Suppose the probability that a baby born in a certain hospital will speak in the next day is 1/3. What is the probability that at least 3 babies out of a cluster of 7 babies will speak tomorrow?


In Mrs. Reed's English class, students are currently reading an $840$-page novel. Two friends, Mia and Leo, are sharing the reading workload. Mia reads a page in $60$ seconds, while Leo reads a page in $40$ seconds. They decide that Mia will read from page $1$ to a specific page, and Leo will read from the next page through page $840$. Determine the last page that Mia should read so that both spend equal time reading.


In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 7$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.


Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 8x + 15|$ and $y = 8 - x$.

Solution: \boxed{\frac{7 + \sqrt{21}}{2} + \frac{7 - \sqrt{21}}{2} = 7}


Solve for \(x\) in the quadratic equation \(x^2 - 5x + 12 = 2x + 60\) and find the positive difference between the solutions.


Evaluate the determinant:
\[
\begin{vmatrix} \cos(\alpha + \frac{\pi}{4})\cos(\beta + \frac{\pi}{4}) & \cos(\alpha + \frac{\pi}{4})\sin(\beta + \frac{\pi}{4}) & -\sin(\alpha + \frac{\pi}{4}) \\ -\sin(\beta + \frac{\pi}{4}) & \cos(\beta + \frac{\pi}{4}) & 0 \\ \sin(\alpha + \frac{\pi}{4})\cos(\beta + \frac{\pi}{4}) & \sin(\alpha + \frac{\pi}{4})\sin(\beta + \frac{\pi}{4}) & \cos(\alpha + \frac{\pi}{4}) \end{vmatrix}
\]


A rectangular picture frame is made from two-inch-wide pieces of wood. The area of just the frame is $36$ square inches, and one of the outer edges of the frame is $7$ inches long. What is the sum of the lengths of the four interior edges of the frame?


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


In the diagram, $ABCD$ is a square piece of paper with a side length of 8 cm. Corner $A$ is folded such that it coincides with $G$, the midpoint of $\overline{BC}$. If $\overline{HF}$ represents the crease created by the fold, with $F$ on $AB$, what is the length of $\overline{FB}$? Describe your solution using common fractions.

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


Eight students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Adam ate the most apples and Zoe ate the fewest. Calculate how many more apples Adam ate compared to Zoe and determine the total number of apples eaten by all participants.

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
for( int i = 0 ; i < 10 ; ++i ){

yaxis[i] = (0,i);

draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,9));
draw((17,9)--(17,0));

fill((1,0)--(1,3)--(2,3)--(2,0)--cycle,grey);
fill((3,0)--(3,5)--(4,5)--(4,0)--cycle,grey);
fill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);
fill((7,0)--(7,6)--(8,6)--(8,0)--cycle,grey);
fill((9,0)--(9,7)--(10,7)--(10,0)--cycle,grey);
fill((11,0)--(11,4)--(12,4)--(12,0)--cycle,grey);
fill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);
fill((15,0)--(15,5)--(16,5)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students/Participants",(8.5,0),S);
label("Results of an Apple Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Apples Eaten",(-1,4.5),W);
[/asy]


The graph of \( y = g(x) \) is depicted, with \( 1 \) unit between grid lines. Assume \( g(x) \) is defined only on the domain shown. Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.

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

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

import graph;

real i;

if(complexplane) {

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

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

} else {

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

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

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

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

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

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

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

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

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

}

if(useticks) {

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

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

} else {

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

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

}
};
rr_cartesian_axes(-6,6,-7,7);
real g(real x) {return (x-4)*(x-2)*(x+2)*(x+4)/48-2;}
draw(graph(g,-4.5,4.5,operator ..), blue);
[/asy]


Consider the sequence starting at $8820$, formed by repeatedly dividing by 5. How many integers are in this sequence?


Let \( a \), \( b \), and \( c \) be solutions of the equation \( x^3 - 6x^2 + 11x = 6 \).

Compute \( \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b} \).


Let
\[
f(x) = 3x + 4, \quad g(x) = (\sqrt{f(x)} - 3)^2, \quad h(x) = f(g(x))
\]
Determine the value of $h(3)$.

Solution: \boxed{70 - 18\sqrt{13}}


The sum of the first 1000 terms of a geometric sequence is 100. The sum of the first 2000 terms is 190. Find the sum of the first 3000 terms.


What is the greatest common factor of 90, 135, and 225?


The polynomial $x^{106} + Cx + D$ is divisible by $x^2 + x + 1$ for some real numbers $C$ and $D$. Find $C + D$.


In a chess tournament each participant played exactly once against all others. The winner of each game received $1$ point, the loser $0$ points, and each scored $\frac{1}{2}$ point if the game was a draw. At the end of the tournament, it was found that exactly half of the points each player obtained came from playing against the twelve players with the lowest scores. Notably, each of these twelve lowest scoring participants got half of their points from games among themselves. Determine the total number of participants in the tournament.


The equation $y = -16t^2 + 100t$ describes the height (in feet) of a projectile launched from the ground at 100 feet per second. At what $t$ will the projectile reach 50 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth.


When $\sqrt[4]{2^9 \cdot 5^2}$ is fully simplified, the result is $\alpha\sqrt[4]{\beta}$, where $\alpha$ and $\beta$ are positive integers. What is $\alpha + \beta$?


Let $Q$ be the plane passing through the origin with normal vector $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$. Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u}$, $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto plane $Q$.

Solution: \boxed{\begin{pmatrix} \frac{5}{7} & \frac{1}{7} & -\frac{3}{7} \\ \frac{1}{7} & \frac{13}{14} & \frac{3}{14} \\ -\frac{3}{7} & \frac{3}{14} & \frac{5}{14} \end{pmatrix}}


Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD.$ If $\angle BAC = 40^\circ,$ determine what $\angle BDC$ is.

Solution: \boxed{\angle BDC = 20^\circ}


The base $8$ representation of a positive integer is $AC$ and its base $6$ representation is $CA$. What is the integer expressed in base $10$?


A wizard is preparing a magical elixir. He has four different magical roots and six different mystical minerals to choose from. However, two of the minerals are incompatible with one of the roots, and one mineral is incompatible with two other roots. How many valid combinations can the wizard use to prepare his elixir?


A stock investment increased by 30% in the first year. In the following year, it further increased by 10%. What percent would it need to decrease in the third year to return to its original price at the beginning of the first year?


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal unit vectors, such that
\[\mathbf{a} = p (\mathbf{b} \times \mathbf{c}) + q (\mathbf{c} \times \mathbf{a}) + r (\mathbf{a} \times \mathbf{b})\] for some scalars $p,$ $q,$ and $r,$ and $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = 1.$  Find $p + q + r.$


A new license plate in a particular region consists of 4 digits and 3 letters. The digits are not necessarily distinct and the same applies to the letters. The three letters must appear consecutively. How many distinct license plates are possible?


In a cross country meet between 2 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 no ties occur among the runners, how many different winning scores are possible?


My friend Sara only likes numbers that are divisible by 8, such as 24, or 8,016. How many different ones digits are possible in numbers that Sara likes?


In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written?


Suppose the function $g(x)$ is defined on the domain $\{u_1, u_2, u_3\}$, and the graph of $y=g(x)$ consists of just three points. Suppose those three points form a triangle of area $50$ square units.

The graph of $y = 3g\left(\frac{x}{4}\right)$ also consists of just three points. What is the area of the triangle formed by these three points?


Evaluate $|\omega^2 + 4\omega + 40|$ if $\omega = 5 + 3i$.


Let $Q$ be the point on line segment $\overline{CD}$ such that $CQ:QD = 3:5.$ Assume that $\overline{CD}$ is defined in three-dimensional space with $\overrightarrow{C}$ and $\overrightarrow{D}$ given. Then
\[\overrightarrow{Q} = p \overrightarrow{C} + q \overrightarrow{D}\]
for some constants $p$ and $q.$ Determine the ordered pair $(p, q).$

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


How many two-digit numbers have a difference of exactly two between the tens digit and the ones digit?


What is the remainder when $2021 \cdot 2022 \cdot 2023 \cdot 2024 \cdot 2025$ is divided by 23?


The arithmetic mean of seven numbers is 45. If three numbers $x$, $y$, and $z$ are added to the list, the mean of the ten-member list becomes 58. What is the mean of $x$, $y$, and $z$?


How many positive integer multiples of $143$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers, and $0 \leq i < j \leq 99$?


Find the area of triangle $ABC$ given below:

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy]


Calculate $\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}^4.$

Solution: \boxed{\begin{pmatrix} -8 & 8\sqrt{3} \\ -8\sqrt{3} & -8 \end{pmatrix}}


Determine the remainder when \( (x^5 - 1)(x^3 - 1) \) is divided by \(x^2 + x + 1\).


What is the smallest possible median for the five-number set $\{x, 2x, 9, 7, 4\}$ if $x$ can be any positive integer?


A stack contains three layers of square blocks, each block having sides of $10 \text{ cm}$. The stack is arranged in a stepped pyramid fashion where the top layer contains one block, the middle layer contains two blocks, and the bottom layer contains three blocks. Find the total height, $h$, of this pyramidal stack of blocks.

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


What is the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{GH} \perp \overline{FG}$, $EF=12$ cm, $GH=7$ cm, and $FG=15$ cm?

Solution: \boxed{P = 34 + 5\sqrt{10} \text{ cm}}


Forty slips are placed into a hat, each bearing a number 1 through 8, with each number entered on five slips. Four slips are drawn from the hat at random and without replacement. Let $p'$ be the probability that all four slips bear the same number. Let $q'$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b\ne a$. What is the value of $q'/p'$?


What is the smallest natural number that can be added to 25,751 to create a palindrome?


The angles of quadrilateral $PQRS$ satisfy $\angle P = 3\angle Q = 4\angle R = 6\angle S$. What is the degree measure of $\angle P$, rounded to the nearest whole number?


