Rectangle $ABCD$ has sides $\overline{AB}$ of length 5 and $\overline{CB}$ of length 4. Divide $\overline{AB}$ into 200 congruent segments with points $A=P_0, P_1, \ldots, P_{200}=B$, and divide $\overline{CB}$ into 200 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{200}=B$. For $1 \leq k \leq 199$, draw the segments $\overline{P_kQ_k}$. Repeat this construction on the sides $\overline{AD}$ and $\overline{CD}$, and then draw the diagonal $\overline{AC}$. Compute the sum of the lengths of the 399 segments drawn in this way.


Determine how many of the first $30$ rows of Pascal's triangle, excluding row $0$ and $1$, consist entirely of even numbers except for the $1$s at each end.


Tim is choosing between two internet plans for his new apartment. Plan X does not have a base charge but costs 25 cents per GB of data used. Plan Y has an initial setup fee of $15 and charges 15 cents per GB of data used. How many gigabytes of data must Tim use for Plan Y to become the more cost-effective option?


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


Suppose $x$, $y$, and $z$ are positive numbers such that $xyz = 1$, $x + \frac{1}{z} = 7$, and $y + \frac{1}{x} = 31.$ Find $z + \frac{1}{y}.$


Let $f(x)$ be a function piecewise defined as \[f(x) = \left\{
\begin{array}{cl}
-x & x< 0, \\
3x-12 & x \geq 0.
\end{array}
\right.\] If $k$ is negative, find $k$ so that $f(f(f(4)))=f(f(f(k)))$.

Solution: \boxed{k = -\frac{20}{3}}


There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ unique integers $b_k$ ($1\le k\le s$) with each $b_k$ either $1$ or $-1$ such that
\[
b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 2010.
\]
Find $m_1 + m_2 + \cdots + m_s$.


The function $g$ defined by $g(x)= \frac{px+q}{rx+s}$, where $p$, $q$, $r$, and $s$ are nonzero real numbers, has the properties $g(23)=23$, $g(101)=101$ and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.


The lengths of two sides of a triangle are 24 units and 37 units. The third side also has an integral length and is the largest side of the triangle. What is the least possible even number of units in the perimeter of the triangle?


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


How many non-similar triangles have angles whose degree measures are distinct positive integers in a geometric progression?


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)}


Let $\mathbf{X}$ and $\mathbf{Y}$ be matrices such that
\[\mathbf{X} + \mathbf{Y} = \mathbf{X} \mathbf{Y}.\]
If $\mathbf{X} \mathbf{Y} = \begin{pmatrix} 7 & 2 \\ -4 & 5 \end{pmatrix},$ find $\mathbf{Y} \mathbf{X}.$

Solution: \boxed{\begin{pmatrix} 7 & 2 \\ -4 & 5 \end{pmatrix}}


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

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


If $B$ is an angle such that $\tan B - \sec B = -2$, find all possible values of $\cos B.$


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


I had $50 in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on video games?

[asy]
size(150);
pair A, B, C, D, E, O, P, Q, R, S;
O=(0,0);
A=(1,0);
B=(0,1);
C=(-1,0);
D=(0,-1);
E=(0.707,-0.707);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--E);
P=(0.6,0.3);
label("Books", P, N);
label("$\frac{1}{4}$", P, S);
Q=(-0.6,0.3);
label("Video Games", Q, N);
R=(-0.8,-0.2);
label("Snacks", R, E);
label("$\frac{1}{5}$", R, W);
S=(0.3,-0.7);
label("Crafts", S, E);
label("$\frac{3}{10}$", S, NW);
[/asy]


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


A standard deck of 52 cards is arranged randomly. What is the probability that the top three cards alternate in color, starting with a red card, then a black card, followed by another red card?


Calculate $\sqrt[4]{\sqrt[5]{0.00032}}$. Express your answer as a decimal to the nearest thousandth.


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


Below is a portion of the graph of a function, \( y = v(x) \):

[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-3.25,ymax=3.25;

pen cqcqcq=rgb(0.75,0.75,0.75);

/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return -x^3 + 2*sin(x*pi/2);}
draw(graph(f2,-3.25,3.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]

What is the exact value of \( v(-1.75) + v(-0.5) + v(0.5) + v(1.75) \)?


Two congruent cones with radius 8 cm and height 10 cm each are placed within a cylinder. The base of each cone is touching a different base of the cylinder. If the height of the cylinder is 30 cm, compute the volume of space in the cylinder not occupied by the cones. Express your answer in terms of $\pi$.

Solution: \boxed{\frac{4480\pi}{3} \text{ cubic cm}}


Given a function \(g(x)\) that satisfies:
\[ g\left(\frac{x}{y}\right) = \frac{y}{x} g(x) \]
for all positive real numbers \(x\) and \(y\). If \(g(50) = 4\), find \(g(25)\).


Ramon sells one enchilada and four tacos for $\$$2.30 and sells four enchiladas and one taco for $\$$3.10. Assuming a fixed price per item, what is the cost, in dollars, of four enchiladas and five tacos?


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 both exactly four?


In Pascal's Triangle, each entry is the sum of the two entries immediately above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $2:3:4$?


A box contains nine cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and three of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is red? Express your answer as a common fraction.

Solution: \boxed{\frac{6}{8} = \frac{3}{4}}


Po is now trying to solve a modified equation by completing the square: $$49x^2 + 70x - 121 = 0.$$ He aims to rewrite the equation in the form \((ax + b)^2 = c\), where \( a \), \( b \), and \( c \) are integers and \( a > 0 \). What is the value of \( a + b + c \)?


Given the polynomial $f(x) = 2x^3 - x^2 + 4x - 3$, find a cubic polynomial $g(x) = x^3 + bx^2 + cx + d$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Determine the ordered triple $(b, c, d)$.

Solution: \boxed{\left(\frac{15}{4}, \frac{5}{2}, -\frac{9}{4}\right)}


In triangle $ABC,$ $a = 8,$ $b = 10,$ and $c = 6.$ Let $I$ be the incenter and $G$ be the centroid of the triangle.
\[
\overrightarrow{I} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}
\]
where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$ Determine the ordered triple $(x, y, z)$ representing the incenter $I$ and verify if the coordinates match those of the centroid $G$, given that the centroid divides each median in the ratio $2:1$.

Solution: \boxed{\left(\frac{1}{3}, \frac{5}{12}, \frac{1}{4}\right)}


In quadrilateral $PQRS$, sides $\overline{PQ}$ and $\overline{QR}$ both have length 12, sides $\overline{RS}$ and $\overline{SP}$ both have length 20, and the measure of angle $RSP$ is $60^\circ$. What is the length of diagonal $\overline{PR}$?


When the base-10 integers 150 and 950 are expressed in base 2, how many more digits does 950 have than 150 (after being converted)?


Mrs. Reed's English class is now reading a different novel that is $900$ pages long. Alice, Bob, and Chandra are still reading at different speeds: Alice finishes a page in $30$ seconds, Bob in $50$ seconds, and Chandra in $25$ seconds. Bob and Chandra decide to use the same team reading strategy as before. Determine the last page that Chandra should read so that both she and Bob spend the same amount of time reading their respective parts of the novel.


The mean (average) of 8, 12, and 24 is equal to the mean (average) of 16 and $z$. What is the value of $z$?


Let $g(x) : \mathbb{R} \to \mathbb{R}$ be a function such that
\[\frac{g(x) g(y) - g(xy)}{4} = x + y + 3\]for all $x,$ $y \in \mathbb{R}.$  Find $g(x).$

Solution: \boxed{x + 4}


Find the last two digits of the following sum: $$6! + 12! + 18! + \cdots + 96!$$


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent coin. What is the probability that at least the penny, dime, and 50-cent coin all come up heads?


Mr. Greene has 8 children, including one set of twins who are of the same gender. Assuming that for the other 6 children, the gender of each is determined independently with equal likelihood of male and female, calculate the probability that Mr. Greene has more sons than daughters or more daughters than sons.


The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5.


Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^2 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(16) = 16 \), then compute \( [g(4)]^4 \).


Determine the sum of all integers in the set $\{1, 2, \ldots, 10\}$ that are primitive roots modulo $11$.


Triangle $PQR$ has a perimeter of 3010 units. The sides have lengths that are all integer values with $PQ < QR \leq PR$. What is the smallest possible value of $QR - PQ$?


Let $\mathcal{S}$ be the set $\{1, 2, 3, \ldots, 12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by 2000.


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


If $y$ is a real number and $\lceil y \rceil = 15,$ how many possible values are there for $\lceil y^2 \rceil$?


Multiply \(333\) by \(111\) and express the result.


Simplify
\[
\frac{\sin 20^\circ + \sin 40^\circ + \sin 60^\circ + \sin 80^\circ + \sin 100^\circ + \sin 120^\circ + \sin 140^\circ + \sin 160^\circ}{\cos 10^\circ \cos 20^\circ \cos 40^\circ}.
\]


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


How many three-digit positive integers $x$ satisfy $4953x + 532 \equiv 1387 \pmod{29}$?


$ABC$ is a triangle where $A=(0,0)$, $B=(30,16)$, and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have?


Determine the sum of all real numbers $x$ satisfying
\[(x^2 - 4x + 3)^{x^2 - 5x + 4} = 1.\]


A Senate committee has 7 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.)


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

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


For what values of $x$ is $x^3 - 10x^2 + 15x > 0$?

Solution: \boxed{(0, 5-\sqrt{10}) \cup (5+\sqrt{10}, \infty)}


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


In polar coordinates, the point $\left(-5, \frac{5\pi}{6}\right)$ is equivalent to what other point, in the standard polar coordinate representation? Enter your answer in the form $(r, \theta)$, where $r > 0$ and $0 \leq \theta < 2\pi$.

