaning $f^{-1}(x)$ exists for all $x$ in the range of $f$.

If the graphs of $y=f(x^3)$ and $y=f(x^6)$ are drawn, at how many points do they intersect?


Express the quotient $1452_7 \div 14_7$ in base $7$.


Evaluate the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm.$  Enter your answer in the form "\sqrt[a]{b}".


The diagram shows the miles traveled by cyclists Alex and Bella. After six hours, how many more miles has Alex biked than Bella? The diagram includes two lines representing each cyclist's journey, with Alex's line steeper than Bella's. Assume Alex travels at a constant rate of 12 miles per hour and Bella at a rate of 10 miles per hour starting from the same point.


Linda constructs a cone for a science exhibit. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Linda needs to find the smallest cube-shaped box to accommodate the cone so that it can be transported vertically. What is the volume of this box, in cubic inches?


I had \$50 in allowance money and spent it according to the percentages indicated in the pie graph shown. How many dollars did I spend on books?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.866,.5);
B=(-.5,.866);
C=(-.866,-.5);
D=(.766,-.642);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Games", W, N);
label("$\frac{1}{4}$", W, S);
X=(-.6, 0);
label("Books", X, S);
Y=(-.17,-.7);
label("Snacks", Y, N);
label("$\frac{3}{20}$",  Y, S);
Z=(.5, -.15);
label("Clothes", Z, N);
label("$\frac{2}{5}$", Z, S);
[/asy]


The value of $y$ varies inversely as $x^2$, and when $x=4$, $y=1$. What is the value of $x$ when $y=0.25$?


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $40$ faces, consisting of $30$ triangular faces and $10$ pentagonal faces. How many space diagonals does polyhedron $Q$ have?


To bake 24 cookies, I use 1 gallon of milk. There are 4 quarts in a gallon. How many quarts of milk do I need to bake 6 cookies?

Solution: \boxed{1 \text{ quart}}


A circle with center $O$ has a radius of $12$ units, and a circle with center $P$ has a radius of $4$ units. The circles are externally tangent to each other at point $Q$. A line segment $TS$ is a common external tangent to the circles $O$ and $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.


Let \( Q = \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \left( 1 - \frac{1}{5} \right) \dotsm \left( 1 - \frac{1}{n+1} \right) \). What is the value of \( Q \) if \( n = 2010 \)? Express your answer as a common fraction.


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


Find the $4 \times 4$ matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{u} = -2 \mathbf{u}\] for all four-dimensional vectors $\mathbf{u}$, and
\[\mathbf{M} \mathbf{e_4} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ -2 \end{pmatrix},\]
where $\mathbf{e_4} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$ is the fourth standard basis vector.

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


Except for the first two terms, each term of the sequence $500, x, 500 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length?


Evaluate the range of the function $g(x) = \arcsin x + \arccos x + 2\arcsin x.$ All functions are in radians. Note that $g(x)$ is still defined only for $-1 \le x \le 1.$

Solution: \boxed{\left[-\frac{\pi}{2}, \frac{3\pi}{2}\right]}


What is the sum of all of the positive even factors of $630$?


Consider two functions $f(x)$ and $g(x)$ defined on $\mathbb{R}$ such that the range of $f$ is $[1, 6]$ and the range of $g$ is $[-4, -1]$. Determine the largest possible value of $b$ where $b$ is the maximum value in the range of the product $f(x) \cdot g(x)$.


Compute $\tan 30^\circ + 4 \sin 30^\circ.$

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


The second hand on a clock is 8 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Additionally, calculate the angular distance the second hand covers in degrees during this period. Express your answer in terms of \(\pi\) for the linear distance.


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


Find the smallest positive solution to
\[\tan 2x + \tan 4x = \sec 4x\] in radians.


The average of the numbers $1, 2, 3, \dots, 50, y$ is $51y$. What is $y$?


A custom deck of 65 cards includes an extra set of all spades (14 spades instead of 13), maintaining all other suits at 13 cards each. What is the probability that the first card drawn is an Ace of any suit and the second card drawn is any King?


Solve the equation \(9^{x+8} = 10^x\) and express \(x\) in the form \(x = \log_b 9^8\). What is \(b\)?


If $n$ is a positive integer, then let $g(n)$ be the sum of the digits of $\frac{1}{3^n}$ for the numbers to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 20$?


Points $A$, $B$, $R$, $D$, and $C$ lie on a circle, with $B$ and $D$ diametrically opposite each other. The measures of arcs $BR$ and $RD$ are $72^\circ$ and $108^\circ$, respectively. Find the sum of the measures of angles $P$ and $R$ in degrees.
[asy]
import graph;

unitsize(2 cm);

pair A, B, C, D, P, R;

A = dir(160);
B = dir(90);
C = dir(190);
D = dir(-90);
P = extension(A,B,C,D);
R = dir(0);

draw(Circle((0,0),1));
draw(B--P--D);
draw(A--R--C);

label("$A$", A, NW);
label("$B$", B, N);
label("$C$", C, SW);
label("$D$", D, S);
label("$P$", P, E*1.5);
label("$R$", R, E);
[/asy]


In a certain region, the rate of taxation is as follows: \(x\%\) tax is collected for an income of \(x+10\) thousand dollars. What income, in dollars, will yield the greatest take-home pay?


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


Call a $4$-digit number geometric if it has $4$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric $4$-digit numbers.


Given that $x$ is real and $x^4+\frac{1}{x^4}=240$, find $x^2+\frac{1}{x^2}$.


Find the remainder when $4x^5 - 9x^4 + 3x^3 + 5x^2 - x - 15$ is divided by $x - 4$.


If $\sec x + \tan x = \frac{5}{3},$ then find $\sec x - \tan x.$


Solution: \boxed{20x^5(4 - 9x^4)}


Solve for $y$: $\log_3 y + \log_9 y = 5$.


Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in a 1-foot length of this pattern? Express your answer in terms of $\pi$.


The graph of $y = h(x)$ is derived by modifying the graph of $y = f(x)$. The operations performed are reflecting the graph across the y-axis, reflecting again across the x-axis, and then shifting it 6 units to the right. Write the function $h(x)$ in terms of $f(x)$.

Solution: \boxed{-f(6 - x)}


How many integers between 1 and 300 are multiples of both 4 and 6 but not of either 5 or 8?


Solve the quadratic equation $x^2 - 6x + 18 = 28$. Find the roots $a$ and $b$, with $a \geq b$, and compute $a+3b$.

Solution: \boxed{12 - 2\sqrt{19}}


In a modified version of SHORT BINGO, a $5\times5$ card is filled with the middle square marked as WILD and 24 other numbers placed in the remaining squares. The numbers are placed as follows: 5 distinct numbers from $1-10$ in the first column, 5 distinct numbers from $11-20$ in the second column, 4 distinct numbers from $21-30$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $35-44$ in the fourth column, and 5 distinct numbers from $41-50$ in the last column. Moreover, the number 38 must appear in the second row of the fourth column.

How many distinct possibilities are there for the values in the fourth column of this modified SHORT BINGO card?


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


Lines $m$ and $n$ are parallel, and the measure of $\angle 1$ is $\frac{1}{6}$ the measure of $\angle 2$. If $\angle 3$ is equal in measure to $\angle 1$, and $\angle 2$ and $\angle 4$ form a straight line, what is the degree measure of $\angle 5$?


If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn?


Find the smallest positive integer that satisfies the following system of congruences:
\[
\begin{align*}
5x &\equiv 25 \pmod{20} \\
3x + 1 &\equiv 4 \pmod{7} \\
2x-3 &\equiv x \pmod{13} \\
\end{align*}
\]


Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 3$ and $QS = 9$, what is $RS$?


Calculate the number of diagonals in a regular nonagon (9-sided polygon). Also, for context, provide the number of diagonals for a regular octagon and a regular decagon.


Solve the equation:
\[
\frac{x^2 + 3x + 5}{x+3} = x + 6
\]


Calculate the product of one-half, two-thirds, three-quarters, and one and one-half.


My school's math club has 7 boys and 10 girls. I need to select a team of 5 students to send to a regional math competition. The selected team must include at least 2 girls. In how many ways can I select the team?


Find the range of the function
\[ f(x) = \frac{\cos^3 x + 6 \cos^2 x + \cos x + 2 \sin^2 x - 8}{\cos x - 1}, \]
as \( x \) ranges over all real numbers such that \( \cos x \neq 1 \).


The curve $y = \sin x$ cuts the line whose equation is $y = \sin 30^\circ$ into segments having the successive ratios
\[\dots p : q : p : q \dots\] with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p, q)$.

Solution: \boxed{(1, 2)}


What is the product of the largest odd one-digit prime and the two largest two-digit primes?


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


The modified Fibonacci sequence $F_n$ is defined by $F_0 = 4, F_1 = 3$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Find the units digit of $F_{F_{10}}$.


An equilateral triangle has an area of $81\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is increased by 3 cm, by how many square centimeters is the area increased?

Solution: \boxed{29.25\sqrt{3} \text{ cm}^2}


Let $x$ and $y$ be two positive real numbers such that $x + y = 20.$ Find the ordered pair $(x, y)$ that maximizes the value of $x^3y^5$.

Solution: \boxed{(7.5, 12.5)}


The expression $x^2 - 18x + 72$ can be written in the form $(x - c)(x - d)$, where $c$ and $d$ are both nonnegative integers and $c > d$. What is the value of $2d - c$?


