A band has less than 100 members and is arranged in a rectangular formation with $m$ members in each of $r$ rows, leaving three members out. When the director rearranges the formation by adding 2 more members in each row while reducing the number of rows by 1, the new formation exactly accommodates all the band members. What is the largest number of members the band could have?


Solution: \boxed{6 - 3y + x}


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


Points $R$ and $S$ are one third and two thirds along the respective sides of the square. Calculate the fraction of the interior of the square that is not shaded when connecting $R$ and $S$ to the opposite vertices forms a white region within the square.

[asy]
filldraw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(1,3)--(2,3)--(3,1)--(3,2)--(2,0)--(1,0)--(0,2)--cycle,white,linewidth(1));
label("R",(0,1),W);
label("S",(1,3),N);
[/asy]


Eight 8-sided dice are rolled. What is the probability that exactly four of the dice show a 2? Express your answer as a decimal rounded to the nearest thousandth.


What is the greatest four-digit number which is a multiple of 17?


In Ultratown, the streets are all $25$ feet wide, and the blocks they enclose are rectangular with lengths of $500$ feet and widths of $300$ feet. Hannah runs around the block on the longer $500$-foot side of the street, while Harry runs on the opposite, outward side of the street. How many more feet than Hannah does Harry run for every lap around the block?


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}

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


In triangle $XYZ$, $\angle X = 90^\circ$, $YZ = 30$, and $\tan Z = 3\sin Z$. What is $XY$?


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


Let $g$ be a linear function for which $g(8) - g(4) = 20$. What are $g(16) - g(4)$ and $g(20) - g(12)$?


A right pyramid has a square base that measures 8 cm on each side. Its peak is 10 cm directly above the center of its base. Calculate the sum of the lengths of all edges of the pyramid, rounding off to the nearest whole number.

Solution: \boxed{78 \text{ cm}}


Consider an ellipse that is tangent to both the $x$-axis and the $y$-axis. If the foci of the ellipse are located at $(4, -6 + 2\sqrt{5})$ and $(4, -6 - 2\sqrt{5})$, determine the length of the major axis.


Find all real numbers \( x \) such that \[ \frac{x-2}{x-4} \ge 3, \] with \( x \neq 2 \).

Solution: \boxed{(4, 5]}


Suppose that \( c \) and \( d \) are integers such that \( 4d = 10 - 3c \). How many of the first six positive integers must be divisors of \( 3d + 15 \)?


The graph of $y = ax^2 + bx + c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.

[asy]
size(140);
Label f;

f.p=fontsize(4);

xaxis(-3,3,Ticks(f, 1.0));

yaxis(-4,4,Ticks(f, 1.0));

real f(real x)

{

return -3x^2+6x+2;

}

draw(graph(f,-1,3),linewidth(1),Arrows(6));
[/asy]


A regular polygon has a perimeter of 180 cm and each side measures 15 cm. How many sides does this polygon have? Also, calculate the measure of one internal angle.


A triangle has side lengths which are different prime numbers greater than 5, and the length of its perimeter is also prime. What is its smallest possible perimeter?


Find the value of $n$ that satisfies $\frac{1}{n+1} + \frac{2}{n+1} + \frac{n}{n+1} + \frac{1}{n+2} = 4$.

Solution: \boxed{n = \frac{-3 + \sqrt{6}}{3} \text{ or } n = \frac{-3 - \sqrt{6}}{3}}


A sphere is inscribed in a cube. If one edge of the cube is 10 inches, what is the volume of the inscribed sphere? Express your answer in terms of \(\pi\).


Define $n!!$ as in the original problem. Evaluate $\sum_{i=1}^{5} \frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms.


At the end of the day, a bakery had $\frac{8}{9}$ of a pie left over. Louie, Dewey, and Huey decided to split the pie, but this time Louie took double the amount of pie that Dewey or Huey took individually. What fraction of the pie did Louie take home?


Compute without using a calculator: $10! - 9!$.


Express $7.\overline{036}$ as a common fraction in lowest terms.


Find the maximum number of real roots for the polynomial
\[ x^n - x^{n-1} + x^{n-2} - \dots + (-1)^{n-1} x + (-1)^n = 0, \]
where $n$ is a positive integer.


Find all real solutions to $x^3 + (x+2)^3 + (x+4)^3 = (x+6)^3$ such that $x^2 + 4x + 4 > 0$.

Solution: \boxed{x = 6}


In triangle $PQR,$ points $M$ and $N$ are on $\overline{PQ}$ and $\overline{PR},$ respectively, such that $PM = 2,$ $MQ = 6,$ $PN = 3,$ and $NR = 9.$ The angle bisector $\overline{PS}$ intersects $\overline{MN}$ at $T.$ Compute $\frac{PT}{PS}.$

Solution: \boxed{\frac{5}{20} = \frac{1}{4}}


Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$?


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


A company incurs a fixed daily maintenance fee of $550$, an additional one-time daily setup cost of $200$, and pays each worker $18$ per hour. Each worker now makes 6 widgets per hour, which are then sold at $3.50$ each. What is the least number of workers the company must hire to make a profit in an 8-hour workday?


John earned scores of 78, 85, and 92 on his first three English exams. If he scores 95 on his fourth exam, then by how much will his average increase?


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

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


Using only the digits 5, 6, and 7, how many positive seven-digit integers can be made that are palindromes and have an odd number in the middle?


Square \(EFGH\) is constructed along diameter \(EF\) of a semicircle. Point \(P\) is the midpoint of arc \(EF\) of this semicircle. The length of the diameter \(EF\) is 10 centimeters. What is the length of line segment \(PG\)? Express your answer in simplest radical form.


Let \(x\) and \(y\) be real numbers such that
\[
  3 < \frac{x - y}{x + y} < 6.
\]
If \(\frac{x}{y}\) is an integer, what is its value?


A train arrives randomly at some time between 2:00 PM and 4:00 PM, waits for 30 minutes, and then leaves. If Susan also arrives randomly between 2:00 PM and 4:00 PM, what is the probability that the train will be there when Susan arrives?


The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The minimum value of $y = ax^2 + bx + c$ is -3, which occurs at $x = -2$. (2) The graph passes through the point $(1,10)$. If the graph passes through the point $(3,n)$, then what is the value of $n$?


The expression $1728x^3 + 27$ can be written as $(ax+b)(cx^2+dx+e)$. Find $a+b+c+d+e$.


Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \).


Let $a$ and $b$ be positive integers such that $a-b=8$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 16$. Find the smallest possible value of $b$.


The graph of \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] has its foci at $(0,\pm 5),$ while the graph of \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] has its foci at $(\pm 7,0).$ Determine the exact values of $a$ and $b$.

Solution: \boxed{a = 2\sqrt{3}, b = \sqrt{37}}


If the consecutive integers from $60$ to $1$ were written as $$6059585756...,$$ what would be the $79^{\text{th}}$ digit to be written?


A Senate committee has 6 Democrats and 5 Republicans. How many ways can they sit around a circular table such that no two Republicans sit next to each other?


A line described by the equation $bx + (b-1)y = b^2-1$ passes through the point $(2, -5)$. Determine the value of $b$.

Solution: \boxed{\frac{-3 - \sqrt{33}}{2}}


Triangles $ABE$ and $BEC$ are right triangles with $AB = 15$ units, $BE = 20$ units, and $BC = 25$ units. What is the area, in square units, of quadrilateral $ABEC$?

Solution: \boxed{300 \text{ square units}}


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


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


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


How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{10}$?


Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?


The sum of the first 1000 terms of a geometric sequence is 500. The sum of the first 2000 terms is 950. Find the sum of the first 3000 terms.


Let $C$ be the circle with equation $x^2 - 8y - 5 = -y^2 - 6x$. Find the values of $a$, $b$, and $r$, where $(a,b)$ is the center of $C$ and $r$ is its radius, and compute $a+b+r$.

Solution: \boxed{a+b+r = 1 + \sqrt{30}}


Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 6,$ $\|\mathbf{c}\| = 3,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}, \quad \mathbf{a} \cdot \mathbf{c} = 0.\]
If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$


Alice's password consists of a two-digit number, followed by a symbol from the set {$!, @, #, $, %}, followed by another two-digit number. Calculate the probability that Alice's password consists of an even two-digit number followed by one of {$, %, @}, and another even two-digit number.


Three years ago, there were 30 trailer homes on Maple Street with an average age of 15 years. At that time, a group of brand new trailer homes was added to Maple Street. Today, the average age of all the trailer homes on Maple Street is 12 years. How many new trailer homes were added three years ago?


There are four distinct positive integers $a, b, c, d$ less than $12$ which are invertible modulo $12$. Find the remainder when $(abc + abd + acd + bcd + a + b + c + d)(abcd)^{-1}$ is divided by $12$.


A space explorer encounters alien writings that reflect a number $2154_6$. In base 10, what number does this represent?


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

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


The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 4. What is the value of the largest number?


What is the units digit of the sum of the eleven terms of the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 10! + 10, \, 11! + 11$?


What is $72_8 - 45_8 + 23_8?$ Express your answer in base 8.


Five concentric circles are drawn with radii of 2, 4, 6, 8, and 10. The inner circle is painted white, the ring around it is black, the next ring is white, the fourth ring is black, and the outermost ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction.


Consider the lines:
\begin{align*}
y &= 4x + 1, \\
y &= -5x - 2, \\
y &= 4x + 3, \\
y &= \frac{1}{5}x + 4, \\
y &= -5x + 1.
\end{align*}
Let's say that a pair of lines is $\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many pairs are good?


John draws a regular hexagonal shape in the sand, placing a different sea shell at each of its 6 corners and at the midpoint of each of the 6 sides, making 12 distinct points in total. In how many ways can he place 12 different sea shells, if only rotations of the arrangement are considered equivalent (reflections are not considered)?