Let $C$ be a point not on line $AE$, and $D$ a point on line $AE$ such that $CD \perp AE$. Also, $B$ is a point on line $CE$ such that $AB \perp CE$. Introduce a point $F$ on line $AE$ such that $BF \perp AE$. If $AB = 5$, $CD = 10$, $BF = 3$, and $AE = 6$, what is the length of $CE$?


The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, and $z$ are $10$, $6$, and $4$, respectively. What is the value of $x^2+y^2+z^2$?


Consider a sequence of positive real numbers where \( a_1, a_2, \dots \) satisfy
\[ a_n = 9a_{n-1} - n \]
for all \( n > 1 \). Find the smallest possible value of \( a_1 \).


Find the number of intersection points for the graphs of the equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]


Isabelle has $n$ candies, where $n$ is a three-digit positive integer. If she buys 7 more, she will have an amount that is divisible by 9. If she loses 9 candies, she will have an amount divisible by 7. What is the smallest possible value of $n$?


When a class of music students arranged themselves for a group photo, they noticed that when they tried to form rows of 3, there were 2 students left over. When they attempted to form rows of 5, there were 3 students left over, and when they organized themselves in rows of 8, they had 5 students left over. What is the smallest possible number of students in the class?


Solution: \boxed{\frac{7 \times 3437}{23 \times 3439}}


Given that $$(x+y+z)(xy+xz+yz)=24$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)+xyz=15$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Determine the area of the circle described by the equation \(3x^2 + 3y^2 - 15x + 9y + 27 = 0\) in terms of \(\pi\).


The nine horizontal and nine vertical lines on a $6\times6$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


It takes Jana 30 minutes to walk one mile at a steady pace. However, today she starts walking at half her usual pace for the first half of her journey (15 minutes) and doubles her usual pace for the second half (15 minutes). How far will she travel in the first 20 minutes of her walk today?

Solution: \boxed{\frac{7}{12} \text{ miles}}


In the diagram provided, three semicircles are drawn on the straight line segment $AC$ of length 6 units, where $B$ is the midpoint of $AC$. Arc $ADB$ and arc $BEC$ have a radius of 2 units each. Point $D$, point $E$, and point $F$ are the midpoints of arc $ADB$, arc $BEC$, and arc $DFE$ respectively. If arc $DFE$ is also a semicircle with a radius of 1 unit, and a square $DPEF$ inscribes within arc $DFE$ such that $P$ lies at the intersection of lines $DE$ and $BF$, calculate the area of the square $DPEF$.

[asy]
unitsize(0.5inch);
path t=(2,2)..(3,1)--(1,1)..cycle;
draw(t);
path r=shift((3,1))*t;
path s=shift((2,2))*t;
draw(s);
fill(s,gray(0.7));
fill((2,1)--(2,2)--(4,2)--(4,1)--cycle,gray(0.7));
fill(t,white);
fill(r,white);
draw(t);
draw(r);
dot((1,1));
dot((2,2));
dot((3,3));
dot((4,2));
dot((3,1));
dot((5,1));
label("$A$",(1,1),W);
label("$B$",(3,1),S);
label("$C$",(5,1),E);
label("$D$",(2,2),NW);
label("$E$",(4,2),NE);
label("$F$",(3,3),N);
[/asy]


Tom flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that he flips exactly 4 heads?


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

Find the number of possible values of $f(3)$, denote it by $n$. Let $s$ be the sum of all possible values of $f(3)$. Compute $n \times s.$


Given that
\[\frac{\sin^4 \beta}{\sin^2 \alpha} + \frac{\cos^4 \beta}{\cos^2 \alpha} = 1,\]
find the sum of all possible values of
\[\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta}.\]


In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF = 2$ and $GC = 1.5$. Lines $AF$ and $BG$ intersect at $E$. Calculate the area of $\triangle AEB$.


Factor $x^2 - 6x + 9 - 49x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a<d$.


Calculate the result of $655_6 - 222_6 + 111_6$ in base-6.


If $w$ is a complex number such that
\[
  w + w^{-1} = -\sqrt{3},
\]
what is the value of
\[
  w^{2011} + w^{-2011} \, ?
\]


Calculate the average value of the series $4z$, $6z$, $9z$, $13.5z$, and $20.25z$.


A rectangular prism box has dimensions length $x+3$, width $x-3$, and height $x^2+9$. Find the number of positive integer values of $x$ for which the volume of the box is less than 500 units.


How many even integers between 5000 and 9000 have all unique digits?


Calculate the value closest to a whole number for the ratio $\frac{10^{3000} + 10^{3003}}{10^{3001} + 10^{3002}}$.


A 12-foot by 15-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total area of the floor that is shaded.

Solution: \boxed{180 - 45\pi \text{ square feet}}


Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b) = 120$ and $\gcd(a,c) = 1001$. If $b = 120x$ and $c = 1001y$ for integers $x$ and $y$, what is the smallest possible value of $\gcd(b,c)$?


Circle $C$ has a circumference of $18\pi$ meters, and segment $AB$ is a diameter. If the measure of angle $CAB$ is $60^{\circ}$, what is the length, in meters, of segment $AC$?


Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?


Given that $b$ is an even multiple of $1177$, find the greatest common divisor of $3b^2 + 34b + 76$ and $b + 14$.


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

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


One line is parameterized by
\[\begin{pmatrix} -2 + s \\ 4 - ks \\ 2 + ks \end{pmatrix}.\]
Another line is parameterized by
\[\begin{pmatrix} t/3 \\ 2 + t \\ 3 - t \end{pmatrix}.\]
Determine the value of \(k\) for which the lines are coplanar.


A triangle is made of wood sticks of lengths 12, 25, and 30 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the length of the smallest piece that can be cut from each of the three sticks to make this happen?


Andrew's grandfather is fifteen times older than Andrew. If Andrew's grandfather was 60 years old when Andrew was born, how old is Andrew now?


Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.).


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


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.


Let \(A\) and \(B\) be two points on the parabola \(y = 4x^2\), such that the tangents at \(A\) and \(B\) intersect at an angle of \(45^\circ\). Determine the \(y\)-coordinate of their point of intersection, \(P\).


Let $a, b, t,$ and $p$ be real numbers such that $a + b = t$ and $ab = p.$ Find the minimum value of $a^2 + ab + b^2$ given these conditions.


Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+15|$ and $y=8-x$.


What is the remainder when $1582 \cdot 2031$ is divided by $600$?


The Grunters play the Screamers 5 times. The Grunters are the much better team, and are $60\%$ likely to win any given game. What is the probability that the Grunters will win at least 4 games? Express your answer as a common fraction.


In how many different ways can five students stand in a straight line if two of the students refuse to stand next to each other, and one student must always stand at either end of the line?


In a geometric diagram, a regular hexagon has an equilateral triangle inside it, sharing one side with the hexagon. Determine the degree measure of $\angle XYZ$, where $X$ is a vertex of the hexagon, $Y$ is a shared vertex between the triangle and hexagon, and $Z$ is the next vertex of the triangle clockwise from $Y$.


Alan, Maria, and Nick are going on a bicycle ride. Alan rides at a rate of 6 miles per hour. If Maria rides $\frac{3}{4}$ as fast as Alan, and Nick rides $\frac{4}{3}$ as fast as Maria, how fast does Nick ride?

Solution: \boxed{6 \text{ miles per hour}}


The product of two positive integers plus their sum equals 131. The integers are relatively prime, and each is less than 25. What is the sum of these two integers?


What is the eighth term in the arithmetic sequence $\frac{1}{2}, \frac{4}{3}, \frac{7}{6}, \dots$?


The Screamers are coached by Coach Yellsalot. The team has expanded to 15 players, including Bob, Yogi, and Moe. No starting lineup of 5 players can contain any two among Bob, Yogi, and Moe. How many possible starting lineups can Coach Yellsalot assemble under these new conditions?


Find all positive integer values of $k$ for which the equation ${k}x^2 + 24x + 4k = 0$ has rational solutions.


**
Triangle $DEF$ has vertices $D(0, 10)$, $E(4, 0)$, and $F(10, 0)$. A line through $E$ cuts the area of $\triangle DEF$ in half. Find the sum of the slope and $y$-intercept of this line.

**


If $y < 0$, find the range of all possible values of $y$ such that $\lceil y \rceil \cdot \lfloor y \rfloor = 132$. Express your answer using interval notation.

Solution: \boxed{(-12, -11)}


The base of a triangular sheet of paper $ABC$ is $15 \text{ cm}$ long. The triangle is folded over its base, with the crease $DE$ parallel to the base, resulting in a portion projecting below the base. The area of this projected triangle is $25\%$ of the area of the original triangle $ABC$. Determine the length of $DE$.

Solution: \boxed{7.5 \text{ cm}}


On the Cartesian plane where each unit is one meter, a dog is tied to a post at the point $(5,5)$ with a $12$ meter rope. How far can the dog move from the origin?

Solution: \boxed{12 + 5\sqrt{2} \text{ meters}}


Find the least positive integer such that when its last digit is deleted, the resulting integer is 1/17 of the original integer.


The number \( m \) is a four-digit positive integer and is the product of the three distinct prime factors \( x \), \( y \), and \( 10x + y \), where \( x \) and \( y \) are prime numbers less than 10, and \( x > y \). What is the largest possible value of \( m \)?


Clark writes down all the six-digit numbers that contain each of the digits 1, 2, 3, 4, 5, and 6 exactly once. What is the smallest number in Clark's list that is divisible by 6?


Find the area contained by the graph of
\[|x + y| + |x - y| \le 6.\]


If $a$ is a multiple of $2142$, find the greatest common divisor of $a^2+11a+28$ and $a+6$.