The local theater has two ticket windows operating. In how many ways can six people line up and choose exactly one ticket window each to buy a ticket?


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?


A rectangular picture frame is made from two-inch-wide pieces of wood. The area of just the frame is $30$ square inches, and one of the outer edges of the frame is $7$ inches long. What is the sum of the lengths of the four interior edges of the frame?


An ellipse has its foci at $(3, -2)$ and $(11, 6)$. Find the coordinates of its center and calculate the length of the major axis, given that the distance between a point on the ellipse and each focus sums to 20.


A polynomial of degree $15$ is divided by $d(x)$ to give a quotient of degree $8$ and a remainder of $5x^2 + 3x - 9$. What is $\deg d$?


Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 72,$ then find $PQ^2 + PR^2 + QR^2.$


Calculate the value of $\left(7^2 - 3^2\right)^4$.


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


The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum?


A positive two-digit number is even and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?


Fifteen stones are arranged in a circle. They are counted clockwise as $1,2,3, \ldots, 15$, and then the count continues with the first stone as 16 and so on in a repeating pattern. Find the original stone number that is counted as 162.


Let \(\mathbf{A}\) and \(\mathbf{B}\) be matrices such that
\[
\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.
\]
If \(\mathbf{A} \mathbf{B} = \begin{pmatrix} 5 & 2 \\ -3 & 9 \end{pmatrix},\) find \(\mathbf{B} \mathbf{A}.\)

Solution: \boxed{\begin{pmatrix} 5 & 2 \\ -3 & 9 \end{pmatrix}}


A bag contains 6 red marbles, 4 white marbles, and 8 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that they are all the same color?


What is the smallest positive integer $x$ that, when multiplied by $720$, produces a product that is a multiple of $1250$?


Triangle $ABC$ has $AC = 600$ and $BC = 400$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 240$, find $LP$.

Solution: \boxed{LP = 96}


The distance between two cities on a map is 20 inches. For the first 10 inches, the scale is 0.5 inches = 10 miles, and for the remaining distance, the scale is 0.25 inches = 5 miles. How many miles apart are the actual cities?

Solution: \boxed{400 \text{ miles}}


A wooden block is 6 inches long, 3 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into eighteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?


What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{347}$?


When you simplify $\sqrt[3]{54a^6b^8c^{14}}$, what is the sum of the exponents of the variables that are outside the radical?


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


Let \( a \) and \( b \) be real numbers. One of the roots of the polynomial
\[ x^3 + ax^2 + 3x + b = 0 \]
is \( 2 - 3i \). Find the ordered pair \( (a, b) \).

Solution: \boxed{\left(-\frac{3}{2}, \frac{65}{2}\right)}


The curve \( y = \sin x \) cuts the line whose equation is \( y = \sin 60^\circ \) into segments having the successive ratios
\[ \dots p : q : p : q \dots \] with \( p < q \). Compute the ordered pair of relatively prime positive integers \( (p, q) \).

Solution: \boxed{(1, 5)}


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


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


Two historians were each born within the last 1000 years. Each lives (or will live) to be 150 years old, then dies. Each historian is equally likely to be born at any point during those 1000 years. What is the probability that they were contemporaries for any length of time?


Let $x = (3 + 2\sqrt{2})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


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


Solve for $z$ in the equation $2 - 3iz = 4 + 5iz$ where $i^2 = -1$.

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


How many non-congruent triangles with perimeter 11 have integer side lengths?


What is the greatest integer less than or equal to \[\frac{4^{100} + 3^{100}}{4^{95} + 3^{95}}?\]


If $x^{2y} = 81$ and $x = 9$, what is the value of $y$? Express your answer as a common fraction.


How many integers are common solutions to these three inequalities?
\[
\begin{array}{cccc}
(1) & -4y & \geq & y + 9 \\
(2) & -3y & \leq & 15 \\
(3) & -5y & \geq & 2y + 20
\end{array}
\]


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 158 hours, 55 minutes, and 32 seconds, the time will be $A:B:C$. What is the value of $A + B + C$?


Given that $(x - y)^2 = 25$ and $xy = -10$, find the value of $x^2 + y^2$.


A cylindrical container is one-third full of a mixture of oil and water. The ratio of oil to water in the mixture is 1:5. If the container is 9 inches tall and has a diameter of 3 inches, what is the volume of oil in the container? Express your answer as a decimal to the nearest hundredth.


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


A park has five pairs of different birds, one male and one female for each. The caretaker decides to photograph the birds in a specific pattern: each time he photographs a single bird, the next one he photographs must be of a different gender. If he starts by photographing the male peacock, how many ways can he photograph all the birds?


Consider three squares: Square C, Square D, and Square E. Square C has side lengths each measuring $x$ inches, Square D has side lengths each measuring $3x$ inches, and Square E has side lengths each measuring $6x$ inches. What is the ratio of the area of Square C to the area of Square E, and what is the ratio of the area of Square D to the area of Square E? Express your answer as a common fraction.


What is the sum of every third odd number between $100$ and $300$?


Define a function $g$ on the set of ordered pairs of positive integers such that it fulfills the following conditions:
\begin{align*}
g(x,x) &= x^2,\\
g(x,y) &= g(y,x),\\
(x + y) g(x,y) &= yg(x,x + y).
\end{align*}
Calculate $g(2,12) + g(5,25)$.


Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.00001, 0.0001).$


Determine the sum of distances from the point $P = (5, 1)$ to the vertices of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(12,0)$, and $C$ is at $(4,6)$. If the sum of distances is expressed as $x\sqrt{a} + y\sqrt{b}$, where $x$, $y$, $a$, and $b$ are integers, find $x + y$.


Determine the value of $2a - 3bi$, where $a = 5 - 3i$ and $b = 2 + 4i$.

Solution: \boxed{22 - 12i}


The decimal representation of $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits $3, 7,$ and $5$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.


Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $5(5x+2) \equiv 3 \pmod{p}$.


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


Let $M$ be the product of all odd primes less than $2^4$. What remainder does $M$ leave when divided by $2^5$?


Solution: \boxed{-\frac{5}{17} - \frac{14}{17}i}


Find the area of triangle $DEF$ below.

[asy]
unitsize(1inch);
pair D, E, F;
D = (0,0);
E = (2*sqrt(3),0);
F = (0,2);
draw (D--E--F--D, linewidth(0.9));
draw(rightanglemark(E,D,F,3));
label("$D$", D, S);
label("$E$", E, S);
label("$F$", F, N);
label("$8$", F/2, W);
label("$30^\circ$", (1.5, 0), N);
[/asy]


Consider the sum of an arithmetic sequence $$7 + 14 + 21 + \cdots + 77 + 84 + 91$$ where the sequence is considered modulo $10$, and we are to find a number $m$ such that $$7+14+21+\cdots+91 \equiv m \pmod{10},$$ with $0 \leq m < 10$. What is the value of $m$?


Completely factor the following expression:
\[(20y^4 + 100y - 10) - (5y^3 - 15y + 10).\]

Solution: \boxed{5(4y^4 - y^3 + 23y - 4)}


In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{2}{3}$ and $\sin C = \frac{1}{4}$, then what is $DC$?

[asy]
pair A,B,C,D;
A = (0,0);
B = (10*sqrt(5),20);
D = (10*sqrt(5),0);
C = (10*sqrt(5) + 80*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]


Determine the area of the circle described by the graph of the equation
\[r = 4 \cos \theta - 3 \sin \theta.\]


Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\] is true if and only if either $x < -4$ or $|x-25| \le 2$. Given that $a < b,$ find the value of $a + 2b + 3c.$


Let $I$ be the incenter of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $IBC$, $ICA$, and $IAB$, respectively. If the area of triangle $ABC$ is 36, find the area of triangle $G_1 G_2 G_3$.


The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{3}{2} \\ -2 \end{pmatrix}\]lie on a line. Determine the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = -\frac{3}{4} x - \frac{25}{8}}


Find all values of $m$ for which the sum of the cubes of the solutions of the equation
\[3x^2 + 6x + m = 0\]
equals twice the product of these solutions.


How many positive four-digit integers of the form $\_\_50$ are divisible by 50?


The square of \(a\) and the square root of \(b\) vary inversely. If \(a=3\) when \(b=16\), then find \(a\) when \(ab=24\).


A cube has side length $8$. Its vertices are alternately colored green and red. What is the volume of the tetrahedron whose corners are the red vertices of the cube?


In triangle $ABC$, we have $AB=1$ and $AC=3$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$? Express your answer in simplest radical form.


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


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


Determine the value of $x$ if $\sqrt{3 + \sqrt{4x - 5}} = \sqrt{10}$.


The points $(1,3)$ and $(5,6)$ are adjacent vertices of a square. What is the area of the square?


Five standard, six-sided dice are to be rolled. If the product of their values turns out to be an odd number, what is the probability their sum is even? Express your answer as a common fraction.


Find the solution to \( x|x| = 3x + 4 \) which has the smallest value.


A sphere is inscribed in a larger cube with an edge twice the original length. Compute the ratio of the volume of the sphere to the volume of the cube, and additionally compare the ratio of their surface areas. Express your answers as common fractions in terms of $\pi$.


In right triangle $DEF$, $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?