Given two real numbers $p>1$ and $q>1$ so that $\frac{1}{p} + \frac{1}{q} = 1$ and $pq = 9$, what is $q$?

Solution: \boxed{\frac{9 - 3\sqrt{5}}{2}}


In the diagram, what is the perimeter of polygon $ABCDE$?

[asy]
import olympiad;
size(6cm);

pair A = (0, 8);
pair B = (4, 8);
pair C = (4, 4);
pair D = (9, 0);
pair E = (0, 0);

draw(A--B--C--D--E--cycle);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, NE);
label("$D$", D, SE);
label("$E$", E, SW);

label("$4$", A + (B - A) / 2, 2 * N);
label("$8$", E / 2, W);
label("$9$", D / 2, S);

draw(rightanglemark(A, E, D));
draw(rightanglemark(E, A, B));
draw(rightanglemark(A, B, C));
[/asy]

Solution: \boxed{25 + \sqrt{41}}


The digits of a four-digit positive integer add up to 18. The sum of the two middle digits is 11, and the thousands digit minus the units digit is one. If the integer is divisible by 11, what is the integer?


For what values of $x$ is $x^2 - 48x + 576 \leq 16$? Express your answer in interval notation.

Solution: \boxed{[20, 28]}


In triangle $ABC$, where $A$ is at the origin $(0,0)$, $B$ is now at $(12,0)$, and $C$ remains at $(3,5)$. If Wanda now guesses that the Fermat point $P$ is at $(5,3)$, calculate the sum of the distances from $P$ to the vertices of $\triangle ABC$. If she obtains $m\sqrt{a} + n\sqrt{b}$, where $m$, $n$, $a$, and $b$ are integers, find $m + n$.


Let $x$ and $y$ be two positive real numbers such that $x + y = 50.$  Determine the ordered pair $(x,y)$ for which $x^2 y^5$ is maximized.

Solution: \boxed{\left(\frac{100}{7}, \frac{250}{7}\right)}


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


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


How many integers are between $(9.8)^3$ and $(10.1)^3$?


A circle is tangent to the lines $6x - 5y = 40$ and $6x - 5y = -20$. The center of the circle lies on the line $3x + 2y = 0$. Find the center of the circle.

Solution: \boxed{\left(\frac{20}{27}, -\frac{10}{9}\right)}


How many distinct digits can appear as the second to last digit (penultimate digit) of an integral perfect square number?


If \(x^5 - x^4 + x^3 - px^2 + qx - 8\) is divisible by \((x + 3)(x - 2),\) find the ordered pair \((p, q)\).

Solution: \boxed{\left(-\frac{67}{3}, -\frac{158}{3}\right)}


A circular pie with a diameter of $20\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. Calculate $l^2$.


A girl has the following six coins in her pocket: $3$ pennies, $1$ nickel, $1$ dime, and $1$ quarter. She takes out two coins, records the sum of their values, and then puts them back with the other coins. She repeats this process several times. How many different sums can she record at most?


A rectangular prism has dimensions 15 inches by 25 inches by 15 inches. Calculate the length, in inches, of the diagonal connecting point A and point B on opposite corners of the prism. Express your answer in simplest radical form.

Solution: \boxed{5\sqrt{43} \text{ inches}}


Calculate $\sqrt{15q} \cdot \sqrt{8q^2} \cdot \sqrt{14q^3}$ . Express your answer in simplest radical form in terms of $q$.

Solution: \boxed{4q^3 \sqrt{105}}


A puppy, a smaller cat, and a larger cat together weigh 30 pounds. The puppy and the larger cat together weigh three times as much as the smaller cat, and the puppy together with the smaller cat weighs five pounds less than the larger cat. Determine the weight of the puppy.


How many integers between 400 and 700 have the sum of their digits equal to 14?


The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex?


Let the line $q$ be the perpendicular bisector of points $A = (-8, 15)$ and $B = (16, -3)$. Determine the point $C = (x, y)$ where $AB$ meets $q$. What is $3x - 5y$?


Solution: \boxed{472 + 76\sqrt{19}}


If $x = \frac{3}{4}$ and $y = \frac{4}{3}$, find the value of $\left(\frac{1}{3} x^7 y^6\right) \cdot 4$.


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 8 and altitude 10, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


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


Determine the radius of the circle formed by the set of points with spherical coordinates of the form:
\[(\rho, \theta, \phi) = \left( 1, \theta, \frac{\pi}{4} \right).\]


Choose integers $r$ and $k$ such that $-4 < r < 7$ and $2 < k < 9$. What is the probability that the division $r \div k$ results in an integer? Present your answer as a reduced fraction.


Determine all values of $x$ for which $\sqrt[3]{7 - x} = -\frac{5}{3}$.


A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet, and each stripe is 50 feet long. Find the distance, in feet, between the stripes.

Solution: \boxed{24 \text{ feet}}


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

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


Let \(x\), \(y\), and \(z\) be positive numbers such that \(xy = 60\), \(xz = 90\), and \(yz = 150\). Calculate the value of \(x + y + z\).


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


In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 5$, and $XZ = \sqrt{34}$. What is $\tan X$?


If $a$ and $b$ are positive integers such that $\gcd(a, b) = 18$, what is the smallest possible value of $\gcd(12a, 20b)$?


Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\).


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

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

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

import graph;

real i;

if(complexplane) {

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

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

} else {

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

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

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

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

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

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

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,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((2,1),red+5bp);
dot((4,3),red+5bp);
dot((6,5),red+5bp);
dot((8,7),red+5bp);
dot((1,8),red+5bp);
dot((3,7),red+5bp); // Modified
dot((5,4),red+5bp);
dot((7,3),red+5bp); // Modified
dot((9,0),red+5bp);

[/asy]

Larry writes the number 3 on his pinky finger. He then applies $f$ to 3 and writes the output on his ring finger. If Larry continues this process of applying $f$ and writing the output on a new finger, what number will Larry write on his tenth finger?


Simplify the expression \(3x + 6x + 9x + 12x + 15x + 20 + 4y\).

Solution: \boxed{45x + 20 + 4y}


In a particular region, the tax rate is calculated as follows: \( z\% \) tax is collected for an income of \( x \) thousand dollars, where \( z = x + 10 \). What income, in dollars, will yield the greatest take-home pay? (Take-home pay is the income minus the tax on that income.)


When the expression $6(x^3-2x^2+x-3)-5(x^4-4x^2+3x+2)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?


Let $A$ be the greatest common factor and let $B$ be the least common multiple of 18, 24, and 36. What is the value of $A + B$?


Call an integer $n$ oddly powerful if there exist positive odd integers $a$ and $b$, where $b > 1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $3000$?


Winnie has 37 kiwi lollipops, 108 raspberry lollipops, 8 lemon lollipops, and 254 mint lollipops. With no preference for flavor, she divides the lollipops equally among her 13 friends as fairly as possible. How many lollipops does Winnie keep for herself after the distribution?


A bag contains 12 red marbles and 8 blue marbles. Three marbles are selected at random and without replacement. What is the probability that two marbles are red and one is blue?


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 5 \\ -2 \end{pmatrix}.$

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


Two rays with common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=2$. What is the maximum possible length of $\overline{OB}$?


$ABCDEFGH$ is a rectangular prism with dimensions length $a$, width $b$, and height $c$. Points $A$, $B$, $C$, $D$ are on the ground, and points $E$, $F$, $G$, $H$ are directly above them, respectively. Given $a = 2$, $b = 2$, and $c = 3$ in some units, find $\sin \angle HAC$.


Simplify the expression $18 \cdot \frac{8}{15} \cdot \frac{1}{12}$.


Lee can make 18 cookies using 2 cups of flour and 1 cup of sugar. How much sugar is needed if he uses 5 cups of flour?

Solution: \boxed{2.5 \text{ cups of sugar}}


Let $O$ be the origin. There exists a scalar $m$ such that for any points $E,$ $F,$ $G,$ and $H$ satisfying the vector equation:
\[4 \overrightarrow{OE} - 3 \overrightarrow{OF} + 2 \overrightarrow{OG} + m \overrightarrow{OH} = \mathbf{0},\]
the four points $E,$ $F,$ $G,$ and $H$ are coplanar. Find the value of $m.$


In a triangle with integer side lengths, one side is twice as long as the second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?


In a school event, 540 students need to be seated in rows with an equal number of students in each row. There must be at least 12 rows and each row must have at least 20 students but no more than 30 students. What is the sum of all possible values for the number of students per row, $x$?


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


Express $\sin 3x + \sin 7x$ as a product of trigonometric functions.

Solution: \boxed{2 \sin 5x \cos 2x}


Let \( A \), \( M \), and \( C \) be nonnegative integers such that \( A+M+C=15 \). Determine the maximum value of the expression \[ A\cdot M\cdot C + A\cdot M + M\cdot C + C\cdot A. \]


Lisa decides to invest money into several accounts with annual compounds. Initially, she invests $500 in one account with a 2% interest rate, which she plans to add an additional $500 each year. The interest accumulates annually. How much will she have after 3 years?


What is the area enclosed by the graph of $|x| + |3y| = 12$?


A standard deck of 52 cards is modified to include an extra suit of stars ($\star$), making it 65 cards in total, each rank now having 5 suits. The deck is randomly arranged. What is the probability that the top card is a $\star$?


June and Julia live 2.5 miles apart. It takes June 10 minutes to ride her bike directly to Julia's house. At the same rate, how many minutes would it take June to ride the 4 miles from her own house to Bernard's house, considering a headwind slows her return trip by 20%?


Determine the number of values of $x$ for which the expression $\frac{x^2-4}{(x^2-2x-8)(x-4)}$ is undefined.


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two.  If $SP = SQ = 12$ and $SR = 8$, then what is the volume of pyramid $SPQR$?

