In triangle $DEF$, $\angle D = 90^\circ$ and $\sin E = \frac{3}{5}$. Find $\cos F$.


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

Solution: \boxed{0, 1, 2}


Determine the slope \(m\) of the asymptotes for the hyperbola given by the equation
\[
\frac{y^2}{16} - \frac{x^2}{9} = 1.
\]


A rectangle has a perimeter of 100 meters and each side has an integer length. How many non-congruent rectangles meet these criteria?


Find the $3 \times 3$ matrix $\mathbf{M}$ such that for any $3 \times 3$ matrix $\mathbf{N}$, $\mathbf{M} \mathbf{N}$ results in the first row being tripled and swapped with the second row, which is scaled by a factor of 2, and the third row being tripled. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} 2d & 2e & 2f \\ 3a & 3b & 3c \\ 3g & 3h & 3i \end{pmatrix}.
\]

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


Triangle $PQR$ has vertices $P(0, 10)$, $Q(4, 0)$, and $R(10, 0)$. A line through $Q$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.


For how many values of the digit $B$ is it true that $45$ is divisible by $B$ and $451{,}B7$ is divisible by $3$?


Rationalize the denominator of $\frac{7}{\sqrt{63}}$.


A portion of the graph of \( g(x) = ax^2 + bx + c \) is shown below. The distance between grid lines on the graph is 1 unit.

What is the value of \( a+2b+c \)?

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-4,3,-2,9);
real g(real x) {return 6-(x+2)^2;}
draw(graph(g,-3.9,0.16,operator ..), red);
[/asy]


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


Find the matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 4 \\ -3 \\ 7 \end{pmatrix} \times \mathbf{v}\] for all vectors $\mathbf{v}.$

Solution: \boxed{\begin{pmatrix} 0 & -7 & -3 \\ 7 & 0 & -4 \\ 3 & 4 & 0 \end{pmatrix}}


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


Determine the number of integer values of \( x \) for which the square root of \( 3x \) is greater than 4 and less than 6.


Express the following as a common fraction: $\sqrt[3]{8 \div 12.75}$.


Determine the value of \( k \) if the roots of the quadratic equation \( 9x^2 + 5x + k \) are \( \frac{-5 \pm i\sqrt{371}}{18} \).


If the system of equations
\[
4x - 3y = c,
\]
\[
6y - 8x = d.
\]
has a solution, find \(\frac{c}{d},\) assuming \(d \neq 0.\)


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix}^n = \mathbf{I}.\]


Calculate the sum of the coefficients in the polynomial \( 5(2x^8 - 3x^5 + 4) + 6(x^6 + 9x^3 - 8) \) after it is completely expanded.


How many three-digit perfect cubes are divisible by both $8$ and $9$?


Each of four, standard, six-sided dice is rolled once. What is the probability that there is at least one pair and no more than two pairs (that is, there are two dice showing the same value or two different pairs, but no three dice show the same value)?


For \(\mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 7 \\ 5 \end{pmatrix}\),
\[
\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -14 \\ -10 \end{pmatrix}.
\]
Find \(y\).


How many 12-step paths are there from point $E$ to point $G$ which pass through point $F$ on a grid, where $E$ is at the top-left corner, $F$ is 5 steps to the right and 2 steps down from $E$, and $G$ is at the bottom-right corner, 6 steps to the right and 5 steps down from the top-left corner?


Let $ABCDEFGH$ be a rectangular prism where $AB = 2$, $AD = 4$, and $AE = 3$. Find $\sin \angle GAC$.

Solution: \boxed{\sin \angle GAC = \frac{3}{\sqrt{29}}}


A play has three different male roles, three different female roles, and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If six men and seven women audition, in how many ways can the eight roles be assigned?


If $x$ is a real number and $x^3 - 9x^2 + 23x - 15 < 0$, what are the possible values for $x$? Use interval notation to express your answer.

Solution: \boxed{(-\infty, 1) \cup (3, 5)}


Find the point on the line
\[ y = \frac{x + 3}{2} \]
that is closest to the point \((4,0)\).

Solution: \boxed{\left(\frac{13}{5}, \frac{14}{5}\right)}


What is the smallest positive integer $m$ such that $m^2 + m$ is divisible by some but not all integer values of $k$ when $0\leq k \leq m$?


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

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


In the diagram, $BP$ and $BQ$ trisect $\angle ABC$. $BM$ trisects $\angle PBQ$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$.


Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{2} (AL) + \log_{2} (AM) &= 6, \\
\log_{2} (ML) + \log_{2} (MR) &= 8, \\
\log_{2} (RA) + \log_{2} (RL) &= 10.
\end{align*}
Compute the value of the product $ARML$.


A bridge is structured by aligning a wooden plank between two isosceles triangular supports. Here’s the diagram:
```
         A         D
        / \       / \
       /   \     /   \
      B-----C   E-----F
```
Given $AB = AC$ and $DE = DF$, and with $\angle BAC = 15^\circ$ and $\angle EDF = 45^\circ$, determine the sum $\angle DAC + \angle ADE$.


What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect squares?


In quadrilateral $ABCD$, $\angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to both $\overline{CD}$ and $\overline{AB}$, with $AB = 15$, $BC = 28$, and $CD = 17$. Calculate the total perimeter if there are two such quadrilaterals $ABCD$.

Solution: \boxed{120 + 2\sqrt{1298}}


The values of a function $g(x)$ are given in the table below.

\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $g(x)$ & 4 & 3 & 1 & 5 & 2 \\ \hline
\end{tabular}
If $g^{-1}$ exists, then what is $g^{-1}(g^{-1}(g^{-1}(3)))$?


A library has eight identical copies of a particular book. Some of these books can be either at the library or checked out at any given time. 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 one book must be in the library and at least one must be checked out?


Find the number of real solutions of the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = 1.\]


Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated (each exactly once), and digits cannot be repeated?


The length of the segment between the points $(3a, 2a-5)$ and $(5, 0)$ is $3\sqrt{10}$ units. What is the product of all possible values for $a$?


What is the smallest integer $k$ such that $k > 1$ and $k$ has a remainder of 1 when divided by any of $19$, $14$, and $9$?


Find the point in the plane $4x - 3y + 6z = 24$ that is closest to the point $(2,1,4).$

Solution: \boxed{\left(\frac{102}{61}, \frac{76}{61}, \frac{214}{61}\right)}


A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.


What is the probability of spinning a spinner with eight sections labeled as 3, 6, 1, 4, 8, 10, 2, and 7, and getting a number that is either a prime or an odd number? Express your answer as a common fraction.


The positive integers $A, B,$ and $C$ form an arithmetic sequence, while integers $B, C,$ and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{4}$, what is the smallest possible value of $A + B + C + D$?


If $x$, $y$, and $z$ are positive integers such that $\gcd(x,y) = 224$ and $\gcd(x,z) = 546$, what is the smallest possible value of $\gcd(y,z)$?


Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. Both $f$ and $g$ intersect at the point $(50, -50)$, and the minimum value of $f$ is the same as $g$. Determine the value of $a + c$.


How many three-digit positive integers \( y \) satisfy \( 5327y + 673 \equiv 1850 \pmod{17} \)?


Consider the "Modulo $m$ graph paper" with a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<m$. Graph the congruences $$y\equiv 7x+3\pmod{20}$$ and $$y\equiv 13x+17\pmod{20}$$ on modulo $20$ graph paper. Find the sum of the $x$-coordinates of the points where these graphs intersect.


A cylindrical tank is one-third full of a solution. The ratio of salt to water in the solution is 1:5. If the tank is 9 feet tall and has a diameter of 3 feet, what is the volume of salt in the tank? Express your answer in cubic feet to the nearest hundredth.


The National University’s Math Department is arranging a group activity for their new batch which contains fewer than $400$ students. When the students are arranged in columns of $17$, $15$ are in the last column. When the students arrange themselves in columns of $19$, there are $10$ students in the last column. Determine the total number of students in this new batch.


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


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


Marius is participating in another wildlife photo contest, this time with nine unique leopards, each of different heights. For a better aesthetic effect, he decides that the two shortest leopards, as well as the tallest leopard, should be positioned at the ends (two on one end, one on the other). In how many ways can he arrange these leopards?

Solution: \boxed{2880 \text{ ways}}


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 $x < -2$ or $|x-30| \le 2.$ Given that $a < b,$ find the value of $a + 2b + 3c.$


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

Solution: \boxed{\left(1, -\frac{23}{8}\right)}


In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\).


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 1000 units?


A box contains tiles, numbered 1, 2, 3,..., 98, 99, 100. Only tiles which are marked with a number congruent to $3 \pmod{7}$ are green. One tile is chosen randomly from the box. What is the probability that the tile is green?


What is the remainder when the sum \(1^2 + 2^2 + 3^2 + \dots + 20^2\) is divided by 13?


The number $0.6125$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


What is the smallest positive integer that is a multiple of both 42 and 56 but not a multiple of 18?


Find the sum of the rational roots of $h(x) = x^3 - 6x^2 + 9x - 3$.


Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 10,$ and $y + \frac {1}{x} = 5.$ Find $z + \frac {1}{y}.$


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


If the two roots of the quadratic equation \(6x^2 + 5x + m\) are \(\frac{-5 \pm i\sqrt{231}}{12}\), what is \(m\)?


For what values of $x$ does the inequality $x^3 - 12x^2 + 36x > 0$ hold?

Solution: \boxed{(0,6) \cup (6,\infty)}


Expand $-(5 - 2c)(c + 3(5 - 2c))$. What is the sum of the coefficients of the expanded form?


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


It is currently $5\!:\!45\!:\!00\text{ a.m.}$ What time will it be in $10,000$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)


Consider polynomials $Q(x)$ of degree at most $3$, each of whose coefficients is an element of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $Q(1) = 11$?


The numbers $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 60, $a$ is three times $b$. Find the value of $b$ when $a=12$.