Arthur walks 8 blocks west and then 10 blocks south. If each block is one-fourth of a mile, how many miles did he walk?


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $720$?


Find integers $D, E$ such that both roots of the polynomial $x^2 - x - 1$ are roots of the polynomial $x^6 - Dx - E$. Determine the product $DE$.


Chandra has five bowls, each one a different color: red, blue, yellow, green, and purple. She also has one glass of each corresponding color. If she chooses one bowl and one glass from her cupboard, how many different pairings are possible?


Let $M$ be the largest integer whose square has exactly $4$ digits when written in base 7.

What is $M$, expressed in base 7?


Let $0 \leq p, q, r, s \leq 1$. Find the possible values of the expression
\[
\sqrt{p^2 + (1 - q)^2} + \sqrt{q^2 + (1 - r)^2} + \sqrt{r^2 + (1 - s)^2} + \sqrt{s^2 + (1 - p)^2}.
\]

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


What is the area enclosed by the graph of $|6x| + |2y| = 12$?


Rodney is trying to guess a secret number based on the following clues:
1. It is a two-digit integer.
2. The tens digit is odd.
3. The units digit is one of the following: 2, 4, 6, 8.
4. The number is greater than 50.

What is the probability that Rodney will guess the correct number? Express your answer as a common fraction.


Bertha has 8 daughters. Some of her daughters have 4 daughters each, and the rest have none. Bertha has a total of 40 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


A circle with center $C$ is tangent to the positive $x$-axis and $y$-axis and externally tangent to the circle centered at $(5,0)$ with radius $3$. What is the sum of all possible radii of the circle with center $C$?


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


In Pascal's Triangle, each entry is the sum of the two entries above it. Determine the smallest row of Pascal's Triangle where three consecutive entries are in the ratio $4: 5: 6$. The top row of Pascal's Triangle has only a single $1$ and is the $0$th row.


Assume $a$ and $b$ are nonzero real numbers such that $a^2 + b^2 = 5ab$. Determine the value of $\left|\frac{a+b}{a-b}\right|$.


Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between if one specific friend, named Alex, must always be one of the neighbors?


Consider the graph of the equation
\[\sqrt{(x-4)^2 + (y-3)^2} + \sqrt{(x+6)^2 + (y-5)^2} = 24.\]
Determine the distance between its foci.


A cube is painted so that one side is black, two sides are yellow, and three sides are white. Determine how many different ways this cube can be painted, considering two cubes identical if one can be rotated to match the other.


Line segments $\overline{AB}$ and $\overline{BC}$ are such that $B$ is between $A$ and $C$. $\overline{BC}$ is extended past $C$ to a point $P$ such that $BP:PC = 7:3$. Then
\[
\overrightarrow{P} = x \overrightarrow{B} + y \overrightarrow{C}
\]
for some constants $x$ and $y$. Find the ordered pair $(x,y)$.

Solution: \boxed{\left(-\frac{3}{4}, \frac{7}{4}\right)}


The Howlers, coached by Coach Loudmouth, have 15 players. However, three of them, Leo, Max, and Neo, refuse to play together in any combination. How many starting lineups (of 6 players) can Coach Loudmouth make if the starting lineup cannot contain Leo, Max, and Neo playing together or any two of them playing together?


I have 17 distinguishable socks in my drawer: 4 white, 4 brown, 5 blue, and 4 black. In how many ways can I choose a pair of socks, provided that I get two socks of different colors?


Given positive integers \(a\) and \(b\) such that \(\frac{1}{a} + \frac{1}{3b} = \frac{1}{6}\), what is the least possible value of \(ab\)?


Consider the graph of the function $y=f(x)$, defined piecewise through several line segments as follows:

- From $x = -4$ to $x = -2$, the line segment follows $y = 2x + 3$.
- From $x = -2$ to $x = -1$, the line segment follows $y = -2x - 3$.
- From $x = -1$ to $x = 1$, the line segment follows $y = -2x + 1$.
- From $x = 1$ to $x = 2$, the line segment follows $y = 2x - 1$.
- From $x = 2$ to $x = 4$, the line segment follows $y = 2x - 3$.

Find the sum of $x$-coordinates where $f(x) = 2.5$.


If $5x \equiv 10 \pmod{25}$ and $4x \equiv 20 \pmod{25}$, find the remainder when $x^2$ is divided by $25$.


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


What is the largest three-digit integer whose digits are distinct and form a geometric sequence and the hundreds digit is 8?


What is the result of $24_8 + 53_8 - 17_8$? Express your answer in base 8.


How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A and E are vowels; any word is valid, not just English language words; and letters may be used more than once.)


Given the equation $2a + b = 42$, where $a$ and $b$ are positive integers and $a$ is even, how many distinct ordered-pair solutions $(a, b)$ exist?


If 5 knicks = 3 knacks and 4 knacks = 6 knocks, how many knicks are equal to 36 knocks?


Choose the largest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{3} + \frac{1}{4}, \ \ \frac{1}{3} + \frac{1}{5}, \ \ \frac{1}{3} + \frac{1}{6}, \ \ \frac{1}{3} + \frac{1}{9}, \ \ \frac{1}{3} + \frac{1}{8}$$


How many four-digit positive integers have at least one digit that is a 5 or a 7?


Choose the smallest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{3} + \frac{1}{4}, \ \ \frac{1}{3} + \frac{1}{5}, \ \ \frac{1}{3} + \frac{1}{6}, \ \ \frac{1}{3} + \frac{1}{7}, \ \ \frac{1}{3} + \frac{1}{8}$$


Given the graph of the function \( f(x) \) below, determine how many values of \( x \) satisfy \( f(f(x)) = 5 \). The function is defined piecewise, as seen on the graph which shows three segments of lines connecting points \( (-4, 4), (-1, 1), (3, 5) \), and \( (5, 3) \).


Find the distance between the foci of the ellipse
\[\frac{x^2}{35} + \frac{y^2}{7} = 5.\]

Solution: \boxed{4 \sqrt{35}}


How many diagonals can be drawn in an octagon if one vertex is removed?


The modified sequence consists of all those positive integers which are powers of 5 or sums of distinct powers of 5. Find the $50^{\text{th}}$ term of this sequence.


List all positive integers in increasing order whose digits add up to 15. What is the fifteenth number in that list?


How many perfect squares are factors of 450?


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


What is the result when $\frac{2}{3}$ is subtracted from $2\frac{1}{4}$?


In triangle $XYZ$, $\angle X = 90^\circ$, $YZ = 16$, and $\tan Z = 3\sin Z$.  What is $XZ$?


If the price of a notebook is $45$ cents, what is the maximum number of notebooks that can be purchased with $\$40.50$?


Compute the sum $15 e^{\pi i/6} + 15 e^{5 \pi i/6}$, and express it in the form $re^{i \theta}$. Find the ordered pair $(r, \theta)$.

Solution: \boxed{\left( 15, \frac{\pi}{2} \right)}


Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 4,$ $\|\mathbf{w}\| = 5,$ and $\mathbf{v} \cdot \mathbf{w} = 12,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$.


Two different integers are randomly chosen from the set $$\{ -3, -7, 0, 5, 3, -1 \}$$. What is the probability that their product is non-positive? Express your answer as a common fraction.


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


Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \end{pmatrix}}


Mrs. Thompson awards extra credit on tests to her students with test grades that exceed the class median. Given that 50 students take the same test, what is the largest number of students who can be awarded extra credit?


How many integers are between $(10.1)^3$ and $(10.4)^3$?


It is currently $7\!:\!45\!:\!00 \text{ a.m.}$ What time will it be in $8888$ seconds?

Solution: \boxed{10\!:\!13\!:\!08 \text{ a.m.}}


Let $a, b, c$ be real numbers such that $a + b + 2c = 0$. Find the set of all possible values of $ab + ac + bc$.

Solution: \boxed{(-\infty, 0]}


What is the greatest common factor of all three-digit palindromes that end with the digit 1? (Note: A palindrome reads the same forwards and backwards.)


Let $n$ be a positive integer, and let $b_0, b_1, \dots , b_n$ be a sequence of real numbers such that $b_0 = 45$, $b_1 = 81$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{5}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$.


Let $\omega$ be a complex number such that $\omega^9 = 1$ and $\omega \ne 1.$ Compute
\[
\omega^{18} + \omega^{21} + \omega^{24} + \dots + \omega^{63}.
\]


How many distinct three-digit positive integers are divisible by 3 and have digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}?


A $3 \times 4$ rectangle and a $4 \times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the rectangles. What is the smallest possible area of this square?


Let $R'$ be the set of all possible remainders when a number of the form $3^n$, where $n$ is a nonnegative integer, is divided by 500. Let $S'$ be the sum of the elements in $R'$. Find the remainder when $S'$ is divided by 500.


At the school cafeteria, Michael wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Michael's options in the cafeteria. However, due to dietary restrictions, if Michael chooses a meat-based main course, he cannot select a soda as his beverage. How many distinct possible lunches can he buy from these options?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages}&\textbf{Snacks} \\ \hline
Hamburger & Water & Apple Pie \\ \hline
Veggie Burger & Soda & Fruit Cup \\ \hline
Chicken Sandwich& & \\ \hline
Pasta & & \\ \hline
\end{tabular}


If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
  \binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]
Compute the value of
\[
  \frac{\binom{-1/2}{1007} \cdot 2^{1007}}{\binom{2014}{1007}} \, .
\]


A $\frac{1}{1004}$-array is structured similarly to a $\frac{1}{2008}$-array, but each element in the first row starts with $\frac{1}{1004}$ and each subsequent row's first element is $\frac{1}{2 \times 1004}$ times the first element of the previous row. Each subsequent term in a row is $\frac{1}{1004}$ times the previous term. If the sum of all the terms in this $\frac{1}{1004}$-array can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $1004$.