An ellipse has foci at $(1, 1)$ and $(7, 1)$, and it passes through the point $(0, 8)$. Given this, we can write the equation of the ellipse in standard form as
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,
\]
where \(a, b, h, k\) are constants, and \(a\) and \(b\) are positive. Find the ordered quadruple \((a, b, h, k)\).

Solution: \boxed{(6\sqrt{2}, 3\sqrt{7}, 4, 1)}


A, B, C, D, and E are collinear in that order such that \( AB = BC = 2, CD = 3, \) and \( DE = 5 \). If P can be any point in space, what is the smallest possible value of \( AP^2 + BP^2 + CP^2 + DP^2 + EP^2 \)?


For what base is the representation of $157_{10}$ a four-digit number whose final digit is odd?


Cynthia has deposited $\$9,\!000$ into an account that pays $5\%$ interest compounded annually.

David has deposited $\$12,\!000$ into an account that pays $4\%$ simple annual interest.

After 25 years, Cynthia and David compare their respective balances. To the nearest dollar, what is the positive difference between their balances?


Let $k$ and $m$ be real numbers, and suppose that the roots of the equation \[x^3 - 9x^2 + kx - m = 0\] are three distinct positive integers. Compute $k + m.$


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 must be a girl, the vice-president must be a boy, and the secretary must be of different gender from the vice-president?


In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 6$, and $\angle ABO = \text{arc } CD = 45^\circ$. Find the length of $BC$.


In the given 4x4 square board, some squares are divided into triangles. Specifically, the top left 2x2 region is composed of four unit squares, each diagonally divided into two triangles, and the bottom right 2x2 region is divided the same way. One half of each divided square in the top left and bottom right squares is shaded. Calculate the fraction of the entire board that is shaded.


John ran a total of $1732_7$ miles. Convert this distance from base seven to base ten.


Find the minimum value of \[\frac{4x^2}{x - 10}\] for \(x > 10\) where \(x\) is an integer.


In a pentagon $FGHIJ$, two interior angles $F$ and $G$ are $90^\circ$ and $70^\circ$, respectively. Angles $H$ and $I$ are equal, and the fifth angle $J$ is $20^\circ$ more than twice $H$. Find the measure of the largest angle.


A circle is tangent to the lines $5x - 2y = 20$ and $5x - 2y = -40.$ The center of the circle lies on the line $3x + y = 0.$ Find the center of the circle.

Solution: \boxed{\left(-\frac{10}{11}, \frac{30}{11}\right)}


For all real numbers \(x\) except \(x = 0\) and \(x = 2\), the function \(g(x)\) is defined by
\[g\left(\frac{x}{x - 2}\right) = \frac{1}{x}.\]
Suppose \(0 \leq t \leq \frac{\pi}{2}\). What is the value of \(g(\csc^2 t)\)?

Solution: \boxed{\frac{\cos^2 t}{2}}


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


Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-4x+3|$ and $y=\frac{25}{4}-x$.


In the given diagram, lines \( p \) and \( q \) are parallel, and the transversal \( r \) intersects them. If angle \( a \) formed by \( r \) and \( q \) is \( 45^\circ \) and the angle \( b \) immediately adjacent to it on the same side of the transversal is \( 120^\circ \), find the measure of angle \( x \) which is formed on the inside of lines \( p \) and \( q \) and opposite to \( b \) across line \( p \). Assume all intersections form as shown in the following diagram:

```text
|p
|
|             / r
|            /
|           /
|          /
|         /b
|--------x---------
|q      /a

```


What is the smallest positive integer $n$ such that $5n \equiv 1846 \pmod{26}?$


The function $f$ satisfies
\[
f(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3
\]
for all real numbers $x, y$. Determine the value of $f(10)$.


Let \( A \) equal the number of four-digit odd numbers divisible by 3, and \( B \) equal the number of four-digit multiples of 7. Find \( A+B \).


The graph of $y = f(x)$ is described as follows:
1. From $x = -3$ to $x = 0$, $y = -2 - x$.
2. From $x = 0$ to $x = 2$, $y = \sqrt{4 - (x - 2)^2} - 2$.
3. From $x = 2$ to $x = 3$, $y = 2(x - 2)$.

Consider the graph of $y = g(x)$ obtained by reflecting $f(x)$ across the $y$-axis and shifting it 4 units to the right and 3 units up. Define $g(x)$ in terms of $f(x)$.

Solution: \boxed{g(x) = f(4 - x) + 3}


How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct?


Calculate the product of $0.\overline{08}$ and $0.\overline{36}$ and express it as a common fraction.


A hexagon is constructed by connecting the points (0,0), (1,4), (3,4), (4,0), (3,-4), and (1,-4) on a coordinate plane. Calculate the area of this hexagon.


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


A puppy, two cats, and a rabbit together weigh 40 pounds. The sum of the squares of the weights of the puppy and the larger cat is four times the weight of the smaller cat, and the sum of the squares of the weights of the puppy and the smaller cat equals the square of the weight of the larger cat. How much does the puppy weigh?


Suppose that a real number $y$ satisfies \[\sqrt{64-y^2}-\sqrt{36-y^2}=4.\] What is the value of $\sqrt{64-y^2}+\sqrt{36-y^2}$?


A ball bounces back up $\frac{3}{4}$ of the height from which it falls. If the ball is dropped from a height of $256$ cm, after how many bounces does the ball first rise less than $30$ cm?


The side of a square has the length $(x-4)$, while a rectangle has a length of $(x-5)$ and a width of $(x+6)$. If the area of the rectangle is three times the area of the square, what is the sum of the possible values of $x$?


The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $2:3:4$ that can be expressed in the form \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Find the sum of all possible values of $n+k$.


Let $f(x) = \displaystyle \frac{1}{2ax + 3b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = -1$. Express your answer in terms of $a$ and/or $b$.

Solution: \boxed{x = \frac{1}{-2a + 3b}}


Convert $312_{10}$ to base 3. Let $x$ be the number of zeros, $y$ be the number of ones, and $z$ be the number of twos in base 3. What is the value of $z-y+x$?


Suppose in a right triangle \(PQR\) with \(\angle PQR = 90^\circ\), it is given that \(\cos Q = \frac{3}{5}\). What is the length of \(PR\) if \(QR = 3\)?


Calculate the mean height of the players on Oak Ridge High School boys' basketball team presented in a modified stem and leaf plot. (Note: “6|2” represents 62 inches.)

Height of the Players on the Basketball Team (inches)

$5|8\; 9$

$6|0\;1\;2\;3\;5\;5\;8$

$7|0\;1\;4\;6\;8\;9$

$8|1\;3\;5\;6$

Solution: \boxed{70.74 \text{ inches}}


Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all five slips bear the same number. Let $q$ be the probability that three slips bear a number $a$ and the other two bear a number $b\ne a$. What is the value of $q/p$?


A cake is cut into thirds and stored. On each trip to the pantry, you eat half of the current cake portion left. How much of the cake is left after 4 trips?


Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\).


If $c$ and $d$ are integers with $c^2 > d^2$, what is the smallest possible positive value of $\frac{c^2+d^2}{c^2-d^2} + \frac{c^2-d^2}{c^2+d^2}$?


What is the sum of all the solutions of the equation $x = \left|2x - |100 - 2x|\right|$?


Consider a 10x10 chessboard with 100 squares. A person labels each square from $1$ to $100$. On each square $k$, they place $3^k$ grains of rice. Calculate how many more grains of rice are placed on the $15^{th}$ square than on the first $10$ squares combined.


Let $f(x) = 18x + 4$. Find the product of all $x$ that satisfy the equation $f^{-1}(x) = f((2x)^{-1})$.


In triangle $XYZ$, angle $XZY$ is 60 degrees, and angle $YZX$ is 80 degrees. Let $D$ be the foot of the perpendicular from $X$ to $YZ$, $O$ the center of the circle circumscribed about triangle $XYZ$, and $E$ the other end of the diameter which goes through $X$. Find the angle $DXE$, in degrees.


If $m$ and $n$ are positive integers such that $\gcd(m, n) = 15$, what is the smallest possible value of $\gcd(14m, 20n)$?


The quadratic $4x^2 - 40x + 100$ can be written in the form $(ax+b)^2 + c$, where $a$, $b$, and $c$ are constants. What is $2b-3c$?


Let \( x, y, \) and \( z \) be nonnegative numbers such that \( x^2 + y^2 + z^2 = 1 \). Find the maximum value of
\[ 2xy \sqrt{2} + 6yz. \]


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


Find the product of all positive integral values of $m$ such that $m^2 - 40m + 399 = q$ for some prime number $q$. Note that there is at least one such $m$.


Compute the value of:
\[
\left\lfloor \frac{2021! + 2018!}{2020! + 2019!} \right\rfloor.
\]


How many of the first $30$ rows in Pascal's triangle, excluding row $0$ and row $1$, consist of numbers that are exclusively even, not counting the $1$ at each end of the row?


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


Find the point in the plane $2x + 3y - 6z = 18$ that is closest to the point $(2, -1, 1)$.

Solution: \boxed{\left(\frac{144}{49}, \frac{20}{49}, -\frac{89}{49}\right)}


Let \( f(x) = 3x - 2 \). Find the sum of all \( x \) that satisfy the equation \( f^{-1}(x) = f(x^{-2}) \).


The square quilt block shown is made from nine unit squares, two of which are divided in half diagonally to form triangles, and one square is divided into four smaller equal squares, one of which is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction.

[asy]size(75);
fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray(.6)); // shading one small square
fill((1,1)--(1,2)--(2,2)--cycle,gray(.6));  // shading the triangle
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,gray(.6)); // shading the square
fill((1,2)--(0,2)--(1,3)--cycle,gray(.6)); // shading the triangle
draw(scale(3)*unitsquare);
for (int i = 1; i <= 2; ++i) {
    draw((i,0)--(i,3));
    draw((0,i)--(3,i));
}
draw((0.5,2.5)--(1,3)--(1,2.5)--(0.5,2)--cycle);
[/asy]


As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $15n+4$ and $9n+2$?