Consider the graph of $y=g(x)$ for $-2\le x\le 4$. The graph passes through the points as follows: $(-2, -1)$, $(-1,-1)$, $(0,0)$, $(1,0)$, $(2,1)$, $(3,1)$, and $(4,2)$. Each interval is a line segment connecting consecutive points. Determine the range of $g(x) - x$ on this interval. Express your answer in interval notation.

Solution: \boxed{[-2, 1]}


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


The equation
\[(x - \sqrt[3]{20})(x - \sqrt[3]{70})(x - \sqrt[3]{120}) = \frac{1}{2}\] has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$


For how many integers $n$ between 1 and 200 is the greatest common divisor (gcd) of 21 and $n$ equal to 3?


Find the matrix **N** such that
**N**
\[\begin{pmatrix} 2 & -5 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} -21 & -2 \\ 13 & 1 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} \frac{71}{14} & -\frac{109}{14} \\ -\frac{43}{14} & \frac{67}{14} \end{pmatrix}}


Evaluate $\lfloor -3.75 \rfloor + \lceil 34.2 \rceil + \frac{1}{2}$.


For how many different digits $n$ is the three-digit number $25n$ divisible both by $n$ and by $5$?


Four of the five vertices of a polygon shaped as a right trapezoid are (5, 11), (16, 11), (16, -2), and (5, -2); it includes a semicircle with a diameter along the bottom base, centered at (10.5, -2) and radius 5.5. Determine the total area bounded by the lower base of the trapezoid and the semicircle.


Mrs. Thompson recorded the following scores from a recent test taken by all of her students. Calculate the average percent score for these $100$ students.

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&10\\\hline
85&15\\\hline
75&20\\\hline
65&25\\\hline
55&15\\\hline
45&10\\\hline
35&5\\\hline
\end{tabular}


Six points are placed inside or on a circle with radius $1$. Determine the smallest possible number $b$, such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$.


Find the remainder when $8x^4 + 5x^3 - 3x^2 - 7x + 15$ is divided by $3x - 9.$


Let $a = \pi/1004$. Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\] is divisible by $2$.


Consider a cube $ABCDEFGH$ where each edge measures 1 unit. Calculate the volume of the pyramid formed by the vertices $ABDH$.


How many odd integers between 1500 and 3000 have all three of the numbers 18, 24, and 36 as factors?


Let \( p, q, r, s \) be real numbers such that
\[
\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{7}.
\]
Find the sum of all possible values of
\[
\frac{(p - r)(q - s)}{(p - q)(r - s)}.
\]


A right circular cone is sliced into five pieces by planes parallel to its base. Each slice has the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?


A rectangular solid has a volume of \(512 \text{ cm}^3\) and a total surface area of \(384 \text{ cm}^2\). The dimensions of the solid are in geometric progression. Determine the sum of the lengths in cm of all the edges of this solid.

Solution: \boxed{96 \text{ cm}}


Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2 + y^2 = 16x + 36y$. What is the minimum value of $y$?

Solution: \boxed{18 - \sqrt{388}}


Thirty teams play a tournament in which each team plays every other team exactly once. As before, no ties occur with each team having a 50% chance to win any given match. The probability that no two teams win the same number of games is expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $\log_2 n.$


What are the rightmost three digits of $7^{2023}$?


How many vertical asymptotes does the equation $y=\frac{x^2-4x+3}{x^3+4x^2-11x-30}$ have?


In triangle $ABC,$ $\cot A \cot C = \frac{1}{3}$ and $\cot B \cot C = \frac{1}{12}.$  Find $\tan C.$

Solution: \boxed{\tan C = \sqrt{21}}


What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~33$ is an arithmetic sequence?


Max picks two different cards without replacement from an enlarged 65-card deck, where the deck is equally divided into five suits (each suit has 13 cards). What is the probability that the cards are of different suits?


Define a new operation $A \spadesuit B = 4A + 3B + 6$. Find the value of $A$ for which $A \spadesuit 5 = 59$.


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


In a row of 10 chairs, Mary and James each choose their seats at random but are not allowed to sit in the first or the last chair (chairs #1 and #10). What is the probability that they do not sit next to each other?


Find the fraction that equals \( 0.\overline{568} \).


What is the smallest positive perfect square that is divisible by both 5 and 7?


A gardener plants four maple trees, five oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.


Let b be a real number randomly selected from the interval $[-9,9]$. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation $x^4 + 16b^2 = (2b^2 - 8b)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.


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


The increasing sequence $1, 3, 4, 9, 10, 12, 13, \dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\text{th}}$ term of this sequence.


A right triangle with legs \(15\) and \(12\) is similar to another right triangle with corresponding legs \(y\) and \(9\), respectively. What is the value of \(y\)? Express your answer as a decimal to the nearest hundredth.


How many integers between 1 and 300 are multiples of both 6 and 8 but not of either 4 or 11?


An ordinary $6$-sided die has numbers from $1$ to $6$. How many ways can I paint three faces of the die red, ensuring that the sum of the numbers on these faces doesn't equal $11$?


Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, such that $\mathbf{u} \cdot \mathbf{v} = 0$. Let $\mathbf{w} = \mathbf{u} \times \mathbf{v} + 2\mathbf{u}$. Compute the value of $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.


Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $6$ steps at a time, there are $4$ steps left at the top. When he does it $7$ steps at a time, there are $3$ steps left at the top. What is the smallest possible value of $n$ that is greater than $20$?


A circle has an area of \(x\) square units and a circumference of \(y\) units. The sum of \(x\) and \(y\) is \(100\pi\). Determine the radius of this circle in units.

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


A pentagon is formed by connecting the points $(0,0)$, $(2,0)$, $(3,2)$, $(2,3)$, and $(0,2)$, then returning to $(0,0)$. The perimeter of the pentagon can be written in the form $a+b\sqrt{2}+c\sqrt{3}+d\sqrt{5}$, where $a$, $b$, $c$, and $d$ are whole numbers. Find $a+b+c+d$.


Find the value of $k$ such that the equation
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 5 \\ -8 \end{pmatrix} = \begin{pmatrix} 0 \\ -1 \end{pmatrix} + s \begin{pmatrix} -2 \\ k \end{pmatrix}\]
does not have any solutions in $t$ and $s$.


Calculate $\sqrt{12q} \cdot \sqrt{8q^2} \cdot \sqrt{9q^5}$ and express your answer in simplest radical form in terms of $q$.

Solution: \boxed{12q^4 \sqrt{6}}


I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction.


At the SuperPup Training Center, dogs can learn to do three tricks: jump, fetch, and bark. Of the dogs at the center:
\begin{tabular}{l@{\qquad}l}
40 dogs can jump & 15 dogs can jump and fetch \\
35 dogs can fetch & 10 dogs can fetch and bark \\
30 dogs can bark & 14 dogs can jump and bark \\
7 dogs can do all three & 5 dogs can do none
\end{tabular}
How many dogs are at the center?


Imagine an 11-sided die with 5 red faces, 4 yellow faces, and 2 blue faces. What is the probability of rolling a yellow face followed by a blue face?


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


The interior of a right, circular cone is 12 inches tall with a 4-inch radius at the opening. The interior of the cone is filled with vanilla ice cream, and the cone has a hemisphere of chocolate ice cream exactly covering the opening of the cone. Additionally, a smaller right cone with a height of 3 inches and a base radius of 2 inches is placed symmetrically on top of the hemisphere. Calculate the total volume of ice cream.

Solution: \boxed{\frac{332}{3}\pi \text{ cubic inches}}


Calculate the area of the ellipse described by the equation $3x^2 - 12x + y^2 + 4y + 4 = 0.$


Let \(x, y, z\) be nonzero complex numbers such that \(x + y + z = 10\) and
\[
(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz.
\]
Find \(\frac{x^3 + y^3 + z^3}{xyz}\).


The first few rows of a sequence are arranged as follows, where each row number's corresponding value increases in steps of $3$ starting from $3$, and each row contains as many numbers as the cube of the row index. What is the value of the $80^{\mathrm{th}}$ number if the sequence is continued?

$\bullet$ Row 1: $3$

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

$\bullet$ Row 3: $9,$ $9,$... (total of $27$ times)


For how many integer values of $m$ between 1 and 594 inclusive does the decimal representation of $\frac{m}{595}$ terminate?


Determine the remainder when \(y^{20} + y^{15} + y^{10} + y^5 + 1\) is divided by \(y^3 + y^2 + y + 1\).


A company needs to hire a number of new employees so that 95 of the new employees will be able to handle customer service, 80 of the new employees will be able to handle technical support, and exactly 30 of the new employees will be able to handle both. (These 30 are included in the 95 and 80 mentioned above.) What is the minimum number of employees that need to be hired?


Find the minimum value of $w$ if $w = 2x^2 + 3y^2 - 12x + 9y + 35$.


Given six positive integers from a list of nine positive integers which are $4, 5, 3, 7, 9, 6$, what is the largest possible value of the median of this list of nine positive integers?


Let $\mathcal{S}$ be the set $\{1, 2, 3, \ldots, 12\}$. Let $n$ be the number of sets consisting of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder when $n$ is divided by $1000$.


In a 4 by 4 grid where each small square measures 3 cm by 3 cm, each square is shaded gray. The grid is covered by three small circles, each with a diameter equal to a side of a square and one large circle with a diameter equal to the length of two squares, all unshaded. Calculate the area of the visible shaded region in the form $A-B\pi$ square cm and determine the value of $A+B$.


Both $c$ and $d$ are positive integers and $d > 1$. When $c^d$ is the greatest possible value less than 800, what is the sum of $c$ and $d$?


Harry has a quadratic of the form $x^2+bx+24$, where $b$ is a specific positive number. Using his knowledge of how to complete the square, Harry is able to rewrite this quadratic in the form $(x+n)^2+16$. What is $b$?