Let $x$ be a positive integer such that $7x \equiv 1 \pmod{26}$.
What is the remainder when $13 + 3x$ is divided by $26$?


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


In a modified ellipse situation, the center is now at $(2, -1)$, one focus is at $(2, -3),$ and one endpoint of a semi-major axis is at $(2, 2).$ Find the semi-minor axis of this ellipse.


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


A right triangle has sides 3, 4, and 5. Calculate the distance from the midpoint of the hypotenuse to one of the triangle's vertices and the circumradius of the triangle.


Suppose \( b \) is a positive integer greater than or equal to 2. When 256 is converted to base \( b \), the resulting representation has 4 digits. What is the number of possible values for \( b \)?


Given that $a$ is a multiple of $1632$, find the greatest common divisor of $a^2+13a+36$ and $a+6$.


Let $a$, $b$, $c$, $d$, $e$, and $f$ be integers such that $729x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ holds for all $x$. Compute $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$.


What is the domain of the function $$v(x) = \frac{1}{\sqrt{x} + \sqrt[3]{x}}~?$$ Express your answer in interval notation.

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


If I roll a fair, regular six-sided die five times, what is the probability that I will roll the number $2$ exactly four times, and that the last roll is not a $2$?


How many positive integers \( n \) less than 150 have a corresponding integer \( m \) divisible by 4 such that the roots of \( x^2 - nx + m = 0 \) are positive integers two apart?


How many pairs of positive integers $(m,n)$ satisfy the inequality $m^2 + 3n < 40$?


What is the smallest positive integer $n$ such that $11n-4$ and $8n - 5$ share a common factor greater than $1$?

Solution: \boxed{n = 15}


How many even three-digit numbers exist such that the sum of the tens and units digits is 12, and the hundreds digit is greater than the units digit?


Using the digits $1$, $2$, $3$, $5$, and $6$ exactly once, the five-digit positive integers are formed and arranged in ascending order. What is the $60^{\text{th}}$ integer in this list?


Let $p,$ $q,$ $r,$ and $s$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(p,0,0),$ the midpoint of $\overline{AC}$ is $(0,q,0),$ and the midpoint of $\overline{AB}$ is $(0,0,r).$ Suppose the centroid of triangle $ABC$ is at the point $(0,0,s)$. Find
\[
\frac{AB^2 + AC^2 + BC^2}{p^2 + q^2 + r^2}.
\]


Choose the smallest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{4} + \frac{1}{5}, \ \ \frac{1}{4} + \frac{1}{6}, \ \ \frac{1}{4} + \frac{1}{9}, \ \ \frac{1}{4} + \frac{1}{8}, \ \ \frac{1}{4} + \frac{1}{7}$$


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


For $\mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ -0.6 \end{pmatrix}.\] Find $y$.

Solution: \boxed{y = -5.6}


Simplify $18 \cdot \frac{8}{12} \cdot \frac{1}{9} \cdot 4$.


Find the difference between $\frac{5}{19}$ and $\frac{2}{23}$. Express your answer in its simplest form.


Determine the number of non-zero digits to the right of the decimal point when the expression $\frac{180}{2^4 \cdot 5^6 \cdot 3^2}$ is converted into a decimal.


A fair 8-sided die is rolled. What is the probability that the number rolled is a divisor of 12?


How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $19,$ $7,$ and $3?$


Let triangle $PQR$ be a right triangle in the xy-plane with a right angle at $R$. The lengths of legs $PR$ and $RQ$ are given as 24 and 73 respectively, and $PQ$ (hypotenuse) becomes 75. The median through $P$ lies along the line $y = 3x + 4$ and the median through $Q$ lies along the line $y = -x + 5$. Find the area of triangle $PQR$.


Let $ABCDE$ be a convex pentagon, and let $G_A,$ $G_B,$ $G_C,$ $G_D,$ $G_E$ denote the centroids of triangles $BCDE,$ $ACDE,$ $ABDE,$ $ABCE,$ and $ABCD,$ respectively. Find $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$.


Find the number of positive integers $n \leq 500$ such that $21n$ is a perfect square.


In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.


The volume of a rectangular solid is $343\text{ cm}^3$, its total surface area is $294\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.

Solution: \boxed{84 \text{ cm}}


The volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\mathbf{a} + 2\mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $2\mathbf{c} - 5\mathbf{a}.$


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


If $x+\frac{1}{x}=5$, then what is the value of $x^3+\frac{1}{x^3}$?


Completely factor the following expression: \[(4b^3 + 126b^2 - 9) - (-9b^3 + 2b^2 - 9)\]

Solution: \boxed{b^2(13b + 124)}


How many of the 729 smallest positive integers written in base 9 use 6 or 7 (or both) as a digit?


Let $\mathbf{B} = \begin{pmatrix} e & f & g \\ f & h & i \\ g & i & j \end{pmatrix}$. Given that $\mathbf{B}$ is symmetric and orthogonal, find $e^2 + f^2 + g^2 + h^2 + i^2 + j^2.$


Find $XY$ in the triangle below.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$6$", (0.5,0), S); // Base of the triangle
label("$30^\circ$", (0.1,0.1), NE);
[/asy]


Consider the function \( g(x) = x^2 + 3\sqrt{x} \). Evaluate \( 3g(3) - g(9) \).

Solution: \boxed{-63 + 9\sqrt{3}}


Write the expression $\frac{6 + 4d}{9} + \frac{3}{2}$ as a single fraction.

Solution: \boxed{\frac{39 + 8d}{18}}


Winnie has 60 cherry lollipops, 145 wintergreen lollipops, 10 grape lollipops, and 295 shrimp cocktail lollipops. She wants to equally share these lollipops among her 13 friends, distributing as many whole lollipops as possible. How many lollipops does Winnie have left after sharing?


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\sqrt{2}$ square feet and $675\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


In a specific sequence, the first term is $a_1 = 1007$ and the second term is $a_2 = 1008$. The values of subsequent terms are set such that
\[ a_n + a_{n+1} + a_{n+2} = 2n \]
for all $n \geq 1$. Determine the value of $a_{1000}$.


How many distinct sequences of four letters can be made from the letters in "DEMAND" if each sequence must begin with D, end with M, and no letter can appear in a sequence more than once?


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


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


Solution: \boxed{2x^3 - x^2 + 23x - 3}


Both $c$ and $d$ are positive integers and $d > 1$. When $c^d$ is the greatest possible value less than 500, what is the sum of $c$ and $d$?


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


The hypotenuse of a right triangle measures 15 cm and one angle is $45^{\circ}$. What is the number of square centimeters in the area of the triangle?


Solution: \boxed{(10 - 5x + 4y)(10 + 5x - 4y)}


If 25,197,629 hot dogs are packaged in sets of 6, how many will be left over?


If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$


A sample of helium gas is held in a 3.4 liter container at a pressure of 8 kPa. If the gas is transferred to a new container with a volume of 8.5 liters at the same temperature, what is the new pressure of the helium gas in kPa?

Solution: \boxed{3.2 \text{ kPa}}


One hundred more than the square of a number is the same as the square of the quantity that is \( 20 \) less than the number. What is the number?


Determine the value of $(6 + x)(21 - x)$ where $\sqrt{6 + x} + \sqrt{21 - x} = 8$.


Sixty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, with each number entered on five cards. Five cards are drawn from the box at random and without replacement. Let $p$ be the probability that all five cards bear the same number. Let $q$ be the probability that four of the cards bear a number $a$ and the other bears a number $b$ that is not equal to $a$. What is the value of $q/p$?


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


In the triangle shown, $n$ is a positive integer, and $\angle B > \angle A > \angle C$. How many possible values of $n$ are there? [asy]
draw((0,0)--(1,0)--(.4,.5)--cycle);
label("$A$",(.4,.5),N); label("$B$",(1,0),SE); label("$C$",(0,0),SW);
label("$4n + 6$",(.5,0),S); label("$3n - 3$",(.7,.25),NE); label("$2n + 7$",(.2,.25),NW);
[/asy]


Given that the sum of the interior numbers in the sixth row of Pascal's Triangle is 30, calculate the sum of the interior numbers in the eighth row.


What is the greatest common factor of 120, 180, and 300?


On each of the first five days of January, there is a $\frac{1}{5}$ chance that it will snow where Alice lives. On each of the next five days, there is a $\frac{1}{3}$ chance that it will snow. What is the probability that it snows at least once during the first ten days of January?


The diagonals of a regular octagon have different lengths. Determine the ratio of the shorter diagonal to the longer diagonal. Express your answer as a common fraction in simplest form.


**Find the number of functions \( f : \mathbb{R} \to \mathbb{R} \) such that**
\[ f(xy) + f(xz) - f(x)f(yz) > 1 \]
**for all real numbers \( x, y, \) and \( z. \)**


Determine how many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are "spacy", where a spacy set contains no more than one out of any three consecutive integers.