Triangle $ABC$ has a perimeter of 3010 units. The sides have lengths that are all integer values with $AB < AC \leq BC$. What is the smallest possible value of $AC - AB$?


Compute the sum of the squares of the roots of the equation \[x^{10} + 3x^7 + 5x^2 + 404 = 0.\]


Find $q(x)$ if the graph of $\frac{x^3-2x^2-5x+6}{q(x)}$ has vertical asymptotes at $2$, $-2$, and $1$, and no horizontal asymptote, and $q(4) = 24$.

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


Find the number of real solutions to the equation
\[\frac{x}{50} = \cos x.\]


Define a $\textit{good word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$, where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?


Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt{144}$ and $\angle BAC = 45^\circ,$ then compute the area of triangle $ABC.$


Suppose Alice and Bob played a game where they both picked a complex number. If the product of their numbers was \(48 - 16i\), and Alice picked \(7 + 4i\), what number did Bob pick?

Solution: \boxed{\frac{272}{65} - \frac{304}{65}i}


Suppose the function $g$ is graphed below, where 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;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

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

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

} else {

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

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

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

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

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

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

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

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

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

}

if(useticks) {

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

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

} else {

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

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

}
};
rr_cartesian_axes(-1,9,-1,9);

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

[/asy]

Samantha writes the number 2 on her thumb. She then applies $g$ to 2 and writes the output on her index finger. If Samantha continues this process of applying $g$ and writing the output on a new finger, what number will Samantha write on her tenth finger?


Let \( f(x) = \frac{1}{ax^2 + bx + c} \), where \(a\), \(b\), and \(c\) are nonzero constants. Find all solutions to \( f^{-1}(x) = 0 \). Express your answer in terms of \(a\), \(b\), and \(c\).

Solution: \boxed{x = \frac{1}{c}}


Given that \( b \) is an even multiple of \( 5959 \), find the greatest common divisor of \( 4b^2 + 73b + 156 \) and \( 4b + 15 \).


A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Find the probability that the bug moves to its starting vertex on its eighth move, expressed as a reduced fraction.


Below is the graph of $y = 3 \csc(2x - \pi)$ for some positive constants. Find the new value of **a** in this adjusted equation considering the graph's behavior:

[asy]
import TrigMacros;
size(500);

real h(real x) {
    return 3*csc(2*x - pi);
}

draw(graph(h, -pi + 0.1, pi - 0.1),blue+linewidth(1));
limits((-pi,-7),(pi,7),Crop);
draw((-pi/4,-7)--(-pi/4,7),dashed);
draw((pi/4,-7)--(pi/4,7),dashed);
trig_axes(-pi,pi,-7,7,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);

label("$3$", (0,3), E);
label("$-3$", (0,-3), E);
[/asy]


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 product of the numbers on the marbles?


In a chess tournament, each player played exactly one game against each of the other players. A win granted the winner 1 point, the loser 0 points, and each player 1/2 point if there was a tie. It was found that precisely half of the points each player earned came against the twelve players with the lowest number of points. What was the total number of players in the tournament?


Find all real numbers \( b \) such that the equation
\[ x^3 - 2bx^2 + bx + b^2 - 2 = 0 \]
has exactly one real solution in \( x \).

Solution: \boxed{0 \text{ and } 2}


A book with 73 pages numbered 1 to 73 has its pages renumbered in reverse, from 73 to 1. How many pages do the new page number and old page number share the same units digit?


Determine the smallest possible value of $xy$ for positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{6}$.


Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, $32$ of which are triangular and $12$ of which are hexagonal. Determine the number of space diagonals of $Q$.


A point $P$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\triangle DFP$ has a greater area than each of $\triangle DEP$ and $\triangle EFP$?


Consider two lines parameterized as follows:
\[\begin{pmatrix} -2 + s \\ 4 - ks \\ 2 + ks \end{pmatrix},\]
and
\[\begin{pmatrix} 2t \\ 2 + 2t \\ 3 - 2t \end{pmatrix}.\]
Determine the value of $k$ if the lines are coplanar.


Twelve 6-sided dice are rolled. What is the probability that exactly five of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.


In a class of 150 students, all students took the same test. What is the maximum number of students who can receive a grade above the class average?


Find the quotient when \(x^6 - 8\) is divided by \(x - 2.\)

Solution: \boxed{x^5 + 2x^4 + 4x^3 + 8x^2 + 16x + 32}


If $\mathbf{B} = \begin{pmatrix} x & y \\ z & w \end{pmatrix},$ then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} x & z \\ y & w \end{pmatrix}.\]Given that $\mathbf{B}^T = 2\mathbf{B}^{-1},$ find $x^2 + y^2 + z^2 + w^2.$


**New Problem Statement**: The lengths of two sides of a triangle are 35 units and 43.5 units. If the third side also has an integral length, what is the least possible number of units in the perimeter of the triangle?


Let $D$ be the circle with equation $x^2 + 16y + 81 = -y^2 - 12x$. Determine the center $(c,d)$ and radius $s$ of $D$, and find the value of $c+d+s$.

Solution: \boxed{-14 + \sqrt{19}}


A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 3 that can be expressed as a 3-digit palindrome in a different base? Provide your response in base 3.


Each letter represents a non-zero digit. Determine the value of \( t \) given the following equations:
\begin{align*}
a + b &= x \\
x + d &= t \\
t + a &= y \\
b + d + y &= 16
\end{align*}


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


Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 10 inches, and $EG = 15$ inches. The length of segment $AB$ is equal to one-third the length of segment $AD$. What is the area of rectangle $ABCD$?


In trapezoid $EFGH$, the sides $EF$ and $GH$ are both equal and parallel. The distances between the ends of the parallel sides are 5 units and $EG = 20$ units. If the vertical height from $F$ to line $GH$ is 6 units, compute the perimeter of trapezoid $EFGH$.
```plaintext
[asy]
pen p = linetype("4 4");
draw((0,0)--(5,6)--(20,6)--(25,0)--cycle);
draw((5,0)--(5,6), p);
draw((4.5,0)--(4.5, .5)--(5.0,0.5));

label(scale(0.75)*"E", (0,0), W);
label(scale(0.75)*"F", (5,6), NW);
label(scale(0.75)*"G", (20, 6), NE);
label(scale(0.75)*"H", (25, 0), E);
label(scale(0.75)*"15", (12.5,6), N);
label(scale(0.75)*"25", (12.5,0), S);
label(scale(0.75)*"6", (5, 3), E);
[/asy]
```

Solution: \boxed{40 + 2\sqrt{61}}


How many positive integers \(n\) satisfy \[\dfrac{n+900}{80} = \lfloor \sqrt{n} \rfloor?\]


A triangle forms from sticks of lengths 7, 24, and 26 inches joined end-to-end. If you either add or subtract the same integral length $x$ to/from each of the sticks, find the smallest value of $x$ such that the three resulting lengths can no longer form a triangle.


Given that $a$ is an even multiple of $947$, find the greatest common divisor (GCD) of $3a^2 + 47a + 101$ and $a + 19$.


Triangle $PQR$ has vertices $P(-3, 9)$, $Q(4, -2)$, and $R(10, -2)$. A line through $Q$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.


Given the function \( y = f(x) \) defined as follows:
- \( f(x) = -x - 1 \) for \( -4 \leq x \leq 0 \)
- \( f(x) = \sqrt{9 - (x - 3)^2} - 3 \) for \( 0 \leq x \leq 6 \)
- \( f(x) = 3(x - 6) \) for \( 6 \leq x \leq 7 \)

The graph of \( y = h(x) \) is a transformation of \( y = f(x) \). Determine \( h(x) \) in terms of \( f(x) \) if \( h(x) \) is \( f(x) \) reflected across the y-axis and then shifted 5 units to the right.

Solution: \boxed{h(x) = f(5 - x)}


Let $t$ be a parameter that varies over all real numbers. Any parabola of the form
\[ y = 4x^2 + tx - t^2 - 3t \]
passes through a fixed point. Find this fixed point.

Solution: \boxed{(3, 36)}


Allen and Christie each arrive at a conference at a random time between 1:00 and 2:00. Each stays for 20 minutes, then leaves. What is the probability that Allen and Christie see each other at the conference?


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


The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{12} \le 2005$ such that $a_i - i$ is odd for $1 \le i \le 12$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000.


Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 600, what is the sum of $a$ and $b$?


A school estimates its student population to be between $1000$ and $1200$ students. Each student studies either German or Russian, and some study both. The number studying German is between $70\%$ and $75\%$ of the student population, while those studying Russian are between $35\%$ and $45\%$. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Calculate $M - m$.


Let \( x, y, z \) be complex numbers such that:
\[
xy + 3y = -9,
yz + 3z = -9,
zx + 3x = -9.
\]
Determine all possible values of \( xyz \).


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


Let \( a, b, c \) be real numbers such that \( 2a + 3b + c = 6 \). Find the maximum value of
\[ ab + ac + bc. \]


How many rows of Pascal's Triangle contain the number $41$?


Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.


Triangle $PQR$ has vertices $P(0, 10)$, $Q(2, 0)$, $R(10, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{PQ}$ at $V$ and line segment $\overline{PR}$ at $W$, forming $\triangle PVW$ with area 20. Compute $s$.

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


Both \(a\) and \(b\) are positive integers and \(b > 2\). When \(a^b\) is the greatest possible value less than 500, what is the sum of \(a\) and \(b\)?


In a circle with center $O$, $AD$ is a diameter, $PQR$ is a chord, $AP = 8$, and $\angle PAO = \text{arc } QR = 45^\circ$. Find the length of $PQ$.


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3\cdot5^5}$?


Find the sum of the coefficients of the $x^3$ and $x^5$ terms in the expansion of $(x+1)^{50}$.