A $\textit{composite number}$ is a number that has two or more prime factors. The number 103 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such composite numbers contributing to this sum?


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


Find a polynomial $q(x)$ if the graph of $\frac{x^3-2x^2-5x+3}{q(x)}$ has vertical asymptotes at $1$ and $-3$, no horizontal asymptote, and $q(4) = 20$.

Solution: \boxed{q(x) = \frac{20}{21}x^2 + \frac{40}{21}x - \frac{20}{7}}


How many different four-letter arrangements can be formed using the letters $A, B, C, D, E, F, G,$ and $H$, if the first letter must be $D$, two of the other letters must be $A$ and $B$, and no letter can be used more than once in the arrangement?


For some constants $x$ and $a$, the fourth, fifth, and sixth terms in the expansion of $(x + a)^n$ are 210, 420, and 630, respectively. Find $n.$


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

Determine the number of possible values of $f(-2),$ and the sum of all possible values of $f(-2).$ Return the product of these two quantities.


Let \( n \) be a positive integer. The sequence \( (y_k) \) is defined by \( y_0 = 0 \), \( y_1 = 1 \), and
\[
y_{k + 2} = \frac{(n - 2) y_{k + 1} - (n - k - 1) y_k}{k + 1}
\]
for \( k \ge 0 \). Find \( y_0 + y_1 + y_2 + \dotsb \) as a function of \( n \).

Solution: \boxed{2^{n - 2}}


The eleven horizontal and eleven vertical lines on a $10\times10$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


Determine the tens digit of $13^{2023}$.


A line contains the points $(8,10)$, $(-3, k)$, and $(-12, 6)$. What is the value of $k$?


A rectangle is called super cool if the number of square units in its area equals three times the number of units in its perimeter. This rectangle must also have integer side lengths. What is the sum of all different possible areas of super cool rectangles?


Given six points in the plane \((-\frac{5}{2}, 2), (0,0), (0,3), (4,0), (4,3), (2,4)\), with no three points collinear, there is a unique conic section passing through all these points. Propose that this conic section is an ellipse centered at \((2, \frac{3}{2})\) with its axes parallel to the coordinate axes. Find the length of the minor axis of this ellipse.


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


In the expression $x \cdot y^z - w$, the values of $x$, $y$, $z$, and $w$ are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the expression?


Two spinners, $C$ and $D$, are spun. Spinner $C$ is divided into four equal sections numbered 2, 5, 7, 6. Spinner $D$ is divided into three equal sections numbered 11, 3, 6. What is the probability that the product of the two spinners' numbers is a multiple of 3? Express your answer as a common fraction.


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{2}{5} \end{pmatrix}}


If $\mathbf{B}^{-1} = \begin{pmatrix} 3 & -1 \\ 0 & 5 \end{pmatrix},$ then find the inverse of $\mathbf{B}^3.$

Solution: \boxed{\begin{pmatrix} 27 & -49 \\ 0 & 125 \end{pmatrix}}


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{37}$?


In trapezoid $JKLM$ with $JK$ parallel to $LM$, the diagonals $JL$ and $KM$ intersect at $O$. Given that the area of triangle $JKO$ is 75 square units and the area of triangle $JMO$ is 45 square units, find the area of trapezoid $JKLM$.


For how many positive integers $n$ does $1 + 2 + \cdots + n$ evenly divide $10n$?


How many obtuse interior angles can a quadrilateral have?


What is the area, in square units, of the rectangle with vertices at $P\ (1, 2)$, $Q\ (-3, 3)$, $R\ (-2, 7)$, and $S\ (2, 6)$?

Solution: \boxed{17 \text{ square units}}


Determine the value of the infinite product $(3^{1/2})(9^{1/4})(27^{1/8})(81^{1/16}) \dotsm.$  Enter your answer in the form "\sqrt[a]{b}", which stands for $\sqrt[a]{b}.$


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

Let $m$ be the number of possible values of $g(4),$ and let $t$ be the sum of all possible values of $g(4).$ Find $m \times t.$


The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. Find \(b^2\).


Ten identical crates each with dimensions $2\mathrm{ft}\times 3\mathrm{ft}\times 5\mathrm{ft}$ are to be stacked one on top of the other. Each crate's orientation is chosen at random, and placed flat on the previous crate. Calculate the probability that the total height of the stack is exactly $38\mathrm{ft}$. Express the probability in its simplest form, $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Find $m$.


What is the matrix $\mathbf{N}$ that performs the transformation which sends square $EFGH$ to square $E'F'G'H'$? Assume that $E$ goes to $E'$, and so on, by rotating $30^\circ$ and scaling by $2$.

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


If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 6, \\
a+b+d &= -3, \\
a+c+d &= 0, \text{ and} \\
b+c+d &= -9,
\end{align*}
what is $ac + bd$?


What is the minimum value of $3x^2 - 18x + 2023$?


Let
\[ p(x,y) =
\begin{cases}
x + 2y &\quad \text{if } x > 0 \text{ and } y > 0, \\
x - 3y &\quad \text{if } x < 0 \text{ and } y < 0, \\
4x + 2y &\quad \text{if } x \geq 0 \text{ and } y \leq 0, \\
x^2 + y^2 &\quad \text{if } x = y.
\end{cases}
\]
What is $p(p(2,-2),p(-3,-1))$?


Triangle $PQR$ has side lengths $PQ = 7$, $QR = 8$, and $PR = 9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?


Name the greatest whole number less than $200$ that is a perfect square and has exactly $9$ positive factors.


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle, dividing the triangle into a square and two smaller right triangles. If the area of one of the smaller right triangles is $n$ times the area of the square, find the ratio of the area of the other smaller right triangle to the area of the square. Express your answer as a common fraction in terms of $n$.


The parabola $y = x^2 + 4$ and the hyperbola $y^2 - mx^2 = 1$ are tangent. Find $m$.

Solution: \boxed{m = 8 + 2\sqrt{15}}


Many states are considering a new design for their license plates with a sequence of four letters followed by four digits. The new pattern is designed to have each four-letter four-digit arrangement as a palindrome (reading the same left-to-right as right-to-left). Given that each arrangement is equally likely, find the probability that such a license plate will contain a palindrome in both the letter and digit sequences. Express your answer as a fraction $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and find $p+q.$


A region is bounded by quarter circular arcs constructed on each side of a square, where each side of the square measures $4/\pi$. What is the perimeter of this region?


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

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


Given that the point $(3,5)$ is on the graph of $y=f(x)$, determine one point that must be on the graph of $2y=4f(3x)-3$. What is the sum of the coordinates of that point?


Keiko and Ephraim each toss two pennies. What is the probability that they get the same number of heads? Express your answer as a common fraction.


A square surrounds four identical circles, each with a radius of 7 inches. The circles are arranged such that each circle touches two neighboring circles and two sides of the square. Calculate the area of this square.


Sandy plans to cover one exterior wall and two identical roof faces of a playhouse with a weather-resistant siding. The siding is sold in 10-foot by 15-foot sections at a cost of $\$30.50$ each. The wall measures 10 feet by 8 feet, and each of the roof faces measures 10 feet by 6 feet. How much will it cost Sandy to purchase the necessary siding?


Consider a game involving two standard decks of 52 cards each mixed together, making a total of 104 cards. Each deck has 13 ranks and 4 suits, with the suits retaining their colors as in a standard deck. What is the probability that the top card of this combined deck is the King of $\diamondsuit$?


The polynomial $x^{104} + Cx + D$ is divisible by $x^2 + x + 1$ for some real numbers $C$ and $D$. Find $C + D$.


A right cylinder with a base radius of 4 units is inscribed in a sphere of radius 7 units. Find the total volume, in cubic units, of the space inside the sphere and outside the cylinder, expressed as $X\pi$. Compute $X$ as a common fraction.

Solution: \boxed{\frac{1372}{3} - 32\sqrt{33}}


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


Factor the expression $4x(x-5) + 7(x-5) + 12(x-5)$.

Solution: \boxed{(4x + 19)(x-5)}


What is $\frac{1}{6}$ of $\frac{1}{4}$ of $\frac{1}{5}$ of 120? Express the result as the number of halves.


In $\triangle ABC$, where $\angle A = 90^\circ$ and $\tan C = 5$, find $\cos C$.


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


Yan is somewhere between his home and the stadium. To reach the stadium, he can either walk directly to the stadium or walk home and then ride his bicycle to the stadium. He rides 4 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?


A circle centered at $O$ is circumscribed about $\triangle ABC$ as follows:
- $\angle AOB = 120^\circ$
- $\angle BOC = 130^\circ$
What is the measure of $\angle BAC$, in degrees?


Let $a$ and $b$ be positive real numbers such that
\[\frac{1}{a + 3} + \frac{1}{b + 3} = \frac{1}{4}.\]Find the minimum value of $a + 3b.$

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


Line segment $\overline{AB}$ is a diameter of a circle with $AB = 30$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. Determine the exact area of the region bounded by this curve.


What is the smallest five-digit integer equivalent to 5 mod 9?


Julie has an infinite number of balls and empty boxes that can each hold up to seven balls. The boxes are arranged in a row from left to right. At the first step, she places a ball in the leftmost box. At each subsequent step, she places a ball in the first box from the left that still has room, and empties any box to its left. Determine the total number of balls in the boxes as a result of Julie's $1729$th step.


Find the largest value of $c$ such that $2$ is in the range of $f(x)=x^2-7x+c$.


Solve the congruence \( 13n \equiv 9 \pmod{53} \).


Find the radius of the circle with equation $2x^2 - 8x + 2y^2 + 4y + 10 = 0$.


In the country of Yonder, license plates are made up of two letters, followed by three digits, and finishing with another letter. How many valid license plates are possible?


Given a line that passes through the points $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$, find the value of $b$ where the line's direction vector is of the form $\begin{pmatrix} b \\ -1 \end{pmatrix}$.