Solution: \boxed{\left(5, \frac{11\pi}{6}\right)}


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


Compute the smallest base-10 positive integer greater than 15 that is a palindrome when written in both base 3 and base 5.


How many positive whole numbers have cube roots that are less than $20$? For example, $7000$ would be counted since $\sqrt[3]{7000} < 20$.


Let $a_1, a_2, a_3, \dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 24$, what is $a_3a_6$?


Janet can buy pens, pencils, and notebooks for \$3, \$4, and \$10 each, respectively. What is the largest number of notebooks she can purchase if she must buy at least one of each item and plans to spend exactly \$100?


In the following diagram, $AB=80$. Find $AX$.

[asy]
import markers;

real t=.67;
pair A=(0,0);
pair B=(3,-2);
pair C=(1.5,1.5);
pair X=t*A+(1-t)*B;

draw(C--A--B--C--X);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SW);

markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));
markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));

label("$72$",.5*(B+C),E);
label("$36$",.5*(A+C),NW);
[/asy]


Let \(a\), \(b\), \(c\), and \(d\) be nonnegative numbers whose sum is 150. Find the largest possible value of
\[
ab + bc + cd.
\]


Find the integer $n$, $-90 < n < 90$, such that $\tan n^\circ = \tan 348^\circ.$


Let $\mathbf{v} = \begin{pmatrix} 12 \\ -8 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -9 \\ 6 \end{pmatrix}$. Compute $\text{proj}_{\mathbf{w}} \mathbf{v}$.

Solution: \boxed{\begin{pmatrix} 12 \\ -8 \end{pmatrix}}


A biased die shows a 6 with a probability of $\frac{1}{2}$, any number from 1 to 5 with equal probability. Rolling a 6 gains $\$5$, and rolling any number from 1 to 4 gains $\$1$, but rolling a 5 loses $\$10$. What is the expected value of rolling this die?


A digit is added to the right of the units digit of $816$. If the resulting four-digit number is divisible by $5$, how many possibilities are there for the digit that was added?


Find integers $b, c$ such that both roots of the polynomial $x^2 - 2x - 1$ are also roots of the polynomial $x^5 - bx - c$. Determine the product $bc$.


Three dice with faces numbered 1 through 6 are stacked differently from the previous problem, exposing eight of the eighteen faces while ten remain hidden. The visible faces show the numbers 1, 2, 2, 3, 5, 4, 5, and 6 respectively. What is the total number of dots NOT visible in this arrangement?


Each letter represents a non-zero digit. What is the value of $p?$
\begin{align*}
a + b &= k \\
k + m &= p \\
p + a &= r \\
b + m + r &= 18
\end{align*}


Suppose $\alpha$ is inversely proportional to $\frac{1}{\beta}$. If $\alpha = 8$ when $\beta = 2$, find $\alpha$ when $\beta = 5$.


Given the graph of the equation $y = \frac{x}{x^3 + Ax^2 + Bx + C}$, where $A, B, C$ are integers, and noting that the graph has vertical asymptotes at $x = -1, 0, 3$, find $A + B + C$.


Jordan needs to unlock his six-digit bike lock. He remembers that the code uses digits from 1 to 6 and alternates between even and odd digits. How many possible combinations can Jordan's code be?


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


Let \(a, b, c\) be real numbers, all greater than 4, so that
\[
\frac{(a + 3)^2}{b + c - 3} + \frac{(b + 5)^2}{c + a - 5} + \frac{(c + 7)^2}{a + b - 7} = 45.
\]
Enter the ordered triple \((a,b,c)\).

Solution: \boxed{(12, 10, 8)}


The increasing sequence $1, 3, 4, 9, 10, 12, 13\cdots$ 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.


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

Solution: \boxed{\sqrt{89 + 40\sqrt{2}}}


Identify the smallest abundant even number greater than 12.


Find a positive integer that is divisible by 15 and has a square root between 33 and 33.5.


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


In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $k$?


Solution: \boxed{x = 1 \text{ or } x = 9}


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


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


What is the last digit of the decimal expansion of $\frac{1}{2^{15}}$?


In the diagram, four circles of radius 6 units intersect at the origin. What is the number of square units in the area of the shaded region? Express your answer in terms of $\pi$. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
fill(Arc((1,0),1,90,180)--Arc((0,1),1,270,360)--cycle,gray(0.6));
fill(Arc((-1,0),1,0,90)--Arc((0,1),1,180,270)--cycle,gray(0.6));
fill(Arc((-1,0),1,270,360)--Arc((0,-1),1,90,180)--cycle,gray(0.6));
fill(Arc((1,0),1,180,270)--Arc((0,-1),1,0,90)--cycle,gray(0.6));
draw((-2.3,0)--(2.3,0)^^(0,-2.3)--(0,2.3));
draw(Circle((-1,0),1)); draw(Circle((1,0),1)); draw(Circle((0,-1),1)); draw(Circle((0,1),1));
[/asy]

Solution: \boxed{72\pi - 144}


A ball is initially dropped from a height of 160 feet and rebounds to three-quarters of the distance it falls each time it bounces. How far will the ball have traveled when it hits the ground for the fifth time?

Solution: \boxed{816.25 \text{ feet}}


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


Sally bought 30 items each priced at 50 cents, 5 dollars, or 10 dollars. If her total purchase price was \$100.00, how many 50-cent items did she purchase?


Square EFGH has its center at $(10, -5)$ 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 (0,0) 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{(36, -21)}


Factor the following expression: $276a^2b^2 + 69ab - 138ab^3$.

Solution: \boxed{69ab(4ab + 1 - 2b^2)}


Determine the $2 \times 2$ matrix $\mathbf{N}$ such that $\mathbf{N} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 7 \\ -2 \end{pmatrix}$ and $\mathbf{N} \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 11 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{8}{7} & \frac{11}{7} \\ \frac{31}{14} & -\frac{15}{7} \end{pmatrix}}


The product of two 2-digit numbers is $4500$. What is the smaller of the two numbers?


Let $c$ and $d$ be positive real numbers such that each of the equations $x^2 + cx + 3d = 0$ and $x^2 + 3dx + c = 0$ has real roots. Find the smallest possible value of $c + d.$


The energy stored by a pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges are initially placed at the corners of a square with each side length $d$. This configuration stores a total of $20$ Joules of energy. How much energy, in Joules, would be stored if two of these charges are moved such that they form a new square with each side doubled (i.e., side length $2d$)?


The Grunters play the Screamers 5 times. The Grunters are the much better team, and are $80\%$ likely to win any given game. What is the probability that the Grunters will win at least 4 out of the 5 games? Express your answer as a common fraction.


Evaluate the value of the expression \[(2^{1000} + 5^{1001})^2 - (2^{1000} - 5^{1001})^2\] and express it as $k \cdot 10^{1000}$ for some positive integer $k$.


Let \(x\) and \(y\) be positive real numbers. Find the maximum value of
\[
\frac{(x + y)^3}{x^3 + y^3}.
\]


How many pairs of positive integers $(x, y)$ satisfy the equation $x^2 - y^2 = 77$?


The operation $\star$ is now defined as $a \star b = a^2 + \frac{a}{b}$. Calculate the value of $5 \star 2$.


The number $1027$ can be written as $21q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $2q - r$?


In polar coordinates, what is the equivalent point in the standard polar coordinate representation for the point $\left( -5, \frac{5 \pi}{7} \right)$? Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$

Solution: \boxed{\left( 5, \frac{12 \pi}{7} \right)}


Calculate the value of $\sqrt{5} \times 5^{\frac{1}{2}} + 20 \div 4 \times 3 - 8^{\frac{3}{2}} + 5$.

Solution: \boxed{25 - 16\sqrt{2}}


Find the perimeter of a triangle with points $A(2,3)$, $B(2,9)$, and $C(7,6)$ on a Cartesian coordinate plane. Also, determine if this triangle is a right-angled triangle.

Solution: \boxed{P = 6 + 2\sqrt{34}}


Assume \(x\), \(y\), and \(z\) are real numbers such that:
\[ \cos x + \cos (y + \frac{\pi}{3}) + \cos (z - \frac{\pi}{3}) = \sin x + \sin (y + \frac{\pi}{3}) + \sin (z - \frac{\pi}{3}) = 0. \]
Determine the sum of all possible values of \(\cos 2x + \cos 2y + \cos 2z\).


In the diagram, $\triangle ABF$, $\triangle BCF$ and $\triangle CDF$ are right-angled, with $\angle ABF=\angle BCF = \angle CDF = 45^\circ$, and $AF=32$. Find the length of $CF$.


Let $n \ge 2$ be an integer, and let $T = \{2,4,8,\ldots,2^n\}$. Find the smallest value of $n$ such that for every partition of $T$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.


How many integers $n$ are there such that $2 \leq n \leq 12$ and the number $121_n$ (which is an expression of the number 121 in base $n$) is a perfect square?


Two vectors $\begin{pmatrix} -6 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v}$. Find the resulting projection vector $\mathbf{p}$.

Solution: \boxed{\begin{pmatrix} -\frac{12}{17} \\ \frac{54}{17} \end{pmatrix}}


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


The graphs of $y = 5 - 2x^2 + x^3$ and $y = 3 + 2x^2 + x^3$ intersect at multiple points. Find the maximum difference between the $y$-coordinates of these intersection points.


Let $x$ and $y$ be nonzero real numbers such that $x \neq y$. Determine the maximum value and minimum value of the expression:
\[
Z = \frac{|x^2 - y^2|}{|x^2| + |y^2|}
\]
and compute the difference between the maximum and the minimum value.


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


A scaled map of a city where Dora, Eli, and Felix live is depicted on the Cartesian plane. Dora is at point $(8, -15)$, and Eli is at point $(2, 18)$. They decide to meet at a point equidistant from their locations and then walk vertically to reach Felix, who is situated at $\left(5, 7\right)$. Calculate how many units they must walk upward to meet Felix.