In an isosceles triangle $ABC$ with $AB=AC$, point $D$ is the midpoint of $\overline{AC}$ and $\overline{BE}$, where $\overline{BE}$ is 13 units long. Determine the length of $\overline{BD}$. Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--(3,13^.5)--(6,0)--cycle);
draw((0,0)--(-3,13^.5)--(6,0));
label("$A$", (6,0), SE);
label("$B$", (0,0), SW);
label("$C$", (3,13^.5), N);
label("$D$", (1.5,6.5), N);
label("$E$", (-3,13^.5), NW);
[/asy]


Robin decided to trek along Alpine Ridge Trail, which she completed in five days. The first three days she hiked a total of 30 miles. The second and fourth days she averaged 15 miles per day. On the last two days, she covered a total of 28 miles. The total distance she traveled on the first and fourth days was combined to 34 miles. How long in total was the Alpine Ridge Trail?

Solution: \boxed{58 \text{ miles}}


Suppose $C$ is a point not on line $AF$, and $D$ is a point on line $AF$ such that $CD \perp AF$. Additionally, let $B$ be a point on line $CF$ but not midpoint such that $AB \perp CF$. Given $AB = 6$, $CD = 9$, and $AF = 15$, determine the length of $CF$.


Let $a,$ $b,$ and $c$ be constants, and suppose the inequality \[\frac{(x-a)(x-b)}{x-c} \geq 0\] is true if and only if either $x < -6$ or $20 \leq x \leq 23.$ Given that $a < b,$ find the value of $a + 2b + 3c.$


Let $\mathbf{u}$, $\mathbf{v}_1$, and $\mathbf{v}_2$ be unit vectors, all mutually orthogonal. Let $\mathbf{w}$ be a vector such that $\mathbf{u} \times (\mathbf{v}_1 + \mathbf{v}_2) + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}_1 + \mathbf{v}_2$. Compute $\mathbf{u} \cdot (\mathbf{v}_1 + \mathbf{v}_2 \times \mathbf{w})$.


Consider the equation
\[
(x - \sqrt[3]{23})(x - \sqrt[3]{63})(x - \sqrt[3]{113}) = \frac{1}{3}
\]
which has three distinct solutions \( r, s, \) and \( t \). Calculate the value of \( r^3 + s^3 + t^3 \).


Let $c$ and $d$ be nonzero complex numbers such that $c^2 - cd + d^2 = 0$. Evaluate
\[
\frac{c^6 + d^6}{(c - d)^6}.
\]


Ella is attempting to calculate the sum of distances from a point $Q = (3,3)$ to the vertices of $\triangle DEF$, where $D$ is at the origin, $E$ is at $(7,2)$, and $F$ is at $(4,5)$. Solve for the sum of the distances from point $Q$ to each vertex of the triangle $\triangle DEF$, and write your answer in the form $p + q\sqrt{r}$, where $p$ and $q$ are integers. What is $p + q$?


Find the angle $\theta$ in degrees for the expression
\[
\text{cis } 55^\circ + \text{cis } 65^\circ + \text{cis } 75^\circ + \dots + \text{cis } 145^\circ
\]
expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$.


Given dots spaced one unit apart, both horizontally and vertically, find the number of square units enclosed by the following polygon:
[asy]
size(1inch,1inch);
pair a=(0,0), b=(10,0), c=(30,0), d=(40,0);
pair e=(0,10), f=(10,10), g=(30,10), h=(40,10);
pair i=(0,20), j=(10,20), k=(30,20), l=(40,20);
pair m=(0,30), n=(10,30), o=(30,30), p=(40,30);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
dot(g);
dot(h);
dot(i);
dot(j);
dot(k);
dot(l);
dot(m);
dot(n);
dot(o);
dot(p);
draw(a--b--f--j--n--m--i--e--a);
[/asy]


The integer $z$ has 18 positive factors. The numbers 16 and 18 are factors of $z$. What is $z$?

Solution: \boxed{2^5 \cdot 3^2 = 288}


Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$.


The number $6\,21H\,408\,3G5$ is divisible by $6$. If $H$ and $G$ each represent a single digit, what is the sum of all distinct possible values of the product $HG$?


How many ways are there to distribute 4 distinguishable balls into 3 indistinguishable boxes?


The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers?


A point $(x, y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 5$. What is the probability that $x + y \leq 5$? Express your answer as a common fraction.


Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $c,a,b$ is a geometric progression. What is the smallest possible value of $c$?


A region \( S \) in the complex plane is defined by
\[
S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\}.
\]
A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). What is the probability that \( \left(\frac{1}{2} + \frac{1}{2}i\right)z \) is also in \( S \)?


For certain real values of $a, b, c,$ and $d_{}$, the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$ has four non-real roots. The sum of two of these roots is $5 + 2i$ and the product of the other two roots is $10 - i$, where $i^2 = -1$. Find $b$.


Simplify the following expression:
\[(2 + 2\cot (A+\alpha) - 2\csc (A+\alpha))(2 + 2\tan (A+\alpha) + 2\sec (A+\alpha)).\]


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 $7$?


A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?


For real numbers $x,$ $y,$ and $z,$ consider the matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
Determine if the matrix is invertible and, if not, list all possible values of
\[\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y}.\]


The graph of the function \( g(x) \) is defined on the interval from \( -3 \) to \( 5 \), inclusive, on \( x \). Let \( g(x) \) be defined as follows: \( g(x) = -\frac{1}{2}x^2 + x + 3 \) when \( -3 \leq x \leq 1 \), and \( g(x) = \frac{1}{3}x^2 - 3x + 11 \) when \( 1 < x \leq 5 \). Determine how many values of \( x \) satisfy \( g(g(x)) = 3 \).


Find all values of $x$ such that $0 \le x < 2\pi$ and $\sin x - \cos x = \sqrt{2}$.


A solid rectangular block is created by gluing together \(N\) 1-cm cube units. When this block is situated such that three faces are visible, \(462\) of the 1-cm cubes cannot be seen. Determine the smallest possible value of \(N\).


Solution: \boxed{16 \sqrt[3]{4}}


For how many integer Fahrenheit temperatures between 32 and 2000 inclusive does the original temperature equal the final temperature when only temperatures where $F-32 \equiv 2 \pmod{9}$ are converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer?


Given the volume of the parallelepiped determined by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6, find the volume of the parallelepiped determined by the vectors $\mathbf{a} + 2\mathbf{b}$, $\mathbf{b} + 4\mathbf{c}$, and $\mathbf{c} - 3\mathbf{a}$.


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

1. $0 < x^2 < 4$
2. $x^2 > 6$
3. $-2 < x < 0$
4. $0 < x < 2$
5. $-2 < x - x^2 < 3$

What is the maximum number of these statements that can be true for any value of $x$?


Find the maximum \( y \)-coordinate of a point on the graph of \( r = \sin 3\theta \).


On a 9x9 chessboard, each square is labeled with the reciprocal of the sum of its row and column indices, given by $\frac{1}{r+c-1}$, where $r$ and $c$ are the row and column indices starting from 1. Nine squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of the nine chosen squares.


How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?


Alice throws five identical darts. Each hits one of four identical dartboards on the wall. After throwing the five darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible?


A regular dodecagon $Q_1 Q_2 \dotsb Q_{12}$ is drawn in the coordinate plane with $Q_1$ at $(1,0)$ and $Q_7$ at $(3,0)$. If $Q_n$ represents the point $(x_n,y_n)$, compute the numerical value of the product:
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).\]


Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac{1}{z} = 6,$ and $y + \frac{1}{x} = 30.$ Find $z + \frac{1}{y}.$


Determine the intersection points and the ratio of the segments formed when the curve $y = \cos x$ intersects the line $y = \cos 60^\circ$. Assume the successive ratios $\dots p : q : p : q \dots$ with $p < q$. Compute the ordered pair of relatively prime positive integers $(p, q)$.

Solution: \boxed{(1, 2)}


Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2 + 5a + 108$ and $3a + 9$.


A cylinder has a height that is increased by $2$ units, resulting in a $72\pi$ cubic units increase in volume. Likewise, increasing the radius by $2$ units also increases the volume by $72\pi$ cubic units. If the original radius is $3$, find the original height.


Elmo makes $N$ sandwiches for a school event. For each sandwich he uses $C$ slices of cheese at $3$ cents per slice and $T$ slices of tomato at $4$ cents per slice. The total cost of the cheese and tomato to make all the sandwiches is $\$3.05$. Assume that $C$, $T$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the tomatoes Elmo uses to make the sandwiches?


How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 4, 5, 5, 5, 6, 6\}$ if no digit may be used more times than it appears in the given set of available digits?


The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?


Let $a$ and $b$ be angles such that $\cos a + \cos b = 1$ and $\sin a + \sin b = \frac{1}{2}$. Additionally, assume that $\tan(a - b) = 1$. Find \[\tan \left( \frac{a + b}{2} \right).\]


A truck travels due west at $\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truck. Determine the average time $\frac{1}{2}(t_1 + t_2)$ during which the truck is within the storm circle, where $t_1$ is the time the truck enters and $t_2$ is the time the truck exits the storm circle.


Let $p$ and $q$ be positive integers such that \[\frac{6}{11} < \frac{p}{q} < \frac{5}{9}\] and $q$ is as small as possible. What is $q-p$?


Mark had a box of chocolates. He consumed $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to his friend Lucy. Mark and his father then each ate 20 chocolates from what Mark had left. Finally, Mark's sister took between five and ten chocolates, leaving Mark with four chocolates. How many chocolates did Mark start with?


On a 24-hour clock, an elapsed time of five hours looks the same as an elapsed time of $25$ hours. Because of this, we can say that five hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $5$ hours and is "clock equivalent'' to its square number of hours?


What is the 24th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{13}$ and $\frac{1}{8}$?