Solution: \boxed{192 \text{ cubic units}}


Given a $3 \times 3$ chess board, how many ways can you place three distinct pawns on the board such that no row or column contains more than one pawn?


The sum of two positive integers $a$ and $b$ is 1008. What is the largest possible value of $\gcd(a, b)$?


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


Given two hyperbolas \[\frac{x^2}{9} - \frac{y^2}{16} = 1\] and \[\frac{y^2}{32} - \frac{x^2}{M} = 1\], find the value of $M$ if they have the same asymptotes.


Let $p,$ $q,$ $r$ be three distinct positive real numbers such that $p,$ $q,$ $r$ form a geometric sequence, and
\[\log_r p, \ \log_q r, \ \log_p q\] form an arithmetic sequence. Find the common difference of the arithmetic sequence.


If I roll a fair, regular six-sided die six times, what is the probability that I will roll the number $2$ exactly four times?


Evaluate the expression \[\frac{y^{1} \cdot y^{3} \cdot y^{5} \cdots y^{19}}{y^{2} \cdot y^{4} \cdot y^{6} \cdots y^{12}}\] if \( y = 3 \).


Determine how many integers satisfy the following inequalities:
\[
\begin{array}{cccc}
(1) & -4x & \geq & 2x + 13 \\
(2) & -3x & \leq & 15 \\
(3) & -5x & \geq & 3x + 23
\end{array}
\]


Suppose in a right triangle $PQR$, where $\angle Q = 90^\circ$, we know $\sin R = \frac{3}{5}$. If $PQ = 9$, find the length of $PR$.


**New Problem Statement**:
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 110^\circ$ and $\angle AOB = 140^\circ$. Determine the degree measure of $\angle ABC$.


What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence?


A sports conference has 16 teams split into two divisions of 8 teams each. How many games are there in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once?


A region is bounded by semicircular arcs constructed on each side of a rectangle whose length is $4/\pi$ and width is $2/\pi$. Calculate the perimeter of this region.


Twelve people sit down at random seats around a round table. Five of them are math majors, four are physics majors, and three are biology majors. What is the probability that all five math majors sit in consecutive seats?


One angle of a triangle is three times another, and the sides opposite these angles have lengths 10 and 20. Compute the length of the third side of the triangle.


Find the smallest three-digit palindromic prime where the hundreds digit is 3.


Find the value of $x$ such that $\sqrt{9 - 2x} = 8$.


If \(x + 5\) is a factor of \(cx^3 + 23x^2 - 5cx + 55\), find the constant \(c\).


If \( B = 3 + 2i \), \( Q = -5 \), \( R = -2i \), and \( T = 1 + 5i \), and you need to multiply the result of their sum by 2, find \( 2(B - Q + R + T) \).

Solution: \boxed{18 + 10i}


A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $7$ and $9$ through $20$ and has one blank face. When the two dice are rolled, what is the probability that the sum of the two numbers facing up will be $25$? Express your answer as a common fraction.


A box contains ten cards. Four of the cards are black on both sides, three cards are black on one side and red on the other, two cards are red on both sides, and one card is blue on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?


A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 4$?


Let $A,$ $B,$ and $C$ be points such that $\frac{AC}{AB} = \frac{1}{3}$ and $\frac{BC}{AB} = \frac{2}{3}$. If $B = (2, 5)$ and $C = (5, 8)$, then what is the sum of the coordinates of $A$?


Let $f(x) = x^4 + 10x^3 + 34x^2 + 40x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.


You are given a sequence of $60$ terms; each term has the form $Q + n$ where $Q$ is the product $2 \times 3 \times 5 \times \ldots \times 59$ of all prime numbers less than or equal to $59$, and $n$ takes successively the values $1, 2, 3, \ldots, 60$. Determine the number of primes appearing in this sequence.


If $\sqrt{9 + x} + \sqrt{16 - x} = 8$, what is the value of $(9 + x)(16 - x)$?


What is the 58th digit after the decimal point in the decimal representation of $\frac{1}{17}$?


A Senate committee is comprised of 6 Democrats and 6 Republicans. If all members are distinguishable, in how many ways can they sit around a circular table without restrictions? Assume seating arrangements are identical if one can be rotated into the other.


How many real numbers \( x \) satisfy the equation \[ |x-2| = |x-5| + |x-8| \]?


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


The line $y = -\frac{1}{2}x + 8$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


A sphere is inscribed in a cube, and the cube has a surface area of 96 square meters. A second cube is then inscribed within the sphere. Compute the surface area and the volume of the inscribed cube.

Solution: \boxed{\frac{64\sqrt{3}}{9} \text{ cubic meters}}


The vertices of a $6 - 8 - 10$ right triangle are the centers of three mutually externally tangent circles. Calculate the sum of the areas of these circles.


Two students, A and B, each decide to study in the library. They arrive independently at random times between 9 a.m. and 11 a.m., and each stay for exactly $n$ minutes. The probability that one arrives while the other is still in the library is $50\%$. Express $n$ in the form $n = x - y\sqrt{z}$, where $x, y,$ and $z$ are positive integers, and $z$ is not divisible by the square of any prime. Find $x + y + z$.


Determine $x + y + z$ if $x$, $y$, and $z$ are positive numbers and satisfy the equations $xy = 40$, $xz = 80$, and $yz = 160$.


Given vectors $\mathbf{v}$, $\mathbf{w}$, and $\mathbf{u}$ such that $\|\mathbf{v}\| = 5$, $\|\mathbf{w}\| = 8$, $\mathbf{v} \cdot \mathbf{w} = 20$, $\|\mathbf{u}\| = 6$, and $\mathbf{v} \cdot \mathbf{u} = 15$, find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$ and $\|\operatorname{proj}_{\mathbf{u}} \mathbf{v}\|$.


Suppose that $x$ is inversely proportional to $y$. If $x = 5$ when $y = 15$, find $x$ when $y = 30$.


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


Find the number of real solutions to
\[\cos (\arcsin( \tan (\arccos x) ) ) = x.\]

Solution: \boxed{x = 1}


John and Chris each roll a sheet of 5-inch by 7-inch paper to form a cylindrical tube. John tapes the two 7-inch sides together without overlap. Chris tapes the two 5-inch sides together without overlap. What is $\pi$ times the positive difference of the volumes of the two tubes?


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

Solution: \boxed{-1416 + 6\sqrt{3}}


A hemisphere with radius 10 cm is submerged in a liquid such that the top 5 cm of the hemisphere is above the liquid. Including the area of its circular top, what is the total exposed surface area? Express your answer in terms of $\pi$.


What is the largest six-digit integer whose digits have a product equal to the product $(8)(7)(6)(5)(4)(3)(2)(1)$?


Tim plans to secretly switch Nathan's spoon every day for a week as a joke. On Monday, he convinces Peter to do it. On Tuesday, he has three friends (Sam, Jane, or Luke) who could be convinced to do it. On Wednesday, none of these four can participate, but he has six new friends who might help. On Thursday, none of these six are available, but four others are willing. Finally, on Friday, Tim is considering whether to do it himself or to convince one of two new friends (Anne or Bob). How many different combinations of people could Tim involve to switch the spoon each day?


The solution of the equation $5^{x+7} = 6^x$ can be expressed in the form $x = \log_b 5^7$. What is $b$?


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


The following line is parameterized, so that its direction vector is of the form $\begin{pmatrix} 3 \\ b \end{pmatrix}.$  Find $b.$

[asy]
unitsize(0.4 cm);

pair A, B, L, R;
int i, n;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0),Arrows(6));
draw((0,-8)--(0,8),Arrows(6));

A = (-3,3);
B = A + (4,3);
L = extension(A, B, (-8,0), (-8,1));
R = extension(A, B, (0,8), (1,8));

draw(L--R, red);

label("$x$", (8,0), E);
label("$y$", (0,8), N);
[/asy]


Determine the greatest common divisor of $12345$ and $6789$.

Solution: \boxed{\gcd(12345, 6789) = 3}


For how many values of the digit $A$ is it true that $72$ is divisible by $A$ and $273{,}1A2$ is divisible by $4$?


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

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

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

c) When divided by 7, the remainder is 6.


Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6.


In tetrahedron \(PQRS,\)
\[\angle PRS = \angle PSQ = \angle QSR = 90^\circ.\] Also, \(a = \cos \angle PQS\) and \(b = \cos \angle PSR.\) Express \(\sin \angle PQS\) in terms of \(a\) and \(b\).

Solution: \boxed{\sin \angle PQS = \sqrt{1 - a^2}}


**Fifty teams** play a tournament where every team plays every other team exactly once. No ties occur, and each team has a $50\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Find $\log_2 n$.


For how many non-negative real values of $x$ is $\sqrt{256-\sqrt[4]{x}}$ an integer?


Let $O$ be the origin and $P = (1,1,1)$. A variable plane passes through $P$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A,$ $B,$ and $C,$ respectively. If the normal vector to the plane is in the direction of vector $(1,1,1)$, determine the value of
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2},\]
where $(p,q,r)$ is the centroid of triangle $ABC.$


In a pentagon $ABCDE$, two interior angles $A$ and $B$ are $75^{\circ}$ and $95^{\circ}$ respectively. Angles $C$ and $D$ are not equal, with angle $D$ being $10^{\circ}$ more than angle $C$. The fifth angle $E$ is $20^{\circ}$ more than twice $C$. Find the measure of the largest angle.


Let $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where $a, b, c, d$ are real numbers. If $\mathbf{A} + 2\mathbf{A}^{-1} = \mathbf{0}$, find $\det \mathbf{A}$.


A study reported that 55% of men and 75% of women surveyed show a positive attitude towards adopting new technology for environmental sustainability. This study surveyed 150 men and 850 women. What is the overall percentage of people surveyed who showed a positive attitude?