When the graph of $y = 3x^2 - x + 7$ is shifted five units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.


Isosceles triangle $XYZ$ has an area of 180 square inches and is intersected by line $\overline{MN}$, creating an isosceles trapezoid and a smaller isosceles triangle above it. The area of the trapezoid is 108 square inches. If the altitude of triangle $XYZ$ from $X$ is 30 inches, determine the length of $\overline{MN}$.

Solution: \boxed{\frac{12\sqrt{10}}{5} \text{ inches}}


Tamika selects two different numbers at random from the set $\{10, 11, 12\}$ and adds them. Carlos takes two different numbers at random from the set $\{4, 5, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.

Solution: \boxed{\frac{3}{9} = \frac{1}{3}}


How many even perfect square factors does $2^6 \cdot 7^9 \cdot 3^2$ have?


Five positive integers are divided by $30$, and the remainders are $7,$ $11,$ $18,$ $5,$ and $9$, respectively.

When the sum of these five integers is divided by $30$, what is the remainder?


A right triangle has legs of lengths 5 cm and 12 cm. The hypotenuse of a similar triangle is 39 cm. What is the perimeter of the larger triangle?


By partial fractions, decompose the following expression:
\[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4} + \frac{F}{x + 5}\]
for some constants $A, B, C, D, E, F$. Find $A + B + C + D + E + F$.


The expression \[(x+y+z)^{2008}+(x-y-z)^{2008}\] is simplified by expanding it and combining like terms. How many terms are in the simplified expression?


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


Let $\mathbf{B}$ be a matrix such that
\[
\mathbf{B} \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} -12 \\ 4 \end{pmatrix}.
\]
Find $\mathbf{B}^5 \begin{pmatrix} 3 \\ -1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} -3072 \\ 1024 \end{pmatrix}}


What is the sum of the digits of the base $8$ representation of $999_{10}$?


Determine the remainder when $7\cdot10^{20} + 2^{20} + 5$ is divided by 9.


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


How many even perfect square factors does \(2^6 \cdot 7^{10} \cdot 3^2\) have?


In a setup with three concentric circles, the radius of the middle circle is three times the radius of the innermost circle, and the radius of the outermost circle is twice the radius of the middle circle. If the width between the innermost and the middle circle is 3 feet, calculate the area of the region between the middle and the outermost circle in square feet. Express your answer in terms of $\pi$.


Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 3,$ and $x \ge -1,$ $y \ge -2,$ and $z \ge -4.$ Find the maximum value of
\[\sqrt{4x + 4} + \sqrt{4y + 8} + \sqrt{4z + 16}.\]


An isosceles triangle has its base at $(0, 5)$ and $(8, 5)$ with the base angle being $60^\circ$. If the third vertex lies in the first quadrant, what is the y-coordinate of this vertex?

Solution: \boxed{5 + 4\sqrt{3}}


In Elaina's class of 48 students, 18 prefer chocolate pie, 12 prefer apple pie, and 8 prefer blueberry pie. The remaining students' preferences are split among three different pies: cherry, lemon, and peach, each equally preferred by the students. Calculate the number of degrees that should represent cherry pie on Elaina's pie chart.


The sides of a triangle have lengths of $13$, $14$, and $15$. Find the length of the shortest altitude.


Find the monic quadratic polynomial with real coefficients which has $-3 + 2i$ as a root and whose discriminant is a multiple of 4.

Solution: \boxed{x^2 + 6x + 13}


Solve the inequality
\[ |x - 2| + |x + 3| < 8. \]

Solution: \boxed{\left(-\frac{9}{2}, \frac{7}{2}\right)}


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


There are 61 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by a prime number of chips, what is the greatest possible number of large chips?


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

Solution: \boxed{(0, -1)}


How many of the 300 smallest positive integers are congruent to 3 (mod 8)?


The arithmetic mean of an even number of consecutive integers is $z$. Find the sum of the smallest and largest integers in terms of $z$.


Add $56.238$ to $75.914$ and round your answer to the nearest thousandth.


The area of the parallelogram produced by vectors $\mathbf{a}$ and $\mathbf{b}$ is 12. Determine the area of the parallelogram formed by the vectors $3\mathbf{a} - \mathbf{b}$ and $4\mathbf{b} - 2\mathbf{a}.$


Find the number of complex solutions to
\[
\frac{z^4 - 1}{z^3 + 2z^2 - z - 2} = 0.
\]


Find the volume of the region in space defined by
\[ |x + 2y + z| + |x + 2y - z| \le 12 \]
and $x, y \geq 0$, $z \geq -2$.


A fair twenty-faced die has 19 of its faces numbered from $2$ to $20$ and has one blank face. Another fair twenty-faced die has 19 of its faces numbered from $1$ to $19$ (skipping number $11$) and has one blank face. When the two dice are rolled, what is the probability that the sum of the two numbers facing up will be $30$? Express your answer as a common fraction.


Determine the slope of line $\overline{CD}$ where circles given by $x^2 + y^2 - 6x + 4y - 5 = 0$ and $x^2 + y^2 - 10x + 16y + 24 = 0$ intersect at points $C$ and $D$.


Find the value of \(m + n\) where \(m\) and \(n\) are integers defined as follows: The positive difference between the roots of the quadratic equation \(5x^2 - 9x - 14 = 0\) can be expressed as \(\frac{\sqrt{m}}{n}\), with \(m\) not divisible by the square of any prime number.


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


Lisa has 12 friends and 60 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble, no two friends receive the same number of marbles, and the number of marbles left after distribution is a prime number?


How many of the positive divisors of 5880 are multiples of 4?


Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 45$, what is $a_3a_6$?


If the least common multiple of two 5-digit integers has 9 digits, what is the maximum number of digits that their greatest common divisor can have?


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


The table shown is partially filled in with the results of a survey done by the radio station KMIX. How many of the males surveyed do not listen to this station?

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males & 70 & \\ \hline
females & & 110 \\ \hline
total & 145 & 160 \\
\hline
\end{tabular}


Alicia needs to clear snow from a rectangular path leading to her garage. The path is 30 feet long and 3 feet wide. The depth of the snow increases linearly from 0 feet at the start of the path to 1 foot at the end of the path. Calculate the volume of snow Alicia needs to clear.


For the quadrilateral shown, how many different whole numbers could be the length of the diagonal (dotted line)?

[asy]
draw((0,0)--(4,4)--(10,0)--(5,-7)--cycle,linewidth(0.7));
draw((0,0)--(10,0),dashed);
label("7",(2,2),NW);
label("9",(7,2),NE);
label("14",(7.5, -3.5),SE);
label("10",(2.5,-3.5),SW);
[/asy]


The number of students in Carlos' graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Carlos' graduating class?


A teacher challenges her class with a new problem, asking for the value of $\left(7^2 - 3^2 + 2\right)^3$. What is the value of the expression?


Determine the number of solutions to
\[3 \sin^2 x - 7 \sin x \cos x + 2 \cos x = 0\] in the range $0 \le x \le 2\pi.$


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


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


An equilateral triangle has two vertices at $(2,3)$ and $(10,3)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form.

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


Rationalize the denominator of $\displaystyle\frac{36}{\sqrt[3]{12}}$.


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\beta,$ and the angle between $\mathbf{c}$ and $\mathbf{a} \times \mathbf{b}$ is $\beta.$  If $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{2},$ find the smallest possible value of $\beta,$ in degrees.


Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$.


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


The number 15 is the sum of three consecutive integers. What is the largest number of positive, consecutive integers whose sum is 15?


Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that vertex B is one of the bottom vertices? There are three bottom vertices in total in a dodecahedron.


John has $25$ dollars and $45$ cents. He wants to buy as many books as he can with his money from a bookstore where each book costs $2$ dollars and $85$ cents. If John buys more than 10 books, the price of each additional book drops to $2$ dollars and $50$ cents. How many books can John buy?


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{14}\right)\left(5^{16}\right)\left(7^{9}\right)$?


What is the least positive integer that satisfies the following conditions?

a) When divided by 3, the remainder is 2.

b) When divided by 4, the remainder is 3.

c) When divided by 5, the remainder is 4.

d) When divided by 9, the remainder is 8.


Solution: \boxed{\frac{1}{10} \text{ days}}


Consider a parallelogram with vertices at $(12,50)$, $(12,120)$, $(30,160)$, and $(30,90)$. A line through the origin cuts this parallelogram into two congruent polygons. Find the slope of the line in the form $m/n$, where $m$ and $n$ are coprime integers, and determine the sum $m+n$.


Let \( p, q, r, s, t \) be real numbers such that
\[ p^2 + q^2 + r^2 + s^2 + t^2 = 5. \]
Find the maximum value of \( p^3 + q^3 + r^3 + s^3 + t^3 \).


Find the inverse of the matrix
\[
\begin{pmatrix} 5 & -3 \\ 2 & 1 \end{pmatrix}.
\]
If the inverse does not exist, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} \frac{1}{11} & \frac{3}{11} \\ -\frac{2}{11} & \frac{5}{11} \end{pmatrix}}


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


Solution: \boxed{3 - x}


Triangle $ABC$ has vertices $A$, $B$, and $C$ on the sides of a rectangle 6 units by 9 units. Point $A$ is located at (6,2), point $B$ at (0,9), and point $C$ at (3,0). Calculate the area of triangle $ABC$ in square units.