In a similar geoboard setup, points are evenly spaced vertically and horizontally. Segment $DE$ is drawn using two points, as shown. Point $F$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $DEF$ being isosceles if $DE$ is three units long? [asy]
draw((0,0)--(0,6)--(6,6)--(6,0)--cycle,linewidth(1));
for(int i=1;i<6;++i)
{for(int j=1;j<6;++j)
{dot((i,j));}
}
draw((2,2)--(5,2),linewidth(1));
label("D",(2,2),SW);
label("E",(5,2),SE);
[/asy]


Both roots of the quadratic equation $x^2 - 65x + k = 0$ are prime numbers. Find the number of possible values of $k.$


If \(n\) is the smallest positive integer for which there exist positive real numbers \(a\) and \(b\) such that
\[(a + bi)^n = 2(a - bi)^n,\]
compute \(\frac{b}{a}.\)


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 10 & 6 \\ -4 & 2 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$

Solution: \boxed{\begin{pmatrix} 10 & 6 \\ -4 & 2 \end{pmatrix}}


A truncated cone has horizontal bases with radii 20 and 5. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


Define a function $A(m, n)$ with the same recursive definition as provided:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned} \right.\]
Compute $A(3, 2).$


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


Solution: \boxed{\mathbf{B}^{200} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}}


Find all real values of \( x \) that satisfy \( \frac{x(x+1)}{(x-3)^2} \ge 10 \). (Give your answer in interval notation.)

Solution: \boxed{\left[\frac{61 - \sqrt{481}}{18}, 3\right) \cup \left(3, \frac{61 + \sqrt{481}}{18}\right]}


Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_{10} \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_{10} \frac{511}{512}.\]


The exact amount of fencing that enclosed the six congruent equilateral triangular corrals is reused to form one large equilateral triangular corral. What is the ratio of the total area of the six small corrals to the area of the new large corral? Express your answer as a common fraction.


Five dice are stacked in a pyramid as shown, with two on the bottom and three on top. Fourteen of their thirty faces are visible. The visible numbers are 1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 1, 6, 6, and 5. What is the total number of dots NOT visible in this view?


For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{525}$ terminate?


The function $g$ is defined for all integers and satisfies the following conditions:
\[ g(n) = \begin{cases}
n-3 & \mbox{if } n \geq 500 \\
g(g(n+4)) & \mbox{if } n < 500
\end{cases} \]
Find $g(56)$.


Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits, starting with an 8.


Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 4030.


A solid rectangular block is constructed by sticking together $N$ identical 1-cm cubes face to face. When the block is viewed such that three of its faces are visible, exactly $378$ of the 1-cm cubes are not visible. Determine the smallest possible value of $N$.


What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{4}{3}x - \frac{2y}{5}\right)^8$?


The cost of three pencils and four pens is \$3.20, and the cost of two pencils and three pens is \$2.50. What is the cost of one pencil and two pens?


The polynomial equation \[x^4 + ax^2 + bx + c = 0,\] where \(a\), \(b\), and \(c\) are rational numbers, has \(3-\sqrt{5}\) as a root. It also has a sum of its roots equal to zero. What is the integer root of this polynomial?


How many integers \( n \) are there such that \( 200 < n < 300 \) and \( n \) leaves the same remainder when divided by \( 7 \) and by \( 9 \)?


Given the points $P = (2, -3, 1)$, $Q = (4, -7, 4)$, $R = (3, -2, -1)$, and $S = (5, -6, 2)$ in space, determine if they form a flat quadrilateral and find its area.


Let \(x\), \(y\), and \(z\) be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of \(x^2 + y^2 + z^2.\)


Let \(w, x, y, z\) be distinct real numbers that sum to \(0\). Compute \[ \dfrac{wy+xz}{w^2+x^2+y^2+z^2}. \]


In the sequence of natural numbers listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers, all of which are nonprime?


Compute the number of real solutions $(x, y, z, w)$ to the system of equations:
\begin{align*}
x + 1 &= z + w + zwx, \\
y - 1 &= w + x + wxy, \\
z + 2 &= x + y + xyz, \\
w - 2 &= y + z + yzw.
\end{align*}


How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[\left\{ \begin{aligned} x+2y&=2 \\ \left| |x| - |y| \right| &= 2 \end{aligned}\right.\]


Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[P(1) = P(3) = P(5) = P(7) = a\] and
\[P(2) = P(4) = P(6) = P(8) = -a.\] Additionally,
\[P(0) = 0 \] and
\[P(9) = a.\]
What is the smallest possible value of $a$?


The real numbers $a,$ $b,$ and $c$ satisfy the equation
\[a^3 + b^3 + c^3 + a^2 + b^2 + 1 = d^2 + d + \sqrt{a + b + c - 2d}.\] Find $d$.

Solution: \boxed{1 \text{ or } -\frac{4}{3}}


How many integers $n$ satisfy the condition $150 < n < 250$ and have the same remainder when divided by $7$ or by $9$?


Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side.


Six couples are at a gathering. Each person shakes hands exactly once with everyone else except for their spouse and one additional person of their choice. How many handshakes were exchanged?


An annulus is formed by two concentric circles with radii $R$ and $r$, where $R > r$. Let $\overline{OA}$ be a radius of the outer circle, $\overline{AB}$ be a tangent to the inner circle at point $B$, and $\overline{OC}$ be the radius of the outer circle that passes through $B$. Let $x = AB$, $y = BC$, and $z = AC$. Determine the area of the annulus. Express your answer in terms of $\pi$ and at most one of the variables $x, R, r, y, z$.

Solution: \boxed{\pi x^2}


Sides $\overline{AM}$ and $\overline{CD}$ of regular dodecagon $ABCDEFGHIJKL$ are extended to meet at point $P$. What is the degree measure of angle $P$?


Define the function $f(n)$ to return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, given that $k$ is of the form $3^n * 2^n$. What is the number of positive integer divisors of $f(2010)$?


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 3$ and
\[f(x^2 + y^2) = (x + y)(f(x) + f(y))\] for all real numbers $x$ and $y.$

Determine the value of $f(5)$.

Solution: \boxed{f(5) = 45}


A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $300$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.


An isosceles triangle $ABC$ with $AB = AC$ is given. A point $P$ is chosen at random within the triangle. Determine the probability that $\triangle ABP$ has a smaller area than both $\triangle ACP$ and $\triangle BCP$.


Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy]


In a triangle $ABC$, $BP$ and $BQ$ trisect $\angle ABC$. A line $BM$ bisects $\angle PBQ$ and another line $CN$ bisects $\angle QBC$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle NCQ$.


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


Let $g(x) = \left\lceil \frac{\cos(x)}{x + 3} \right\rceil$ for $x > -3$, and $g(x) = \left\lfloor \frac{\cos(x)}{x + 3} \right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Determine if there is an integer that is not in the range of $g(x)$.


Calculate the number of roots of unity that are also roots of the quadratic equation $z^2 + az - 1 = 0$ for some integer $a$.


For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have?


Let \( m \) be the smallest integer whose cube root is of the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \( 1/500 \). Find \( n \).


The cost of three notebooks and two pens is $\$7.40$, and the cost of two notebooks and five pens is $\$9.75$. What is the cost of one notebook and three pens?


Suppose
\[
\frac{1}{x^3 - x^2 - 17x + 45} = \frac{A}{x + 5} + \frac{B}{x - 3} + \frac{C}{x + 3}
\]
where $A$, $B$, and $C$ are real constants. Determine the value of $A$.

Solution: \boxed{A = \frac{1}{16}}


Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).


Let \( c \) be a complex number. Suppose there exist distinct complex numbers \( r \), \( s \), and \( t \) such that for every complex number \( z \), we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)
\]
where \( r \), \( s \), \( t \) are the third roots of unity. Compute the number of distinct possible values of \( c \).


Find the number of real solutions to the equation
\[
\frac{x}{50} = \cos x.
\]


A $4 \times 2$ rectangle and a $5 \times 3$ rectangle are contained within a square. One of the rectangles must be rotated 90 degrees relative to the other, and the sides of the square are parallel to one of the rectangle's sides at a time. What is the smallest possible area of the square?


In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$.


Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 48.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 3z^2.\]


In an acute triangle \( \triangle ABC \), altitudes \( \overline{AD} \) and \( \overline{BE} \) intersect at point \( H \). If \( HD = 6 \) and \( HE = 3 \), calculate \( (BD)(DC) - (AE)(EC) \).


What is the largest possible median for the seven number set $\{x, 2x, y, 3, 2, 5, 7\}$ if $x$ and $y$ can be any integers?


The least common multiple of two positive integers is divided by their greatest common divisor, yielding a result of 24. If one of these integers is 36, what is the smallest possible value of the other integer?


Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 3$, $\|\mathbf{b}\| = 4$, and $\|\mathbf{c}\| = 5$. Find all possible values of $\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c}$.

Solution: \boxed{[-32, 32]}


A basketball team has 12 members, of which only 4 are tall enough to play as centers, while the rest can play any position. Determine the number of ways to choose a starting lineup consisting of one point guard, one shooting guard, one small forward, one center, and one power forward, given that the shooting guard and small forward can't be the same person who plays as the point guard.


The graph of the function $f(x)$ is shown. How many values of $x$ satisfy $f(f(x)) = 5$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;

for(int i = -4; i <= 5; ++i) {
	draw((i,-1)--(i,6), dashed+mediumgrey);
}

for(int i = 1; i <= 6; ++i) {
	draw((-4,i)--(5,i), dashed+mediumgrey);
}

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

xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2));

label("$f(x)$",(-3.52,4.6),SE*lsf);

dot((-4.32,4.32),ds); dot((5.56,2.44),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

[/asy]


Each of \(b_1, b_2, \dots, b_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \le i < j \le 150} b_i b_j.
\]


Let $x,$ $y,$ $z$ be real numbers, all greater than 4, such that
\[\frac{(x + 3)^2}{y + z - 3} + \frac{(y + 5)^2}{z + x - 5} + \frac{(z + 7)^2}{x + y - 7} = 45.\]
Enter the ordered triple $(x,y,z).$

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


Consider the sequence of numbers defined recursively by $t_1=2$ and for $n>1$ by $t_n=n+t_{n-1}$ when $n$ is even, and by $t_n=\frac{t_{n-1}}{n}$ when $n$ is odd. Given that $t_n=\frac{1}{221}$, find $n$.


How many numbers are in the list $ -22, -15, -8, \ldots, 43, 50?$


The lengths of the sides of a triangle are $\log_{10}15$, $\log_{10}90$, and $\log_{10}m$, where $m$ is a positive integer. Determine how many possible values there are for $m$.


Given that $y$ is a multiple of $3456$, what is the greatest common divisor of $g(y) = (5y+4)(9y+1)(12y+6)(3y+9)$ and $y$?


Let $l, m, n,$ and $p$ be real numbers, and let $A, B, C$ be points such that the midpoint of $\overline{BC}$ is $(l,p,0),$ the midpoint of $\overline{AC}$ is $(0,m,p),$ and the midpoint of $\overline{AB}$ is $(p,0,n).$ Find the value of:
\[
\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2 + p^2}.
\]


Find the units digit of $n$ given that $mn = 34^5$ and $m$ has a units digit of 6.


Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$.


A recently released album has a cover price of \$30. The music store offers a \$5 discount and two percentage discounts: 10% and 25%. A savvy shopper wants to maximize savings by choosing the optimal order of applying these discounts. Calculate how much more she will save by choosing the best order over the worst one. Express your answer in cents.

Solution: \boxed{162.5\text{ cents}}


Find the reflection of the vector $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ over the normalized vector of $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 4.6 \\ -2.8 \end{pmatrix}}


Three mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 9:30 a.m., and each stays for exactly $n$ minutes. The probability that any one arrives while another is in the cafeteria is $50 \%,$ and $n = d - e\sqrt {f},$ where $d, e,$ and $f$ are positive integers, and $f$ is not divisible by the square of any prime. Find $d + e + f.$


Consider a sequence $y_1,$ $y_2,$ $y_3,$ $\dots$ defined by
\begin{align*}
y_1 &= \sqrt[4]{4}, \\
y_2 &= (\sqrt[4]{4})^{\sqrt[3]{4}},
\end{align*}
and in general,
\[y_n = (y_{n - 1})^{\sqrt[3]{4}}\] for $n > 1.$ What is the smallest value of $n$ for which $y_n$ is an integer?


The operation $\oplus$ is defined by
\[a \oplus b = \frac{a + b}{1 + ab}.\]
Compute
\[1 \oplus (2 \oplus (3 \oplus (\dotsb \oplus (999 \oplus 1000) \dotsb))).\]


The coordinates of the vertices of isosceles trapezoid $PQRS$ are all integers, with $P=(10,50)$ and $S=(11,51)$. The trapezoid has no horizontal or vertical sides, and $\overline{PQ}$ and $\overline{RS}$ are the only parallel sides. Determine the sum of the absolute values of all possible slopes for $\overline{PQ}$.


Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, the triangle formed by the points 0, $z_1,$ and $z_2$ is an equilateral triangle but rotated, such that $z_2 = \omega z_1$, where $\omega = e^{2\pi i/3}$. Find $\frac{a^2}{b}$.


Circle $\Gamma$ is the incircle of $\triangle ABC$ and also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, the point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A = 50^\circ$, $\angle B = 70^\circ$, let $\angle C$ be computed based on this data. What is the measure of $\angle YZX$?


Solution: \boxed{AP = 2}


Shown below are rows 1, 2, and 3 of Pascal's triangle.

\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((d_i), (e_i), (f_i)\) be the sequence, from left to right, of elements in the 2010th, 2011th, and 2012th rows, respectively, with the leftmost element occurring at \(i = 0.\) Compute
\[
\sum_{i = 0}^{2010} \frac{e_i}{f_i} - \sum_{i = 0}^{2009} \frac{d_i}{e_i}.
\]


In the United Kingdom, coins have the following thicknesses: 1p, 1.65 mm; 2p, 2.05 mm; 5p, 1.85 mm; 10p, 1.95 mm. If a stack of these coins is precisely 19 mm high, consisting of 2p and 10p coins only, how many coins are in the stack?

Solution: \boxed{10 \text{ coins}}


The graph of the function $y=g(x)$ is given. For all $x > 3$, it is true that $g(x) > 0.3$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, find $D+E+F$. Assume the graph shows vertical asymptotes at $x = -3$ and $x = 2$, and the horizontal asymptote is below $y = 1$ but above $y = 0.3$.


Express 1000 as a sum of at least two distinct powers of 2. What would be the least possible sum of the exponents of these powers?


Let's consider two mathematicians, Alice and Bob, each thinking of a polynomial. Each polynomial is monic, has a degree of 4, and both polynomials have the same positive constant term and the same coefficient of $x^3$. The product of their polynomials is given by \[x^8 + 2x^7 + 3x^6 + 2x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 9.\] What is the constant term of Bob's polynomial?


Let $a < b < c$ be three integers such that $a, b, c$ is an arithmetic progression and $c, a, b$ is a geometric progression. Further assume that $a, b, c$ are multiples of 5. What is the smallest possible value of $c$ where all numbers are positive?


Determine the number of solutions to
\[ |\sin x| = \left(\frac{1}{2}\right)^x \]
on the interval \((0, 200\pi)\).


Find the smallest possible value of the expression $$\frac{(a+b)^2+(b+c)^2+(c-a)^2}{c^2},$$ where $c > b > a$ are real numbers, and $c \neq 0.$


Let \( S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\} \). A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). Compute the probability that the transformation \( \left(\frac{1}{2} + \frac{i}{2}\right)z \) results in a number that remains within \( S \).


In triangle $XYZ$, sides $XY$, $YZ$, and $ZX$ are tangent to an external circle with center $A$. Given that $\angle XYZ = 80^\circ$ and $\angle YZA = 20^\circ$, find the measure of angle $\angle YXZ$, in degrees.


The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:

1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.

How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively.


Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(a + b + c = 3\) and no one of these numbers is more than three times any other. Find the minimum value of the product \(abc.\)


Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid. Suppose each edge of $DABC$ has length 10, 24, or 24, but no face of $DABC$ is equilateral. Determine the surface area of $DABC$.


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

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


What is the smallest positive integer that is a multiple of both 36 and 45 but not a multiple of 10?


The numbers $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ are equal to the numbers $1, 2, 3, \dots, 9$ in some order. Each triplet $(a_1, a_2, a_3)$, $(b_1, b_2, b_3)$, and $(c_1, c_2, c_3)$ forms an arithmetic sequence. Find the smallest possible value of
\[
a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.
\]


In circle $O$, four points $P, A, B, Q$ lie on the circumference such that tangents to $O$ at points $A$, $B$, and $Q$ meet at point $P$ outside the circle, and $\angle APB = 60^\circ$. Determine $\angle AQB$.


Let \(x, y, z\) be nonzero real numbers such that \(x + y + z = 0\) and \(xy + xz + yz \neq 0\). Find all possible values of
\[
\frac{x^6 + y^6 + z^6}{xyz (xy + xz + yz)}.
\]


Find the common ratio of the infinite geometric series: $$\frac{4}{7} - \frac{16}{21} - \frac{64}{63} - \dots$$


In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?


In triangle $ABC$, $\angle C = 90^\circ$, $AC = 7$ and $BC = 24$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 10$, then what is the length of $BD$?


Determine the product of the real parts of the complex solutions to the equation $x^2 + 2x = -i$.


Let $p<q<r$ be three integers such that $p,q,r$ is a geometric progression and $p,r,q$ is an arithmetic progression. Find the smallest possible value of $r$.


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


Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$.


What is the least whole number that is divisible by 9, but leaves a remainder of 1 when divided by any integer 2 through 6 and 8?


In triangle $ABC$, $AB = 4$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Additionally, the area of triangle $ABC$ is $12$ square units. Find $\cos \angle BAD.$


Linda bought 50 items each priced at 50 cents, 2 dollars, or 4 dollars. If her total purchase price was $\$$50.00, how many 50-cent items did she purchase?


Let $AB$ be a diameter of a circle centered at $O$.  Let $F$ be a point on the circle such that $F$ is not on the semicircle containing $E$. Let the tangent at $B$ intersect the tangent at $F$ and $AF$ at points $C'$ and $D'$ respectively. If $\angle BAF = 30^\circ$, find $\angle C'ED'$, in degrees.


Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 6 units from $\overline{BG}$ and 3 units from $\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n$.


Let $T$ be the set of complex numbers of the form $x + yi$, where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{3}}{2} \le x \le \frac{2}{\sqrt{3}}.\]
Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = 1.$


Determine the number of angles between 0 and $2 \pi,$ excluding integer multiples of $\frac{\pi}{2},$ such that $\sin^2 \theta,$ $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order.


For each positive integer $n$, let $f(n) = n^4 - 400n^2 + 600$. What is the sum of all values of $f(n)$ that are prime numbers?


Given the property for any positive integer $n$, the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}^n$ is equal to $\begin{pmatrix} G_{n + 1} & 2G_n \\ 2G_n & G_{n - 1} \end{pmatrix}$, where $G_n$ corresponds to a sequence defined by $G_{n} = 2G_{n - 1} + G_{n - 2}$ with $G_1 = 1$ and $G_2 = 2$. Calculate $G_{100} G_{102} - 4G_{101}^2$.


The inscribed circle of triangle $XYZ$ is tangent to $\overline{XY}$ at $P,$ and its radius is $15$. Given that $XP=18$ and $PY=24$, and the height from $Z$ to $\overline{XY}$ is $36$, find the area of triangle $XYZ$.


In a slightly larger weekend softball tournament, five teams (A, B, C, D, E) are participating. On Saturday, Team A plays Team B, Team C plays Team D, and Team E will automatically advance to the semi-final round. On Sunday, the winners of A vs B and C vs D play each other (including E), resulting in one winner, while the remaining two teams (one from initial losers and Loser of semifinal of E's match) play for third and fourth places. The sixth place is reserved for the loser of the losers' game. One possible ranking of the teams from first place to sixth place at the end of this tournament is the sequence AECDBF. What is the total number of possible six-team ranking sequences at the end of the tournament?


I have 5 shirts, 5 pairs of pants, and 7 hats. The pants come in tan, black, blue, gray, and green. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which the shirt and pants are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


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


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the small right triangles is $n$ times the area of the square. What is the ratio of the area of the other small right triangle to the area of the square? Express your answer as a common fraction in terms of $n$.


In rectangle $JKLM$, points $N$ and $P$ bisect $\overline{JK}$ and points $Q$ and $R$ bisect $\overline{LM}$. Rectangle $JKLM$ has dimensions $JK = 6$ and $JL = 4$. Find the area of quadrilateral $STUV$ formed by connecting midpoints of sides in the following manner: $S$ is the midpoint of $\overline{JN}$, $T$ is the midpoint of $\overline{KP}$, $U$ is the midpoint of $\overline{LQ}$, and $V$ is the midpoint of $\overline{MR}$.


Find the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity.


Let \(a, b, c\) be nonzero real numbers, and define
\[
x = \frac{b}{c} + 2\frac{c}{b}, \quad y = \frac{a}{c} + 2\frac{c}{a}, \quad z = \frac{a}{b} + 2\frac{b}{a}.
\]
Simplify \(x^2 + y^2 + z^2 - xyz\).


In rectangle $ABCD$, $AB=7$ and $BC =4$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 2$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.


Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?


Let $f(n)$ be the sum of all the divisors of a positive integer $n$. Determine the number of positive integers $n$ for which $f(f(n)) = n + 3$.


Given a matrix for projecting onto a certain line $m$, which passes through the origin, is represented by the matrix
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{5}{21} & -\frac{2}{21} & -\frac{2}{7} \\ -\frac{2}{21} & \frac{1}{42} & \frac{1}{14} \\ -\frac{2}{7} & \frac{1}{14} & \frac{8}{14} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $m$ in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a, b, c$ are integers, $a > 0$, and $\gcd(|a|,|b|,|c|) = 1$.

Solution: \boxed{\begin{pmatrix} 5 \\ -2 \\ -6 \end{pmatrix}}


How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)?


When the digits in the number $8019$ are reversed, we obtain the number $9108.$ Factorize $9108$ such that it is the product of three distinct primes $x,$ $y,$ and $z.$ How many other positive integers are the products of exactly three distinct primes $q_1,$ $q_2,$ and $q_3$ such that $q_1 + q_2 + q_3 = x+y+z$?

Solution: \boxed{6 \text{ distinct integers}}


In rectangle $ADEH$, points $B$ and $C$ bisect $\overline{AD}$, and points $G$ and $F$ bisect $\overline{HE}$. In addition, $AH=AC=3$. What is the area of quadrilateral $WXYZ$?


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


For all integers $x$ and $y$, define the operation $\diamond$ such that $x \diamond 0 = x$, $x \diamond y = y \diamond x$, and $(x + 2) \diamond y = (x \diamond y) + 2y + 3$. Calculate the value of $8 \diamond 3$.


Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all the stickers across the sheets of paper, given that only the number of stickers on each sheet matters?


Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\overline{QR}, \overline{RP}$, and $\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 60$, and $20$, respectively. Find the perimeter of the triangle whose sides lie on lines $m_P, m_Q$, and $m_R$.


Let $\triangle ABC$ have side lengths $AB = 40$, $BC = 24$, and $AC = 32$. Point $Y$ lies in the interior of $\overline{AC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABY$ and $\triangle BCY$, respectively. Find the minimum possible area of $\triangle BI_1I_2$ as $Y$ varies along $\overline{AC}$.


The matrix for projecting onto a certain line $m,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{18} & \frac{1}{9} & -\frac{1}{6} \\ \frac{1}{9} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{6} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $m.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

Solution: \boxed{\begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}}


Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a regular octagon. His leash is exactly long enough to reach the midpoint of each side of this octagonal yard. If the fraction of the area of Fido's yard that he can reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of $a \times b$?


Find all real values of $x$ that satisfy \(\frac{1}{x(x+1)} - \frac{1}{(x+1)(x+2)} < \frac{1}{4}\). (Give your answer in interval notation.)

Solution: \boxed{(-\infty, -2) \cup (-1, 0) \cup (2, \infty)}


Let \(x\), \(y\), and \(z\) be real numbers such that
\[3x^2 + 9xy + 6y^2 + z^2 = 1.\]
Let \(n\) and \(N\) be the minimum and maximum values of \(x^2 + 4xy + 3y^2 + z^2\), respectively. Find the product \(nN.\)


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$.


Find the least positive integer $n$ such that
$$\frac{1}{\sin 30^\circ \sin 31^\circ} + \frac{1}{\sin 32^\circ \sin 33^\circ} + \cdots + \frac{1}{\sin 148^\circ \sin 149^\circ} = \frac{a}{\sin n^\circ},$$
where $a = 2\sin 1^\circ$.


How many distinct two-digit numbers can appear as the last two digits of an integral perfect-cube number?


There are two cubes. The mass of the second cube is $25\%$ less than the mass of the first cube, and the edge length of the second cube is $25\%$ greater than that of the first cube. By what percentage does the density of the second cube differ from the density of the first cube?


An urn contains 9 balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number, with the number on José's ball being the tens digit and the number on Maria's ball being the units digit. What is the probability that this number is even?


At the World Meteorological Conference, each participant took turns announcing the average monthly temperature in their home city. At that moment, all other participants recorded the product of the temperatures in his and their cities. A total of 50 positive and 60 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?


A reception hall in the palace of the thirteenth kingdom comprises points on a plane whose coordinates satisfy the condition \(4|x| + 5|y| \leq 20\). How many identical tiles of double-sided parquet, each in the shape of a right triangle with legs 1 and \(5/4\), are required to tile the floor of the hall? Tiling is considered to be done without any gaps, overlaps, and without extending beyond the area.


Let \( R \) be a point on the curve such that \( OMRN \) is a square. If \( r \) is the \( x \)-coordinate of \( R \), find the value of \( r \).

Solution: \boxed{r = 1}


Find the sum of all four-digit numbers (written in base 10) which contain only the digits 1, 2, 3, 4, and 5, and contain no digit more than once.


Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.

The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?


In a summer camp, there are 100 campers, each of whom can sing, dance, or perform at least one skill. Some campers have more than one skill, but no one has all three skills. If 42 campers do not sing, 65 campers do not dance, and 29 campers do not perform, how many campers have at least two skills?
(A) 16
(B) 25
(C) 36
(D) 49
(E) 64


Find the probability that a randomly selected two-digit number is divisible by either 2, 5, or both.


Let \( x \) and \( y \) be the integral and fractional parts of \( \sqrt{37 - 20 \sqrt{3}} \). Find the value of \( x + y + \frac{4}{y} \).


In an $8 \times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected?


As shown in the figure, $A$ and $B$ are the two endpoints of a diameter of a circular track. Three miniature robots, labeled as A, B, and C, start at the same time from point $A$ and perform uniform circular motion on the track. Robots A and B start from $A$, while robot C starts from $B$. Robot B moves in a clockwise direction, while robots A and C move in a counterclockwise direction. After 12 seconds, robot A reaches $B$. After an additional 9 seconds, robot A catches up with robot C for the first time and simultaneously meets robot B for the first time. Calculate how many seconds after robot C first reaches $A$ it will take for robot B to reach $B$ for the first time.


The sought-after three-digit number ends with the digit 1. If this digit is erased and then reattached as the first digit of the number, the resulting new three-digit number will be smaller than the original by \(10 a^{\log _{\sqrt{a}} 3}\). Find this number.


The product of three natural numbers is equal to 60. What is the largest possible value of their sum?


In a regular hexagon with an area of $144 \mathrm{~cm}^{2}$, all its diagonals were drawn. The hexagon thus "split" into several triangles and quadrilaterals.

a) Determine into how many parts the diagonals have divided the hexagon.

b) Determine the area of the regular hexagon formed by combining all the quadrilateral parts of the given hexagon.

Solution: \boxed{48 \text{ cm}^2}


In a convex pentagon \(ABCDE\), \(\angle A = 60^\circ\), and the other angles are equal to each other. It is known that \(AB = 6\), \(CD = 4\), and \(EA = 7\). Find the distance from point \(A\) to the line \(CD\).


Donald Duck and Mickey Mouse are competing in a 10,000-meter race. Mickey Mouse runs at a speed of 125 meters per minute, while Donald Duck runs at a speed of 100 meters per minute. Donald Duck has an electronic remote control that can force Mickey Mouse to move backward. When this remote control is used for the \( n \)-th time, Mickey Mouse will move backward for one minute at a speed that is \( n \times 10\% \) of his original speed, then continue running forward at his original speed. What is the minimum number of times Donald Duck needs to use the remote control in order to win the race?


Three positive integers greater than 1000 satisfy the condition that the unit digit of the sum of any two of them is equal to the unit digit of the third number. What are the possible values for the last three digits of the product of these three numbers?

Solution: \boxed{000, 250, 500, 750}


In the plane, there are 2020 points, some of which are black and the rest are green.

For each black point, there are exactly two green points that are at a distance of 2020 from this black point.

Determine the minimum possible number of green points.


Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)


Compute the limit of the function:

$$\lim _{x \rightarrow 0} \frac{\ln \left(x^{2}+1\right)}{1-\sqrt{x^{2}+1}}$$