Let $\alpha$ and $\beta$ be complex numbers such that $\alpha + \beta$ and $i(\alpha - 3 \beta)$ are both positive real numbers. If $\beta = 2 + 3i,$ compute $\alpha.$

Solution: \boxed{6 - 3i}


Let $a,$ $b,$ $c,$ and $k$ be nonzero real numbers with $k > 0$. Find the number of real roots of the equation:
\[\begin{vmatrix} x & kc & -kb \\ -kc & x & ka \\ kb & -ka & x \end{vmatrix} = 0.\]


A pyramid $PABCD$ has a rectangular base $ABCD$. Let $AB = 10$, $BC = 5$, $\overline{PA}$ is perpendicular to both $\overline{AD}$ and $\overline{AB}$, and the distance from vertex $P$ to vertex $B$ (line $PB$) is 20. Determine the volume of the pyramid $PABCD$.

Solution: \boxed{\frac{500\sqrt{3}}{3} \text{ cubic units}}


Gabriella created a Venn diagram showing the number of students in her physics class who like three different genres of music: Rock, Jazz, and Classical. There are 40 students in her class. Also, she knows $\frac{1}{4}$ of students like Rock, $\frac{1}{5}$ like Jazz, 8 students like other genres (not these three), and 6 students do not like any of these three genres. How many students like all three genres (Rock, Jazz, and Classical)? [asy]unitsize(50);
import graph;
pair A = (0,-1); pair B = (sqrt(3)/2,1/2); pair C = (-sqrt(3)/2,1/2);
draw(Circle(A,1.2) ^^ Circle(B,1.2) ^^ Circle(C,1.2));
label("12",A); label("4",B); label("5",C); label("$z$",(0,0)); label("$w$",(B+C)/2); label("$y$",(A+B)/2); label("$x$",(A+C)/2);
label("Classical",1.5C,C); label("Other Genres",2B,C); label("Rock", 1.7A,A);[/asy]


John draws a regular six-pointed star (hexagram) in the sand. At each of the 12 points, he places one of twelve different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?


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


A function \(f\) is defined by \(f(z) = -i\overline{z}\), where \(i^{2} = -1\) and \(\overline{z}\) is the complex conjugate of \(z\). Find how many values of \(z\) satisfy both \(|z| = 3\) and \(f(z) = z\).


What is the product of the greatest common divisor of \(210\) and \(4620\) and three times the least common multiple of \(210\) and \(4620\)?


A card is chosen at random from a standard deck of 52 cards, which includes two jokers making the total 54 cards. Then, the card is replaced and another card is chosen. What is the probability that at least one of the cards is a heart or a joker?


For how many integer values of \( n \) between 1 and 180 inclusive does the decimal representation of \( \frac{n}{180} \) terminate?


Let $a$, $b$, $c$, and $d$ be positive real numbers. Calculate the smallest possible value of \[(a+b+c+d)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+d}+\frac{1}{c+d}\right).\]


For some real numbers $a$ and $b$, the equation $7x^3 + 3ax^2 + 6bx + 2a = 0$ has three distinct positive roots. If the sum of the base-3 logarithms of the roots is 5, and the sum of the squares of the roots is 49, find the value of $a$.


Integrate the product $(x^2 + 2x - 3)(4x^2 - x + 1)$ from $x=0$ to $x=2$.


In a class of 40 students, 20 students have dogs, $\frac{1}{4}$ of the students have cats, and 10 students have other types of pets. Five students don't have any pets at all. In a Venn diagram, the number of students who have only dogs is labeled as 15, only cats as 4, and only other pets as 5. The numbers of students who have exactly two types of pets are labeled as follows: dogs and cats ($x$), dogs and other pets ($y$), and cats and other pets ($z$). How many students have all three types of pets?


In Mrs. Leung's English class, students are engaged in reading a 900-page novel. Among them, Emma, Liam, and Noah decide to form a team reading group. Emma reads a page in 15 seconds, Liam reads a page in 45 seconds, and Noah reads a page in 30 seconds. Emma and Liam, both holding a copy of the book, plan to split the novel such that Emma starts from page 1 and reads up to a certain page, and Liam starts from the following page and finishes the book. Noah, however, will read the last 200 pages of whatever portion Liam reads. Find the last page that Emma should read so that both she and Liam, excluding the pages read by Noah, spend the same amount of time reading.


A right circular cone is inscribed in a rectangular prism. The rectangular base of the prism has dimensions 3 units by 4 units, and the height of the prism is $2h$ where $h$ is the height of the cone. Find the ratio of the volume of the cone to the volume of the prism.


In a modified version of SHORT BINGO, a $5\times5$ card is also filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. However, for the first column, 5 distinct numbers are chosen from the set $1-15$. How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card? (The placement on the card matters, so the order of the numbers matters.)


Let \( g(x) \) be a function such that for all real numbers \( x \) and \( y \):
\[ g(xy) + x = xg(y) + g(x). \]
If \( g(-1) = 5 \), determine \( g(-1002) \).


Find the domain of the expression $\frac{\sqrt{x-3}}{\sqrt{7-x} \cdot (x-1)}$.

Solution: \boxed{[3, 7)}


Fully factor the following expression: $3x^3 - 27x$


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


How many positive factors of 180 are perfect squares?


For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\]


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

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


If $y\%$ of five-digit numbers have at least two digits that are the same (the repeated digits do not need to be adjacent), what is $y$? Express your answer as a decimal to the nearest tenth.


Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$.


The graph of the line $2x - y = c$ is a perpendicular bisector of the line segment from $(2, 5)$ to $(8, 11)$. What is the value of $c$?


An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?


Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws six socks. What is the probability that she ends up with exactly two pairs, each of a different color?


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


If $a$, $b$, and $c$ are positive integers satisfying $ab+c = bc+a = ac+b = 31$, what is the value of $a+b+c$?


What is the smallest whole number that has a remainder of 2 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 3 when divided by 5?


A repair crew needs to reach a point $C$ at the center of a circular fountain. They place a 20-foot plank from point $A$ to $B$ and then a 12-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. What is the area of the circular base of the fountain if $CD$ is perpendicular to $AB$? Express your answer in terms of $\pi$.

Solution: \boxed{244\pi \text{ square feet}}


Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{48}$ and $\frac{\pi}{72}$ radians, respectively, with the positive $x$-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{11}{50}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.


Isosceles triangle $PQR$ has an area of 180 square inches and is cut by line $\overline{ST}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $PQR$ from $P$ is 30 inches, what is the length of $\overline{ST}$?

Solution: \boxed{6 \text{ inches}}


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


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


At a new airport, there are sixteen gates arranged in a circle with exactly $75$ feet between adjacent gates. Dina’s departure gate is assigned at random. After waiting at that gate, Dina is told the departure gate has been changed to a different gate, again at random. Calculate the probability that Dina walks $300$ feet or less to the new gate, given the new setup. Let the probability be a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Bob's new password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number. What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a single-digit prime number?


Solution: \boxed{\left(\frac{5}{2}, 3\right)}


Let $\mathbf{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}.$ Calculate the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}$. Also, determine the vector $\mathbf{a} + \mathbf{b}$.


Leila enjoys juggling and coding in a day. We can model her utility as $$\text{Utility} = \text{hours of juggling} \times \text{hours of coding}.$$ Leila juggles for $s$ hours and codes for $12 - s$ hours on Wednesday. On Thursday, to keep herself engaged, she juggles for $6-s$ hours and codes for $s+4$ hours. If her utility remains the same on both days, determine the value of $s$.


If $\frac{5}{2} (x + y + z + w) = 25$, what is the average of $x$, $y$, $z$, and $w$?


In triangle \( ABC \), altitudes \( AD \), \( BE \), and \( CF \) intersect at the orthocenter \( H \). If \( \angle ABC = 58^\circ \) and \( \angle ACB = 20^\circ \), find the measure of \( \angle BHC \), in degrees.


Find the positive value of $x$ which satisfies
\[
\log_4 (x - 1) + \log_{\sqrt{4}} (x^2 - 1) + \log_{\frac{1}{4}} (x - 1) = 2.
\]


The consecutive angles of a pentagon form an arithmetic sequence. If the largest angle measures $140^\circ$, what is the measure of the smallest angle?


What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome and that palindrome is at least 200 more than 4321?


What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-2|\le1$?


Find \(q(x)\) if the graph of \(\frac{2x^3 - 3x^2 - 4x + 6}{q(x)}\) has vertical asymptotes at \(1\) and \(3\), no horizontal asymptote, and \(q(2) = 10\).

Solution: \boxed{-10x^2 + 40x - 30}


Piravena needs to travel from city $A$ to city $B$, from $B$ to city $C$, and from $C$ back to $A$ completing a triangle. The triangle formed by the cities is right-angled at $C$, with $CA = 4000$ km and $AB = 4500$ km. Transportation choices are either bus or airplane, where the bus costs $\$0.20$ per kilometer, and the airplane costs $\$120$ booking fee plus $\$0.12$ per kilometer. As usual, Piravena will choose the cheapest option for each leg of the journey. What is the total cost of her trip?


The positive four-digit integers that use each of the four digits $2,$ $3,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $20^\text{th}$ integer in the list?


A line contains the points $(8,10)$, $(0, k)$, and $(-8, 3)$. What is the value of \( k \)?


Let there be a constant \(d\), so that among all chords \(\overline{AB}\) of the parabola \(y = x^2\) passing through \(D = (0,d),\) the sum
\[
s = \frac{1}{AD^2} + \frac{1}{BD^2}
\]
is a fixed constant. Find the constant \(s\).


How many integers are solutions to the equation \((x-3)^{(36-x^2)}=1\)?


Let $a \equiv 23 \pmod{60}$ and $b \equiv 95 \pmod{60}$. Determine the smallest integer $n$ within the range $\{150,151,152,\ldots,190,191\}$ such that $$a - b \equiv n \pmod{60}.$$


What is the digit in the thousandths place of the decimal equivalent of $\frac{7}{32}$?