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

Find all possible functions $f(x)$ that satisfy this equation.

Solution: \boxed{f(x) = 0}


Compute \( A(3, 1) \) with the function \( A(m, n) \) defined as:
\[ A(m,n) = \begin{cases}
n+1 & \text{if } m = 0 \\
A(m-1, 1) & \text{if } m > 0 \text{ and } n = 0 \\
A(m-1, A(m, n-1)) & \text{if } m > 0 \text{ and } n > 0
\end{cases} \]


The infinite sequence $T=\{t_1,t_2,t_3,\ldots\}$ is defined by $t_1=3$ and $t_n=3^{t_{n-1}}$ for each integer $n>1$. What is the remainder when $t_{50}$ is divided by $7$?


If $x = 3$ and $y = 4$, what is the value of $\frac{x^5 + 2y^3}{8}$?


Rationalize the denominator: $\sqrt{\frac{5}{3 - \sqrt{2}}}$.

Solution: \boxed{\frac{\sqrt{105 + 35\sqrt{2}}}{7}}


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

[asy]
draw((0,0)--(6,6)--(13,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(13,2),dashed);
label("9",(3,3),NW);
label("11",(9.5,4),NE);
label("17",(10.5, -2.5),SE);
label("13",(4,-2.5),SW);
[/asy]


In a kennel with 80 dogs, 18 dogs like watermelon, 58 dogs like salmon, and 7 like both salmon and watermelon. Additionally, 16 dogs like chicken, with 6 dogs liking both chicken and salmon, and 4 liking both chicken and watermelon, but only 3 dogs like all three foods. How many dogs in the kennel will not eat any of the three foods?


What is the nearest integer to $(3+\sqrt5)^4$?


A map of a city where Daniel, Emma, and Fiona live is represented by a Cartesian plane. Daniel is located at $(5, -15)$ and Emma is at $(2, 20)$. They agree to meet at the closest point that is equidistant from their locations and then walk upwards together to reach Fiona's location at $\left(\frac{7}{2}, 5\right)$. How many units do Daniel and Emma need to walk upwards together to meet Fiona?


Simplify the expression: $$\dfrac{\sqrt{450}}{\sqrt{200}} + \dfrac{\sqrt{252}}{\sqrt{108}} + \dfrac{\sqrt{88}}{\sqrt{22}}.$$

Solution: \boxed{\frac{21 + 2\sqrt{21}}{6}}


The graph of $y = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are quadratic polynomials, has a vertical asymptote at $x = -3$ and a horizontal asymptote at $y = -4$. Additionally, there is a hole in the graph at $x = 4$. Given that the graph passes through the point $(2,0)$, determine $\frac{p(2)}{q(2)}$.


In a cross country meet between two teams of 6 runners each, a runner who finishes in the $n$th position contributes $n$ points to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?


Determine the value of \( x \) in the equation \( \frac{3}{x} + \frac{1}{x} \div \frac{5}{x} + \frac{1}{2x} = 1.25 \).


A bag contains 6 red marbles, 7 white marbles, 8 blue marbles, and 9 green marbles. Four marbles are drawn from the bag without replacement. What is the probability that all drawn marbles are the same color?


The mean of the numbers 3, 7, 10, and 15 is twice the mean of $x$, 20, and 6. What is the value of $x$?


In right triangle $ABC$, we have $\sin A = 1$ (i.e., $\angle A = 90^\circ$) and $\cos B = \frac{3}{5}$. Find $\sin C$.


The values of $x$ and $y$ are positive, with the relationship that $x^3$ and $y$ vary inversely. Given that $y$ is 8 when $x$ is 2, find the value of $x$ when $y$ is 1000.


The graph of the equation $y = |x| - 4$ is translated four units to the right and five units up. What are the coordinates of the minimum point of the new graph?

Solution: \boxed{(4, 1)}


Calculate $\sqrt{48y} \cdot \sqrt{18y} \cdot \sqrt{50y}$. Express your answer in simplest radical form in terms of $y$.

Solution: \boxed{120y \sqrt{3y}}


Given the square of an integer $x$ is 2209, what is the value of $(2x+1)(2x-1)$?


The admission fee for a concert is $ \$30$ per adult and $ \$15$ per child. On a certain day, the concert collected $ \$2550$ in admission fees with at least one adult and one child attending. Of all the possible ratios of adults to children on that day, which one is closest to $1$?


Solution: \boxed{3x^{18} - 6x^{10} - 24x^3 + 3}


Let \( f \) be a function taking positive integers to positive integers, such that:

(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) = m \)

Find all possible values of \( f(60) \).


For real numbers $x,$ $y,$ and $z,$ the matrix
\[\begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix}\]
is not invertible. If $x + y + z = k$ for some constant $k$, find
\[\frac{x}{y + z - k} + \frac{y}{z + x - k} + \frac{z}{x + y - k}\]
assuming $y + z \neq k, z + x \neq k, x + y \neq k$.


Given that \(-5 \leq x \leq -3\) and \(1 \leq y \leq 3\), what is the largest possible value of \(\frac{x+y}{x-1}\)?


Triangle $PQR$ has vertices $P(-2, 10)$, $Q(3, -2)$, $R(10, -2)$. A line through $Q$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.


An equilateral triangle and a rectangle have the same perimeter of 24 inches. If the length of the rectangle is twice its width, what is the ratio of the side length of the triangle to the width of the rectangle? Express your answer as a common fraction.


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 15$, what is $b_3b_6$?


Erika, who is 18 years old, flips a fair coin whose sides are labeled 15 and 20, and then she adds the number on the top of the flipped coin to the number she rolls on a standard die. What is the probability that the sum equals her age in years? Express your answer as a common fraction.


If $x$ is a real number and $\lceil x \rceil = 13,$ how many possible values are there for $\lceil x^2 \rceil$?


In the diagram, each circle is divided into two equal areas and $O$ is the center of the largest circle. The area of the largest circle is $81\pi.$ What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),3,180,360)--cycle,mediumgray);fill(Arc((0,1.5),1.5,0,180)--cycle,mediumgray);
draw(Circle((0,0),3));
draw(Circle((0,1.5),1.5));

dot((0,0)); label("$O$",(0,0),N);
draw((-3,0)--(3,0)); draw((-1.5,1.5)--(1.5,1.5));
[/asy]


Compute \[
\left\lfloor \frac{2015! + 2012!}{2014! + 2013!}\right\rfloor.
\]
(Note that \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).)


A collection of seven positive integers has a mean of 5, a unique mode of 4, and a median of 5. If a 9 is added to the collection, what is the new median?


Given that \( n \) is an integer and \( 3 < 5n < 40 \), what is the sum of all possible integer values of \( n \)?


If six distinct points $A, B, C, D, E, F$ are to be selected from $2004$ points evenly spaced around a circle. What is the probability that chord $\overline{AB}$ intersects with chord $\overline{CD}$, assuming that none of the points are collinear?


Let $a$, $b$, and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 3$. Find the maximum value of \[2ab + 2bc \sqrt{3}.\]


A scalene triangle has side lengths which are consecutive odd prime numbers, and the length of its perimeter is also prime. What is the smallest possible perimeter?


A **composite number** is a number with two or more prime factors. The number 101 can be expressed as the sum of two composite numbers in several ways. What is the minimum positive difference between two such numbers?


Add 75.126 to 8.0034 and round to the nearest thousandth.


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

Solution: \boxed{0, 1, 2}


A circle centered at $O$ is circumscribed about $\triangle ABC$ as follows: [asy]
pair pA, pB, pC, pO;
pO = (0, 0);
pA = pO + dir(-30);
pB = pO + dir(90);
pC = pO + dir(210);
draw(pA--pB--pC--pA);
draw(pO--pA);
draw(pO--pB);
draw(pO--pC);
label("$O$", pO, S);
label("$120^\circ$", pO, NE);
label("$140^\circ$", pO, NW);
label("$A$", pA, SE);
label("$B$", pB, N);
label("$C$", pC, SW);
draw(circle(pO, 1));
[/asy] What is the measure of $\angle BAC$, in degrees?


What is the product of the solutions of the equation $8 = -2x^2 - 6x$?


Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


Find the integer \(n\), \(0 \leq n \leq 10\), such that \[n \equiv 123456 \pmod{11}.\]


Six numbers are given: 1877, 1999, 2039, 2045, 2119, and 2131. Four of these numbers have a mean of 2015. What is the mean of the other two numbers?


The first digit of a string of 2500 digits is a 2. Any two-digit number formed by consecutive digits within this string is divisible by 23 or 29. What is the largest possible last digit in this string?


Max picks three different cards sequentially without replacement from a standard 52-card deck. What is the probability that each card is of a different suit?


What is the maximum value of the expression $-3r^2 + 30r + 24$ for real $r$?


Consider the set of numbers 1879, 1997, 2023, 2029, 2113, and 2125. Four of these six numbers have a mean (average) of 2018. What is the mean (average) of the other two numbers?


What is the greatest number of Mondays that can occur in the first $45$ days of a year?


Find the product of all positive integral values of $n$ such that $n^2-31n+240= p$ for some prime number $p$. Note that there is at least one such $n$.


Find the largest value of $c$ such that $2$ is in the range of $g(x)=x^2-5x+c$.


Seven couples are at a gathering. Each person shakes hands with every person of the opposite gender except their spouse, and does not shake hands with anyone of the same gender. How many handshakes were exchanged?


Let $\triangle ABC$ be an isosceles right triangle with $AB = 6$ and $\angle B = 90^\circ$. Find the length of $BI$ if $I$ is the incenter of $\triangle ABC$.

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


How many paths are there from point $A$ to point $B$ in a $7 \times 6$ grid, if every step must be up or to the right, and you must not pass through the cell at position $(3,3)$?
[asy]size(4cm,4cm);int w=7;int h=6;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label("$A$", (0,0), SW, p);label("$B$", (w-1,h-1), NE, p);fill((3,3)--(4,3)--(4,4)--(3,4)--cycle, grey);[/asy]


Four standard dice are tossed, and the numbers $a, b, c, d$ are seen on the upper faces of these dice. What is the probability that the product $abcd$ equals 5?


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


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


Two circles of radius 8 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) of the two circles. Express your answer in simplest radical form.

Solution: \boxed{8\sqrt3 \text{ cm}}


Rationalize the denominator of $\frac{5}{4\sqrt{7} + 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?


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


In a recent survey, $75.3\%$ of the respondents believed that squirrels caused power outages. Of these, $38.2\%$ incorrectly believed that squirrels frequently caused internet disruptions. Given that these 29 respondents were incorrect, how many people in total did the survey include?


Find the sum of the coefficients in the polynomial \(5(2x^8 - 3x^3 + 4) - 6(x^6 + 4x^3 - 9)\) when it is fully simplified.


A square has vertices at \((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\). The line \( y = -\frac{x}{2} \) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \( 2a \). Express your answer in simplified radical form.

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


Using only the digits 6, 7, 8, and 9, how many positive nine-digit integers can be made that are palindromes?


How many positive multiples of 7 that are less than 150 have a units digit of 7?


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


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


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


Compute $4 \begin{pmatrix} 3 \\ -9 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -8 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 6 \\ -12 \end{pmatrix}}


What is the greatest common factor of all three-digit palindromes that are multiples of 3?


Consider a monic quartic polynomial $Q(x)$ with real coefficients, having four complex roots: $\cos \alpha + i \sin \alpha$, $\cos \alpha - i \sin \alpha$, $\cos(\alpha + \frac{\pi}{6}) + i \sin(\alpha + \frac{\pi}{6})$, and $\cos(\alpha + \frac{\pi}{6}) - i \sin(\alpha + \frac{\pi}{6})$, where $0 < \alpha < \frac{\pi}{6}$. The area of the quadrilateral formed by plotting these roots in the complex plane is equal to a quarter of $Q(0)$. Find $Q(0)$ and the sum of the four roots of this polynomial.

Solution: \boxed{Q(0) = 1}


For what digit $d$ is the five-digit number $5678d$ a multiple of 9?


Find all solutions $x$ of the inequality $$\frac{3}{20} + \left|2x - \frac{5}{40}\right| < \frac{9}{40}.$$ Express your answer in interval notation, simplifying all fractions in your answer.

Solution: \boxed{\left(\frac{1}{40}, \frac{1}{10}\right)}


What is the sum of the last two digits of this modified Fibonacci Factorial series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!+144!$?


When simplified, what is the value of $\sqrt{2} \times 2^{\frac{1}{2}} + 18 \div 3 \times 3 - 8^{\frac{3}{2}}$?

Solution: \boxed{20 - 16\sqrt{2}}


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point \((-12, 9)\)?


Using only the digits 0, 1, 7, 8, and 9, how many positive nine-digit integers can be made that are palindromes?


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


What is the value of $n$ such that $10^n = 10^{-3}\times \sqrt{\frac{10^{81}}{0.0001}}$?


What is the sum of the squares of the lengths of the **medians** of a triangle whose side lengths are 13, 14, and 15?


If \(a\), \(b\) are nonzero real numbers such that \(a^2 + b^2 = 10ab\), find the value of \(\left|\frac{a+b}{a-b}\right|\).


Let $g(x)$ be the function defined on $-2 \leq x \leq 2$ by the formula $$g(x) = 2 - \sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?


How many integers \( n \) satisfy \( (n+1)(n-5) < 0 \)?


You recently purchased over 100 eggs for a large culinary event. The eggs are packed in containers that can each hold exactly 10 eggs. After inspection, you find that three containers only have 9 eggs each, while all other containers are full. What is the smallest number of eggs you could have?


Determine how many distinct four-digit positive integers exist such that the product of their digits equals 18.


For positive integers $n$, let $g(n)$ return the smallest even positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $g(2010)$ have?


A curve is described parametrically by
\[(x,y) = (3 \sin t, \cos t + 2 \sin t).\] The graph of the curve can be expressed in the form
\[ax^2 + bxy + cy^2 = 4.\] Find the ordered triple \((a,b,c)\).

Solution: \boxed{\left(\frac{20}{9}, -\frac{16}{3}, 4\right)}


Find the sum of all positive divisors of $72$ that are also divisors of $24$.


Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum he has to the number of pieces of apple gum will be exactly the same as if he instead finds 4 packs of apple gum. Find $y$.


For a given positive integer $n > 2^3$ and an integer $m$, what is the greatest common divisor of $n^3 + m^2$ and $n + 2$?

Solution: \boxed{\text{gcd}(n^3 + m^2, n+2) = 1}


The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of sequences of three adjacent cards where the first two cards are both black, and the third card is red?


In parallelogram $EFGH$, $EF = 42$ cm, $FG = 4y^2 + 1$ cm, $GH = 3x + 6$ cm, and $HE = 28$ cm. Determine the product of $x$ and $y$.


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

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


Sam is hiking in Colorado as well. He starts at his campsite, walks eastward five miles, then turns $45$ degrees northward and walks eight miles. How far is he from his starting point? Express your answer in simplest radical form.

Solution: \boxed{\sqrt{89 + 40\sqrt{2}}}


Let's consider two individuals, Alex and Taylor, who each think of a polynomial. Each polynomial is monic, has degree 3, and has the same positive constant term and the same coefficient for $x^2$. The product of their polynomials is \[x^6 + 2x^5 + 3x^4 + 4x^3 + 3x^2 + 2x + 9.\] What is the constant term of Taylor's polynomial?


Find the largest integer $n$ for which $18^n$ evenly divides $30!$.


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


A rectangular band formation is a formation with $n$ band members in each of $r$ rows, where $n$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has three members left over. If he increases the number of members in each row by 1 and reduces the number of rows by 3, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have?


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha^2}{\beta}$ is a real number and $|\alpha - \beta| = 4$. Find $|\alpha|$.


A student walks along a corridor of lockers numbered from $1$ to $2048$. Initially, all lockers are closed. He starts by opening every locker. On his return trip, he skips one locker and opens the next. When he reaches the beginning again, he skips two lockers and opens the next closed locker, and continues this pattern. He repeats this pattern, increasing the skip by one each time until all lockers are open. What is the number of the last locker he opens?


A function $g$ from the integers to the integers is defined as:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 35.$ Find $m$.


Marius is now arranging his nine snow leopards for a larger wildlife photo contest. Among these, the shortest two have a complex and must be placed at either end of the row. Additionally, the tallest leopard insists on being placed right in the middle. In how many ways can Marius arrange these leopards following their demands?

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


Evaluate the number of digits to the right of the decimal point when $\frac{2^8 \cdot 5^2}{8^3 \cdot 1250}$ is expressed as a decimal.


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


A sequence $a_1, a_2, a_3, \dots,$ is defined recursively by $a_1 = 2, a_2 = 2,$ and for $k \ge 3,$
\[a_k = \frac{2}{5} a_{k - 1} + \frac{1}{6} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$


Two standard dice are rolled. What is the expected number of 2's obtained? Express your answer as a common fraction.


One of the roots of \( z^2 = -75 + 100i \) is \( 5 + 10i \). What is the other root?

Solution: \boxed{-5 - 10i}


How many positive two-digit integers have exactly five positive factors?


The sum of two numbers is 10. The difference of their squares is 24. What is the positive difference between the two numbers?


For which positive integer values of $k$ does $kx^2 + 22x + k = 0$ have rational solutions? Express your answers separated by commas and in increasing order.


If $\frac{80}{2^4 \cdot 5^6}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


What is the value of $x + y$ if the sequence $3, ~7, ~11, \ldots, ~x,~y, ~35$ is an arithmetic sequence?


Find $y$ such that $\lceil y \rceil + y = 150$.


Five consecutive positive integers have a product of 2520. What is the largest of these five integers?


If $x^{2y}= 64$ and $x = 8$, what is the value of $y$? Express your answer as a common fraction.


What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7?


Find the remainder when \(x^{2025} + 1\) is divided by \(x^{12} - x^8 + x^4 - 1.\)

Solution: \boxed{x^9 + 1}


If $x, y, z > 0$, find the smallest possible value of
\[
\left\lfloor{\frac{x+y+z}{x}}\right\rfloor + \left\lfloor{\frac{x+y+z}{y}}\right\rfloor + \left\lfloor{\frac{x+y+z}{z}}\right\rfloor.
\]


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


Given that $b$ is a multiple of $570$, find the greatest common divisor of $5b^3 + 2b^2 + 6b + 95$ and $b$.


If $m = 2^{12} \cdot 3^{10} \cdot 5^{9}$, how many of the natural-number factors of $m$ are multiples of 200?


Mr. and Mrs. Smith have three children. When they travel in their family SUV, two people sit in the front seats and three sit in the back seats. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible?


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


How many different positive three-digit integers can be formed using only the digits in the set $\{1, 3, 3, 4, 4, 4, 7\}$ if no digit may be used more times than it appears in the given set?


I cycle at a consistent speed, and it takes me 30 minutes to reach the library, which is 5 miles away from home. After spending 15 minutes at the library, I head to the community center, which is an additional 3 miles away. How long will it take me to reach the community center from my house?

Solution: \boxed{63\text{ min}}


Calculate the sum of the squares of the coefficients of $6(x^3 + 4x^2 + 2x + 3)$.


What is the result of subtracting 2 from the product of $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{6}$ of 72?


If $x^{3y}= 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


The product of four different positive integers is equal to $11^4$. What is the sum of these four integers?


The number $m$ is a three-digit positive integer and is the product of the three factors $x$, $y$, and $10x+y$, where $x$ and $y$ are distinct single-digit numbers under 10. Additionally, both $10x+y$ and $x+y$ must be prime numbers. What is the smallest possible value of $m$?

Solution: \boxed{m = 138}


How many odd integers are there between $\frac{23}{5}$ and $\frac{45}{2}$?


Matt's five cousins are coming to visit. There are five identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms?


Let $b_n = 5^{n} - 7^{n}$. Determine the remainder upon dividing $b_{40}$ by $49$.


Define $X \circ Y$ as $X \circ Y = \frac{(X+Y)}{4}$. What is the value of $(3 \circ 9) \circ 6$?


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


Suppose that $x$ is a positive multiple of $4$. If $x$ cubed is less than $5000$, what is the greatest possible value of $x$?


Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. The distances are $PQ=3$ units, $QR=7$ units, and $PS=18$ units. What is the ratio of $PR$ to $QS$?


Points \( E \), \( F \), \( G \), and \( H \) lie on a line, in that order. If \( EF = 3 \) units, \( FG = 6 \) units, and \( EH = 16 \) units, what is the ratio of \( EG \) to \( FH \)?


What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{113}$?


Find the units digit of the following within the indicated number base: \(65_7 + 34_7\).


Suppose that $uv = 24 - 10i$, and $|u| = 5$. What is $|v|$?


What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $7(5x-3) \equiv 14 \pmod{10}$?


What is the largest $2$-digit prime factor of $n = {210\choose 105}$?


Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$ Find $\sin \phi.$


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 4$. Find $|\alpha|$.


What is the smallest positive integer $n$ such that $13n \equiv 567 \pmod{5}$?


What is the base-10 integer 746 expressed in base 7?


The cinema has enough seats for $312_5$ people. If 3 people are supposed to sit in one row, how many rows does the cinema have?


What is the sum of the last two digits of this modified Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+144!$?


How many four-digit numbers are multiples of neither 3 nor 4?


For each value of $x$, $g(x)$ is defined to be the minimum value among the four numbers $2x + 2,$ $\frac{1}{2} x + 1,$ $-\frac{3}{4} x + 7,$ and $3x + 4$. Find the maximum value of $g(x)$.


Define a sequence $c_1, c_2, \ldots$ where $c_1 = 3$, $c_2 = 9$, and for $n \geq 3$, $c_n = c_{n-1} \cdot c_{n-2}$. Calculate $c_{10}$.


Factor completely: $x^8 - 4x^6 + 6x^4 - 4x^2 + 1$.


Let \( d \) be a positive number such that when \( 143 \) is divided by \( d \), the remainder is \( 11 \). Compute the sum of all possible two-digit values of \( d \).


What is the area, in square units, of a triangle whose vertices are at $(2, 3)$, $(2, -4)$, and $(7, -4)$?


In a classroom, there are 12 students including Fiona. Fiona decides to observe the interactions but focuses only on pairs that include her best friend, Alice. How many such pairs can Fiona observe?


The set of all solutions of the system
$$
\begin{cases}
x+y\leq 4 \\
2x+y\geq 1 \\
x\geq 0 \\
y\geq 0
\end{cases}
$$
is a quadrilateral region. Find the number of units in the length of the longest side. Express your answer in simplest radical form.


In a certain sequence, the first term is $a_1=2023$ and the second term is $a_2=2024$. Furthermore, the values of the remaining terms are chosen so that $a_n+a_{n+1}+a_{n+2}=n+2$ for all $n \geq 1$. Determine $a_{1000}$.


For how many positive integers \( n \) does \( \frac{1}{n} \) yield a terminating decimal with a non-zero tenths digit?


A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and radius equal to the upper base of the frustum attached to the top of the frustum. Calculate the total surface area excluding the bases.

Solution: \boxed{39\pi\sqrt{5} + 20\pi}


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


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


For each pair of real numbers $x \ne y$, define the operation $\diamond$ as \[
(x \diamond y) = \frac{x + y}{x + y - 2xy}.
\] What is the value of $((3 \diamond 4) \diamond 5)$?


For each integer $n$, let $h(n) = n^4 - 3n^3 + 360n^2 - 3n + 401$. What is the sum of all values of $h(n)$ that are prime numbers?


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that for all real numbers \( x \), \( y \), and \( z \):
\[ f(x^2 + yf(z) + z) = xf(x) + zf(y) + f(z). \]
Find the number of possible values of \( f(5) \), and let \( s \) be the sum of all possible values of \( f(5) \). Compute \( n \times s \).


A rectangular picture frame is constructed with 2-inch-wide pieces of wood. The exposed area of the frame alone is 30 square inches, and one of the outer edges of the frame measures 6 inches. Determine the sum of the lengths of the four interior edges of the frame.

Solution: \boxed{7 \text{ inches}}


For what value of $y$ does $5^{2y^{2} - 3y + 5} = 5^{2y^{2} + 9y - 7}$? Express your answer as a common fraction.


How many even perfect square factors does $2^6 \cdot 7^3 \cdot 3^8$ have?


Find the product of the roots of the equation $12x^2 + 28x - 315 = 0$.


All positive integers whose digits add up to 12 are listed in increasing order. What is the thirteenth number in that list?


When $\frac{1}{101}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point?


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

Solution: \boxed{p(x) = x^2 + 1}


In right triangle $ABC$, angle $B$ is a right angle, angle $A$ is $45^\degrees$, and $AB = 10\sqrt{2}$. Calculate the area of triangle $ABC$ and find the length of the segment $BD$, where $D$ is the foot of the perpendicular from $B$ to $AC$.


How many positive divisors do 9240 and 8000 have in common?


For how many non-negative real values of $x$ is $\sqrt{196-\sqrt[3]{x}}$ an integer?


How many integers $n$ satisfy the condition $150 < n < 300$ and the condition $n$ has the same remainder when it is divided by $7$ and by $9$?


Let $a$ and $b$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + ax - b = 0\] are three distinct positive integers. Compute $a + b.$


Bryan has some 3 cent stamps, 4 cent stamps, and 5 cent stamps. What is the least number of stamps he can combine so the value of the stamps is 47 cents, ensuring he uses at least two different types of stamps?


What is the harmonic mean of \(\frac{3}{4}\) and \(\frac{5}{6}\)? Express your answer as a common fraction.


What is the largest $2$-digit prime factor of the integer $n = {250\choose 125}$?


A hurricane in Miami, USA, caused $45 million in damage. That estimate was in American dollars. At that time, 2 American dollars were equivalent to 3 Canadian dollars. Determine the number of Canadian dollars of damage the hurricane caused.


Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2$.  Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-3,9)$, $(1,1)$, and $(4,16)$, find the sum of the Euclidean distances from the focus of $\mathcal{P}$ to all four intersection points.


If $S$, $H$, and $E$ are all distinct non-zero digits less than $6$ and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$. $$\begin{array}{c@{}c@{}c@{}c} &S&H&E_6\\ &+&H&E_6\\ \cline{2-4} &S&E&S_6\\ \end{array}$$


Solution: \boxed{18 - 14\sqrt{2}}


A rectangular block is entirely made by gluing $N$ congruent 1-cm cubes edge to edge. When observed such that one can see three of its side faces, exactly $270$ of the 1-cm cubes are hidden from view. Determine the smallest possible value of $N.$


Both roots of the quadratic equation \( x^2 - 101x + k = 0 \) are prime numbers. How many possible values of \( k \) are there?


A bookstore has a sale on days of the month that are multiples of 4 (like July 4, July 8...). A shoe store has a sale every 8 days, starting from July 2. How many times in July do both stores have sales on the same date?


If $z$ is a complex number such that
\[
  z + z^{-1} = -\sqrt{3},
\]
what is the value of
\[
  z^{1001} + z^{-1001} \, ?
\]


If the least common multiple of \(X\) and \(Y\) is \(180\) and the ratio of \(X\) to \(Y\) is \(2:5\), then what is their greatest common divisor?


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


The graph of the parabola $x = 3y^2 - 9y + 4$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


Let $a>0$ and $b>0,$ and define two operations:
$$a \nabla b = \dfrac{a + b}{1 + ab}$$
$$a \Delta b = \dfrac{a - b}{1 - ab}$$
Calculate $3 \nabla 4$ and $3 \Delta 4$.


Consider the polynomial $x^3 - 2500x + m$ with integer roots $p$, $q$, and $r$. Find $|p| + |q| + |r|$.


At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups?


An arithmetic sequence begins with a first term of $3$ and has a common difference of $10$. A second sequence starts with $5$ and has a common difference of $8$. In the range of $1$ to $150$, find the largest number common to both sequences.


The line in the coordinate plane can be parameterized so that its direction vector is of the form $\begin{pmatrix} 3 \\ b \end{pmatrix}.$ Find $b$ if the line passes through the points $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 4 \end{pmatrix}.$


How many degrees are in the measure of the smaller angle formed by the hour and minute hands of a clock when the time is 3:30 p.m.?


On a balance scale, $4$ red balls balance $8$ blue balls, $3$ orange balls balance $7$ blue balls, and $5$ blue balls balance $3$ silver balls. How many blue balls are needed to balance $5$ red, $3$ orange, and $4$ silver balls?


Compute $\arccos(\cos 9).$ All functions are in radians.

Solution: \boxed{9 - 2\pi}


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

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


What is the smallest positive integer \(x\) that satisfies the congruence \(5x \equiv 17 \pmod{31}\) and where \(31\) is also a prime number?


Find all $b,$ $0^\circ < b < 360^\circ,$ such that $\cos b,$ $\cos 2b,$ and $\cos 3b$ form a geometric sequence, in that order. Enter the solutions, separated by commas, in degrees.


A $\textit{composite number}$ is a number that has two or more prime factors. The number 93 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?


In a gathering, there are 25 elves and 18 dwarves. The dwarves, due to an old rivalry, refuse to shake hands with each other, but they will shake hands with all of the elves. The elves are sociable and shake hands with each other and with all the dwarves. Calculate the total number of handshakes at this gathering.


What is the smallest positive integer $n$ such that $n^2$ is divisible by 24 and $n^3$ is divisible by 1024?


What is the value of $a$ for which $\frac{1}{\text{log}_5a} + \frac{1}{\text{log}_6a} + \frac{1}{\text{log}_{10}a} = 1$?


Two rectangles have integer dimensions, and both have a perimeter of 180 cm. What is the greatest possible difference between the areas of two such rectangles?

Solution: \boxed{2025 - 89 = 1936 \text{ square cm}}


Find the value of $a$ such that $394_{a} + 586_{a} = 980_{a}$.


The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$?


Let $n'$ be the number of ordered quadruples $(x_1', x_2', x_3', x_4')$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i' = 74.$ Find $\frac{n'}{100}.$


The values of $x$, $y$, $z$, and $w$ are 7, 8, 9, and 10, but not necessarily in that order. What is the largest possible value of the sum of the four products $xy$, $yz$, $zw$, and $xw$ if $x < y < z < w$?


John is thinking of a number. He gives the following 3 clues: "My number has 125 as a factor. My number is a multiple of 30. My number is even and between 1000 and 3000." What is John's number?


A rectangle has dimensions given by $x - 3$ by $2x + 3$. It is known that the area of this rectangle is $4x - 9$. Find the value of $x$.


If $\frac{120}{2^3 \cdot 5^6}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


William is participating in a 30-question, multiple-choice competition. Each question offers six answer choices. Suppose William guesses randomly on the last six questions. What is the probability that he will get at least one of these last six questions right?


Sasha has $\$4.85$ in U.S. coins. She has the same number of quarters and nickels, but she also has twice as many dimes as nickels. What is the greatest number of quarters she could have?


A new telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 5 meters from its base. If Leah walks 3 meters from the base of the pole towards the point where the cable is anchored on the ground, and her head just touches the cable at that point, knowing Leah's height is now 1.75 meters. How tall is the pole?

Solution: \boxed{4.375\text{ meters}}


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


Given that $a$, $b$, $c$, and $d$ are nonzero real numbers, find all possible values of the expression
\[
\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{d}{|d|} + \frac{abcd}{|abcd|}.
\]

Solution: \boxed{5, 1, -3}


Marius is participating in a different wildlife photo contest and this time, he has eight unique snow leopards of different heights. The shortest and the tallest leopards, however, have developed a peculiar relationship and insist on being placed at the two ends of the row. In how many different ways can Marius arrange the leopards?

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


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


What is the smallest whole number $b$ such that 80 can be expressed in base $b$ using only three digits?


Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=3x-2$. Find $g^{-1}(f(-10))$.


Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\]on the interval $(0,150 \pi).$


The Bank of Springfield's Super High Yield savings account compounds annually at a rate of three percent. If Bart invests $5000 in one of these accounts, how much interest will he earn after ten years? (Give your answer to the nearest dollar.)


Two angles form one branch of an X. The angles are complementary and their measures are in a ratio of $4:1$. What is the measure, in degrees, of the smaller angle?


If $x\%$ of three-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.


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 845^\circ.$


Consider the polynomial
\[ b_{6} x^{6} + b_{5} x^{5} + b_{4} x^{4} + b_{3} x^{3} + b_{2} x^{2} + b_{1} x + b_{0} = 0 \]
with integer coefficients, where all its roots \( s_1, s_2, \ldots, s_{6} \) are integers. Additionally, the roots of the polynomial
\[ b_{0} x^{6} + b_{1} x^{5} + b_{2} x^{4} + b_{3} x^{3} + b_{4} x^{2} + b_{5} x + b_{6} = 0 \]
are also \( s_1, s_2, \ldots, s_{6} \). Find the number of possible multisets \( S = \{s_1, s_2, \ldots, s_{6}\} \).


Assume that $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$.


The quadratic $5x^2 + 20x - 25$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms.


Determine how many terms of the arithmetic sequence 100, 96, 92, $\dots$ appear before the term $-4$ is reached and ensure that the sum of these terms exceeds 1000.


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?


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point \((12, -5)\)?


Let $d$ be a complex number. Suppose there exist distinct complex numbers $u$, $v$, $w$, and $x$ such that for every complex number $z$, we have
\[
  (z - u)(z - v)(z - w)(z - x) = (z - du)(z - dv)(z - dw)(z - dx).
\]
Calculate the number of distinct possible values of $d$.


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


Determine the product of all real values of \( r \) such that the equation \(\frac{1}{4x} = \frac{r - 2x}{13}\) has exactly one real solution.


Suppose that \(\{b_n\}\) is an arithmetic sequence with
\[
b_1 + b_2 + \cdots + b_{150} = 150 \quad \text{and} \quad b_{151} + b_{152} + \cdots + b_{300} = 300.
\]
What is the value of \(b_2 - b_1\)? Express your answer as a common fraction.


Determine the sum of the last two digits of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!+144!$.


There are pairs of real numbers $(x,y)$ that satisfy the equation $x + y = 3xy = 6$. Given that the solutions for $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers, and the expression is completely simplified, what is the value of $a + b + c + d$?


Determine the sum of all single-digit replacements for $z$ such that the number ${35{,}z62}$ is divisible by 6.


How long is the segment whose endpoints are (1,2) and (10,8)?


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


Four concentric circles are drawn with radii of 2, 4, 6, and 8. The innermost circle is painted black, the ring around it is white, the next ring is black, and the outermost ring is white. Calculate the ratio of the black area to the white area and express your answer as a common fraction.


In a school course, 30 students are enrolled. Every student except one attends both math and history classes. Six students received an A in history and 15 received an A in math, including three that received an A in both courses. Moreover, one additional student received an A only in math but never attended history. How many students did not receive an A in either class?


Given that $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = -3$, find
\[
\begin{vmatrix} p & 5p + 4q \\ r & 5r + 4s \end{vmatrix}.
\]


Calculate $7! - 6 \cdot 6! - 2\cdot 6!$.


For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $15+i$ and the sum of the other two roots is $2+5i,$ where $i^2 = -1.$ Find $b.$


Two transformations are applied to the complex number $-1 - 2i$:
1. A $60^\circ$ rotation around the origin in the counter-clockwise direction.
2. A dilation, centered at the origin, with scale factor $2$.
What is the resulting complex number?

Solution: \boxed{2\sqrt{3} - 1 - (2 + \sqrt{3})i}


Find all values of $x$ such that $x + \lceil x \rceil = 21.3$ and $x$ is greater than $0$.


Determine the value of \(n\) such that the expression
\[\frac{\tan \frac{\pi}{7} + i}{\tan \frac{\pi}{7} - i}\]
is a fourteenth root of unity, i.e., it equals \(\cos \frac{2n \pi}{14} + i \sin \frac{2n \pi}{14}\) for some integer \(n\) between 0 and 13 inclusive.


Evaluate the expression $\frac{10! \cdot 7! \cdot 3!}{9! \cdot 8!}$.


If $x \star y = xy - 3x + y$, what is the value of $(6 \star 5) - (5 \star 6)$?


Calculate the total number of days in the years from 2006 to 2010.


A square shares one side with an isosceles right triangle. If one angle of the isosceles right triangle is 45°, what is the degree measure of $m\angle CDB$? Assume side CD is common to both the square and the triangle, with point B at the vertex opposite the right angle of the triangle. [asy] pair A,E,C,D,B; D = dir(45); C = dir(0); E = (0,-1); B = C+E; draw(B--D--C--B--E--A--C--D--A); label("D",D,N); label("C",C,E); label("B",B,dir(0)); [/asy]


Determine how many values of $a$ satisfy the following conditions:
(1) $a$ is a positive integer such that $1 \le a \le 100$.
(2) The quadratic equation $2x^2 + (3a+1)x + a^2 = 0$ has two distinct integer solutions.


What is the modulo $7$ remainder of the sum $1+2+3+...+200?$


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


Ben starts with a number $8^7$ and creates a sequence by alternatively dividing by 4 and multiplying by 3. What is the number in the sequence after the 10th operation? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as possible.

Solution: \boxed{2^{11} \cdot 3^5}


Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm, and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square and the distance from vertex $R$ (opposite the right angle) to the hypotenuse, in cm? Express your answer as a common fraction.

Solution: \boxed{\frac{36}{5} \text{ cm}}


Find the remainder when $8x^4 - 10x^3 + 7x^2 - 5x - 30$ is divided by $2x - 4$.


The expression \(2y^2 - 5y - 12\) can be written as \((2y + a)(y + b)\), where \(a\) and \(b\) are integers. What is \(a - b\)?


Find the coefficient of $x$ when $5(x - 6) + 6(9 - 3x^2 + 3x) - 10(3x - 2)$ is simplified.


Let $a = \underbrace{3333333}_{\text{7 digits}}$ and $b = \underbrace{666666666}_{\text{9 digits}}$. What is $\gcd(a, b)$?


The function $g(x)$ satisfies
\[g(x + y) = g(x) + g(y)\]for all real numbers $x$ and $y,$ and $g(3) = 4.$  Find $g(6).$


On the door, BIOLOGY is spelled out with 7 magnets, one letter per magnet. Two vowels and two consonants fall off and are put away in a bag. If the O's are indistinguishable, how many distinct possible collections of letters could be put in the bag?


Triangle $QRS$ is an isosceles triangle with an area of 180 square inches. A line segment $MN$ cuts the triangle, forming a smaller isosceles triangle and an isosceles trapezoid, where $MN$ is parallel to the base $RS$. The area of the isosceles trapezoid is 120 square inches. If the altitude of triangle $QRS$ from vertex $Q$ is 30 inches, determine the length of $\overline{MN}$.

Solution: \boxed{4\sqrt{3} \text{ inches}}


A particle is located on the coordinate plane at $(8,0)$. Define a 'move' for the particle as a counterclockwise rotation of $\frac{\pi}{3}$ radians about the origin followed by a translation of $8$ units in the positive $x$-direction. Find the particle's position after $180$ moves.

Solution: \boxed{(8, 0)}


In isosceles triangle $\triangle ABC$, the sides $AB = AC = 5$. The altitude from vertex $B$ intersects $\overline{AC}$ at point $H$. Given that $AH = 2(HC)$, determine the length of $BC$.


In right triangle $DEF$ where $\angle D = 90^\circ$, the lengths of sides $DE = 8$ and $EF = 17$. Calculate $\sin F$.

Solution: \boxed{\sin F = \frac{8}{17}}


In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=30$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, dir(0));
label("E", E, NE);
[/asy] Find the perimeter of quadrilateral $ABCD.$

Solution: \boxed{26.25\sqrt{3} + 33.75}


Altitudes $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at $H$. If $\angle BAC = 30^\circ$ and $\angle ABC = 80^\circ$, then what is $\angle AHB$?

[asy]
size(150); defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (1.8,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$D$",P,S); label("$E$",Q,E); label("$H$",H,NW);
draw(rightanglemark(C,P,H,3.5));
draw(rightanglemark(H,Q,C,3.5));
[/asy]


Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 45.$


If \(\lfloor{x^2}\rfloor = 16\), how many possible integer values of \(x\) are there?


What is the 25th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{8}$ and $\frac{1}{11}$?


These two spinners are divided into halves and fifths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be a multiple of 3? Express your answer as a common fraction.

[asy]
size(5cm,5cm);
draw(Circle((0,0),1));
draw(Circle((3,0),1));
draw((0,0)--(0,1));
draw((0,0)--(0,-1));
draw((3,0)--(3.2,0.8),Arrow);
draw((0,0)--(0.2,0.8),Arrow);
label("$3$",(0,0.5));
label("$4$",(0,-0.5));
label("$2$",(2.6,-0.4));
label("$3$",(3.5,0.4));
label("$5$",(2.6,0.4));
label("$6$",(3.5,-0.4));
label("$9$",(3,0));
[/asy]


The points $P$, $Q$, and $R$ lie on the surface of a sphere with center $S$ and radius $25$. It is given that $PQ=15$, $QR=20$, $RP=25$, and that the distance from $S$ to $\triangle PQR$ is $\frac{x\sqrt{y}}{z}$, where $x$, $y$, and $z$ are positive integers with $x$ and $z$ relatively prime and $y$ not divisible by the square of any prime. Find $x+y+z$.


How many positive integers less than $350$ are multiples of $6$, but not multiples of $18$?


How many rows of Pascal's Triangle contain the number $53$?


How many four-digit numbers have at least one 6 or at least one 8 as digits?


Triangle $DEF$ has vertices $D(2, 10)$, $E(1, 0)$, and $F(11, 0)$. A line through $E$ cuts the area of $\triangle DEF$ in half. Determine the sum of the slope and the $y$-intercept of this line.


What is the units digit of the sum of the cubes of the first 3000 odd positive integers?


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

[asy]
pair A, B, C;
A = (0,0);
C = (20,0);
B = (0,12);
draw(B--C--A--B);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$C$",C,SE);
label("$B$",B,N);
label("$12$", B/2,W);
label("$20$", (A+C)/2,S);
[/asy]


Given that $9^{-1} \equiv 43 \pmod{97}$, find $81^{-1} \pmod{97}$, as a residue modulo 97. (Provide an answer between 0 and 96, inclusive.)


Solution: \boxed{\mathbf{B}^{2023} = \begin{pmatrix} \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ -\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix}}


The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). What is the greatest possible value of \( x \)?


How many different 8-digit positive integers exist if the last digit must be an odd number?


How many two-digit numbers have digits whose sum is a perfect square, now considering sums that include the perfect square $25$ as well?


If $(x,y)$ is a solution to the system
\begin{align*}
xy &= 12, \\
x^2 y + xy^2 + x + y &= 99,
\end{align*}
find $x^2 + y^2$.

Solution: \boxed{x^2 + y^2 = \frac{5745}{169}}


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


On the Saturday of a weekend soccer tournament, Team A plays Team B, Team C plays Team D, and Team E plays Team F. On Sunday, the three Saturday winners play each other in a round-robin format to determine the first, second, and third places, while the three Saturday losers play another round-robin to determine the fourth, fifth, and sixth places. There are no ties in any matches. What is the total number of possible six-team ranking sequences at the end of the tournament?


If $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 5$, then find all integer values of $n$.


Both roots of the quadratic equation $x^2 - 65x + k = 0$ are consecutive prime numbers. How many possible values of $k$ are there?


The 5 a.m. temperatures for eight consecutive days were $-7^\circ$, $-4^\circ$, $-4^\circ$, $-5^\circ$, $1^\circ$, $3^\circ$, $2^\circ$, and $4^\circ$ Celsius. What is the mean 5 a.m. temperature for these eight days in degrees Celsius?

Solution: \boxed{-1.25^\circ \text{ Celsius}}


Construct the matrix $\mathbf{N}$ that swaps the rows of any matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that
\[
\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & d \\ a & b \end{pmatrix}.
\]
If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

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


What is the area, in square units, of a trapezoid bounded by the lines $y = 2x$, $y = 12$, $y = 8$, and the $y$-axis?


Let the first term of a geometric sequence be $\frac{5}{8}$, and let the second term be $50$. What is the smallest $n$ for which the $n^{th}$ term of the sequence is divisible by one billion ($10^9$)?


In a conference, 15 out of 22 participants voted in favor of adopting a new resolution. What is the ratio of affirmative votes to total participants, rounded to the nearest tenth?


How many $x$-intercepts does the graph of the parabola $x = -3y^2 + 2y + 3$ have?


What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 45\cdot 2x + 45^2$ is a multiple of 43?


Let $m$ be the smallest integer whose fourth root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/10000$. Find $n$.


Our club has 24 members, 12 boys and 12 girls. In how many ways can we choose a president, a vice-president, and a secretary if the president and the vice-president must be of different genders and the secretary must be of the same gender as the vice-president?


Jessica is hired for a 30-day period. On days that she fully works, she earns $\$$80. For each day that she does not work, $\$$40 is subtracted from her earnings. Additionally, for five days, she only worked half-days and earned half of the daily earnings. At the end of the 30-day period, she received $\$$1600. How many full days did she not work?


If $\cos \theta = \frac{1}{2},$ then find $\cos 2 \theta$ and $\sin 2 \theta.$


A new design involves a rectangle and four identical circles, each tangent to two adjacent sides of the rectangle and its neighboring circles. The design is such that the length of the rectangle is twice its width. If the rectangle has a width of 30 inches, determine the total shaded area when the circles are excluded.

Solution: \boxed{1800 - 225\pi \text{ square inches}}


An artist wants to install a square glass pane (the shaded region) precisely within a square frame for an art installation. The outer perimeter of the frame measures 100 cm on each side. The width of each side of the frame is 15 cm. What is the area of the glass pane?

Solution: \boxed{4900 \text{ cm}^2}


Compute $\arccos (\cos 8)$. All functions are in radians.

Solution: \boxed{8 - 2\pi}


The bacteria in a laboratory culture grow at a rate such that their number quadruples every 12 hours. If there are initially 200 bacteria in the culture, calculate how many hours it will take until there are exactly 819,200 bacteria. Also, determine how many bacteria there will be after 24 hours.


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


A rectangle with integer length and width has a perimeter of 84 units. What is the number of square units in the least possible area?


Rationalize the denominator of $\frac{7}{\sqrt{200}}$.

Solution: \boxed{\frac{7 \sqrt{2}}{20}}


Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - 2n \] for all \( n > 1 \). Find the smallest possible value of \( a_1 \).


Bryan has some 3 cent stamps, 4 cent stamps, and 5 cent stamps. What is the least number of stamps he can combine so the value of the stamps is 50 cents?


The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&32&43&58&&\\
1&05&10&12&15&20\\
2&25&30&55&&\\
3&00&15&30&35&45\\
4&10&12&&&\\
\end{tabular}


Compute $\tan \left(\operatorname{arccot} \frac{3}{5}\right)$ and verify if $\sin \left(\operatorname{arccot} \frac{3}{5}\right) > \frac{1}{2}$.


Find all \(y \in [0, 2\pi]\) for which \[\sin(x+y) \geq \sin(x) - \sin(y)\] for every \(x\) between \(0\) and \(2\pi\), inclusive.

Solution: \boxed{y = 0 \text{ or } y = 2\pi}


Solution: \boxed{4i - 2}


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

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


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{5}.$  Find $|\alpha|.$


Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 73 \times 83 \times 93$ is divided by $5$.


Let $a \star b = \frac{\sqrt{a+b}}{\sqrt{a-b}}$. If $x \star 36 = 9$, find $x$ assuming $x > 0$.


Determine how many integers are multiples of 30 from the smallest positive multiple of 30 that is a perfect square to the smallest positive multiple of 30 that is a perfect cube, inclusive.


The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.


In a group consisting of four members -- Alice, Bob, Carol, and Dave -- in how many ways can three people be chosen and appointed to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job and with the condition that Alice does not want to be the treasurer?


The function $g$ is graphed below. Each small box has width and height 1.

[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);

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

}

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(-1,9,-1,9);

dot((0,0),red+5bp);
dot((2,1),red+5bp);
dot((4,3),red+5bp);
dot((6,5),red+5bp);
dot((8,7),red+5bp);
dot((1,8),red+5bp);
dot((3,6),red+5bp);
dot((5,4),red+5bp);
dot((7,2),red+5bp);
dot((9,0),red+5bp);

[/asy]

Starting with the number 2 written on his thumb, Alex applies $g$ to 2 and writes the output on his next finger. If Alex continues this process of applying $g$ and writing the output on a new finger, what number will Alex write on his eighth finger?


A circle inscribed in $\triangle ABC$ is tangent to each side at points lying on the extensions of sides $AB$, $BC$, and $CA$. Suppose the angles at the points where the inscribed circle is tangent to the sides $BC$, $CA$, and $AB$ measure $120^\circ$, $130^\circ$, and $\theta^\circ$ respectively. What is $\theta$?


Suppose I have 8 shirts, 5 pairs of pants, 4 ties, and 2 belts. If an outfit requires a shirt and pants, and can optionally include either a tie, a belt, or both, how many outfits can I make?


Determine the value of $\frac{c}{b}$ where the quadratic $x^2 + 1500x + 2400$ can be written in the form $(x+b)^2+c$.


A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?


Consider $x$ to be a multiple of $18432$. Determine the greatest common divisor of $g(x) = (5x+3)(8x+2)(11x+7)(3x+5)$ and $x$.


A square and a right triangle are positioned in a plane with their bases on the $x$-axis. The right triangle's hypotenuse starts from the top left vertex of the square and ends at the $x$-axis. The lower right vertex of the square and the lower right vertex of the triangle are at $(12, 0)$. The sides of the square are each $12$ units, and the base of the triangle along the $x$-axis is also $12$ units. Calculate the area of the region formed by the portion of the square below the diagonal of the triangle.

Solution: \boxed{72 \text{ square units}}


In the given arrangement, the rule extends as described:

$\bullet$ Row 1: $2,$ $2$

$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$

$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$

$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$

$\bullet$ Row 5: $10,$ $10,$ $10,$ $10,$ $10,$ $10,$ $10,$ $10,$ $10,$ $10$

Based on the pattern established in earlier rows, determine the value of the $40^\text{th}$ number in this sequence.


An infinite geometric series has a common ratio of $\frac{1}{4}$ and a sum of $10$. What is the second term of the sequence?


Convert the point $(\sqrt{2}, -\sqrt{2}, 2)$ 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( 2\sqrt{2}, \frac{7\pi}{4}, \frac{\pi}{4} \right)}


Find the greatest common divisor of $8!$ and $(6!)^2 \cdot 2^3.$


If $\tan x = \frac{1}{3},$ find $\tan \left( x + \frac{\pi}{3} \right).$

Solution: \boxed{2 + \frac{5 \sqrt{3}}{3}}


Clara has more than $\$2$ but less than $\$20$ worth of quarters. When she arranges her quarters in stacks of 3, she has 1 left over. She places the quarters in stacks of 4 and still has 1 left over. She does the same with stacks of 5, always having 1 left over. How many quarters does Clara have?


Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\]


Find the matrix $\mathbf{M}$ that doubles the first row and triples the second row of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.\] If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


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


John owns 16 pairs of shoes: eight identical black pairs, four identical brown pairs, three identical gray pairs, and one pair of red shoes. If he picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


Let $z$ be a complex number such that
\[|z^2 + 9| = |z(z + 3i)|.\]
Find the smallest possible value of $|z + i|$.


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.


Calculate $\sqrt{42q} \cdot \sqrt{7q} \cdot \sqrt{3q}$, where $q$ is a positive real number. Express your answer in simplest radical form in terms of $q$.

Solution: \boxed{21q \sqrt{2q}}


When $500^{150}$ is expanded out, the result is $1$ followed by how many zeros?


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


Trent is planning a picnic and wants to buy burgers and burger buns. Burgers come in packages of 5, and burger buns come in packages of 7. What is the smallest number of burger packages he can buy in order to have an equal number of burgers and burger buns?


Consider the set of points that are inside or within two units of a rectangular parallelepiped that measures 2 by 3 by 6 units. Calculate the total volume of this set, expressing your answer in the form $\frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n$ and $p$ being relatively prime.

Solution: \boxed{\frac{540 + 164\pi}{3}}


Let $O$ be the origin, and let $(a, b, c)$ be a fixed point not on the origin. A plane passes through $(a, b, c)$ and intersects the $x$-axis at $(2a, 0, 0)$, the $y$-axis at $(0, 2b, 0)$, and the $z$-axis at $(0, 0, 2c)$, all distinct from $O$. Let $(p, q, r)$ be the center of the sphere passing through $O$, $(2a, 0, 0)$, $(0, 2b, 0)$, and $(0, 0, 2c)$. Determine the value of
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]


Call a set of integers "spacy" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy?


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


Find the total volume in cubic centimeters of a structure that consists of a cone on the bottom and a cylinder on the top. The entire structure has a consistent diameter of 8 cm. The cone's height is 10 cm, and the cylinder's height is 4 cm. Express your answer in terms of \(\pi\).

Solution: \boxed{\frac{352}{3} \pi \text{ cm}^3}


Simplify the following expression: $(2x^6 + 3x^5 + x^4 + 3x^3 + 2x + 15) - (x^6 + 4x^5 + 2x^3 - x^2 + 5).$ Express your answer as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{x^6 - x^5 + x^4 + x^3 + x^2 + 2x + 10}


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{500}$?


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


What is the smallest prime divisor of $3^{15} + 11^{13}$?


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


What is the smallest positive integer \(n\) such that \(\frac{n}{n+150}\) is a terminating decimal?


$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\frac{1}{4}$. Find $n$.

Solution: \boxed{n = 5}


A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are possible?


In triangle $XYZ$, altitudes $XM$, $YN$, and $ZO$ intersect at the orthocenter $P$. If $\angle XYZ = 55^\circ$ and $\angle XZY = 25^\circ$, find the measure of $\angle YPZ$, in degrees.


Evaluate the polynomial \[ x^3 - 3x^2 - 9x + 5, \] where \( x \) is the positive number such that \( x^2 - 3x - 9 = 0 \).


Find all solutions to the equation $\sqrt{10 + 3x - y} = 7$.

Solution: \boxed{y = 3x - 39}


How many prime numbers have squares between 10000 and 15625?


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

Solution: \boxed{24x - 25}


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

Solution: \boxed{16x - 59}


In right triangle $ABC$ with $\angle C = 90^\circ$, we have $AB = 15$ and $AC = 9$. Find $\cos B$.


Solution: \boxed{2011 \times 2010}


You have a standard 10-sided die, with faces numbered from 1 to 10. The probability of rolling any specific number is uniform. Calculate the expected value of a roll if each odd number rolled is valued at double its face value, while each even number is valued at its normal face value.


Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]


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


Determine the value of $k$ such that the lines represented by the equations are perpendicular:
\begin{align*}
y &= 3x+7 \\
4y+kx &= 4
\end{align*}


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


What is the least five-digit positive integer which is congruent to 7 (mod 17)?


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


In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 1500$ and $PS = 3000$. Let $\angle P = 37^\circ$, $\angle S = 53^\circ$, and $X$ and $Y$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $XY$.


Two numbers $120$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?


The integers \(r\) and \(k\) are randomly selected, where \(-3 < r < 6\) and \(2 \leq k < 7\) where \(k\) is a prime number. What is the probability that the division \(r \div k\) results in an integer? Express your answer as a common fraction.


How many rows of Pascal's Triangle contain the number $53$?


A monic quartic polynomial $p(x)$ satisfies $p(-2)=-4$, $p(1)=-1$, $p(3)=-9$, and $p(5)=-25$. Find the value of $p(0)$.

Solution: \boxed{p(0) = -30}


Sarah's grandmother is ten times Sarah's age. When Sarah was born, her grandmother was 60 years old. Determine Sarah's current age.


Let set $\mathcal{C}$ be an 80-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{C}$. Find the number of possible values of $S$.


How many integers between 1 and 9 are divisors of the five-digit number 35,424?


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


Let $r(x)$ be a function with domain $\{-2, -1, 0, 1, 2\}$ and range $\{-2, 0, 2, 4, 6\}$. Let $s(x)$ be a function with domain $\{0, 1, 2, 3, 4\}$ defined by $s(x) = x^2 + 1$. What is the sum of all possible values of $s(r(x))$?


What is the 24th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{7}$ and $\frac{1}{9}$?


Express $7.\overline{836}$ as a common fraction in lowest terms.


Let \( x \) be a real number with the property that \( x+\frac{1}{x} = 4 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \).


Triangle $PQR$ has $PQ=13$, $PR=14$, and $QR=15$. Points $S$ and $T$ are located on $\overline{PQ}$ and $\overline{PR}$, respectively, such that $\overline{ST}$ is parallel to $\overline{QR}$ and contains the center of the inscribed circle of triangle $PQR$. Find $ST$ in its simplest form.


Find the volume of the region given by the inequality
\[|x - y + z| + |x - y - z| + |x + y - z| + |-x + y - z| \le 6.\]


Let $b_n$ be the integer obtained by writing all the integers from 1 to $n$ from left to right. For example, $b_4 = 1234$ and $b_{13} = 12345678910111213$. Compute the remainder when $b_{21}$ is divided by $12$.


Consider the sequence where each term after the first is created by the addition of an increasing count multiplied by 2 starting from the first term being $3$ (i.e., $3, 3+2\cdot1, 3+2(1+2), \ldots$). Find the $10^{\text{th}}$ term in this sequence.

Solution: \boxed{a_{10} = 93}


Amerigo Vespucci has a new map of a fictional land on the complex plane, where distances are accurately represented. In this map, City A corresponds to $0$ in the complex plane, and City B corresponds to $3900i$. Meanwhile, City C is located at the point $1170 + 1560i$. Calculate the distance from City C to City A on this complex plane.