Compute the distance between the parallel lines given by
\[\begin{pmatrix} 4 \\ -2 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}\]and
\[\begin{pmatrix} 3 \\ -1 \end{pmatrix} + s \begin{pmatrix} 2 \\ -3 \end{pmatrix}.\]


Let \( S \) be the set comprising all sides and diagonals of a regular hexagon. A pair of elements from \( S \) are selected randomly without replacement. What is the probability that the two chosen segments have the same length?


What is the volume of the region in three-dimensional space defined by the inequalities $|x+y| \le 1$ and $|x+y+z-2| \le 2$?

Solution: \boxed{V = 8}


What is $\frac{5}{6}$ minus $\frac{1}{3}$, then divided by $\frac{2}{9}$?


How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments in a 3x3 grid?

[asy]
unitsize(0.08inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15));
dot((15,0));
label("$A$",(0,15),NW);
label("$B$",(15,0),SE);
[/asy]


Let $x$, $y$, $z$ be the sides of a triangle, and let $\xi$, $\eta$, $\zeta$ be the angles opposite them, respectively. If $x^2 + y^2 = 2023z^2$, find the value of
\[\frac{\cot \zeta}{\cot \xi + \cot \eta}.\]


I have two 12-sided dice that each have 3 red sides, 4 blue sides, 3 green sides, and 2 purple sides. If I roll both dice, what is the probability that they come up showing the same color?


Point $E$ is on side $BC$ of triangle $ABC$. If $\angle BAE = 30^\circ$ and $\angle BEC = 45^\circ$, determine the measure of $\angle EAB$, when $\angle C = 40^\circ$.


Line $l_1$ has equation $4x - 3y = 2$ and goes through point $A = (0, -\frac{2}{3})$. Line $l_2$ has equation $y = 2$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $6$. What is the slope of $l_3$?


For how many non-negative real values of $x$ is $\sqrt{169 - \sqrt[3]{x}}$ an integer?


A fair 8-sided die is rolled once. If I roll $n$, then I normally win $8-n$ dollars. However, if I roll an 8, I triple my winnings amount. What is the expected value of my win, in dollars?


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

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


Determine the value of $d$ if $x\cdot(4x + 3) < d$ if and only when $x \in \left(-\frac{5}{2}, 1\right)$.


Let $n$ be the number of ordered quadruples $(x_1, x_2, x_3, x_4)$ of positive integers that satisfy $\sum_{i = 1}^4 x_i = 100.$ Find $n$.


Let $\alpha$ and $\beta$ be the roots of $x^2 - ax - 1 = 0,$ and let $\gamma$ and $\delta$ be the roots of $x^2 - bx - 1 = 0.$  Express
\[(\alpha - \gamma)^2(\beta - \gamma)^2(\alpha + \delta)^2(\beta + \delta)^2\] in terms of $a$ and $b.$

Solution: \boxed{(b^2 - a^2)^2}


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(4) = 8$ and $g(16) = 512,$ find the largest possible value of $g(8).$


Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter $BC$ intersects side $AC$ at $D$ and intersects side $BC$ at point $E$ where $E$ is not equal to $B$ or $C$. Given that the area of $\triangle ABC$ is $200$ and $AC = 30$, find the length of $BD$.


A circle is centered at $O$ and circumscribes triangle $ABC$. The central angles are $\angle AOB = 120^\circ$ and $\angle BOC = 90^\circ$. What is the measure of $\angle BAC$, in degrees?


Two distinct positive integers $a$ and $b$ are factors of 60. If $a \cdot b$ is also not a factor of 60, what is the smallest possible value of $a \cdot b$?


In triangle $DOG$, $\angle DGO = \angle DOG$ and $\angle DOG = 60^\circ$. If $\overline{DS}$ bisects $\angle DOG$, find the measure of $\angle DSG$.


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

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


Let $\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}$ be three mutually orthogonal unit vectors, such that
\[
\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a})
\]
for some scalars $p, q, \text{ and } r$, and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2$. Find $p + q + r$.


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$?


Let $E$ be the determinant of a matrix whose column vectors are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}.$ Find the determinant of the matrix whose column vectors are $2\mathbf{u} + \mathbf{v},$ $\mathbf{v} + 2\mathbf{w},$ and $2\mathbf{w} + \mathbf{u},$ in terms of $E.$


Find the phase shift of the graph of $y = 2 \cos\left(2x + \frac{\pi}{3}\right).$


Solve the following problem: The equation $x^2 - 6x + 15 = 24$ has two solutions, $a$ and $b$, where $a \geq b$. Calculate the value of $3a + 2b$.

Solution: \boxed{15 + 3\sqrt{2}}


Find the greatest common divisor (GCD) of 1151 and 3079.


Find the first term of an infinite geometric series where the common ratio is \(-\frac{3}{7}\) and the total sum of the series is 20.


What is the largest integer that must divide the product of any $5$ consecutive integers?


Find all solutions $x$ of the inequality $$\frac{3}{16} + \left|x-\frac{17}{64}\right| < \frac{7}{32}.$$ Express your answer in interval notation, simplifying all fractions in your answer.

Solution: \boxed{\left(\frac{15}{64}, \frac{19}{64}\right)}


In triangle $ABC$, angle $A$ is $90^\circ$, $BC = 10$ and $\tan C = 3\cos B$. What is $AB$?


In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.

[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W);
[/asy]


If the three points $(2, a, b), (a, 3, b), (a, b, 4)$ are collinear, what is the value of $a + b$?


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


I have 8 shirts, 5 pairs of pants, and 7 hats. The pants come in red, black, blue, gray, and green. The shirts and hats come in those colors, and also purple and white. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, which include one shirt, one hat, and one pair of pants, are possible?


Factor $2w^4 - 50$ as far as possible, where the factors are monic polynomials with real coefficients.

Solution: \boxed{2(w^2 - 5)(w^2 + 5)}


Find the smallest four-digit integer that is divisible by each of its distinct, non-zero digits.


Find the real root of the equation \[\sqrt{x} + \sqrt{x+3} = 12.\]


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


A retailer offers a hefty discount on all stock at 40% off. Shortly after, an additional 10% off is offered on the new reduced prices. The retailer then advertises that the final price is a 60% reduction from the original. What is the difference, as a percentage of the original price, between the true discount and the retailer's claimed discount?


What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integers and the largest angle is obtuse?


Consider the equation $$64x^2 + 96x - 81 = 0.$$ Rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)?


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


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 3:1$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 12, determine the area of $\triangle ABC$.


Calculate $3 \cdot 7^{-1} + 6 \cdot 13^{-1} \pmod{60}$.

Express your answer as an integer from $0$ to $59$, inclusive.


If the system of equations
\begin{align*}
4x - 2y &= a, \\
6y - 12x &= b.
\end{align*}
has a solution, find $\frac{a}{b},$ assuming $b \neq 0.$


What is the measure of angle $x$ in the triangle below?
[asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(60),Sin(60)), C = (Cos(60) + Sin(60)*sqrt(3),0);
draw(A--B--C--cycle);
label("$45^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy]


Consider polynomials $Q(x) = ax^3 + bx^2 + cx + d$ where the degree is at most 3 and each coefficient is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $Q(1) = 1$?


Find the smallest positive solution to the equation
\[\tan 3x + \tan 4x = \sec 4x\]
in radians.


Suppose $u$ and $v$ are complex numbers such that $|u+v| = 2$ and $|u^2+v^2| = 17.$ Find the smallest possible value of $|u^3+v^3|.$


Find the second smallest perimeter of a scalene triangle where all sides are different prime numbers, and the perimeter itself is also a prime number.


In a certain ellipse, the endpoints of the major axis are $(-9,3)$ and $(11,3).$ Also, the ellipse passes through the point $(9,6).$ Find the area of the ellipse.

Solution: \boxed{50 \pi}


How many distinct four-digit positive integers are such that the product of their digits equals 18, and the number does not end with the digit 0?


Let \( p, q, \) and \( r \) be real numbers such that \( p + q + r = 5 \) and \( p^2 + q^2 + r^2 = 27 \). What is the difference between the maximum and minimum possible values of \( r \)?


Compute the product of the complex numbers $(5 \operatorname{cis} 30^\circ)$ and $(-3 \operatorname{cis} 50^\circ)$ in polar form, expressing the result as $r \operatorname{cis} \theta$ where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Enter the ordered pair $(r, \theta)$.

Solution: \boxed{(15, 260^\circ)}


Let \(\theta\) be the angle between the planes \(3x + 2y - z + 4 = 0\) and \(9x + 6y + 3z - 7 = 0.\) Find \(\cos \theta.\)

Solution: \boxed{\cos \theta = \frac{6}{7}}


In Vermont, 520 people were surveyed to determine their preference for what they call soft drinks. The results are shown in a pie chart. The central angle for the ``Soda'' sector of the chart is $278^\circ$, to the nearest whole degree. Determine how many of the surveyed people preferred calling it ``Soda''. Give your answer as a whole number.


Calculate $\left(4\frac{2}{7} \times 5\frac{1}{2}\right) - \left(3\frac{1}{3} + 2\frac{1}{6}\right)$.


A penny, nickel, dime, quarter, fifty-cent coin, and one-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, rounded to the nearest tenth of a cent.