Suppose we roll two standard fair 6-sided dice. What is the probability that the sum of the numbers rolled is a prime number?


Consider the function $g(x) = 3x^3 - 2x^2 + x + 1$. Evaluate $3g(2) + 2g(-2)$.


What is the maximum value of the expression $-7r^2 + 50r - 20$ for real $r$?


Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\]


Let \( a, b, c \) be real numbers satisfying \( 9a^2 + 4b^2 + 36c^2 = 4 \). Find the minimum value of \( 3a + 6b + 12c \).


If $n$ is a positive integer, let $g(n)$ be the sum of the digits of $2^n + 3^n$. What is the smallest positive integer $n$ such that $g(n) > 15$?


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{77}$ and $a^{-1}$ is also undefined $\pmod{88}$?


The arithmetic mean of 15 scores is 85. When the highest and lowest scores are removed, the new mean becomes 88. If the highest of the 15 scores is 99, what is the lowest score?


Regular pentagon $ABCDE$ and regular octagon $AEFGHIJK$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of the exterior angle $DEF$?


Monica tosses a fair 8-sided die. If the roll is a prime number, then she wins that amount in dollars. If the roll is composite, she loses that amount in dollars. If she rolls a 1, she neither wins nor loses anything, and if she rolls an 8, she is awarded an extra $5. What is the expected value of her winnings on one die toss?


A basketball team consists of 16 players, including a set of 4 quadruplets: Ben, Bill, Bob, and Brad. In how many ways can we choose 5 starters if at most two of the quadruplets are in the starting lineup?


A number 198 is multiplied by 53. The product is then required to be expressed in base-8. Before finding the product, subtract 3 from 198 and add 7 to 53. What is the unit's digit of this new product in base-8 representation?


A right pyramid has a square base measuring 12 cm on each side. Its peak is 15 cm above the center of the base. Calculate the sum of the lengths of the pyramid's eight edges.

Solution: \boxed{48 + 12\sqrt{33}}


The sum of three numbers $a, b$, and $c$ is 120. If we decrease $a$ by 9, we get the value $M$. If we increase $b$ by 9, we also get the value $M$. If we multiply $c$ by 9, we obtain the value $M$. What is the value of $M$?


How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$? (Consider $B$, $C$, and $D$ as consonants. Words can be any combination of letters, not restricted to valid English words, and letters may be used more than once.)


What is the area, in square units, of a trapezoid bounded by the lines \(y = 2x\), \(y = 6\), \(y = 3\) and the \(y\)-axis?


On a whiteboard, BIOLOGY is written with magnetic letters, one letter per magnet. Two vowels and four consonants are randomly selected and placed in a box. If all the O's are indistinguishable, how many distinct collections of letters could be put in the box?


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


Four aluminum cans are required to make a new can. How many new cans can eventually be made from 256 aluminum cans? Subsequent recycling of the created cans should be considered for making newer cans. Do not include the original 256 cans in your count.


In a circle with center $O$, triangle $ABC$ is circumscribed such that $\angle BOC = 110^\circ$ and $\angle AOB = 150^\circ$. Determine the degree measure of $\angle ABC$.


What is the greatest divisor of 540 that is smaller than 60 and also a factor of 180?


Find the arithmetic mean of the prime numbers in this list: 33, 37, 39, 41, 43.


On a 24-hour clock, an elapsed time of six hours looks the same as an elapsed time of 30 hours. Because of this, we can say that six hours is "clock equivalent" to its square number of hours. What is the least whole number of hours that is greater than 6 hours and is "clock equivalent" to its square number of hours?


Determine the integer \( x \) for which the following equation holds true:
\[ 1 \cdot 2023 + 2 \cdot 2022 + 3 \cdot 2021 + \dots + 2022 \cdot 2 + 2023 \cdot 1 = 2023 \cdot 1012 \cdot x. \]


Let $n$ be a positive integer. If $b\equiv (5^{3n}+4)^{-1}\pmod{11}$, what is the remainder when $b$ is divided by 11?


Express the sum as a common fraction: $.3 + .04 + .005 + .0006 + .00007.$


The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:3:6$ that can be written in the form\[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]Find the sum of all possible values of $n+k$.


How many ways are there to put 7 balls into 4 boxes if the balls are not distinguishable and neither are the boxes?


Ms. Hamilton's eighth-grade class has expanded its annual basketball tournament. This year, 25 teams are interested in participating, but the main tournament can only accommodate 16 teams. To qualify, there will be a preliminary single-game elimination round among 9 selected teams to reduce their number to 1. The winning team will join the other 16 originally selected teams to make it a total of 16 competing teams. Then, a traditional single-elimination tournament is held for these 16 teams. How many games in total will be played to determine the winner?


Carlos is the 75th best and also the 75th worst student in his class. How many students are in Carlos's class?


Define a function $A(m, n)$ by
\[
A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned}
\right.
\]
Compute $A(3, 2)$.


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


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


Determine the value of $x$ for which $3^{4x^2 - 9x + 5} = 3^{4x^2 + 3x - 7}$. Express your answer as a common fraction.


The measures of angles \(A\) and \(B\) are both positive, integer numbers of degrees. The measure of angle \(A\) is a multiple of the measure of angle \(B\), and angles \(A\) and \(B\) are supplementary angles. Determine how many measures are possible for angle \(A\).


What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 24] = 120$?


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

Solution: \boxed{\begin{pmatrix} \frac{22}{5} \\ -\frac{6}{5} \end{pmatrix}}


A flag consists of three horizontal strips of fabric, each one being a solid color, chosen from red, white, blue, green, or yellow. No two adjacent strips can have the same color, and the top and bottom strips must also be different colors. How many distinct flags are possible?


I have six bananas and nine pears. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I construct? The bananas are identical, and the pears are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.


Find the modular inverse of $14$, modulo $15$.

Express your answer as an integer from $0$ to $14$, inclusive.


Find the coefficient of the $x^4$ term in the expansion of the product $(2x^3 + 5x^2 - 3x)(3x^3 - 8x^2 + 6x - 9)$.


In the diagram, each of two concentric circles is divided into two equal areas, and $O$ is the center of both circles. The area of the outer circle is $100\pi$. What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),5,180,360)--cycle,mediumgray);fill(Arc((0,0),2.5,0,180)--cycle,mediumgray);
draw(Circle((0,0),5));
draw(Circle((0,0),2.5));

dot((0,0)); label("$O$",(0,0),N);
draw((-5,0)--(5,0)); draw((-2.5,0)--(2.5,0));
[/asy]


The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex together?


The equation $x^2 + bx = -21$ has only integer solutions for $x$. If $b$ is a positive integer, what is the greatest possible value of $b$?


Find the smallest integral value of $y$ which solves: $\frac{1}{4}<{\frac{y}{7}}<{\frac{2}{3}}$


What is the smallest possible number of whole 3-by-4 non-overlapping rectangles needed to cover a square region exactly, without extra overhangs and without gaps?


**A triangle has vertices at coordinates (1,3), (4,7), and (4,3). Calculate the number of units in the length of the longest side and the perimeter of the triangle.**


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


Solution: \boxed{x = -\frac{26}{9}}


Simplify
\[
\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.
\]

Solution: \boxed{2 \csc x}


In the diagram, if $\angle PQR = 60^\circ$, what is the measure of $\angle PMN$? Assume $\triangle PMN$ is isosceles with $PM=PN$, and $PR=RQ$.
[asy]
size(6cm);
pair p = (0, 0); pair m = dir(180 - 30); pair n = dir(180 + 30); pair r = 1.3 * dir(30); pair q = 2 * 1.3 * Cos(60) * dir(-30);
label("$M$", m, W);
label("$R$", r, E);
label("$P$", p, 1.5 * S);
label("$N$", n, E);
label("$Q$", q, E);
draw(m--q--r--n--cycle);
add(pathticks(m--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
add(pathticks(r--q, 2, spacing=0.9, s=4));
[/asy]


Choose two distinct integers, $a$ and $b$, from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. What is the probability that $ab - a - b$ is odd?


If $\sqrt{3\sqrt{t-3}} = \sqrt[4]{8 - t}$, then find $t$.


The graph of \[\frac{x^2}{c^2} + \frac{y^2}{d^2} = 1\] has its foci at $(0,\pm 5),$ while the graph of \[\frac{x^2}{c^2} - \frac{y^2}{d^2} = 1\] has its foci at $(\pm 7,0).$ Compute the value of $|cd|.$


In triangle $XYZ$, $XY = 25$ and $XZ = 20$. Find the largest possible value of $\tan Y$.


Given the expression $15n + 5$ and $9n + 4$, as $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of these two expressions when $n$ is congruent to 1 modulo 3?


A gift shop now offers ten varieties of wrapping paper, five colors of ribbon, and six types of gift cards. Using one variety of wrapping paper, one color of ribbon, and one type of gift card, how many combinations are possible if each combination must include either a silver or gold ribbon among the choices?


*Determine the sum of all single-digit replacements for $y$ such that the number $18{,}y52$ is divisible by 8.*


Emily has 8 blue marbles and 7 red marbles. She randomly selects a marble, notes its color, and returns it to the bag. She repeats this process 6 times. What is the probability that she selects exactly three blue marbles?


Let $x$, $y$, and $z$ be nonzero real numbers. Determine the minimum value $m$ and the maximum value $M$ of
\[
\frac{|x + y + z|}{|x| + |y| + |z|}
\]
and compute $M - m.$


Five standard six-sided dice are rolled. We determine there are exactly two pairs of dice showing the same numbers but different from each other. The fifth die, which does not match any of the numbers in the two pairs, is re-rolled. What is the probability that after re-rolling this die, at least three of the five dice show the same value?


A particle is located on the coordinate plane at $(8,0)$. Define a "move" for the particle as a counterclockwise rotation of $\frac{\pi}{3}$ radians about the origin followed by a translation of $8$ units in the positive $x$-direction. Find the particle's position after $90$ moves.

Solution: \boxed{(8, 0)}


Ted is solving the equation by completing the square: $$64x^2+48x-36 = 0.$$ He aims to write the equation in a form: $$(ax + b)^2 = c,$$ with \(a\), \(b\), and \(c\) as integers and \(a > 0\). Determine the value of \(a + b + c\).


A ball bounces back up $\frac{3}{4}$ of the height from which it falls. If the ball is dropped from a height of $320$ cm, after how many bounces does the ball first rise less than $40$ cm?


A line segment has endpoints at $A(1, 3)$ and $B(7, -1)$. The segment is extended from $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates of point $C$? Express your answer as an ordered pair.

Solution: \boxed{(10, -3)}


A club has 30 members and needs to select 4 officers: president, vice-president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only serve together, either both as officers or neither. How many ways can the club choose its officers?


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


Calculate the total number of days in the years 2010 through 2015 and identify how many of those years were leap years.


The graph of the line \(x-y=c\) is the perpendicular bisector of the line segment from \((2,4)\) to \((6,8)\). What is the value of \(c\)?


Calculate the value of $3^2 \times 5 \times 7^2 \times 11$.


I have a bag with $8$ marbles numbered from $1$ to $8.$ Mathew has a bag with $16$ marbles numbered from $1$ to $16.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles is a prime number and equals the number on his marble?


Let $D$ be the circle with the equation $x^2 - 8x - 4y = -y^2 + 2y + 13$. Determine the values of $a$, $b$, and $r$ for the center $(a, b)$ and radius $r$ of $D$, and compute $a+b+r$.

Solution: \boxed{7 + \sqrt{38}}


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


The terms of the sequence \((a_i)\) are defined by \(a_{n + 2} = \frac{a_n + 2017}{1 + a_{n + 1}}\) for \(n \geq 1\), and are positive integers. Determine the minimum possible value of \(a_1 + a_2\).


In triangle $DEF$, the measure of $\angle D$ is $85$ degrees. The measure of $\angle E$ is $15$ degrees more than four times the measure of $\angle F$. What is the measure, in degrees, of $\angle F$?

Solution: \boxed{16 \text{ degrees}}


Find the slope $m$ of the asymptotes for the hyperbola given by the equation \[\frac{x^2}{144} - \frac{y^2}{81} = 1.\]


Find the integer \( n \), where \(-120 \le n \le 120,\) such that \( \sin n^\circ = \cos 678^\circ. \)


How many positive factors of 240 are perfect squares?


In trapezoid $PQRS$, sides $\overline{PQ}$ and $\overline{RS}$ are parallel, and it is given that $\angle P = 3\angle S$ and $\angle R = \angle Q$. Find $\angle P$.

Solution: \boxed{\angle P = 135^\circ}


A circle with a radius of 6 cm is tangent to the two shorter sides and one of the longer sides of a rectangle. The area of the rectangle is three times the area of the circle. Calculate the length of the shorter side of the rectangle in centimeters, expressing your answer in terms of $\pi$.


What is the remainder when the sum of the seven consecutive odd integers starting from $11063$ is divided by $16$?


The base-10 numbers 301 and 52 are multiplied. The product is then written in base-7. What is the units digit of the base-7 representation?


The Northwest High School math club, which has 150 members, regularly conducts preparation sessions for math competitions. Each member consumes an average of 3 candies during the session. Alice and Bob are in charge of providing Kit rain's colorful candies this year. Their candy package, which serves exactly 18 candies, contains these items:

- 5 sticks of gum
- 2 lolipops
- 9 jellybeans
- 2 packets of 6 chocolates each

Alice and Bob only prepare full packages. This year, due to a parallel music festival, attendance is predicted to drop by 30%. How many full packages of candies should Alice and Bob prepare?


It is currently $10\!:\!00\!:\!00 \text{ a.m.}$ What time will it be in $12345$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)


A region is bounded by semicircular arcs constructed on the sides of a square, where each side measures $1/\pi$. Calculate the perimeter of this region.


Let $a$, $b$, $c$, $p$, $q$, and $r$ be real numbers that satisfy the following three equations:
\begin{align*}
19p + bq + cr &= 0,\\
ap + 29q + cr &= 0,\\
ap + bq + 56r &= 0.
\end{align*}
Suppose that $a \neq 19$ and $p \neq 0$. Find the value of
\[
\frac{a}{a - 19} + \frac{b}{b - 29} + \frac{c}{c - 56}.
\]


In the diagram, rectangle $ABCD$ is divided into four identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?
[asy]
size(4cm);
pair a = (0, 1); pair b = (4, 1); pair c = (4, 0); pair d = (0, 0);
draw(a--b--c--d--cycle);
draw(shift(1) * (a--d)); draw(shift(2) * (a--d)); draw(shift(3) * (a--d));
label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW);
[/asy]

Solution: \boxed{1024 \text{ square centimeters}}


What is the area, in square units, of a triangle with sides of $5, 4, 3$ units? Express your answer in simplest radical form.


According to the updated table below, calculate the median salary among the employees at this company:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$130,000$\\\hline
Vice-President&6&$\$95,000$\\\hline
Director&12&$\$80,000$\\\hline
Associate Director&10&$\$55,000$\\\hline
Administrative Specialist&30&$\$25,000$\\\hline
\end{tabular}


Find the quotient when \(x^6 + 5\) is divided by \(x - 1\).

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


Find the greatest common divisor of 4560, 6080, and 16560.


Find the difference of $652_8$ and $274_8$ in base 8.


An ellipse has foci at $(0, 3)$ and $(4, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair.

Solution: \boxed{\left(\frac{56}{11}, 0\right)}


Calculate the sum of the squares of all integers between -5.7 and 7.1.


Six positive integers from a list of nine positive integers are $3, 4, 6, 5, 9, 7$. What is the largest possible value of the median of this list of nine positive integers?


Find the number of ordered pairs of integers \( (a, b) \) with \( 1 \leq a \leq 50 \) where \( a \) is odd, and \( b \) is a non-negative even number, such that the polynomial \( x^2 + ax + b \) can be factored into the product of two linear factors with integer coefficients.


Suppose $p$, $q$, $r$, and $s$ are integers satisfying:

1. $p - q + r = 7$,
2. $q - r + s = 8$,
3. $r - s + p = 4$, and
4. $s - p + q = 3$.

What is the value of $p + q + r + s$?


Let $p$ and $q$ be positive integers such that \[\frac{4}{7} < \frac{p}{q} < \frac{7}{12}\] and $q$ is as small as possible. What is $q-p$?


One of the asymptotes of a hyperbola has equation $y = 2x$. The foci of the hyperbola have the same $x$-coordinate, which is $4$. Determine the equation of the other asymptote of the hyperbola, and present your answer in the form "$y = mx + b$".

Solution: \boxed{y = -2x + 16}


Simplify $\frac{\sqrt{8}}{\sqrt{5}} \cdot \frac{\sqrt{18}}{\sqrt{7}} \cdot \frac{\sqrt{12}}{\sqrt{11}}$ and rationalize the denominator of the resulting fraction.


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

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


A trapezoid has consecutive angles that form an arithmetic sequence. If the smallest angle measures \(45^\circ\), what is the measure of the largest angle?


The equation $y = -8t^2 - 12t + 72$ describes the height (in feet) of a ball thrown downward at 12 feet per second from a height of 72 feet from the surface of Venus. Determine in how many seconds the ball will hit the ground. Express your answer as a decimal rounded to the nearest hundredth.


Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 6.$ Find the minimum value of
\[\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.\]


A $3\times 5$ rectangle and a $4\times 6$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?


The modified Lucas sequence is defined as 2, 1, 3, 4, 7, $\ldots$ where the first term is 2, the second term is 1 and each term thereafter is the sum of the two preceding terms. What is the remainder when the $150^{\text{th}}$ term of this sequence is divided by 5?


Find the maximum number of real roots to a polynomial of the form
\[x^n - x^{n-1} + x^{n-2} - \dots \pm 1 = 0,\]
where the signs alternate, and $n$ is a positive integer.


When a polynomial $q(x)$ is divided by $x - 2,$ the remainder is 3. When $q(x)$ is divided by $x + 3,$ the remainder is 2. Find the remainder when $q(x)$ is divided by $(x - 2)(x + 3).$

Solution: \boxed{\frac{1}{5}x + \frac{13}{5}}


Find $x$ such that $\lfloor x \rfloor + x = \dfrac{19}{4}$. Express $x$ as a common fraction.


A line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \end{pmatrix}.
\]
A second line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} + u \begin{pmatrix} 4 \\ 2 \end{pmatrix}.
\]
If $\theta$ is the acute angle formed by the two lines, then find $\cos \theta.$


Find the maximum value of
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_1,
\]
over all real numbers $\theta_1, \theta_2, \theta_3, \theta_4$.


Given that $\cos Q = \frac{3}{5}$ in the triangle below, and that the length of $QP$ is 15 units, find the length of $QR$.

[asy]
pair P,Q,R;
P = (0,0);
Q = (6,0);
R = (0,6*tan(acos(3/5)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,18));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$15$",Q/2,S);
[/asy]


Find the radius of the circle with the equation $4x^2 + 8x + 4y^2 - 16y + 20 = 0.$


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


Evaluate the modulus of the product of $(7 - 4i)$ and the conjugate of $(3 + 2i)$.


How many positive integers, not exceeding 150, are multiples of 2 or 5 but not 15?


Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 2020 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 2020 \right).\]


How many integers $n$ satisfy $(n-3)(n+5)<0$?


Calculate the fraction of squares to rectangles formed by the lines on a $9\times9$ checkerboard in its simplest form. Then, find the sum of the numerator and denominator of this fraction.


A sports league consists of 16 teams divided into 2 divisions of 8 teams each. Throughout the season, each team plays every other team in its own division three times and every team in the other division twice. How many games are in a complete season for the league?


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


What is the smallest positive four-digit integer equivalent to 2 mod 11?


Find the smallest three-digit number such that both the number and its reversed version are composite.


How many positive divisors does 8! have?


Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$".

Solution: \boxed{\sqrt[9]{81} + 2}


Keisha's basketball team, after another year, is deciding on a new uniform again. This year, the seventh-graders will pick the color of the shorts (black, gold, or blue) and the eighth-graders will pick the color of the jersey (black, white, or gold). However, they cannot pick a black jersey this year as it was the color last year. With each possible color equally likely to be chosen for shorts and remaining jersey options, what is the probability that the shorts will be a different color than the jersey? Express your answer as a common fraction.


The diagonal of a square is 4 inches long. Calculate the side length of the square and its area.

Solution: \boxed{8 \text{ square inches}}


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


What is the smallest integer $y$ such that $\frac{y}{4} + \frac{3}{7} > \frac{2}{3}$?


The operation $\odot$ is now defined as $a \odot b = a + \frac{3a^2}{2b}$. What is the value of $9 \odot 3$?


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

Solution: \boxed{x^2 + 1}


How many numbers between 100 and 499 have distinct digits?


Consider the given functions:
$$\begin{array}{ccc}
f(x) & = & 6x^2 + 4x - \frac{1}{x} + 2 \\
g(x) & = & x^2 - k + 3x
\end{array}$$
If $f(3) - g(3) = 5$, what is the value of $k$?


Find the coefficient of $x^2$ in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(6x^3 + 7x^2 + 8x + 9).$$


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


Define a new operation $\$N = 2(N+1)$. Calculate $\#(\$(\$(\$5)))$ where $\#N = .5N + 1$ as defined earlier.


When the product $(5x + 3y + 2)(2x + 5y + 3)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?


Faces \(ABC\) and \(BCD\) of tetrahedron \(ABCD\) meet at an angle of \(45^\circ\). The area of face \(ABC\) is \(150\), the area of face \(BCD\) is \(90\), and \(BC = 12\). Find the volume of the tetrahedron.


What is the remainder when the sum of $10 - 2n$ and $4n + 2$ is divided by $6$?


Participation in the local basketball league is 15% higher this year compared to last year. From last year to this year, the number of males increased by 10% and the number of females increased by 25%. Last year, there were 30 males participating in the league. What fraction of the league's participants is female this year? Express your answer as a common fraction.


If $x^2+bx+25$ has at least one real root, find all possible values of $b$. Express your answer in interval notation.

Solution: \boxed{(-\infty, -10] \cup [10, \infty)}


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


There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O?


Let $a_1, a_2, \dots$ be a sequence of positive real numbers such that
\[ a_n = 13a_{n-1} - 2n \] for all $n > 1.$ Find the smallest possible value of $a_1.$


Eugene, Brianna, Katie, and Alex are going on a run. Eugene runs at a rate of 6 miles per hour. If Brianna runs $\frac{3}{4}$ as fast as Eugene, and Katie runs $\frac{4}{3}$ as fast as Brianna, how fast does Alex run if he runs $\frac{5}{6}$ as fast as Katie?

Solution: \boxed{5 \text{ miles per hour}}


In a triangle, the area is numerically one-fourth of the square of the perimeter. What is the radius of the inscribed circle if the perimeter is 24 units?


A circle with center $A$ has radius $9$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $UV$ is the common external tangent to circle $A$ and circle $B$ at points $U$ and $V$, respectively. What is the length of segment $AV$? Express your answer in simplest radical form.


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

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


Solution: \boxed{\sum_{m=1}^{501} \frac{1}{m^2} - 1}


Let $G$ be the centroid of an equilateral triangle $ABC$, and let $Q$ be a point such that vector $\vec{QA}$ is the same as vector $\vec{BC}$. Then there exists a constant $k'$ such that:
\[
QA^2 + QB^2 + QC^2 = k' \cdot QG^2 + GA^2 + GB^2 + GC^2.
\]
Find $k'$.

Solution: \boxed{k' = 3}


Let $x$ and $y$ be two positive real numbers such that $x + y = 100.$ Find the ordered pair $(x, y)$ for which $x^4 y^6$ is maximized.

Solution: \boxed{(40, 60)}


What is the greatest integer value of $c$ such that $-9$ is not in the range of $y = x^2 + cx + 20$?


Four points $P$, $Q$, $R$, and $S$ are on a straight line. The point $T$ is placed such that $\angle PQT = 100^\circ$ and $\angle TRS = 90^\circ$. If the reflex angle at $T$ is $y^\circ$, determine the value of $y$.


An arithmetic sequence with first term $3$ has a common difference of $8$. A second sequence begins with $5$ and has a common difference of $9$. In the range of $1$ to $200$, what is the largest number common to both sequences?


Let $P$ be the incenter of triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is 48, find the area of triangle $G_1 G_2 G_3$.


What is the sum of all the solutions of \( x = \left|2x-|50-2x|\right| \)?


What is the ratio of the volume of a cube with an edge length of 4 inches to the volume of a cube with an edge length of 2 feet? Express your answer as a common fraction.


Thirty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5 or 6, with each number entered on five cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ be the probability that two of the cards bear a number $a$, another two bear a number $b$ (with $b \ne a$). What is the value of $q/p$?


A flag consists of three horizontal stripes, each stripe a unique solid color from the available options: red, white, blue, green, or yellow. No two adjacent stripes can have the same color. How many distinct flags can be created?


Let \( a, b, c, d, e, f \) be positive real numbers such that \( a + b + c + d + e + f = 10 \). Find the minimum value of
\[
\frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{9}{d} + \frac{16}{e} + \frac{25}{f}.
\]


Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$


The sum of the lengths of the twelve edges of a rectangular box is $160$, and the distance from one corner of the box to the farthest corner is $25$. Additionally, the volume of the box is $600$. What is the total surface area of the box?


What is the smallest five-digit palindrome that is divisible by 7?


Let $ABCD$ be a convex quadrilateral, and let $M_{AB},$ $M_{BC},$ $M_{CD},$ $M_{DA}$ denote the midpoints of sides $AB,$ $BC,$ $CD,$ and $DA$, respectively. Find the ratio $\frac{[M_{AB} M_{BC} M_{CD} M_{DA}]}{[ABCD]}.$


How many prime numbers between 50 and 100 have a prime remainder when divided by 12?


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{6}$, find the least possible value of $xy$.


Henry, Liz, Jack, and Tara are cycling on a track. Henry cycles at a rate of 5 miles per hour. If Liz cycles $\frac{3}{4}$ as fast as Henry, and Jack cycles $\frac{6}{5}$ as fast as Liz, how fast does Jack cycle? Additionally, if Tara cycles $\frac{9}{8}$ as fast as Jack, what is Tara's cycling speed?


Solution: \boxed{2 - 12z^2 + z}


The admission fee for an exhibition is \$30 per adult and \$18 per child. Last Wednesday, the exhibition collected \$2340 in admission fees, and there was at least one adult and one child. Among all the possible ratios of adults to children on that particular day, which ratio is closest to 1?


What integer \( n \) satisfies \( 0 \leq n < 17 \) and
\[ 38574 \equiv n \pmod{17} \, ? \]


Sequence $A$ is a geometric sequence starting with $3$. Sequence $B$ is an arithmetic sequence starting with $10$. Each sequence stops as soon as one of its terms is greater than $300$. Find the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B$.

$\bullet$ Sequence $A$: 3, 6, 12, 24, 48, 96, 192, 384 (stops at 384 as it is greater than 300).

$\bullet$ Sequence $B$: 10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250, 270, 290, 310 (stops at 310 as it is greater than 300).


Suppose two distinct integers are chosen from between 3 and 15, inclusive. What is the probability that their sum is even?


Carla rotated point $A$ 330 degrees counterclockwise about point $B$ and it landed at point $C$. Devon rotated the original point $A$ $y$ degrees clockwise about point $B$, and it also landed at point $C$. If $y<360$, what is the value of $y$?


Dr. Green gives bonus points on a test for students who score above the class average. In a class of 150 students, what is the maximum number of students who can score above the average if their scores are integers?


A cylindrical tank is one-third full of a juice mix. The ratio of orange juice to apple juice in the mix is 1:5. If the tank is 9 feet tall and has a diameter of 3 feet, what is the volume of orange juice in the tank? Express your answer as a decimal to the nearest hundredth.


If $x$ is a positive multiple of 7, and $x^2 > 200$ but $x < 30$, what is $x$?


A weather forecast predicts that for each day from Monday to Friday in the first week, there is a $30\%$ chance of sun, a $40\%$ chance of 2 inches of rain, and a $30\%$ chance of 8 inches of rain. For the second week, for each day from Monday to Friday, there is a $20\%$ chance of sun, a $30\%$ chance of 6 inches of rain, and a $50\%$ chance of 12 inches of rain. What is the expected value of the total amount of rain which will fall over these two weeks? Give your answer as a decimal rounded to the nearest tenth.

Solution: \boxed{55.0 \text{ inches total}}


Compute
\[
\sin^6 0^\circ + \sin^6 10^\circ + \sin^6 20^\circ + \dots + \sin^6 90^\circ.
\]


The parabolas defined by the equations $y = 3x^2 - 6x + 3$ and $y = -2x^2 - 4x + 6$ intersect at points $(a,b)$ and $(c,d)$, where $c \ge a$. What is $c-a$?


In the graph of $\frac{x^2-2x-8}{x^3-2x^2-x+2}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asymptotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$.


In triangle $XYZ$, where $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = m \sqrt{2}$, find $m$.


A biased coin flips heads with a probability of $\frac{2}{5}$ and tails with a probability of $\frac{3}{5}$. You play a game where if you flip heads, you win $\$4$, but if you flip tails, you lose $\$1$. What is the expected value, in dollars, of your winnings after two flips?


A standard deck of 52 cards is randomly arranged. What is the probability that the first two cards drawn are both $\spadesuit$, and the first of these is a King?


A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 1000?


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


Let \( p, q, r, s, t, u \) be positive real numbers such that \( p + q + r + s + t + u = 10 \). Find the minimum value of
\[ \frac{1}{p} + \frac{9}{q} + \frac{4}{r} + \frac{16}{s} + \frac{25}{t} + \frac{36}{u}. \]


Consider the odd numbers that are greater than 200 and less than 405. How many such numbers have a sum greater than 20000?


The math competition scores students by awarding 7 points for each correct answer, deducting 2 points for each incorrect answer, and giving 2 points for each unanswered problem. Paul decides to answer the first 18 problems of a 25-question competition and leave the last 7 unanswered. How many of the 18 attempted problems must he solve correctly to score at least 90 points?


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


What is the value of $\sqrt{49+121} \times \sqrt{64-49}$?


Find the smallest positive integer $x$ such that $x + 3457 \equiv 1537 \pmod{15}$.


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


The diameter, in inches, of a sphere with triple the volume of a sphere of radius 6 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.


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


Calculate $a + b$ where $a$ and $b$ are consecutive integers between which $\log_5{1250}$ lies.


In a right-angled triangle $XYZ$, where $Y$ is the right angle, suppose $\cos X = \frac{3}{5}$. Given that $XZ = 10$, calculate the length of $YZ$.


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


Let $c$ and $d$ be the roots of the equation $x^2 - nx + 3 = 0$. Suppose that $c + \frac{1}{d}$ and $d + \frac{1}{c}$ are the roots of the equation $x^2 - rx + s = 0$. Find $s$.


Find the remainder when \(x^5 + 2\) is divided by \(x^2 - 4x + 7.\)

Solution: \boxed{-31x - 54}


How many positive $3$-digit numbers are multiples of $25$, but not of $75$?


Consider the function defined piecewise by
\[ f(x) = \left\{ \begin{aligned}
2x + 1 & \quad \text{if } x < 1 \\
x^2 & \quad \text{if } x \ge 1
\end{aligned} \right. \]
Determine the value of \( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \).

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


Bob ate a total of 55 muffins in four days. Each day after the first day he ate two more muffins than he had eaten on the previous day. How many muffins did Bob eat on the second day?


A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a common fraction.


Evaluate \( \left| \frac{3}{4} + 3i \right| \).


Mr. and Mrs. Anderson have three children. In their family van, they need two people in the front (including the driver), and three in the back. Either Mr. Anderson or Mrs. Anderson must sit in the driver's seat, and the youngest child must sit in the middle seat in the back (the only spot with a child seat). How many seating arrangements are possible?


Suppose $x^2 + 2\sqrt{2}x + 5$ is a factor of $x^4 + Px^2 + Q$. Determine $P + Q$.


Write the expression $\frac{5+4d}{9}+3$ as a single fraction.


Let \( c \) and \( d \) be the roots of the equation \( x^2 - nx + 3 = 0 \). Suppose that \( c + \frac{2}{d} \) and \( d + \frac{2}{c} \) are the roots of the equation \( x^2 - rx + s = 0 \). What is \( s \)?


A circular pie with a diameter of $16\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Determine the square of the length of the longest line segment that can be drawn in one of these pieces.


What is the largest perfect square factor of 2520?


Let $a, b, c$ be a three-term arithmetic series where all terms are positive, with the property that $abc = 216$. Find the smallest possible value of $b$.


The prime factorization of 3920 is $2^3 \times 5 \times 7^2$. How many of its positive integer factors are perfect cubes?


A geometric sequence of positive integers is formed for which the first term is 5 and the fourth term is 320. What is the third term of the sequence?


Find the smallest positive solution to
\[\tan 2x + \tan 5x = \sec 5x\]
in radians.


Four standard dice are tossed, and the numbers \( a, b, c, d \) are obtained. Find the probability that \( a + b + c + d = 5 \).


Five fair coins are to be flipped. What is the probability that at least four coins will show the same face (either all heads or all tails)?


In triangle $ABC$, $D$ is on segment $BC$ such that $BD:DC = 3:2$. Point $E$ bisects $\angle BAC$. Given $BD = 45$ units and $DC = 30$ units, find the length of $BE$.


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


Determine the radius of a cookie if its boundary is described by the equation $x^2 + y^2 + 36 = 6x + 24y$.


Given that $3+\sqrt{5}$ is a root of the equation \[x^3 + ax^2 + bx + 20 = 0\], and that $a$ and $b$ are rational numbers, compute $b$.


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 10 & 5 \\ -4 & 6 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$

Solution: \boxed{\begin{pmatrix} 10 & 5 \\ -4 & 6 \end{pmatrix}}


A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+5$?


Let \( a, b, c \) be real numbers such that \( 9a^2 + 4b^2 + 25c^2 = 1 \). Find the maximum value of
\[ 3a + 4b + 5c. \]


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


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

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


Solve for $z$ in the equation: $4+2iz = 3 - 5iz$.


What is the range of the function $y = \log_2(\sqrt{\sin x \cos x})$ for $0^\circ \leq x \leq 90^\circ$?

Solution: \boxed{(-\infty, -\frac{1}{2}]}


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


What is the value of $x$ for which $(9-x)^2 = x^2$?


Evaluate the expression
\[
\left(d^d - d(d-2)^d + (d-1)!\right)^2
\]
when $d = 4$.


Altitudes $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at $H$. If $\angle BAC = 40^\circ$ and $\angle ABC = 65^\circ$, then what is $\angle AHB$?

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


The point $A$ $(3,4)$ is reflected over the $y$-axis to $B$. Then $B$ is reflected over the line $y=-x$ to $C$. What is the area of triangle $ABC$?


The equation $x^3 - 8x^2 + 15x - 7 = 0$ has three real roots $a$, $b$, $c$. Find $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.


Solution: \boxed{2 \frac{11}{12}}


Let \(T\) be the sum of all the real coefficients of the expansion of \((1 + ix)^{2011}\). What is \(\log_{2}(T)\)?


Four fair eight-sided dice (with faces showing 1 to 8) are rolled. What is the probability that the sum of the numbers on the top faces equals 32?


If 20 bahs are equal to 30 rahs, and 12 rahs are equal in value to 20 yahs, how many bahs are equal in value to 1500 yahs?


What integer $n$ satisfies $0\le n<23$ and $$58294 \equiv n \pmod{23}~?$$


In how many distinct ways can I arrange my five keys on a keychain, if I want my house key, my car key, and my office key to be next to each other in that specific order? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation).


Sally has 12 blue marbles and 8 red marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly five of the marbles that she removes are blue? Express your answer as a decimal rounded to the nearest thousandth.


What is the smallest whole number $b$ such that 150 can be expressed in base $b$ using only three digits?


What is the units digit of the sum of the twelve terms in the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 11! + 11, \, 12! + 12$?


In the diagram below, we have $AB = 25$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{4}{5}$ and $\sin C = \frac{1}{5}$, then what is $DC$, and what is the area of triangle $ABD$?

[asy]
pair A,B,C,D;
A = (0,0);
B = (5*sqrt(5),20);
D = (5*sqrt(5),0);
C = (5*sqrt(5) + 50*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]

Solution: \boxed{DC = 40\sqrt{6}, \text{Area} = 150}


Let
\[
\mathbf{N} = \begin{pmatrix} 3 & 4 \\ x & y \end{pmatrix}.
\]
If $\mathbf{N} \mathbf{N}^T = 25 \mathbf{I}$, determine the ordered pair $(x,y)$.

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


Alice has a pink marble, a blue marble, a white marble, and four identical black marbles. How many different groups of two marbles can Alice choose?


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

Solution: \boxed{\cos^2 t}


Consider a new function $g(x) = \frac{2ax - b}{cx - 2d}$, where $a, b, c, d \neq 0$ and $g(g(x)) = x$ for all $x$ in the domain of $g$. Assume further that $b=3a$ and $c=3d$. What is the value of $2a - 2d$?


Consider the polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
with integer coefficients, which has roots $-3$, $6$, $10$, and $-2/5$. If $e$ is a positive integer, find its smallest possible value.


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

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{12}{5} \end{pmatrix}}


At a regional competition, 25 contestants are each assigned a unique room number from 1 to 25 in a hotel. If the contestants in rooms 15 and 16 did not arrive, what is the median room number of the remaining 23 contestants?


In a high school, 500 students participated in a survey to determine pet ownership. It was found that 120 students own cats, and 200 students own dogs. Moreover, 40 students own both cats and dogs. What percent of the students own only cats?


How many positive integers $n$ less than 300 have a corresponding integer $m$ divisible by 5 such that the roots of $x^2-nx+m=0$ are consecutive odd integers?


A round pizza with a diameter of $16\text{ cm}$ is divided into four equal-sized sector-shaped pieces. Let $m$ represent the length, in centimeters, of the longest line segment that may be drawn in one of these pieces. Compute $m^2$.


The Gnollish language now has four words: "splargh," "glumph," "amr," and "blargh." In forming sentences:
- "splargh" cannot come directly before "glumph"
- "amr" cannot come directly before "blargh"
How many valid 3-word sentences can be formed under these restrictions?


Determine how many numbers less than 50 are abundant numbers.


Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $G$ is one focus of the ellipse. If $OG = 8$ and the diameter of the inscribed circle of triangle $ODG$ is 4, compute the product $(AB)(CD).$


Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half. Find the sum of the slope and $y$-intercept of this line.