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


In New Perfectville, the streets are all $30$ feet wide, and each block encloses a square with side length $500$ feet. If Sarah runs around the block on the $500$-foot side of the street and Sam runs on the street just opposite, calculate how many more feet than Sarah does Sam run for every lap around the block.


In the diagram, one polygon is a regular pentagon and another is a regular octagon. Calculate the sum of the measures of angles $ABC$ and $ABD$, where:
- $A$ and $B$ are common vertices of the pentagon and octagon.
- $C$ is a vertex of the pentagon not shared with the octagon.
- $D$ is a vertex of the octagon not shared with the pentagon.

[Insert appropriate diagram here]


Al, Betty, and Clare split $\$1200$ among themselves to be invested in different ways, each starting with different amounts. At the end of one year, they have a total of $\$1800$. Betty and Clare have both doubled their money, whereas Al's investment has decreased by 25%. What was Al's original portion?


Express as a common fraction: $$\cfrac{ \frac{3}{7}+\frac{5}{8} }{ \frac{5}{12}+\frac{2}{15}}.$$


Find the maximum value of M(k), the number of factors of 2 in the prime factorization of $10\ldots 0648$, where there are k zeros between the 1 and the 6.


Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x - 11$, the remainder is 25, and when $Q(x)$ is divided by $x - 101$, the remainder is 13. What is the remainder when $Q(x)$ is divided by $(x - 11)(x - 101)$?

Solution: \boxed{-\frac{2}{15}x + \frac{397}{15}}


A company incurs a daily maintenance fee of $\$600$ and pays each worker $\$20$ per hour. Each worker produces 7 widgets per hour, and each widget sells for $\$4$. How many workers must the company employ to make a profit during a 10-hour workday?


In right triangle $XYZ$, $XY=15$, $XZ=9$, and $YZ=12$ units. What is the distance from $Z$ to the midpoint of segment $XY$?


A large equilateral triangle is constructed by arranging smaller equilateral triangles made of paper, each colored on both sides with the same color. There are eight different colors available. The large triangle is formed by placing one small triangle in the center and three others at the corners within the outline of the large triangle, as described:

[asy]
draw((0,0)--(10,0)--(5,8.7)--cycle);
draw((5,0)--(2.5,4.3)--(7.5,4.3)--cycle);
[/asy]

Two large triangles are considered distinguishable if it is not possible to place one on the other such that all corresponding small triangles match in color, even using translations, rotations, and reflections. How many distinguishable large equilateral triangles can be formed?


In triangle $PQR$, the midpoint of $\overline{QR}$ is $(2,3,1),$ the midpoint of $\overline{PR}$ is $(1,8,-2),$ and the midpoint of $\overline{PQ}$ is $(3,1,5).$  Find the coordinates of vertex $P.$

Solution: \boxed{(2, 6, 2)}


In a sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. Additionally, the terms of the sequence satisfy the relation:
\[a_n + a_{n+1} + a_{n+2} = n\]
for all $n \ge 1$. Determine $a_{1000}$.


Let $a$, $b$, and $c$ be positive numbers such that $abc = 1$, $a + \frac{1}{c} = 7$, and $b + \frac{1}{a} = 35$. Find $c + \frac{1}{b}$.


A line is parameterized with its direction vector in the form $\begin{pmatrix} 3 \\ c \end{pmatrix}$. Find $c$. The line passes through the points $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$.


Two distinct positive integers \(a\) and \(b\) are factors of 48. If \(a \cdot b\) is not a factor of 48, what is the smallest possible value of \(a \cdot b\)?


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2023,0),(2023,2024),$ and $(0,2024)$. What is the probability that $x > 9y$? Express your answer as a common fraction.


Two circles of radius 12 cm overlap such that each circle passes through the center of the other. Find the length of the common chord between these two circles, expressed in simplest radical form.

Solution: \boxed{12\sqrt{3} \text{ cm}}


How many positive multiples of $11$ that are less than $2000$ end with the digit $7$?


Simplify $4 \cdot \frac{18}{5} \cdot \frac{25}{-72}$.


Find the range of the function $g(x) = \cos^4 x - \cos x \sin x + \sin^4 x$.

Solution: \boxed{\left[0, \frac{9}{8}\right]}


If $x + \frac{1}{x} = -\sqrt{3}$, find the value of $x^{24}$.


Let \( r_1, r_2, \) and \( r_3 \) be the roots of the polynomial equation
\[x^3 - 4x^2 + 10 = 0.\]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3. \)

Solution: \boxed{x^3 - 12x^2 + 270}


An airplane has three sections: First Class (30 seats), Business Class ($20\%$ of the total number of seats) and Economy ($\frac{3}{4}$ of the total number of seats). How many seats does the plane have?


Eight 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?


Let $x$, $y$, $z$, and $w$ be positive real numbers such that $x + y + z + w = 2$ and $xyzw = \frac{1}{16}$. Find the minimum value of
\[
\frac{x + y + z}{xyzw}.
\]


James notices that a 60-foot tree next to him is casting a 20-foot shadow. How tall, in inches, is James if he is casting a 25-inch shadow at the same time?


A $6 \times 9$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction.


The equation $x^2 + ax = -24$ has only integer solutions for $x$. If $a$ is a positive integer, what is the greatest possible value of $a$?


How many three-digit positive integers are congruent to 2 (mod 4)?


Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{24} \rfloor.\]


Below is the graph of $y = a \csc(bx + c)$ for some constants $a > 0$, $b > 0$, and $c$. The function reaches its minimum positive value of 3 at $x = \frac{\pi}{6}$. Find $a$ and $b$.


Consider the function given by the expression $\frac{x^2 - 4x + 4}{x^3 - 2x^2 - x + 2}$. Determine the number of holes ($a$), vertical asymptotes ($b$), horizontal asymptotes ($c$), and oblique asymptotes ($d$) in the graph of the function. Finally, calculate $a + 2b + 3c + 4d$.


Imagine you own 8 shirts, 5 pairs of pants, 4 ties, and 3 different jackets. If an outfit consists of a shirt, a pair of pants, and optionally a tie and/or a jacket, how many different outfits can you create?


Solve $\arcsin x + \arcsin (x-1) = \arccos (1-x).$


Let $R$ be the set of all possible remainders when a number of the form $3^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000.


Fifty-five percent of a train's passengers are men and fifteen percent of those men are in the luxury coach. What is the number of men in the luxury coach if the train is carrying 300 passengers?


I have 7 books on a shelf, three of which are identical copies of one math book and two of which are identical copies of a physics book, and all other books are different. In how many ways can I arrange these books on the shelf?


Find a positive integer that is divisible by 18 and has a square root between 22 and 22.5.


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


The roots of the equation \( x^2 + kx + 7 = 0 \) differ by \( \sqrt{85} \). Determine the greatest possible value of \( k \).


In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 65^\circ$ and $\angle ACB = 20^\circ$, then find the measure of $\angle BHC$, in degrees.


What is the probability that the square root of a randomly selected three-digit whole number is less than 20? Express your answer as a common fraction.


How many different five-letter arrangements can be formed using the eight letters $A, B, C, D, E, F, G,$ and $H$, if the first letter must be $D$, one of the letters must be $E$, and another one of the letters must be $A$ in the last position, and no letter can be used more than once?


A circle centered at $O$ is circumscribed about $\triangle ABC$ as follows: $O$ is the center, $\angle AOB = 130^\circ$, and $\angle BOC = 90^\circ$. Find the measure of $\angle BAC$.


Solution: \boxed{1 - \frac{1}{\log_3 101}}


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

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


It can be shown that for any integer \( n \),
\[
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{-n} = \begin{pmatrix} F_{-n + 1} & F_{-n} \\ F_{-n} & F_{-n - 1} \end{pmatrix},
\]
where \( F_n \) denotes the \( n \)th Fibonacci number defined for all integers \( n \) (with \( F_{-n} = (-1)^{n+1} F_n \)).

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


Consider a quadratic equation of the form $x^2 + ax + \frac{1}{4}$, where $a$ is a specific negative number. Using the method of completing the square, the quadratic is rewritten in the form $(x+n)^2 + \frac{1}{16}$. Determine the value of $a$.


A line whose $y$-intercept is $(0,8)$ intersects the ellipse $4x^2 + 25y^2 = 100.$ Find all possible slopes of this line.

Solution: \boxed{(-\infty, -\sqrt{2.4}] \cup [\sqrt{2.4}, \infty)}


For each value of \(x\), \(g(x)\) is defined to be the minimum value of the three numbers \(3x + 3\), \(\frac{1}{3} x + 2\), and \(-\frac{2}{3} x + 8\). Find the maximum value of \(g(x)\).


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


Find the remainder when $5^{5^{5^5}}$ is divided by 500.


Emma writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them. How many integers are in Emma's list?


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

\begin{tabular}{ |c | c|}
\hline \textbf{Type of Car} & \textbf{$\%$ of Total Cars Sold} \\ \hline
Audi & $10\%$ \\ \hline
Toyota & $20\%$ \\ \hline
Acura & $15\%$ \\ \hline
Ford & $25\%$ \\ \hline
\end{tabular}


The graphs of three functions, labelled (1), (2), and (3), are shown below. Note that the domain of function (1) is $$\{-6, -5, -4, -3, -2, -1, 0, 1\}.$$ Determine the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1, pic2, pic3;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));

real f(real x) {return sqrt(x+3);}
real h(real x) {return -sin(x);}
real k(real x) {return 2/x;}

real x;

draw(pic1,graph(f,-3,5),Arrows(4));
draw(pic2,graph(h,-8,8),Arrows(4));
draw(pic3,graph(k,-8,-0.25),Arrows(4));
draw(pic3,graph(k,0.25,8),Arrows(4));

dot(pic1,(-6,3)); dot(pic1,(-5,2)); dot(pic1,(-4,1)); dot(pic1,(-3,0));
dot(pic1,(-2,1)); dot(pic1,(-1,2)); dot(pic1,(0,3)); dot(pic1,(1,4));

label(pic1,"(1)",(0,-9));
label(pic2,"(2)",(0,-9));
label(pic3,"(3)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(20,-20)*pic3);
[/asy]


In a set of seven consecutive integers, the largest integer is less than three times the average of the seven integers. What is the smallest integer that could be in the set?


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


A number in the set $\{50, 51, 52, 53, ... , 1050\}$ is randomly selected. What is the probability that it is a three-digit number? Express your answer as a common fraction.


Twenty tiles are numbered 1 through 20 and are placed into box $A$. Thirty tiles numbered 10 through 39 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 10 and the tile from box $B$ is either odd or greater than 35? Express your answer as a common fraction.


How many positive integers less than $151$ are multiples of either $6$ or $8$, but not both at once?


Find the smallest solution to the equation:
\[\frac{1}{x-3} + \frac{1}{x-5} = \frac{5}{2(x-4)}.\]

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


A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\lambda\sqrt{k}$, where $\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\lambda + k$.


Max and Maxine each bought coffee, Max bought 14 ounces in a 20-ounce cup, and Maxine bought 16 ounces in an 18-ounce cup. Max drank 4 ounces of his coffee and then added 3 ounces of cream. Maxine added 3 ounces of cream to her coffee, stirred it well, and then drank 5 ounces. What is the resulting ratio of the amount of cream in Max's coffee to that in Maxine's coffee? Express your answer as a fraction.


The graph of a certain quadratic $y = ax^2 + bx + c$ is a parabola with vertex $(2,0)$ which passes through the point $(0,-50)$. What is the value of $a$?


What is the domain of the function $f(x) = \frac{x-3}{\sqrt{x^2 - 5x + 6}}$?

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


The Thunderbolts are coached by Coach Loudvoice. The Thunderbolts have 15 players, but two of them, Alice and Zen, refuse to play together. How many starting lineups (of 6 players) can Coach Loudvoice make, if the starting lineup can't contain both Alice and Zen? (The order of the 6 players in the lineup does not matter; that is, two lineups are the same if they consist of the same 6 players.)


The 5 a.m. temperatures for seven consecutive days were $-10^\circ$, $-5^\circ$, $-5^\circ$, $-7^\circ$, $0^\circ$, $7^\circ$, and $5^\circ$ Celsius. Calculate both the mean and the median 5 a.m. temperatures for the week in degrees Celsius.


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$


Compute $2 \begin{pmatrix} 3 \\ -5 \end{pmatrix} + 4 \begin{pmatrix} -1 \\ 6 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -4 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} -4 \\ 26 \end{pmatrix}}


Let \( Q \) be a point in coordinate space with all positive coordinates. A line from the origin to \( Q \) forms angles \( \alpha', \beta', \) and \( \gamma' \) with the \( x \)-, \( y \)-, and \( z \)-axis respectively. Given \( \cos \alpha' = \frac{2}{5} \) and \( \cos \beta' = \frac{1}{4}, \) find \( \cos \gamma' \).


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,6)$. Find $m + b$.


Find all real values of $x$ that satisfy $x + \frac{36}{x-3} = -9.$


In triangle ABC, angle B is a right angle, and $\sin \angle BAC = \frac{3}{5}$. Point D lies on line segment BC such that $\cos \angle BAD = \frac{4}{5}$. What is $\sin \angle DAC$?


Suppose the function $g(x)$ is defined on the domain $\{a, b, c\}$ such that its graph, $y=g(x)$, contains exactly three points. These points form a triangle with an area of $45$ square units.

Evaluate the area of the triangle formed by the graph of $y = 3g\left(\frac{x}{3}\right)$.


Each letter represents a non-zero digit. What is the value of $s?$ \begin{align*}
a + p &= v \\
v + e &= s \\
s + a &= r \\
p + e + r &= 14
\end{align*}


How many three-digit positive integers are congruent to 2 (mod 4)?


In an expanded professional bowling tournament, the top 6 bowlers proceed to the playoffs. Initially, #6 bowls against #5. The loser receives the 6th prize, and the winner faces #4. The loser of this match receives the 5th prize, and the winner proceeds to bowl against #3. The subsequent loser gets the 4th prize, and the winner bowls #2. The loser of this match takes the 3rd prize, while the winner goes on to compete against #1. Finally, the ultimate winner is awarded the 1st prize, and the loser of this final game receives the 2nd prize. In how many different orders can bowlers #1 through #6 receive the prizes?


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


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?


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


A square carpet of side length 12 feet contains one large shaded square and twelve smaller, congruent shaded squares organized symmetrically as shown. The ratios $12:\text{S}$ and $\text{S}:\text{T}$ are both equal to 4, where $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares. Determine the total shaded area of the carpet.

Solution: \boxed{15.75 \text{ square feet}}


Let $Q$ be a point on line segment $\overline{CD}$ such that $CQ:QD = 3:5$. Then
\[\overrightarrow{Q} = v \overrightarrow{C} + w \overrightarrow{D}\] for some constants $v$ and $w$. Determine the ordered pair $(v,w)$.

Solution: \boxed{\left(\frac{5}{8}, \frac{3}{8}\right)}


How many five-character license plates consist of a vowel, followed by a digit (between 1 and 5), followed by two consonants, and ending with another vowel? (Assume Y is not considered a vowel for this problem.)


A right circular cylinder with a radius of 3 units is inscribed in a hemisphere with a radius of 6 units such that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. Determine how many positive integers $a_1 \leq 2010$ satisfy that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$.


For how many positive integers $n$ less than or equal to 1000 is $$(\cos t - i\sin t)^n=\cos nt - i\sin nt$$true for all real $t$?


Winnie has 36 cherry lollipops, 125 wintergreen lollipops, 8 grape lollipops, and 241 shrimp cocktail lollipops. She decides to share these with her 13 closest friends, giving each as many lollipops as possible. How many lollipops does she keep for herself after distributing them evenly among her friends?


Compute $3 \begin{pmatrix} 3 \\ -9 \end{pmatrix} - 4 \begin{pmatrix} 1 \\ -6 \end{pmatrix}$.

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


Forty-five teams play a tournament where every team plays each other exactly once, and there are no ties, with each game having a 50% chance of winning for any team. Determine the probability that no two teams win the same number of games. Express this probability as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, and find $\log_2 n$.


Find the product of all real values of $s$ for which $\frac{1}{3x}=\frac{s-x}{10}$ has exactly one real solution.


Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $5x$ inches. Square C has side lengths each measuring $2x$ inches. What is the ratio of the area of Square A to the area of Square B, and what is the ratio of the area of Square C to the area of Square B? Express each answer as a common fraction.


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 $2$. What is the sum of all possible radii of the circle with center $C$?


Michael buys a rectangular mirror that fits exactly inside a frame. The outer perimeter of the frame measures 70 cm by 100 cm. The width of each side of the frame is 15 cm. What is the area of the mirror?

Solution: \boxed{2800 \text{ cm}^2}


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}.$
Determine the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$

Solution: \boxed{\begin{pmatrix} 5 \\ 3 \\ 1 \end{pmatrix}}


Phil rolls 8 fair 6-sided dice. What is the probability that at least two dice show the same number?


Consider two polynomials where the coefficients of the polynomial
\[b_{8} x^8 + b_7 x^7 + b_6 x^6 + \dots + b_2 x^2 + b_1 x + b_0 = 0\]
are all integers, and its roots $s_1, s_2, \dots, s_8$ are all integers. Furthermore, the roots of the polynomial
\[b_0 x^8 + b_1 x^7 + b_2 x^6 + \dots + b_6 x^2 + b_7 x + b_8 = 0\]
are also $s_1, s_2, \dots, s_8$. Determine the number of possible multisets $T = \{s_1, s_2, \dots, s_8\}$.


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


The polynomial $Q(x) = x^3 + dx^2 + ex + f$ has the property that the mean of its zeros and the product of its zeros are both equal to the sum of its coefficients minus 1. If the $y$-intercept of the graph of $y=Q(x)$ is 6, find the value of $e$.


A right circular cone with base radius $r$ and height $h$ rests on its side on a horizontal surface. As it rolls on the surface without slipping, the point where the cone’s base meets the surface traces a circular arc centered at the vertex touching the surface. The cone returns to its original position after making $10$ complete rotations. Find the ratio $h/r$ in its simplest form $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime, and calculate $m+n$.


Find the remainder when $x^4 + 1$ is divided by $x^2 - 4x + 6.$

Solution: \boxed{16x - 59}


Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$.


Point $P$ is the center of an ellipse with major axis $\overline{XY}$ and minor axis $\overline{WZ}.$ Point $G$ is one focus of the ellipse. If $PG = 9$ and the diameter of the inscribed circle of triangle $PWG$ is 6, compute the product $(XY)(WZ).$


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

[asy]
for(int i = 0; i < 4; ++i)
{
    for(int j = 0; j < 4; ++j)
    {
        filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));
    }
}
filldraw(circle((1.5,0.5),0.5),white,linewidth(2));
filldraw(circle((3.5,1.5),0.5),white,linewidth(2));
filldraw(circle((1.5,3.5),0.5),white,linewidth(2));
filldraw(circle((0.5,1.5),0.5),white,linewidth(2));
[/asy]


Completely factor the expression: $12x^2 + 4x - 16$.

Solution: \boxed{4(x - 1)(3x + 4)}


A stock investment increased by 40% in the year 2008. Starting at this increased value, what percent would it have to decrease in 2009 to return to its original price at the beginning of 2008?


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


Find the area of triangle $DEF$ below.

[asy]
unitsize(1inch);
pair D, E, F;
D = (0,0);
E = (1,0);
F = (0,1);
draw (D--E--F--D);
draw(rightanglemark(E,D,F,20));
label("$D$",D,SW);
label("$E$",E,SE);
label("$F$",F,N);
label("$4\sqrt{2}$",midpoint(D--E),S);
label("$45^\circ$", (0.2,0.1), NE);
[/asy]


Determine the sum of all integer solutions to $|n^2| < |n - 5|^2 < 16$.


Determine the intersection point of the function $g(x)=4x+b$ and its inverse function at $(-4, a)$ given that both $b$ and $a$ are integers. What is the value of $a$?


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


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


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


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 216$.


In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, how far is point $F$ from the midpoint of segment $DE$?


Given that $$(x+y+z)(xy+xz+yz) = 40$$ and that $$x^2(y+z) + y^2(x+z) + z^2(x+y) = 10$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$.


Determine the $3 \times 3$ matrix $\mathbf{P}$ such that for any $3 \times 3$ matrix $\mathbf{Q},$ $\mathbf{P} \mathbf{Q}$ results in swapping the second row and third row of $\mathbf{Q},$ and tripling the first row of $\mathbf{Q}.$ In other words,
\[\mathbf{P} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} 3a & 3b & 3c \\ g & h & i \\ d & e & f \end{pmatrix}.\]

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


In how many ways can the digits of $47,722$ be arranged to form a 5-digit number, where numbers cannot begin with $0$?


A notebook containing 67 pages, numbered from 1 to 67, is renumbered such that the last page becomes the first one, the second-last becomes the second one, and so on. Determine how many pages have the same units digit in their old and new numbering.


John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property?


You are given a sequence of $53$ terms; each term has the form $Q+m$ where $Q$ represents the product $2 \times 3 \times 5 \times \ldots \times 31$ of all prime numbers less than or equal to $31$, and $m$ takes, successively, the values $3, 4, 5, \ldots, 55$. Determine the number of primes appearing in this sequence.


The difference between two perfect squares is 121. What is the smallest possible sum of the two perfect squares?


Find the quotient when \(x^6 + 6\) is divided by \(x - 1\).

Solution: \boxed{x^5 + x^4 + x^3 + x^2 + x + 1}


In square $ABCD$, points $M$ and $N$ are the midpoints of sides $\overline{BC}$ and $\overline{CD}$ respectively. Point $P$ is the midpoint of side $\overline{AB}$. Find $\cos \phi$ where $\phi$ is the angle $\angle APN$.


The integer $m$ is the smallest positive multiple of 18 such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$.


What is the greatest integer value of \(x\) such that \(\frac{x^2 + 2x + 10}{x-3}\) is an integer?


In Treeland, an ornamental license TreeTag system uses three characters: one numeral, one letter, and one symbol. The numeral comes from the set $\{1, 2, 3, 4\},$ the letter from $\{A, E, I\},$ and the symbol from $\{!, @, \#, \$\}.$

Given the need for more TreeTags, Treeland's authorities decide to add two new characters. One character is a numeral, and the other is a symbol. How many additional TreeTags can be made with this new arrangement?


A truck travels 150 miles using 5 gallons of diesel. How far can it travel on 15 gallons of diesel? Additionally, if a car, which uses gasoline and is 30% more fuel-efficient than the truck, were to use 20 gallons of gasoline, how far would it travel?

Solution: \boxed{780 \text{ miles}}


Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. What is the number of minimally intersecting ordered triples of sets where each set is a subset of $\{1,2,3,4,5,6,7,8\}$? Find the remainder when this number is divided by $1000$.


Find the $3 \times 3$ matrix $\mathbf{M}$ such that for a $3 \times 3$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ is the result of swapping the first row and third row of $\mathbf{N},$ and tripling the second row of $\mathbf{N}.$  In other words,
\[\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} g & h & i \\ 3d & 3e & 3f \\ a & b & c \end{pmatrix}.\]

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


License plates in Florida have a different format, which includes two letters at the beginning, three digits in the middle, and one letter at the end. North Dakota's license plates have three letters at the beginning followed by three digits. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, how many more license plates can Florida issue than North Dakota?


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


Faces $PQR$ and $QRS$ of tetrahedron $PQRS$ meet at an angle of $45^\circ$. The area of face $PQR$ is $150$, the area of face $QRS$ is $50$, and $QR=5$. Find the volume of the tetrahedron.


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

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


I have five distinct mystery novels, four distinct fantasy novels, and six distinct biographies. I'm planning a long vacation and want to take three books of different genres. How many possible groups of three books can I choose?


Let $f(x) = x^2 - 3x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 6$?


Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = 2,$ $\|\mathbf{v}\| = 5,$ and $\|\mathbf{w}\| = 6,$ and $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}.$ Additionally, $\mathbf{u}$ is orthogonal to $\mathbf{v}$. Compute $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}$.


A rectangle measures 16 inches by 10 inches. Inside it, two circles are drawn with diameters of 8 inches each. The circles do not extend beyond the rectangle's boundaries. What is the greatest possible distance between the centers of the two circles if placed to maximize this distance?

Solution: \boxed{2\sqrt{17} \text{ inches}}


Consider a list of nine positive integers: six of them are $6, 7, 2, 4, 8, 5$. If two more integers added are strictly greater than 7 and one integer added is lower than 4, what is the largest possible value of the median of this list?


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


The number $101$ is the smallest three-digit palindromic prime. What is the third-smallest three-digit palindromic prime?


Let \(x\) and \(y\) be nonzero real numbers. Determine the minimum value \(m\) and the maximum value \(M\) of
\[
\frac{|x+y|^2}{|x|^2 + |y|^2}.
\]
Find \(M - m.\)


In triangle $ABC$, $AB = 24$, $BC = 26$, and $AC = 28$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and $AD:DB = 3:1$ and $AE:EC = 3:1$. Find the length of $\overline{DE}$.


A triangle with side lengths in the ratio 5:12:13 is inscribed in a circle of radius 6.5. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth.


Consider the graph of $y = \frac{r(x)}{s(x)}$, where $r(x)$ is linear and $s(x)$ is quadratic. The graph has vertical asymptotes at $x = -2$ and $x = 3$, and passes through the points $(0,0)$ and $(1,-2)$.
Calculate $\frac{r(2)}{s(2)}$.


Evaluate the product of the sequence $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \cdot \ldots \cdot \frac{1}{6561} \cdot \frac{19683}{1}$.


Alice constructs a regular tetrahedral sculpture with a vertex height of 15 inches and an equilateral triangular base where each side measures 13 inches. She wants to find the smallest cube-shaped container to store the sculpture vertically without tilting. What is the volume of this container, in cubic inches?


During a marathon, a runner jogs at a speed of 6 miles per hour initially but slows down to 4 miles per hour as fatigue sets in. If the runner covers a total distance of 52 miles in 11 hours, how many hours did the runner jog at the initial speed? Express your answer as a common fraction.


Except for the first two terms, each term of the sequence $1500, y, 1500 - y,\ldots$ 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 $y$ produces a sequence of maximum length?


In triangle $PQR$, angle $P = 90^\circ$ and $\sin Q = \frac{3}{5}$. Find $\cos R$.


In how many ways can 7 people be seated around a round table if a specific person, Alice, must always be seated between Bob and Carol? (Two seatings are considered the same if one is a rotation of the other.)


Find the integer $n$, $10 \le n \le 15$, such that \[n \equiv 12345 \pmod{7}.\]


How many 3-digit numbers have the property that the units digit is at least three times the tens digit?


What is the smallest four-digit whole number, the product of whose digits is 12?


Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{1553}{1554}.\]


The admission fee for an exhibition is $ \$30$ per adult and $ \$15$ per child. Last Sunday, the exhibition collected $ \$2250$ in admission fees from at least one adult and at least one child. Among all possible ratios of adults to children on that day, which one is closest to $ 1$?


Three identical cylindrical pipes, each with a diameter of 8 cm, are stacked in a triangular arrangement in a crate. Determine the total height, \( h \), of this stack of 3 pipes.

Solution: \boxed{8 + 4\sqrt{3} \text{ cm}}


Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(x+5) = 26x - 130.$$If $p > q$, what is the value of $p - q$?


What is the greatest common divisor of \( 2^{2020} - 1 \) and \( 2^{2000} - 1 \)?


In isosceles trapezoid $ABCD$ where $AB$ (shorter base) is 10 and $CD$ (longer base) is 20. The non-parallel sides $AD$ and $BC$ are extended to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$?


Find the distance between the points (0,24) and (10,0).


Square EFGH has its center at $(5, 5)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(3,3)$ and a scale factor of 3. What are the coordinates of the vertex of the image of square EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(15, 15)}


Let $\mathbf{m},$ $\mathbf{n},$ and $\mathbf{p}$ be unit vectors such that the angle between $\mathbf{m}$ and $\mathbf{n}$ is $\alpha,$ and the angle between $\mathbf{p}$ and $\mathbf{m} \times \mathbf{n}$ is also $\alpha.$ If $\mathbf{n} \cdot (\mathbf{p} \times \mathbf{m}) = \frac{\sqrt{3}}{4},$ find the smallest possible value of $\alpha,$ in degrees.


Let $r$ and $s$ be positive integers such that\[\frac{5}{11} < \frac{r}{s} < \frac{4}{9}\]and $s$ is as small as possible. What is $s - r$?


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.


Suppose we flip three coins simultaneously: a nickel, a dime, and a quarter. What is the probability that both the nickel and the quarter come up tails?


In Pascal's Triangle, the sum of the interior numbers in the sixth row is 30. Using the same pattern, find the sum of the interior numbers in the eighth row.


Let $g(x) = \frac{4}{16^x + 4}$. Compute the following sum:
\[ g\left(\frac{1}{2002}\right) + g\left(\frac{2}{2002}\right) + g\left(\frac{3}{2002}\right) + \dots + g\left(\frac{2001}{2002}\right). \]


Two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are chosen on the graph of $f(x) = \log_2 x$, with $0 < x_1 < x_2$. The points $R$ and $S$ trisect $\overline{PQ}$, with $PR < RQ$. A horizontal line is drawn from $R$ to cut the curve at $T(x_4, y_4)$. Find $x_4$ if $x_1 = 4$ and $x_2 = 16$.


How many distinct four-digit positive integers are there such that the product of their digits equals 18?


I received $\$50$ for birthday money and decided to split it between four categories as shown in the pie graph. Calculate how much I spent on games.

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(1,0);
B=(0,1);
C=(-1,0);
D=(0,-1);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(0.5,0);
label("Books", W, E);
label("$\frac{1}{4}$", W, W);
X=(0,0.5);
label("Games", X, N);
Y=(0,-0.5);
label("Snacks", Y, S);
label("$\frac{3}{10}$", Y, N);
Z=(-0.5,0);
label("Apps", Z, W);
label("$\frac{7}{20}$", Z, E);
[/asy]


For how many ordered pairs of positive integers \( (a, b) \) with \( a < b \) is the harmonic mean of \( a \) and \( b \) equal to \( 5^{10} \)?


Wei redesigns his logo, now using a rectangle and three smaller circles. Each circle is tangent to two sides of the rectangle and the other two circles. The rectangle is 30 inches long and 10 inches wide. Calculate the area in square inches that will be shaded after placing the circles.

Solution: \boxed{300 - 75\pi}


Solve the equation:
\[
\frac{x^2 + x + 1}{x + 1} = x + 3
\]


Eight balls, numbered 1, 2, 3, 4, 5, 6, 7, 8, are placed in a hat. Each ball is equally likely to be chosen. If one ball is chosen, what is the probability that the number on the selected ball is either a prime number or an odd number?


In the triangle illustrated below, suppose $\cos Q = 0.5$. What is the length of $QR$?

[asy]
pair P,Q,R;
P = (0,0);
Q = (8,0);
R = (0,8*tan(acos(0.5)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,18));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$16$",Q/2,S);
[/asy]


Bertha has 8 daughters and no sons. Some of her daughters have 4 daughters, 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?


Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$ and the sum of the digits of $n$ is divisible by 3.


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

Find the number $n$ of possible values of $f(5)$, and let $s$ be the sum of all possible values of $f(5).$ Compute $n \times s.$


In triangle $\triangle ABC$, altitudes $\overline{AP}$ and $\overline{BQ}$ meet at point $H$. Given that $HP = 3$ and $HQ = 7$, calculate $(BP)(PC) - (AQ)(QC)$. Introduce a point $R$ on line segment $AB$ such that $BR = 2AR$.


Suppose the function $f(x)$ is defined on the domain $\{x_1, x_2, x_3\}$ such that the graph of $y=f(x)$ consists of just three points. Those three points form a triangle of area $27$. The graph of $y = 3f(2x)$ also consists of just three points. Determine the area of the triangle formed by these three points.


The point $(2,2,2)$ is rotated $90^\circ$ about the $z$-axis, then reflected through the $xy$-plane, reflected through the $yz$-plane, rotated $90^\circ$ about the $x$-axis, and reflected through the $xy$-plane. Find the coordinates of the point now.

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


License plates in California and Texas follow specific alpha-numeric formats. California's license plates follow a format where each plate has an alternating sequence of letter and digit starting with a letter (i.e., L1L2L3). Texas, on the other hand, uses a format where the first three slots are digits followed by three letters (i.e., 123LLL). Assuming again that all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates California can issue than Texas.


How many positive divisors does \(8!\) have?


Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 9x^2 + 27x + a$$ form an arithmetic progression and one of the roots is real.


If $x\%$ of five-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth.


Consider the sequence $1, 3, 4, 9, 10, 12, 13, \ldots$ which consists of all positive integers that are either powers of 3 or sums of distinct powers of 3. Determine the $150^{th}$ term of this sequence.


When a polynomial is divided by $5x^3 - 8x^2 + 15$, what are the possible degrees of the remainder? List all possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Suppose that \( x_1+2=x_2+3=\cdots=x_{10}+11=\sum_{n=1}^{10}x_n + 20 \). Find the value of \( \left\lfloor |S| \right\rfloor \), where \( S=\sum_{n=1}^{10}x_n \).


Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 12$ and
\[pq + pr + ps + qr + qs + rs = 24.\] Find the largest possible value of $s$.

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


Determine the domain of the function $g(x) = \tan(\arccos(x^3)).$

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


Find the matrix \(\mathbf{N}\) if it satisfies \(\mathbf{N} \mathbf{i} = \begin{pmatrix} -1 \\ 4 \\ 9 \end{pmatrix},\) \(\mathbf{N} \mathbf{j} = \begin{pmatrix} 3 \\ 0 \\ 6 \end{pmatrix},\) and \(\mathbf{N} \mathbf{k} = \begin{pmatrix} 5 \\ -3 \\ 2 \end{pmatrix}.\)

Solution: \boxed{\begin{pmatrix} -1 & 3 & 5 \\ 4 & 0 & -3 \\ 9 & 6 & 2 \end{pmatrix}}


Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $BC$ intersects side $AC$ at $D$. Suppose $AD = 3$ and $BD = 6$. If $AE = 2$, where $E$ is on segment $AC$ and $E$ is not equal to $D$, what is $CD$?


In a regular decagon \(ABCDEFGHIJ\), \(M\) and \(N\) are the midpoints of \(\overline{BC}\) and \(\overline{FG}\) respectively. Compute \([ABMO]/[EDCMO]\), where \([ABCD]\) denotes the area of polygon \(ABCD\).


Joe wants to find all the five-letter words that begin and end with the same letter, and none of the letters in the word can be 'E'. How many combinations of letters satisfy this property?


A $200\times 360\times 450$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


Let
\[
\mathbf{B} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{B}^{60}$.

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


Consider an isosceles trapezoid with longer base 24 units and legs of 11 units each. The smaller base is 10 units. Calculate the length of the diagonal.


The positive integers $A, B$ and $C$ form an arithmetic sequence while the integers $B, C$ and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{4},$ what is the smallest possible value of $A + B + C + D$?


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


Determine the number of positive solutions for the equation
\[\cos(\arctan(\sin(\arccos x))) = x.\]

Solution: \boxed{x = 1}


Fifty slips are placed into a hat, each bearing a number from 1 to 10. Numbers 1 to 5 are each entered on five slips, and numbers 6 to 10 are each entered on three slips. Four slips are drawn at random without replacement. Let $r$ be the probability that all four slips bear the same number, which only happens for numbers 1 to 5. Let $s$ be the probability that two of the slips bear a number $c$ (among the numbers 1 to 5) and the other two slips bear a number $d\ne c$ (among numbers 6 to 10). What is the value of $s/r$?


Each row of a seating arrangement seats either 6 or 9 people. A total of 57 people are to be seated. How many rows seat exactly 9 people if every seat is occupied and there are no more than 8 rows?


Palindromes are numbers that read the same backwards and forwards, like 3443. What is the least possible four-digit palindrome that is divisible by 5?


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

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


A sphere is perfectly inscribed in a cube. The edge of the cube is 10 inches long. Calculate the volume of the sphere inscribed within this cube. Express your answer in terms of $\pi$.


Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z = 15 \). What is the maximum value of
\[ X \cdot Y \cdot Z + X \cdot Y + Y \cdot Z + Z \cdot X? \]


A circle with center $A$ has a radius of $10$ units and circle $B$ has a radius of $3$ units. The circles are externally tangent to each other at point $C$. Segment $DE$ is the common external tangent to circle $A$ and circle $B$ at points $D$ and $E$, respectively. What is the length of segment $AE$? Express your answer in simplest radical form.


The equation $y = -16t^2 + 16t + 50$ describes the height (in feet) of a ball thrown upwards at $16$ feet per second from $50$ feet above the ground. Find the time (in seconds) when the ball will hit the ground. Express your answer as a common fraction.

Solution: \boxed{\frac{2 + 3\sqrt{6}}{4}}


Using the digits 1, 2, 3, 4, 5, how many odd three-digit numbers less than 600 can be formed if each digit can be used more than once?


A pirate is now searching for buried treasure on 8 islands. On each island, there is a $\frac{1}{3}$ chance that the island has buried treasure and no traps, a $\frac{1}{6}$ chance that the island has traps but no treasure, and a $\frac{1}{2}$ chance that the island has neither traps nor treasure. What is the probability that while searching all 8 islands, the pirate will encounter exactly 4 islands with treasure, and none with traps?


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


Two standard dice are rolled. What is the expected number of 6's obtained? Express your answer as a common fraction.


If $\sqrt{2} \cos 20^\circ$ can be written as $\sin \phi - \cos\phi$ for some acute angle $\phi$, what is $\phi?$ (Give your answer in degrees, not radians.)


Phil has 8 green marbles and 4 purple marbles in a bag. He removes a marble at random, records its color, replaces it, and then repeats this process until he has withdrawn 8 marbles in total. What is the probability that exactly five of the marbles that he removes are green? Express your answer as a decimal rounded to the nearest thousandth.


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


Given that $x = \frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction.


Monica tosses a fair 8-sided die. If the roll is a prime number, she wins that amount in dollars. If the roll is a perfect square, she loses that amount in dollars. For any other outcome, she wins nothing. What is the expected value of her winnings on one die toss? Express your answer as a dollar value to the nearest cent.


Find the value of \(k\) so that the line \(4x + 6y + k = 0\) is tangent to the parabola \(y^2 = 32x\).


Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.00005, 0.0005).$


How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)?


The expansion of \( (x+1)^n \) has 3 consecutive terms with coefficients in the ratio \( 2:3:4 \) that 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 \( n+k \).


If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 6,$ then find
\[\begin{vmatrix} a & 5a + 2b \\ c & 5c  +2d \end{vmatrix}.\]


Determine how many digits $C$ can make the four-digit number $10C8$ both a multiple of 3 and an even number.


The sequence begins with 1,024,000 and is formed by repeatedly dividing by 4. What is the last integer in this sequence?


Compute $\sqrt{72} \cdot \sqrt{18} \cdot \sqrt{8}$.


In $\triangle XYZ$, $\overline{MN} \parallel \overline{XZ}$, with $XM = 6$ cm, $MY = 15$ cm, and $NZ = 9$ cm. Calculate the length of $\overline{XY}$.

Solution: \boxed{21 \text{ cm}}


Marcus has two numbers, \( c \) and \( d \). When he divides \( c \) by 60, he gets a remainder of 47. When he divides \( d \) by 45, he gets a remainder of 14. What remainder does he get when he divides \( c+d \) by 15?


How many positive integers $N$ are there less than $2000$ such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


Find the product of all positive integer values of $c$ such that $10x^2 + 18x + c = 0$ has two real roots.


Joe wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property?


What is the sum of all the integers from 100 to 2000?


In triangle $XYZ$, angle bisectors $\overline{XU}$ and $\overline{YV}$ meet at point $Q$. Given $XY = 8$, $XZ = 6$, and $YZ = 4$, find the ratio $\frac{YQ}{QV}$.


If $\frac{1}{8}$ of $2^{36}$ is $4^y$, then what is the value of $y$?


How many integers between 1 and 300 are multiples of both 2 and 7 but not of either 3 or 5?


Calculate the remainder of $10!$ when divided by $13$.


A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle.


The values of \(a\), \(b\), \(c\), and \(d\) are 1, 3, 5, and 7, but not necessarily in that order. What is the largest possible value of the sum of the four products \(ab + ac + bd + cd\)?


Determine how many possible values there are for $|z|$ if the equation $3z^2 - 18z + 55 = 0$ is given.


Find the units digit of the sum $65_8 + 74_8$ and then add $3_8$ to this sum, all calculations being within base 8.


Fifty cards are placed in a box, each bearing a number from 1 to 10, with each number appearing on five cards. Five cards are drawn from the box randomly 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 card bears a number $b$, different from $a$. Compute the value of $q/p$.


Solution: \boxed{\frac{9}{20}\text{ days}}


You have a rectangular prism with dimensions length $2x+4$ units, width $x+3$ units, and height $x^2+16$ units. Determine how many positive integer values of $x$ make the volume of the box less than 1000 units.


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


What is the ratio of the volume of a cube with edge length four inches to the volume of a cube with edge length two feet? Additionally, calculate the ratio of their surface areas.


Let $b_n = \frac{7^n - 1}{6}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. Determine the maximum possible value that $e_n$ can take on.


Add $246_8 + 573_8 + 62_8$. Express your answer in base $8$.


Find the modular inverse of $35$, modulo $37$.

Express your answer as an integer from $0$ to $36$, inclusive.


Find the value of $x$ such that $\sqrt{2x - 3} = 10$.

Solution: \boxed{x = 51.5}


All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list?


Simplify $\displaystyle\frac{2-3i}{1+4i}$, where $i^2 = -1.$

Solution: \boxed{-\frac{10}{17} - \frac{11}{17}i}


The least common multiple of $x$, $8$, and $12$ is $120$. What is the greatest possible value of $x$?


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


David is organizing a wildlife exhibition and wishes to line up his eight uniquely tall kangaroos in a single row for a display. The tallest two kangaroos, however, insist on standing at the ends of the row. How many different ways can David arrange the kangaroos for the exhibition?

Solution: \boxed{1440 \text{ ways}}


When the vectors $\begin{pmatrix} -4 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v}$, the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}$.

Solution: \boxed{\begin{pmatrix} -\frac{39}{29} \\ \frac{91}{29} \end{pmatrix}}


A four-inch wooden cube has all its faces painted blue. It is then precisely cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least two faces?


The graph below is represented by the equation $x = ay^2 + by + c$. Given that the vertex of this parabola is $(5, 3)$ and it passes through the point $(7, 6)$, find the value of $c$, assuming the parabola opens downwards and the graph scales are standard units.


Find the value of $k$ so that the line $4x + 7y + k = 0$ is tangent to the parabola $y^2 = 16x.$


Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients.


Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and red. She randomly draws five socks. What is the probability that she ends up with exactly one pair of socks of the same color?


The function $g(x)$ satisfies
\[g(x) + 3g(1 - x) = 4x^2\] for all real numbers $x.$  Find $g(4).$


Solution: \boxed{\frac{1}{3} \log_6 5}


How many total days were there from the year 2003 through 2008?


An isosceles trapezoid has side lengths as labeled below. Determine the length of segment AC.

\[
\text{[asy]}
\text{unitsize(1.5mm);}
\text{defaultpen(linewidth(.7pt)+fontsize(10pt));}
\text{dotfactor=3;}

\text{pair A=(0,0), B=(26,0), C=(17,12), D=(9,12);}
\text{pair[] dots={A,B,C,D};}

\text{draw(A--B--C--D--cycle);}
\text{dot(dots);}
\text{label("A",A,SW);}
\text{label("B",B,SE);}
\text{label("C",C,NE);}
\text{label("D",D,NW);}
\text{label("12",midpoint(C--D),N);}
\text{label("13",midpoint(D--A),NW);}
\text{label("26",midpoint(A--B),S);}
\text{label("13",midpoint(B--C),NE);}
\text{[/asy]}
\]


Find the product of the divisors of $36$.


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(xg(y) - x) = xy - g(x) \] for all \( x, y \).
Let \( m \) be the number of possible values of \( g(-2) \), and let \( t \) be the sum of all possible values of \( g(-2) \). Find \( m \times t \).


Find the product of all integer divisors of $120$ that also divide $15$. (Recall that the divisors of an integer may be positive or negative.)


For which integer $a$ does $x^2 - x + a$ divide $x^{15} + x^2 + 100$?


In triangle $\triangle DEF$, point $L$ is located on $EF$ such that $DL$ is an altitude of $\triangle DEF$. If $DE = 14,$ $EL = 9$, and $EF = 17$, what is the area of $\triangle DEF$?

Solution: \boxed{\frac{17 \sqrt{115}}{2}}


Maria buys a basket of berries, a carton of milk, a loaf of bread, and a block of cheese for $25. The price of a carton of milk is twice the price of a loaf of bread. The cost of a block of cheese is equal to the price of a basket of berries plus $2. How much does a basket of berries and a block of cheese cost together?


Let $r$ be a real number, $|r| < 6,$ and let $z$ be a complex number such that
\[ z + \frac{9}{z} = r. \]
Find $|z|$.


Convert the point $(0, -4 \sqrt{3}, 4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left( 8, \frac{3 \pi}{2}, \frac{\pi}{3} \right)}


Let $g(n)$ be the product of the proper positive integer divisors of $n$. For how many values of $n$ does $n$ not divide $g(n)$, given that $2 \le n \le 100$?


Compute the distance between the vertices of the parabolas represented by the portions of the graph of the equation \[\sqrt{x^2+y^2} + |y+2| = 4.\]


In triangle $ABC$, $AB = 8$, $AC = 17$, and the length of median $AM$ is 12.  Find the perimeter of triangle $ABC$.

Solution: \boxed{25 + \sqrt{130}}


In triangle $XYZ$, the side lengths are $XY=6$, $YZ=8$, and $XZ=10$. Two ants start simultaneously from vertex $X$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They first meet at point $W$. What is the length of segment $YW$?


The endpoints of a line segment are (3, 4) and (10, 20). What is the sum of the coordinates of the midpoint of the segment?


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

Solution: \boxed{(-2, 5)}


Four years ago, there were 30 trailers in River Park with an average age of 12 years. At that time, a group of brand new trailers was added to River Park. Today, the average age of all the trailers in River Park is 10 years. How many new trailers were added four years ago?


Let
\[\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}.\]
Compute \(\mathbf{C}^{50}\).

Solution: \boxed{\begin{pmatrix} 101 & 50 \\ -200 & -99 \end{pmatrix}}


For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 7?


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


Let \( b \equiv (2^{-1} + 4^{-1} + 8^{-1})^{-1} \cdot 3 \pmod{13} \). What is the remainder when \( b \) is divided by 13?


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


Two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are chosen on the graph of \( f(x) = \ln_5 x \), where \( 1 < x_1 < x_2 \). The points \( C \) and \( D \) trisect \( \overline{AB} \), with \( AC < CB \). Through \( C \), a horizontal line is drawn to cut the curve at \( E(x_3, y_3) \). Find \( x_3 \) if \( x_1 = 5 \) and \( x_2 = 3125 \).


A six-sided die and an eight-sided die are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 12?


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


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


Let $x$, $y$, and $z$ be real numbers such that
\[\cos x + \cos y + \cos z = 1 \quad \text{and} \quad \sin x + \sin y + \sin z = 0.\]
Find the sum of all possible values of $\cos 2x + \cos 2y + \cos 2z.$


If 8 knicks = 3 knacks and 2 knacks = 5 knocks, how many knicks are equal to 30 knocks?

Solution: \boxed{32 \text{ knicks}}


A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $30$ cm across at the top and $10$ cm deep. What was the radius of the ball (in centimeters)?

Solution: \boxed{16.25 \text{ cm}}


The probability of snow decreases linearly over the next four days starting from $\frac{3}{4}$ on the first day to $\frac{1}{2}$ on the fourth day. Calculate the probability that it will snow at least once during these four days. Express your answer as a common fraction.


For how many different negative values of $x$ is $\sqrt{x + 200}$ a positive integer?


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.


For what real values of $b$ is the expression $\frac{b-1}{b^2-9}$ undefined?

Solution: \boxed{-3, 3}


Solution: \boxed{-1 - 8z^2}


Solve for $x$, where $x>0$ and $3x^2 - 7x - 6 = 0$. Express your answer as a simplified common fraction.


The values of $f$, $g$, $h$, and $j$ are 6, 7, 8, and 9, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$, and $fj$?


What is the probability that Fatima gets fewer heads than tails if she flips 12 coins?