Let $H$ be the orthocenter of triangle $ABC.$ Let $M$ be the midpoint of the arc $BC$ that does not contain $A$ on the circumcircle of triangle $ABC$. Determine the value of:
\[MA^2 + MB^2 + MC^2 - MH^2\]
in terms of the side lengths $a, b, c$ and circumradius $R$ of triangle $ABC$.

Solution: \boxed{a^2 + b^2 + c^2 - 4R^2}


How many three-digit numbers are multiples of neither 4 nor 11?


Find the length of a segment from a vertex to the center of a regular hexagon where each side length is equal to 16 units. Express your answer in simplest radical form.


Originally, Terrell lifts two 25-pound weights 10 times. If he decides to use two 20-pound weights instead, how many times must Terrell lift them to lift the same total weight?


How many digits are there in the base-7 representation of $2048$?


Consider the following information regarding Pascal's Triangle: The sum of the interior numbers in the sixth row is 30. What is the sum of the interior numbers of the eighth row?


Find the projection of the vector $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ onto the line
\[\frac{x}{3} = y = \frac{z}{-2}.\]

Solution: \boxed{\begin{pmatrix} \frac{30}{7} \\ \frac{10}{7} \\ -\frac{20}{7} \end{pmatrix}}


How many positive two-digit integers are there in which each of the two digits is prime, and the sum of the digits is an even number?


Find the minimum value of
\[\frac{x^3}{y - 2} + \frac{y^3}{x - 2}\]
for real numbers $x > 2$ and $y > 2.$


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


A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(5,0)$ with radius $2$. What is the sum of all possible radii of the circle with center $C$?


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


Find the least integer value of $x$ such that $2x^2 + 2|x| + 7 < 25$.


Spinners $C$ and $D$ are spun. On spinner $C$, the numbers are 1 to 5. On spinner $D$, the numbers are 1 to 4. What is the probability that the product of the two spinners' numbers is divisible by 3?


Determine the coefficient of $x^4$ in the expression $2(x^3 - 2x^4 + x^2) + 4(x^2 + 3x^4 - x^3 + 2x^5) - 6(2 + x - 5x^4 + 2x^3)$ when simplified.


Given that $\overline{AB}\parallel \overline{CD}$ and $\angle AXF = 135^\circ$, and $\angle FZD = 45^\circ$, find the measure of $\angle FYD$.


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


A function $f$ is defined by $f(z) = -i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Determine the number of values of $z$ that satisfy both $|z| = 4$ and $f(z) = z$.


A company has 63 employees across different positions according to the table below. What is the median salary of these employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$140{,}000$\\\hline
General Manager&7&$\$95{,}000$\\\hline
Senior Manager&8&$\$80{,}000$\\\hline
Manager&4&$\$55{,}000$\\\hline
Staff&43&$\$25{,}000$\\\hline
\end{tabular}


What is the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}?\]


Yan finds himself at a point between his home and the stadium. He has two options to reach the stadium: walk directly to the stadium or walk back home and then ride his bicycle to the stadium. He rides 6 times as fast as he walks. If both routes take the same time, what is the ratio of Yan's distance from his home to his distance to the stadium?


What is the smallest positive integer $n$ such that $5n$ is a perfect square and $3n$ is a perfect cube?


Multiply \(6_6\) by \(21_6\) and express your answer in base \(6\).


The base-10 numbers 314 and 59 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation?


How many different four-digit numbers can be formed by arranging the four digits in 3033?


What is the smallest four-digit palindrome that is divisible by 8? A palindrome reads the same forwards and backwards.


In Mr. Hooper's 25-student English class, no two students have the same initials, and each student's first name and last name begin with the same letter, just like Mrs Vale's class. However, the letter "Y" is not considered in this student list, either as an initial for a first name or a last name. What is the probability of randomly picking a student whose initials are consonants? Assume all other letters besides A, E, I, O, U are valid and consider only English alphabets from AA to ZZ minus YY.


Determine \( x \) such that \( \log_x 256 = \log_4 64 \).


How many four-digit numbers have the property that the second digit is the average of the first and third digits, and the digits are all even?


Naomi, Oliver, and Penelope combine their cookies to gift them in packages. If Naomi has 53 cookies, Oliver has 67 cookies, and Penelope has 29 cookies, and if cookies can only be gifted in packages of 15, how many cookies will be left over after all packages are made?


Josef and Timothy play a game where Josef picks an integer between 1 and 500, and Timothy divides 500 by that integer and states whether or not the quotient is an integer. How many integers could Josef pick such that Timothy's quotient is an integer?


A library has ten identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least two books must be in the library and at least three must be checked out?


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


As $m$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $14m + 4$ and $9m + 2$?


Compute $\arccos (\sin 3).$ All functions are in radians.

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


Let $\mathcal{S}$ be the set $\{1, 2, 3, \dots, 12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Calculate the remainder when $n$ is divided by 500.


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


How many positive, three-digit integers contain at least one $3$ as a digit but do not contain a $5$ or a $7$ as a digit?


Given a line that passes through the points $\begin{pmatrix} -3 \\ 6 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$, and has a direction vector of the form $\begin{pmatrix} b \\ -1 \end{pmatrix}$, find the value of $b$.


Fifty teams play a tournament where each team plays against every other team exactly once. No ties occur, and each game has a winner with a $50\%$ chance for either team to win. Calculate the probability that no two teams win the same number of games, represented as $\frac{m}{n}$ where $m$ and $n$ are relatively prime. Determine $\log_2 n$.


In a right triangle, one leg is 9 units long, and the hypotenuse is 15 units long. Calculate $\cos A$ where angle $A$ is opposite the other leg. Additionally, determine $\sin A$.


Find the area bounded by the graph of $y = \arccos(\sin(x + \frac{\pi}{6}))$ and the $x$-axis on the interval $0 \le x \le 2\pi$.


A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 6 units. Determine the total volume in cubic units, of the space inside the sphere but outside the cylinder. Let $X\pi$ denote this volume. Find $X$ as a common fraction.

Solution: \boxed{X = 288 - 64\sqrt{5}}


Winnie has 60 raspberry lollipops, 135 mint lollipops, 10 orange lollipops, and 300 lemon lollipops. She decides to distribute as many lollipops as she can evenly among her 14 friends. Each friend gets the same number of lollipops. How many lollipops does Winnie have left after distributing them?


If 18 bahs are equal to 30 rahs, and 12 rahs are equal in value to 20 yahs, how many bahs are equal in value to 1200 yahs?


Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be positive real numbers such that $a + b + c + d + e + f = 10.$ Find the minimum value of
\[
\frac{1}{a} + \frac{9}{b} + \frac{25}{c} + \frac{49}{d} + \frac{81}{e} + \frac{121}{f}.
\]


Gage skated for 1 hour each day for 6 days and 2 hours each day for 2 days. How many minutes would he have to skate on the ninth and tenth days to average 100 minutes of skating each day for the overall period?


Compute the largest integer \( k \) such that \( 2010^k \) divides \( 2010! \).


Let \( g(x) \) be a function defined as follows:
\[
g(x) = \left\{
\begin{array}{cl}
-x & \text{if } x \leq 0, \\
3x - 22 & \text{if } x > 0.
\end{array}
\right.
\]
If \( a \) is negative, find \( a \) so that \( g(g(g(7))) = g(g(g(a))) \).

Solution: \boxed{a = -\frac{23}{3}}


If $\sqrt{8 + x} + \sqrt{27 - x} = 9$, what is the value of $(8 + x)(27 - x)$?


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


Tom and Jerry run, bicycle, and roller-skate at equal and constant rates. They both spend different amounts of time on each activity, yet the overall distance they cover is equal due to their constant rates in each activity. Tom runs for $3$ hours, bicycles for $4$ hours, and roller-skates for $2$ hours, covering a total of $104$ kilometers. Jerry rolls-skates for $2$ hours, bicycles for $6$ hours, and runs for $3$ hours, covering a total of $140$ kilometers. Determine the sum of the squares of Tom's running, bicycling, and roller-skating rates assuming all rates are whole numbers of kilometers per hour.


Evaluate the expression $\dfrac{\sqrt[4]{7}}{\sqrt[6]{7}}$ and find 7 raised to what power?


How many three-digit numbers are multiples of neither 3, 5, nor 7?


How many positive three-digit integers less than 600 have at least two digits that are the same?


A traffic light runs through the following cycle: green for 50 seconds, yellow for 5 seconds, and red for 45 seconds. Paul picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching?


What is the least positive integer value of $x$ such that $(3x)^2 + 2\cdot 43\cdot 3x + 43^2$ is a multiple of 53?


If $x^5 - 2x^4 + 3x^3 - px^2 + qx + 12$ is divisible by $(x + 3)(x - 2),$ find the ordered pair $(p,q).$

Solution: \boxed{(-28, -74)}


What is the least possible positive four-digit palindrome that is divisible by 4?


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


Let the function $g(x)$ satisfy
\[ g(x + y) = g(x) g(y) \]
for all real numbers $x$ and $y.$ Additionally, if it is given that $g(1) = g(-1)$ and $g(2x) = 4g(x),$ find all possible values of $g(0).$


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


Find $\tan J$ in the right triangle shown below.

[asy]

pair J,I,K;

J = (0,0);

K = (24,0);

I = (0,10);

draw(I--K--J--I);

draw(rightanglemark(I,J,K,20));

label("$J$",J,SW);

label("$K$",K,SE);

label("$I$",I,N);

label("$26$",(I+K)/2,NE);

label("$24$",K/2,S);

[/asy]


How many of the divisors of $10!$ are larger than $9!$?


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

Solution: \boxed{\begin{pmatrix} 10 \\ -11 \end{pmatrix}}


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


The truncated right circular cone has a large base radius 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid.


A student needs to choose a program of five courses from a list consisting of English, Algebra, Geometry, Calculus, History, Art, Science, and Latin. This program must include English, at least one mathematics course (from Algebra, Geometry, or Calculus), and at least one humanities course (from History, Art, or Latin). How many ways can the student choose such a program?


If \(a\), \(b\), and \(c\) are integers with \(a > b > c\), what is the smallest possible positive value of \(\frac{a+b+c}{a-b-c} + \frac{a-b-c}{a+b+c}\)?


Compute $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.


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

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


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


If two numbers are randomly chosen without replacement from $\{2, 3, 6, 9\}$, what is the probability that their product will be a multiple of 18? Express your answer as a common fraction.


The image of the point with coordinates $(-4, 0)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(2, 6)$.  Find $m + b$.


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


Let $\omega$ be a nonreal root of $z^4 = 1.$ Find the number of ordered pairs $(a, b)$ of integers such that $|a \omega + b| = 1.$


Let \( a \otimes b = a \times \sqrt{b + \sqrt{b + 1 + \sqrt{b + 2 + \sqrt{b + 3 + ...}}}} \). If \( 3 \otimes h = 15 \), find the value of \( h \).


In a quadrilateral $WXYZ$, the angles satisfy $\angle W = 3\angle X = 4\angle Y = 6\angle Z$. Determine the exact degree measure of $\angle W$.


Yan is somewhere between his office and a concert hall. To get to the concert hall, he can either walk directly there, or walk to his office and then take a scooter to the concert hall. He rides 5 times as fast as he walks, and both choices take the same amount of time. What is the ratio of Yan's distance from his office to his distance from the concert hall?


Calculate the value of the following expression: $\frac{1}{5}-\frac{1}{10}+\frac{1}{20}-\frac{1}{40}+\frac{1}{80}-\frac{1}{160}$? Express your answer as a common fraction.


What is the sum of all integer solutions to $|n| < |n-3| < 7$?


Find the smallest two-digit prime number such that reversing the digits of the number produces an even number.


When $m$ is divided by 7, a remainder of 2 is given. What is the remainder when $m+2030$ is divided by 7?


For all values of $x$ for which it is defined, let $g(x) = \cot \frac{x}{3} - \cot x$. This function can be expressed as
\[ g(x) = \frac{\sin mx}{\sin \frac{x}{3} \sin x}.\]
Find the value of $m$.


Determine the values of $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5$ in degrees, where the solutions to $z^5 = -32i$ can be expressed as:
\[
z_1 = r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 = r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 = r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 = r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 = r_5 (\cos \theta_5 + i \sin \theta_5),
\]
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ$.


Let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}.$  Find the determinant of the matrix whose column vectors are $2\mathbf{a} + 3\mathbf{b},$ $3\mathbf{b} + 4\mathbf{c},$ and $4\mathbf{c} + 2\mathbf{a},$ in terms of $D.$


Given that $M(3,7)$ is the midpoint of $\overline{CD}$ and $C(5,-1)$ is one endpoint, what are the product and sum of the coordinates of point $D$?


Bob rolls a fair ten-sided die each morning. If Bob rolls a number that is divisible by either 2 or 3, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected number of times Bob will roll his die?


The region between the graph of $y = g(x)$ and the $x$-axis has an area of 15 square units. What will be the area between the graph of $y = 4g(2x - 4)$ and the $x$-axis?


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


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


The repeating decimal for $\frac{7}{19}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?


The numbers 1, 4, 6, 8, 9, 10 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?

\[
\begin{array}{c||c|c|c|}
\times & p & q & r \\ \hline \hline
s & & & \\ \hline
t & & & \\ \hline
u & & & \\ \hline
\end{array}
\]


What is the sum of all two-digit positive integers whose squares end with the digits 36?


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 5x + 7 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a - 3,$ $b - 3,$ and $c - 3.$

Solution: \boxed{x^3 + 9x^2 + 22x + 19}


What is the sum of the 2009 fractions of the form $\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 2 through 2010? Express your answer as a decimal to the nearest thousandth.


What is the smallest possible area, in square units, of a right triangle with two sides measuring $6$ units and $8$ units?


Suppose \( x-4 \) and \( y+2 \) are multiples of 7.

What is the smallest positive integer, \( n \), for which \( x^2 + xy + y^2 + n \) is a multiple of 7?


What is the 8th term of an arithmetic sequence of 15 terms with the first and last terms being 3 and 42, respectively?


Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible. If the graphs of $y=f(x^2)$, $y=f(x^4)$, and $y=f(x^3)$ are drawn, at how many points do these curves intersect?


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


Solution: \boxed{\sqrt{\sqrt{10} - 1} - 1}


The length of a rectangular swimming pool exceeds three times its width by 10 feet, and the perimeter of the pool is 320 feet. What is the area of the swimming pool in square feet?

Solution: \boxed{4593.75 \text{ square feet}}


In triangle $PQR$ with vertices $P(0,10)$, $Q(5,0)$, and $R(10,0)$, a vertical line intersects $PR$ at $U$ and $\overline{QR}$ at $V$, forming triangle $UVQ$. If the vertical line is such that the distance $VQ = 3$ units, find the absolute difference between the $x$ and $y$ coordinates of point $U$.


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


If $f(x)$ is a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(3)=-9$, and $f(5)=-25$, find $f(0)$.


Let \( g(x) = 2x^2 - x + 2006 \). What is the greatest common divisor of \( g(98) \) and \( g(99) \)?


How many digits are located to the right of the decimal point when $\frac{5^8}{2^5 \cdot 10^6}$ is expressed as a decimal?


Add $7A3_{16} + 1F4_{16}$. Express your answer in base 16, using A for 10, B for 11, ..., F for 15.


A spinner is divided into six congruent sectors, numbered from 1 to 6. Jane and her brother each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins?


An ultra-marathon is 50 miles and 800 yards long. One mile equals 1760 yards.

Michael has completed five ultra-marathons. If the total distance Michael covered is $m$ miles and $y$ yards, where $0 \le y < 1760$, what is the value of $y$?


How many three-digit perfect squares are divisible by 4?


What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side are the same color? The tessellation includes interlocking hexagons and squares as depicted below:

[asy]
unitsize(1cm);
int i;
for(i=0; i < 7; i = i + 2){
    draw(shift(2*i,0)*polygon(6));
    draw(shift(2*(i+1), 0)*scale(0.577)*polygon(4));
}
[/asy]


Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[
\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}.
\]
Find the smallest possible value of $|p| + |q| + |r| + |s|$.


Three congruent isosceles triangles \( EFO \), \( OFG \), and \( OGH \) have \( EF=EO=OG=GH=15 \) and \( FG=EO=OH=18 \). These triangles are arranged to form trapezoid \( EFGH \), as shown. Point \( Q \) is on side \( FG \) such that \( OQ \) is perpendicular to \( FG \).

Point \( Z \) is the midpoint of \( EF \) and point \( W \) is the midpoint of \( GH \). When \( Z \) and \( W \) are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid \( EFZW \) to the area of trapezoid \( ZWGH \) in simplified form, and calculate \( r+s \) where the ratio is \( r:s \).


In pentagon $PQRST$, $QR=RS=ST=3$ units, $\angle T$ is a right angle, and $m \angle Q = m \angle R = m \angle S = 135^\circ$. The length of segment $PT$ can be expressed in simplest radical form as $c+3\sqrt{d}$ units. What is the value of $c+d$?


Let $A = (0,4),$ $B = (4,4),$ and $C = (p, p^2 - 4p + 4)$ be three points on the parabola $y = x^2 - 4x + 4,$ where $0 \le p \le 4.$ Find the largest possible area of triangle $ABC.$


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


Tom is planning a small gathering for two other couples, his wife, and himself. He plans to seat the three couples including himself and his wife at a circular table for 6, and wants no two men to be seated next to each other. How many different seating arrangements can he make, if rotations of the seating arrangement are not considered different, but reflections are different?


For how many values of the digit $A$ is it true that $174$ is divisible by $A$ and $306{,}8A4$ is divisible by $4$?


Two different natural numbers are chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that their greatest common factor is one? Express your answer as a common fraction.


Express $\sqrt{x} \div \sqrt{y}$ as a common fraction, given:

$\frac{ {\left( \frac{1}{3} \right)}^2 + {\left( \frac{1}{4} \right)}^2 }{ {\left( \frac{1}{5} \right)}^2 + {\left( \frac{1}{6} \right)}^2} = \frac{25x}{61y}$


The deli has five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Turkey, Swiss cheese, cheddar cheese, white bread, and chicken are each offered at the deli. If a customer never orders a sandwich with a ham/cheddar cheese combination, a sandwich with a white bread/chicken combination, or a turkey/Swiss cheese combination, how many different sandwiches could the customer order?

Solution: \boxed{194 \text{ different sandwiches}}


Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers.