A circle is circumscribed around quadrilateral $EFGH$ as follows: $EF$ subtends a central angle of $100^\circ$, and $GH$ subtends a central angle of $120^\circ$. Find the sum of the angles $\angle EFG + \angle EHG$.


Find the least positive integer $x$ that satisfies $x + 3649 \equiv 304 \pmod{15}$.


Let \(a\) and \(b\) be the roots of the equation \(x^2 - mx + 6 = 0\). Suppose that \(a + \frac{1}{b}\) and \(b + \frac{1}{a}\) are the roots of the equation \(x^2 - px + r = 0\). Find the value of \(r\).


Line $l_1$ has equation $4x - 3y = 2$ and goes through $P = (-2, -3)$. Line $l_2$ has equation $y = 2$ and meets line $l_1$ at point $Q$. Line $l_3$ has positive slope, passes through point $P$, and meets $l_2$ at point $R$. The area of $\triangle PQR$ is $6$. What is the slope of $l_3$?


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


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


A square and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$?
[asy]
for(int i=0; i <=9; ++i) {
draw(dir(360*i/9+90)--dir(360*(i+1)/9+90));
}
pair A = dir(360*3/9+90);
pair F = dir(360*4/9+90);
pair C = A+dir(-90)*(F-A);
pair D = C+F-A;
pair B = dir(360*2/9+90);

draw(A--C--D--F);

label("$A$",A,NE);
label("$B$",B,NW);
label("$C$",C,S);
label("$D$",F,W);

[/asy]


Find the domain of the function
\[ f(x) = \frac{x^2 + 5x + 6}{|x - 2| + |x + 3| - 5}. \]

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


Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{50}$ and $\frac{\pi}{75}$ 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 $y=\frac{7}{25}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.


Find a positive integer that is divisible by 21 and whose cube root is a number between 9 and 9.1.


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


Betty goes to the store to buy flour and sugar. The amount of flour she purchases, in pounds, must be at least 8 pounds more than three-quarters the amount of sugar, but cannot exceed double the amount of sugar. Determine the least amount of sugar, in pounds, that Betty could buy.


Let $w$ be a complex number. Suppose there exist distinct complex numbers $u$, $v$, such that for every complex number $z$, we have
\[
(z - u)(z - v) = (z - wu)(z - wv).
\]
Compute the number of distinct possible values of $w$.


The large square quilt is composed of nine smaller squares, with four squares each divided into two right triangles. What fraction of the square quilt is shaded if three triangles and one whole square are shaded? Express your answer as a common fraction.

[asy]size(75);
fill(shift(2,1)*unitsquare,gray(.6)); // shaded whole square
fill((0,0)--(1,0)--(0,1)--cycle,gray(.6)); // shaded triangle
fill((1,1)--(2,1)--(1,2)--cycle,gray(.6)); // shaded triangle
fill((2,0)--(3,0)--(2,1)--cycle,gray(.6)); // shaded triangle
draw(scale(3)*unitsquare); // borders of the large square
for (int i=1; i<3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } // grid lines
[/asy]


A cone has a radius of 2 inches and a height of 6 inches. Find the total surface area of a sphere that has the same volume as this cone.

Solution: \boxed{4\pi (6^{2/3})}


The average of 15, 30, $x$, and $y$ is 25. What are the values of $x$ and $y$ if $x = y + 10$?


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


How many two-digit positive integers are congruent to 3 (mod 4)?


What is the greatest number of Sundays that can occur in the first 58 days of a year starting on a Monday?


If \(128^7 = 32^y\), what is the value of \(2^{-3y}\)?  Express your answer as a common fraction.


In a press conference before a major baseball game, ten All-Stars will be taking questions. Four are Cubs, three are Red Sox, two are Yankees, and one is a Dodger. If teammates insist on sitting together, how many ways can the ten All-Stars be seated in a row?


In a parallelogram, adjacent sides are of lengths $s$ units and $3s$ units, forming a 30-degree angle. Calculate the value of $s$ if the area of the parallelogram is $27\sqrt{3}$ square units. Express your answer in simplest radical form.


Determine the value of $2^x$ given the system of equations:
\[
2^{x+1} + 5^y = 14,
\]
\[
2^{x+3} + 5^{y+1} = 58.
\]


During the 2012 Math Olympics, Team Germany won 23 out of 150 medals. In the same event, Team Japan won 27 out of 150 medals. Which one of the following fractions is closest to the fraction of the total medals won by Team Germany and which is closest to Team Japan?
$$
\frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8} \qquad \frac{1}{9}
$$


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

\[ \text{[A graph depicting a function symmetric about the line } y = x + 1 \text{ is implied but not shown]} \]

Suppose we define another function by \(g(x) = f(x + b)\). On the evidence of the graph above, for what choice of \(b\) is it true that \(g(x)\) is identical to its inverse, \(g^{-1}(x)\)?

Solution: \boxed{b = -1}


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


When the decimal point of a certain positive decimal number is moved two places to the right, the new number is nine times the reciprocal of the original number. What is the original number?


A regular polygon has a perimeter of 180 cm and each side has a length of 15 cm. How many sides does this polygon have, and what is the measure of each interior angle?


Berengere and her friend from the USA, Tom, are at a French café looking to buy a dessert. The dessert costs 8 euros, and Tom has a ten-dollar bill. If 1 euro equals 1.10 USD, how many euros does Berengere need to contribute for them to afford the dessert?

Solution: \boxed{0 \text{ euros}}


Consider an arithmetic sequence where the third term is $\frac{2}{11}$ and the eleventh term is $\frac{5}{6}$. What is the seventh term?


In a classroom, each row can seat either 6 or 7 students. If there are a total of 53 students, and every seat is occupied, how many rows seat exactly 7 students?


What is the smallest two-digit integer $n$ such that switching its digits and then multiplying by 3 results in $3n$?


Compute the value of \[M = 120^2 + 119^2 - 118^2 - 117^2 + 116^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2,\] where the additions and subtractions alternate in pairs.


Find the center of the circle given by the equation \(x^2 - 6x + y^2 + 2y - 11 = 0\), and determine if the point \((5, -1)\) lies on this circle.

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


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


Find the number of real solutions to the equation
\[
\frac{2}{x - 1} + \frac{4}{x - 2} + \frac{6}{x - 3} + \dots + \frac{200}{x - 100} = x.
\]


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 317^\circ.$


Find the sum of the coefficients in the polynomial $5(2x^9 + 5x^7 - 4x^3 + 6) + 8(x^8 - 9x^3 + 3)$ when it is fully simplified.


Four circles, each with a radius of 3 inches, are arranged in such a way that they touch each other and the sides of a square. What is the area, in square inches, of the square?

Solution: \boxed{144\text{ in}^2}


Suppose we have a right triangle $DEF$ where the right angle is at $E$ such that $DF = \sqrt{85}$ and $DE = 7.$ A circle with its center on $DE$ is tangent to both $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, what is $FQ$?


Let $N$ be the sum of the divisors of $180$. What is the largest prime factor of $N$?


Which digit will appear in the 2021st place after the decimal point in the decimal representation of $\frac{7}{16}$?


Tommy’s Taco Town offers tacos with the following optional toppings: avocado, salsa, jalapeños, onions, cheese, beans, sour cream, and cilantro. A customer can choose to add double meat or triple meat to their taco, and they can select between a soft shell and a hard shell. How many different kinds of tacos can be ordered?


Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has two sides with distinct integer lengths, what is the area of $EFGH$? Express your answer in simplest radical form.


Let \(f\) be a function mapping nonnegative integers to nonnegative integers, such that
\[3f(a^2 + b^2) = [f(a)]^2 + 2[f(b)]^2\]
for all nonnegative integers \(a\) and \(b\).

Let \(n\) be the number of possible values for \(f(16)\), and let \(s\) be the sum of possible values for \(f(16)\). Find \(n \times s\).


The ratio of the areas of two squares is $\frac{50}{98}$. Calculate the ratio of their side lengths and express it in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are integers. What is the value of the sum $p+q+r$?


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


Alex writes down all the five-digit numbers that contain each of the digits 1, 2, 6, 7, and 8 exactly once. What is the smallest number in Alex's list that is divisible by 6?


Let $x_1 < x_2 < x_3$ be the three real roots of the equation $\sqrt{4050} x^3 - 8101x^2 + 4 = 0$. Find $x_2(x_1+x_3)$.


How many four-digit integers have the property that their digits, read left to right, are in strictly decreasing order, and the last digit is an odd number?


How many distinct four-digit numbers are divisible by 5, and have their last two digits as 45?


The sum of the positive divisors of a positive integer of the form \(2^i3^j5^k\) is equal to \(3600\). What is \(i + j + k\)?


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


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

Solution: \boxed{\begin{pmatrix} 1 \\ 3 \\ -7 \end{pmatrix}}


How many distinct triangles can be constructed by connecting three different vertices of a tesseract? (A tesseract, also known as a 4-dimensional hypercube, has sixteen vertices. Triangles are considered distinct if they occupy different set of vertices.)


In a bag, there are 6 blue chips, 5 red chips, 4 yellow chips, and 3 green chips. One chip is drawn from the bag and then placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


What is the largest prime factor of 1717?


Determine the value of $c + d$ if the three points $(2,c,d),$ $(c,3,d),$ $(c,d,4)$ are collinear.


In right triangle $ABC$, with $AB$ as the hypotenuse, points $E$ and $F$ divide sides $\overline{AC}$ and $\overline{AB}$ in the ratios $1:3$ and $2:1$, respectively. The area of $\triangle ABC$ is 50 square units. What is the area of $\triangle CEF$?


At the end of the year, the Science Club decided to hold an election for which 6 equal officer positions were available. In all, 20 candidates were nominated, of whom 10 were past officers. Of all possible elections of the officers, how many will have at least 2 of the past officers?


Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 5 dogs, and one with 3 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 5-dog group?


In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 800$ and $PS = 1600$. Let $\angle P = 45^\circ$, $\angle S = 45^\circ$, and $X$ and $Y$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $XY$.


There are $30$ different complex numbers $z$ such that $z^{30}=1$. For how many of these is $z^{10}$ a real number?


Let
\[\bold{B} = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 2 & 0 & 1 \end{pmatrix}.\]There exist constants $p$, $q$, and $r$ such that
\[\bold{B}^3 + p \bold{B}^2 + q \bold{B} + r \bold{I} = \bold{0},\]where $\bold{I}$ and $\bold{0}$ are the $3 \times 3$ identity matrix and zero matrix, respectively. Find the ordered triple $(p,q,r)$.

Solution: \boxed{(-3, 3, -9)}


For what value of $n$ is the four-digit number $315n$, with units digit $n$, divisible by 12?


Let $f(x) = ax + b$ and $g(x) = -4x + 3$. Given $h(x) = f(g(x))$ and $h^{-1}(x) = x + 3$, find the value of $a - b$.


If the height of a cylinder is tripled and its radius is increased by $150\%$, by what factor is the cylinder's volume multiplied?


Suppose that $(v_n)$ is a sequence of real numbers satisfying
\[v_{n+2}=3v_{n+1}+v_n + 1\]
and that $v_3=11$ and $v_6=242$. What is $v_5$?


It is required to construct a right triangle in the coordinate plane with its legs parallel to the axes, and the medians to the midpoints of these legs should lie on the lines $y = 2x + 1$ and $y = nx + 3$. The task is to find the number of different constants $n$ for which this configuration is possible.


Find the sum of all positive integers that are divisors of each number in the list $$60, 90, 150, 180, 210.$$


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


Solution: \boxed{\frac{9 \sqrt{10}}{50}}


A digit is written to the right of the units digit of $758$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?


Four faucets can fill a 200-gallon tub in 12 minutes. How long, in seconds, does it take eight faucets to fill a 50-gallon tub? Assume all faucets dispense water at the same rate.

Solution: \boxed{90 \text{ seconds}}


It is required to construct a right triangle in the coordinate plane so that its hypotenuse is parallel to the line $y = mx + b$ and its legs are along the $x$ and $y$ axes respectively. Determine the number of different constants $m$ for which the medians to the hypotenuse and one of the legs can simultaneously fall on the lines $y = 2x + 1$ and $y = 5x + 2$.


If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face-up integers is 17?


How many integers between $200$ and $250$ have three different digits, with the digits in increasing order and the last digit being even?


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


Kim borrows $100$ dollars from Sam with a simple interest rate of $10\%$ per day. Additionally, there's a one-time borrowing fee of $10$ dollars that is added to the debt immediately. What is the least integer number of days after which Kim will owe Sam at least twice as much as she borrowed?

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


Simplify $\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{5}}{\sqrt{8}} \cdot \frac{\sqrt[3]{2}}{\sqrt{9}}$ and rationalize the denominator of the resulting fraction.

Solution: \boxed{\frac{\sqrt{210}\cdot\sqrt[3]{2}}{84}}


Find $t$ given that \[4 \log_3 t = \log_3 (6t).\]


How many nonnegative integers can be written in the form \[ a_7 \cdot 2^7 + a_6 \cdot 2^6 + a_5 \cdot 2^5 + a_4 \cdot 2^4 + a_3 \cdot 2^3 + a_2 \cdot 2^2 + a_1 \cdot 2^1 + a_0 \cdot 2^0, \] where each \( a_i \) can be \(-1, 0, 1\) for \( 0 \leq i \leq 7 \)?


The Falcons play the Eagles in a series of nine basketball games. Each team has an equal chance of winning each game. What is the probability that the Falcons will win at least five games? Express your answer as a common fraction.


Pirate Pete and Pirate Paul have a new arrangement for sharing their treasure. Pete, being the captain, says, ``One for me, one for you,'' distributing coins so he gets one and Paul gets one. He then says, ``Two for me, and two for you,'' adding two more coins to his own pile but making Paul's total pile two coins. This sequence continues in similar fashion, with the command, ``Three for me, three for you," and so on, until Pete gives himself `x` more coins and adjusts Paul's pile to `x` coins in total. After all coins are distributed this way, Pete ends up with exactly five times as many coins as Pirate Paul. How many coins do they share in total?


The integers 234 and 78 are expressed in base 5 and added. What is the resulting sum, expressed in base 5?


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


Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. If $PQ = 3$ units, $QR = 6$ units, and $PS = 20$ units, what is the ratio of $PR$ to $QS$?


Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=20$, find the smallest possible value of $|w^3+z^3|$.


Suppose that all four of the numbers \[2 - \sqrt{3}, \; -2 - \sqrt{3}, \; 4 + \sqrt{5}, \; 4 - \sqrt{5}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Find the coordinates of the center of the hyperbola \[\frac{(3y+3)^2}{7^2} - \frac{(4x-8)^2}{6^2} = 1.\]

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


Find \( a+b+c \) if the graph of the equation \( y=ax^2+bx+c \) is a parabola with vertex \( (3, -2) \), vertical axis of symmetry, and contains the point \( (0,5) \).


The sides of a triangle have lengths $13, 17,$ and $k,$ where $k$ is a positive integer. Determine the number of values for $k$ for which the triangle is obtuse.


A flagpole is supported by a wire that extends from the top of the flagpole to a point on the ground 5 meters from its base. When Ben walks 3 meters away from the base of the flagpole towards the point where the wire is anchored to the ground, his head just touches the wire. Ben is 1.8 meters tall. How tall is the flagpole?

Solution: \boxed{4.5 \text{ meters}}


What is the range of the function \( f(x) = \frac{1}{x^3} \)?

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


One of the roots of $z^2 = -39 - 52i$ is $5 - 7i.$ What is the other root?

Solution: \boxed{-5 + 7i}


A line passes through the points $(2, 5)$, $(-3, m)$, and $(15, -1)$. Determine the value of $m$.


For how many integer values of $n$ between 1 and 1600 inclusive does the decimal representation of $\frac{n}{2310}$ terminate?


How many positive integers smaller than $500,000$ are powers of $2$ but are not powers of $4$?


The surface area of a hemisphere, excluding the base, is $256\pi\text{ cm}^2$. What is the volume, in cubic centimeters, of this hemisphere? Express your answer in terms of $\pi$.


Jenny now eats 25% of the jellybeans in her jar at the beginning of each day. After three days, there are 27 jellybeans remaining. Calculate how many jellybeans were originally in the jar.


The 5 a.m. temperatures for seven consecutive days were $-8^{\circ}$, $-6^{\circ}$, $-3^{\circ}$, $-3^{\circ}$, $0^{\circ}$, $4^{\circ}$, and $-1^{\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?


Define an infinite sequence $T = \{t_1, t_2, t_3, \ldots\}$ by $t_1 = 11$ and $t_n = 11^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{100}$ is divided by $7$?


Mr. Ramirez rewards students on their test performance if their scores are above the class mean. His class consists of 150 students who took the same test. What is the maximum number of students who can receive this reward?


Calculate the sum: $(-45) + (-43) + \cdots + 23$.


A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid?


In a company of 63 employees, the salaries are distributed as shown in the table below. What is the median salary?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$140,000$\\\hline
Vice-President&4&$\$95,000$\\\hline
Director&11&$\$78,000$\\\hline
Associate Director&8&$\$55,000$\\\hline
Administrative Specialist&39&$\$25,000$\\\hline
\end{tabular}


Let \( p, q, r, s, t, u, v, \) and \( w \) be distinct elements in the set \[
\{-8, -6, -4, -1, 1, 3, 5, 12\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


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


Find the length of $PQ$ in the triangle below.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(2),0);
R = (0,sqrt(2));
draw (P--Q--R--P);
draw(rightanglemark(Q,P,R,3));
label("$P$",P,S);
label("$Q$",Q,S);
label("$R$",R,N);
label("$9$",R/2,W);
label("$45^\circ$",(1.25,0),N);
[/asy]


On a Cartesian coordinate plane, points $(1,2)$ and $(4, 6)$ are adjacent points on a square. What is the area of the square?


How many numbers that are multiples of 4 and greater than 300 but less than 700?


Four standard six-sided dice are rolled. The result shows no three-of-a-kind, but does include a pair. These two dice displaying the same number are set aside, leaving two dice to be re-rolled. What is the probability that, after re-rolling these two dice, at least three of four dice show the same value?


What is the base $2$ representation of $123_{10}$?


For which integer $a$ does $x^2 - x + a$ divide $x^{13} - x^5 + 90$?


In a similar playoff scenario, the top 6 bowlers compete in a sequence. First, #6 bowls against #5. The winner bowls #4, and the process continues until the winner bowls #1 in the final game. Prizes are awarded from 6th to 1st based on when a player loses, with the final game’s winner receiving 1st prize and the loser getting 2nd prize. Determine the number of possible orders in which bowlers #1 through #6 can receive the prizes.


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


Find the domain of the rational function \( f(x) = \frac{x^3 - x^2 - 4x + 4}{x^2 - 5x + 6} \). Express your answer as a union of intervals.

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


Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal.


Determine the value of \( a \) for which the graph of the equation \[ x^2 + 9y^2 - 6x + 27y = k \] is a non-degenerate ellipse.


Let $n$ be a number with exactly 15 positive divisors, where $n$ is a product of two distinct odd primes. Find the number of positive divisors of $16n^2$.


Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 7.$  Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\]


There are 7 blue chips, 4 yellow chips, and 5 green chips in a bag. One chip is drawn from the bag, then placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


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


How many of the 128 smallest positive integers written in base 8 use 6 or 7 (or both) as a digit?


Suppose $x, y,$ and $z$ are positive integers such that $(x+y+z)^3 - x^3 - y^3 - z^3 = 504$. Determine $x+y+z$.


Solve for \(y\) in the equation \(4y - 8 = 2y + 18\). Then compute the value of \(3(y^2 + 6y + 12)\).


Calculate the number of terms in the simplified expression of \[(x+y+z)^{2020} + (x-y-z)^{2020},\] by expanding it and combining like terms.


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2012} \times 7$?


Suppose that $c$ and $d$ are positive integers where $c$ has $4$ factors and $d$ has $c$ factors. If $d$ is divisible by $c$, what is the least possible value of $d$?


In $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and $CD \parallel AB$. A point $E$ is placed such that $DE$ is perpendicular to $CD$. What is the number of degrees in $m\angle ECD$?


In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $BC = 2\sqrt{10}$. What is $\tan A$?


Define $A\star B$ as $A\star B = \frac{(A+B)}{4}$. What is the value of $(3\star 11) \star 6$?


There are five unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly three people receive the correct letter?


A square is inscribed in the ellipse
\[\frac{x^2}{5} + \frac{y^2}{10} = 1,\]
such that the diagonals of the square are parallel to the coordinate axes. Find the area of the square.


Find the number of integers $n$ that satisfy
\[25 < n^2 < 144.\]


How many positive integers \( n \) satisfy \( (n + 6)(n - 5)(n - 10)(n - 15) < 0 \)?


In a modified game of Deal or No Deal, participants choose a box at random from a set of $30$ boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$2,000\\\hline\$1&\$10,000\\\hline\$10&\$50,000\\\hline\$100&\$100,000\\\hline\$500&\$200,000\\\hline\$1,000&\$300,000\\\hline\$5,000&\$400,000\\\hline\$7,500&\$500,000\\\hline\$10,000&\$750,000\\\hline\$15,000&\$1,000,000\\\hline\end{tabular} After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate in order to have at least a half chance of holding a box with $\$200,\!000$ or more?


What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal, and $n$ contains the digit 2?


A point \((x, y)\) is a distance of 15 units from the \(x\)-axis. It is a distance of 13 units from the point \((2,7)\). It is a distance \(n\) from the origin. Given that \(x > 2\), what is \(n\)?

Solution: \boxed{n = \sqrt{334 + 4\sqrt{105}}}


Let $a$, $b$, $c$, $d$ be nonzero integers such that
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}^2 = \begin{pmatrix}
11 & 0 \\
0 & 11
\end{pmatrix}.
\]
Find the smallest possible value of $|a| + |b| + |c| + |d|$.


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


A relatively prime date is defined similarly as above. Consider the month of November, which consists of 30 days, numbered from 1 to 30. Calculate the number of days in November that are relatively prime to the number of the month.


Let $C$ equal the number of four-digit numbers divisible by 3. Let $D$ equal the number of four-digit multiples of 7. Find $C+D$.


Four numbers, $c_1$, $c_2$, $c_3$, $c_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 500\}$. Assume that $c_1$ is always a prime number. Three other numbers, $d_1$, $d_2$, $d_3$, are then drawn randomly and without replacement from the remaining set of 496 numbers. Let $q$ be the probability that, after a suitable rotation, a brick of dimensions $c_2 \times c_3 \times c_4$ can be enclosed in a box of dimensions $d_1 \times d_2 \times d_3$, with the sides of the brick parallel to the sides of the box. If $q$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?


Suppose we want to divide 12 puppies into three groups where one group has 4 puppies, one has 6 puppies, and one has 2 puppies. Determine how many ways we can form the groups such that Coco is in the 4-puppy group and Rocky is in the 6-puppy group.


The Eiffel Tower has a height of 984 feet. A scale model of the tower has a height of 6 inches. How many feet of the tower does one inch of the model represent?


The product of three different positive integers is equal to $5^4$. What is the sum of these integers?


In how many ways can the digits of $45,\!502,\!2$ be arranged to form a 6-digit number? (Remember, numbers cannot begin with 0.)


The least common multiple of $x$, $12$ and $18$ is $108$. What is the greatest possible value of $x$?


Assuming that the birth of a boy has a probability of $\frac{2}{3}$ and a girl $\frac{1}{3}$, what is the probability that four children in a family include at least one boy and one girl? Express your answer as a common fraction.


What is the value of
\[
\frac{x^2 \cdot x^3 \cdot x^4 \cdots x^{20}}{x^3 \cdot x^6 \cdot x^9 \cdots x^{18}}
\]
if \(x = 3\)?


Define \( X \star Y \) as \( X \star Y = \frac{(X+Y)}{4} \). What is the value of \( (3 \star 7) \star 6 \)?


In triangle $ABC$, where $\angle A = 90^\circ$, the length of hypotenuse $BC$ is 10 units, and $\tan C = 3\sin C$. Determine the length of side $AB$.


Find a number that is 40 and fewer than 120, that is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6.


A circle centered at $O$ is circumscribed about $\triangle ABC$ as shown below:
- $\angle AOB = 130^\circ$
- $\angle BOC = 90^\circ$

What is the measure of $\angle BAC$, in degrees?


The quadratic equation $x^2 + mx + p = 0$ has roots that are three times those of $x^2 + qx + m = 0$, where none of $m$, $p$, and $q$ is zero. Find the value of $p/q$.


Find all values of $b$ for which the points $(0,0,0),$ $(1,b,0),$ $(0,1,b^2),$ and $(b^2,0,1)$ are coplanar.


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


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


The shaded region shown consists of 18 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. The first part of the shaded region extends from $y=0$ to $y=6$ along the $y$-axis and from $x=0$ to $x=1$ along the $x$-axis. The second part extends from $y=0$ to $y=3$ along the $y$-axis and from $x=1$ to $x=5$ along the $x$-axis. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.


Let
\[
\mathbf{B} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{B}^{99}$.

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


Determine the area enclosed by the graph of the inequality
\[ |x - y| + |2x + 2y| \leq 8. \]


One US dollar is equivalent to 7 Namibian dollars. One British pound is equivalent to 1.4 US dollars. One British pound is also equivalent to 9.8 Namibian dollars. If a painting costs 196 Namibian dollars, how much would it cost in Chinese yuan, given that one US dollar is equivalent to 6 Chinese yuan?


The mean of $7, 11$ and $23$ is equal to the mean of $15$ and $z$. What is the value of $z$?


For a science fair, Lisa constructs a rectangular pyramid with a height of 15 inches. The base of her pyramid measures 8 inches in length and 12 inches in width. Lisa needs to determine the smallest cube-shaped box that can house her pyramid to transport it uprightly to the fair. What is the volume of this cube-shaped box, in cubic inches?


Find the phase shift of the graph of $y = 2 \sin \left( x + \frac{\pi}{4} \right)$.


Solve the equation \( |y-4| + 3y = 14 \) for \( y \).


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{15}\right)\left(7^{7}\right)$?


A Senate committee has 6 Democrats and 4 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without any two Republicans sitting next to each other?


Find the integer $n$, $0 \le n \le 6$, such that \[n \equiv 123456 \pmod{7}.\]


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

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males & 45 & \\ \hline
females & & 87 \\ \hline
total & 115 & 160 \\
\hline
\end{tabular}


On a 12-hour clock, an elapsed time of five hours is "clock equivalent" to its square number of hours. What is the least whole number of hours that is greater than 5 hours and is "clock equivalent" to its square number of hours?


Berengere and her Canadian friend Lucas are at a café in Berlin that accepts both euros and Canadian dollars. They want to buy a pie, but neither of them has enough money. If the pie costs 8 euros and Lucas has a Canadian ten-dollar bill, how many euros does Berengere need to contribute to the cost of the pie if 1 euro = 1.5 CAD?

Solution: \boxed{\frac{4}{3} \text{ euros}}


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


What is the units digit of the sum of the squares of the first 1503 odd, positive integers?


Consider the set of all triangles $OPR$ where $O$ is the origin and $R$ and $S$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $37x + y = 2017$. Find the number of such distinct triangles whose area is a positive integer.


Add $92.8531$ to $47.2694$ and round your answer to the nearest hundredth.


An amateur basketball league records the number of games played by each player as shown below. What is the average number of games played by each player? Express your answer to the nearest whole number. [asy]
size(150);
draw((0,7)--(0,0)--(10,0));
for(int i = 1; i <= 6; ++i){

label((string)i,(2*i,0),S);
}
label("Games Played",(0,-1)--(12,-1),S);
void addDots(int pos, int num){

for(int i = 1; i <= num; ++i){

dot((2*pos,i));

}
}
addDots(1,6); addDots(2,3); addDots(3,1); addDots(4,4); addDots(5,2); addDots(6,6);
picture perpLabel;
label(perpLabel,"Number of Players");
add(rotate(90)*perpLabel,(-1,4));
[/asy]


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


What is the base ten equivalent of $23456_{7}$?


The 5 a.m. temperatures for seven consecutive days were $-6^\circ$, $-3^\circ$, $-3^\circ$, $-2^\circ$, $0^\circ$, $4^\circ$, and $5^\circ$ Celsius. Calculate both the mean and the median 5 a.m. temperature for the week in degrees Celsius.

Solution: \boxed{-2^\circ \text{ Celsius}}


The polynomial
\[ax^4 + bx^3 + cx^2 + dx + e = 0\]
has integer coefficients and roots $-3$, $6$, $10$, and $-\frac{1}{4}$. If $e$ is a positive integer, find its smallest possible value.


Solution: \boxed{3x^{17} - 15x^5 - 24x^2 + 6x}


If $n$ is $49$ less than a multiple of $50$, then what is the remainder when $n^2 + 4n + 5$ is divided by $50$?


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


A ray of light passing through the point \(A = (-2, 8, 10)\), reflects off the plane \(x + y + z = 15\) at point \(B\), and then passes through the point \(C = (4, 4, 7)\). Find the point \(B\).

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


The sum of the first 3000 terms of a geometric sequence is 1000. The sum of the first 6000 terms is 1900. Find the sum of the first 9000 terms.


What is the sum of the digits of the base $8$ representation of $878_{10}$?


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


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

Solution: \boxed{9 \text{ meters}}


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

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


A store offers two promotions on footwear. Promotion A says, "Buy one pair of shoes at full price, get the second pair for half the price." Promotion B says, "Get a 10% discount on the total price if the total before discount exceeds $100." The price of each pair of shoes is $60. Jane wants to buy two pairs and can only use one of the promotions. She wishes to use the promotion that provides the greater discount. How much does Jane save by choosing the better promotion?


How many ways are there to put 6 indistinguishable balls into 3 indistinguishable boxes?


The increasing sequence consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $50^{\mbox{th}}$ term of this sequence.


Sets $C$ and $D$, shown in the Venn diagram, are such that the total number of elements in set $C$ is three times the total number of elements in set $D$. Altogether, there are 4050 elements in the union of $C$ and $D$, and their intersection has 1200 elements. What is the total number of elements in set $C$?

[asy]
label("$C$", (2,67));
label("$D$", (80,67));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label("1200", (44, 45));
[/asy]


In the diagram, $PQR$ is a straight line, and point $T$ is below the line, making triangle $QRT$. What is the value of $x$ if $\angle PQS = 124^\circ$ and $\angle QRT = 76^\circ$? Assume $\angle QRS = x$ and $S$ is a point such that $\triangle QRS$ and $\triangle QRT$ share side $QR$.

[asy]
draw((-2,0)--(8,0),linewidth(0.7)); draw((8,0)--(5,-5.5)--(0,0),linewidth(0.7));
draw((8,0)--(3,-3.8)--(0,0),linewidth(0.7));  // New triangle QRT
label("$P$",(-2,0),W); label("$Q$",(0,0),N); label("$R$",(8,0),E); label("$S$",(5,-5.5),S); label("$T$",(3,-3.8),S);
label("$124^\circ$",(0,0),SW); label("$x^\circ$",(7,0),S);
label("$76^\circ$",(4,-3.8),S);
[/asy]


Solution: \boxed{3w - 9}


A biologist wants to calculate the number of fish in a pond. On April 1, she catches a random sample of 80 fish, tags them, and releases them. On August 1, she catches a random sample of 100 fish and finds that 4 of them are tagged. To calculate the number of fish in the pond on April 1, she assumes that 30% of these fish are no longer in the pond on August 1 (because of death and emigrations), that 50% of the fish were not in the pond on April 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the August 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the pond on April 1?


The integer $y$ has 24 positive factors. The numbers 18 and 30 are factors of $y$. What is the smallest possible value of $y$?


Solve $10x + 3 \equiv 7 \pmod{18}$ for $x$ where $x \equiv a \pmod{m}$ and $a < m$ with $m \geq 2$. Find $a+m$.


Determine the value of $q$ for the quadratic $6x^2 + 5x + q$ if its roots are $\frac{-5 \pm i\sqrt{323}}{12}$.


Find the maximum value of
\[
\sin^2 \frac{\theta}{2} \cdot (1 + \cos^2 \theta)
\]
for \(0 < \theta < \pi.\)


Given a regular polygon with \( n \) sides, where \( 3 \leq n \leq 15 \), find how many values of \( n \) result in both the interior and exterior angles having integer degree measures.


At a large family reunion, there are 8 married couples. Every participant shakes hands exactly once with every other person except for their spouse and any of their family members (defined here as the other three couples they are related to through marriage but not direct siblings). How many handshakes occur among the people at the reunion?


Sides $\overline{AB}$ and $\overline{GH}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$?


Let $c$ and $d$ be real numbers. Consider the following five statements:

$\frac{1}{c} < \frac{1}{d}$
$c^3 > d^3$
$c < d$
$c < 0$
$d < 0$

What is the maximum number of these statements that can be true for any values of $c$ and $d$?


A school has between 200 and 250 students enrolled. The students are separated into eight distinct sections of students for a science project. If two students are absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


How many integers $-12 \leq n \leq 12$ satisfy $(n-3)(n+5)(n+9)<0$?


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


Define the function $f(x) = x^6 + x^4 + 3x^3 + 4x^2 + 8x$. Evaluate $f(3) - f(-3)$.


In triangle $XYZ$, sides $XY$, $YZ$, and $ZX$ are tangent to a circle with center $O$. Given that $\angle XYZ = 80^\circ$ and $\angle YZX = 78^\circ$, find $\angle YXO$, in degrees.


Suppose $a$ and $b$ are angles such that
\[\sin(a - b) = \sin a - \sin b.\]
Find the maximum value of $\sin a.$


One of the roots of \(z^2 = -75 + 40i\) is \(5 + 7i\). What is the other root?

Solution: \boxed{-5 - 7i}


What is the coefficient of the $m^6n^6$ term in the expansion of $(m+n)^{12}$?


Given vectors $\mathbf{a}$ and $\mathbf{b}$, let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\] Among all such vectors $\mathbf{p}$, find constants $s$ and $v$ such that $\mathbf{p}$ is at a fixed distance from $s \mathbf{a} + v \mathbf{b}.$ Determine the ordered pair $(s, v).$

Solution: \boxed{\left( \frac{9}{8}, -\frac{1}{8} \right)}


At the CleverPaws Training Center, dogs can learn four tricks: sit, stay, roll over, and play dead. Of the dogs at the center:
\begin{tabular}{l@{\qquad}l}
60 dogs can sit & 22 dogs can sit and stay \\
35 dogs can stay & 15 dogs can stay and roll over, 5 of whom can also play dead \\
40 dogs can roll over & 20 dogs can sit and roll over \\
10 dogs can do all three basic tricks (sit, stay, roll over) & 10 dogs can do none
\end{tabular}
How many dogs are in the training center? Note that all dogs who can play dead can either roll over or stay.


Solution: \boxed{x = -\frac{1}{17}}


What is the arithmetic mean of the integers from -6 through 5, inclusive? Express your answer as a decimal to the nearest tenth.


For some real numbers $c$ and $d$, the equation $9x^3 + 8cx^2 + 7dx + c = 0$ has three distinct positive roots. If the sum of the base-3 logarithms of the roots is 3, what is the value of $c$?


Determine the greatest possible value of $a+b$ where $a^2 + b^2 = 130$ and $ab = 45$.


In the diagram, $CP$ and $CQ$ trisect $\angle ACB$. $CM$ bisects $\angle PCQ$. Find the ratio of the measure of $\angle MCQ$ to the measure of $\angle ACQ$.


In a club of 25 people, 10 of them are left-handed. 18 of them like jazz music. 4 of them are right-handed and dislike jazz music. How many club members are left-handed and like jazz music? Assume people are either left-handed or right-handed, but not both.


Compared to last year, participation in the local basketball league has increased by $15\%$. From the previous year, the number of males increased by $10\%$ and the number of females increased by $30\%$. Last year, there were $30$ males participating in the league. Find the fraction of the league's participants that is female this year and express your answer as a common fraction.


Tim's quiz scores were 85, 87, 92, 94, 78, and 96. Calculate his mean score and find the range of his scores.


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


What is the remainder when $2984 \cdot 3998$ is divided by $1000$?


Find $3^{-1} \pmod{197}$, as a residue modulo 197. (Give an answer between 0 and 196, inclusive.)


Compute $\arctan ( \tan 75^\circ - 2 \tan 30^\circ )$. (Express your answer in degrees as an angle between $0^\circ$ and $180^\circ$.)


Bill is sent to a bakery to buy exactly eight donuts. The shop now offers five types of donuts, and Bill must get at least two of the first type and at least one of each of the other types. How many different ways can Bill fulfill this order?


From the diagram, find the positive difference in the \( x \)-coordinates when lines \( p \) and \( q \) reach \( y = 20 \). Line \( p \) passes through points \( (0, 10) \) and \( (4, 0) \), and line \( q \) passes through points \( (0, 5) \) and \( (10, 0) \).


Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?


Sophie has decided to write all possible rearrangements of the letters in her name. She can write eighteen rearrangements of her name every minute. How many hours does it take for her to write all the possible rearrangements of her name?

Solution: \boxed{\frac{2}{3} \text{ hours}}


Solution: \boxed{3 - x}


On a tetrahedron with vertices labeled $V_1$, $V_2$, $V_3$, and $V_4$, how many different paths are there from $V_1$ to $V_3$ if each path must traverse exactly 2 edges?


Solution: \boxed{25 \text{ units}}


What is the ratio of the volume of cone $C$ to the volume of cone $D$? Express your answer as a common fraction. Cone $C$ has a radius of 16.4 and height of 30.5, while cone $D$ has a radius of 30.5 and height of 16.4.


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


A "super ball" is dropped from a window 20 meters above the ground. On each bounce, it rises $\frac{2}{3}$ of the distance of the preceding high point. The ball is caught when it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it traveled?


The Rotokas alphabet, consisting of letters A, E, G, I, K, O, P, R, T, U, and V (note S is removed from the original alphabet), is used to design four-letter license plates. How many license plates are possible that start with E, end with O, cannot contain I, and must not have repeated letters?


Triangle $\triangle DEF$ contains a right angle at $F$, $\angle E = 45^\circ$, and $DF=8$.  Find the radius of the incircle of $\triangle DEF$.

Solution: \boxed{8 - 4\sqrt{2}}


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


Similarly, by restricting the domain of the function $f(x) = 3x^2 - 6x - 9$ to an interval, we can make it invertible. What is the largest such interval that includes the point $x=2$?

Solution: \boxed{[1, \infty)}


If the least common multiple of two 7-digit integers has 11 digits, then their greatest common divisor has at most how many digits?


What is the greatest integer less than or equal to \[\frac{2^{50}+5^{50}}{2^{47}+5^{47}}?\]


Let $\mathcal{T}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Find the remainder obtained when $m$ is divided by $500$.


Given the polynomial $p(x) = x^5 - 5x^3 + 4x$, find the coefficient of the $x^2$ term in the polynomial $(p(x))^2$.


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


How many integers $n$ satisfy the inequality $-5e \le n \le 12e$?


In an updated scenario, Alberto and Bjorn participate in a race where Alberto bikes at a rate of 18 miles per hour and Bjorn bikes at 17 miles per hour. After 5 hours, how many more miles has Alberto biked than Bjorn?


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-50^2)} = 0\] have for \( x \)?


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


Ryosuke picks up his friend from work and then stops by a grocery store before finally heading home. The car's odometer reads 74,568 when he picks his friend up, reads 74,580 when they reach the grocery store, and it eventually reads 74,608 when Ryosuke gets home. If Ryosuke's car achieves 32 miles per gallon and the price per gallon of gas is $\$4.30$, what was the total cost of gas for the entire trip?


Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows these rules: at each vertex, it randomly chooses one of the three connecting edges, each with an equal likelihood, and then moves to the opposite vertex. Let $p = \frac{n}{2187}$ be the probability that the bug is back at vertex $A$ after traveling exactly 8 meters. Find the value of $n$.


What are the last two digits in the sum of factorials of the first 50 positive integers?


Identify a four-digit number $m$ which is the product of three distinct prime factors $x$, $y$, and $10x+y$. Moreover, introduce another prime factor $z$, such that $10z + x$ must also be a prime. Here $x$, $y$, and $z$ are distinct primes each less than 10. What is the maximum value of $m$ under these conditions?


A $\textit{palindrome}$ is a number which reads the same forward as backward, like 343 and 1221. What is the least natural number that can be added to 52,425 to create a palindrome?


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


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,7)$. Find $m+b$.


If $(2, 10)$ and $(-6, -6)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $y$-coordinates of the other two vertices?


The repeating decimal for $\frac{4}{13}$ is $0.cdcdcd\dots$ What is the value of the sum $c+d$?


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

Solution: \boxed{2 \sec x}


What is half of the absolute value of the difference of the squares of 20 and 15?


Define the function $g$ on positive integers as follows:
\[
g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if } n < 15, \\
g(n - 3) & \text{if } n \ge 15.
\end{array}
\right.
\]
Find the maximum value of the function $g$.


The number of degrees in the measures of the interior angles of a convex hexagon are six consecutive integers. What is the number of degrees in the largest angle?


A circle is tangent to the lines $3x + 4y = 40$ and $3x + 4y = -20.$ The center of the circle lies on the line $x - 2y = 0.$ Find the center of the circle.


Compute the value of $\dfrac{11!}{7!4!}$ without using a calculator.


For how many positive integers \(x\) is \(x^2 + 4x + 4\) between 40 and 100?


Find the distance between the planes $3x - 9y + 6z = 12$ and $6x - 18y + 12z = 24.$


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


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


What is the value of $n$ such that $10^n = 10^{-2} \times \sqrt{\frac{10^{75}}{0.0001}}$?


Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle of size 3 units by 7 units. Vertices $A$ and $B$ lie at the bottom left and bottom right corners of the rectangle, respectively, and vertex $C$ is at the middle of the top side. What is the area of triangle $ABC$?


Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 2$ and $BD = 3,$ then what is $CD$?


In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.

Note: A point is a lattice point if all its coordinates are integers.


Solve for $x: \ \log_2 x + \log_8 x = 9.$


Given $a \star b = a^2 + 2ab + b^2$, and defining a new operation $\diamond$ such that $a \diamond b = a \star b - ab$, find the value of $3 \diamond 7$.


Let $x$ and $y$ be two positive real numbers such that $x + y = 50$. Determine the ordered pair $(x, y)$ for which $x^4 y^3$ is maximized.

Solution: \boxed{\left(\frac{200}{7}, \frac{150}{7}\right)}


Let $f(x) = 2x^3 - 3x + 4$. What is the value of $f(f(2))$?


A circular cylindrical post with a circumference of 5 feet has a string wrapped around it, spiraling from the bottom to the top. The string evenly loops around the post exactly five times, starting at the bottom edge and finishing at the top edge. The height of the post is 20 feet. Determine the length of the string in feet.


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


Find the product of the divisors of $72$.


Call an integer $n$ strongly oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many strongly oddly powerful integers are less than $1729$?


Suppose \( a, b, \) and \( c \) are three positive numbers that satisfy the equations \( abc = 1, \) \( a + \frac{1}{c} = 8, \) and \( b + \frac{1}{a} = 20. \) Find \( c + \frac{1}{b}. \)


Find the domain of the function $\frac{x^4-4x^3+6x^2-4x+1}{x^3-3x^2+2x}$.

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


When the base-12 integer $2743_{12}$ is divided by $10$, what is the remainder?


If Greg rolls five fair six-sided dice, what is the probability that he rolls more 1's than 6's?


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


In the provided diagram, $F$ and $E$ are midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Find the sum of the slope and $y$-intercept of the line passing through $F$ and $B.$ Points are \(A(0,6)\), \(B(0,0)\), \(C(8,0)\).
- Size diagram correspondingly, adding point $F$ at the midpoint of $\overline{AC}$.
- Add coordination to $E$ at midpoint of $\overline{BC}$.


Consider the equation
\[(x - \sqrt[3]{8})(x - \sqrt[3]{27})(x - \sqrt[3]{64}) = 1\]
and let its roots be $u$, $v$, and $w$. Calculate the sum $u^3 + v^3 + w^3$.


Let \( a \) and \( b \) be nonnegative real numbers such that
\[
\cos (ax + b) = \sin (36x)
\]
for all integers \( x \). Find the smallest possible value of \( a \).


Expand the product ${2(x + (a+2))(x + (a-3))}$.

Solution: \boxed{2x^2 + (4a-2)x + 2a^2 - 2a - 12}


There are four points that are $7$ units from the line $y=9$ and $15$ units from the point $(10,9)$.  What is the sum of the $x$- and $y$-coordinates of all four of these points?


Find the greatest constant $N,$ so that
\[\frac{a^2 + b^2 + ab}{c^2} > N\]whenever $a,$ $b,$ and $c$ are the sides of a triangle.


At a gathering, there are 25 gremlins and 20 imps. Owing to some conflicts, only 10 imps are willing to shake hands with the gremlins, and each of these imps shakes hands with just 15 of the gremlins. All gremlins are in good terms with each other and shake hands amongst themselves as well. Calculate the total number of handshakes at this assembly.


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


A box contains eight cards. Four of the cards are blue on both sides, two cards are blue on one side and yellow on the other, and two of the cards are yellow 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 yellow, what is the probability that the other side is yellow? Express your answer as a common fraction.


The number $852$ can be written as $21q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?


Find all solutions to the inequality \[\frac{x^2 + 5x}{(x-3)^2} \ge 0.\] (Give your answer in interval notation.)

Solution: \boxed{(-\infty, -5] \cup [0, 3) \cup (3, \infty)}


Given that $\lceil x + \epsilon \rceil - \lfloor x + \epsilon \rfloor = 1$, where $0 < \epsilon < 1$, find the value of $\lceil x + \epsilon \rceil - (x + \epsilon)$.

Solution: \boxed{1 - \epsilon}


How many positive factors of 364 are also multiples of both 7 and 13?


Eight positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $2107$, what is the sum of the numbers written on the faces?


Solve the inequality
\[3 - \frac{2}{3x + 2} < 5.\]

Solution: \boxed{(-\infty, -1) \cup \left(-\frac{2}{3}, \infty\right)}


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be either 7 or 8, and its hundreds digit is even? Express your answer as a common fraction.


Define the function $g(x)$ such that:
\[ g(x) = \left\lceil \frac{1}{x+1} \right\rceil \text{ for } x > -1, \]
\[ g(x) = \left\lfloor \frac{1}{x+1} \right\rfloor \text{ for } x < -1. \]
$g(x)$ is not defined at $x = -1$. Which integer is not in the range of $g(x)$?


Calculate the total area enclosed by the graph of the equation $|x| + |3y| = 15$.


Consider two hyperbolas:
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]
\[ \frac{y^2}{8} - \frac{x^2}{M} = 1 \]
Find $M$ such that these hyperbolas have the same asymptotes.


If $A\ \spadesuit\ B$ is defined as $A\ \spadesuit\ B = 4A + 3B + 6$, what is the value of $A$ for which $A\ \spadesuit\ 5 = 79$?


In a modified game, Allie and Betty again take turns rolling a standard die but now the score function has changed to \[f(n) = \left\{
\begin{array}{cl}
9 & \text{if } n \text{ is a multiple of 3}, \\
3 & \text{if } n \text{ is only a multiple of 2}, \\
1 & \text{if } n \text{ is odd but not a multiple of 3}, \\
0 & \text{otherwise}.
\end{array}
\right.\]
Allie rolls the die five times and gets 5, 2, 6, 1, and 3. Betty rolls and gets 6, 4, 1, 2, and 5. Calculate the product of Allie's and Betty's total points.


Suppose $f(x) = ax + b$ and $g(x) = -2x + 7$. If $h(x) = f(g(x))$ and $h^{-1}(x) = x + 9$, find the value of $a - b$.


An ellipse has foci at $F_1 = (0,3)$ and $F_2 = (4,0)$. The ellipse intersects the $x$-axis at the origin, and one other point. Determine the other point of intersection on the $x$-axis.

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


If $y < 0$, find the range of all possible values of $y$ such that $\lceil y \rceil \cdot \lfloor y \rfloor = 72$. Express your answer using interval notation.

Solution: \boxed{(-9, -8)}


How many ways are there to put 6 indistinguishable balls into 2 indistinguishable boxes?


Let \( m = \underbrace{3333333}_{\text{7 digits}} \) and \( n = \underbrace{66666666}_{\text{8 digits}} \).

What is \( \gcd(m, n) \)?


Determine the number of solutions to the equation
\[ \sin x (2\sin^2 x - 1)(\sin x - 2) = 0 \]
in the range $0 \leq x \leq 2\pi.$


Let \[f(x) = \left\{
\begin{array}{cl}
x^2 - 4 &\text{ if }x>6, \\
3x + 2 &\text{ if } -6 \le x \le 6, \\
5 &\text{ if } x < -6.
\end{array}
\right.\] If \( x \) is a multiple of 3, add 5 to \( f(x) \). Find \( f(-8) + f(0) + f(9) \).


Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + b^2) = [g(a)]^2 + [g(b)]^2\] for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $g(16),$ and let $s$ be the sum of the possible values of $g(16).$  Find $n \times s.$


Acme T-Shirt Company charges a $60 set-up fee plus $10 for each shirt printed. For orders of more than 20 shirts, Acme offers a 10% discount on the total cost. Beta T-shirt Company has no set up fee, but charges $15 per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?


Triangle $PQR$ has a perimeter of 2021 units. The sides have lengths that are all integer values with $PQ < QR \leq PR$. What is the smallest possible value of $QR - PQ$?


Thirty tiles are numbered 1 through 30 and are placed into box $A$. Forty tiles numbered 10 through 49 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 20 and the tile from box $B$ is either odd or greater than 40? Express your answer as a common fraction.


Find the integer $m,$ $-180 < m < 180,$ such that $\tan m^\circ = \tan 2530^\circ.$


In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid.


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


Among all the roots of
\[ z^6 - z^4 + z^2 - 1 = 0, \]
the maximum imaginary part of a root can be expressed as $\cos \phi$, where $-90^\circ \le \phi \le 90^\circ$. Find $\phi$.


If $0.\overline{356}$ is written as a fraction $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers, what is $a+b$?


Let $\mathbf{B} = \frac{1}{7} \begin{pmatrix} -5 & p \\ q & r \end{pmatrix}$ be a matrix with integer entries where $\mathbf{B}^2 = \mathbf{I}$. Find the largest possible value of $p + q + r.$


Bryan now has some 5 cent stamps and some 6 cent stamps. What is the least number of stamps he can combine to make the value of the stamps add up to 60 cents?


The product of two positive integers plus their sum is 167. The integers are relatively prime, and each is less than 30. What is the sum of the two integers?


In a right triangle, one of the angles is $3x$ degrees, the right angle is $90^\circ$, and the smallest angle is $3x - 40^\circ$. What is the value of $x$?


The letters of the alphabet are each assigned a random integer value, and $T = 15$. The value of a word comes from the sum of its letters' values. If $MATH$ is 47 points, $TEAM$ is 58 points and $MEET$ is 45 points, what is the value of $M$?


My aunt has 12 pieces of art, including 4 prints by Picasso. If she arranges the pieces of art in a row in a random order, what is the probability that all four Picasso prints will be placed consecutively?


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


There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$.


In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100,$ and $AC = 100$. Find the sum of all possible values of $BC$.

Solution: \boxed{100 \sqrt{2}}


We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. A 5-card poker hand is called a "four of a kind" if it contains four cards of one rank and one card of another rank (such as 4444K). What is the probability that five cards chosen at random form a four of a kind?


A list of five positive integers has all of the following properties:

$\bullet$ The only integer in the list that occurs more than once is $6,$

$\bullet$ its median is $7,$ and

$\bullet$ its average (mean) is $12.$

What is the largest possible integer that could appear in the list?


A rectangular box has a volume of $5400$ cubic inches and a surface area of $2352$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. Find the volume of the box, in cubic inches, if its length, width, and height were each increased by one inch.

Solution: \boxed{6637 \text{ cubic inches}}


An annulus is formed by two concentric circles with radii $r$ and $s$, where $r > s$. Let $\overline{OP}$ be a radius of the larger circle, let $\overline{PQ}$ be tangent to the smaller circle at $Q$, and let $\overline{OR}$ be the radius of the larger circle that contains $Q$. Let $x=PQ$, $u=QR$, and $v=PR$. Calculate the area of the annulus in terms of $\pi$ and at most one of the variables $x, r, s, u, v$.

Solution: \boxed{\pi x^2}


Solution: \boxed{m = 9}


A triangle with sides of 8, 15, and 17 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?


In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{8}.$ Find $\tan F.$


Find the cubic polynomial with real coefficients that has $3+2i$ as one of its roots and where the coefficient of $x^2$ is $-10$.

Solution: \boxed{x^3 - 10x^2 + 37x - 52}


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?


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


A line through the points $(5, -12)$ and $(k, 23)$ is parallel to the line $4x + 6y = 12$. What is the value of $k$?


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

Solution: \boxed{9\pi \text{ cm}}


A frustum of a right circular cone is obtained by cutting the top smaller cone off of a larger cone. The frustum has an altitude of 30 centimeters, the area of its lower base is $400\pi$ square centimeters, and the area of its upper base is $100\pi$ square centimeters. What is the altitude of the small cone that was cut off, and what was the total height of the original cone?

Solution: \boxed{60 \text{ cm}}


Elena decides to earn money through freelance graphic designing for her uncle's business. She agreed on the following rates for her work: for the first five hours, she charges incrementally from $\$3$ to $\$7$ respectively; after five hours, the sequence resets. If she works a total of 47 hours, how much money will she earn?


Given the points (3, 10), (6, 20), (12, 35), (18, 40), and (20, 50), determine the sum of the $x$-coordinates of the points that lie above the line $y = 3x + 5$ in the coordinate plane.


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(8) = 32$ and $g(32) = 2048,$ find the largest possible value of $g(16).$


Given that $ABCDEH$ is a rectangle, where $AD = 4$ and $AB = 2$. Inside rectangle $ABCDEH$, a regular hexagon $FGHIJK$ is inscribed such that its vertices touch the midpoint of each side of the rectangle. Find the total area of the hexagon.


John believes that the amount of study time he invests before a test and his score on that test are directly related. On his first exam, he studied for 3 hours and scored 60 on the exam. To the nearest tenth, how many hours does John believe he must study the night before his second exam so that the average of his two exam scores is a 75?


Rectangle EFGH has its center at $(5, -3)$ and has an area of 8 square units. The longer side of the rectangle is horizontal and is twice as long as the shorter side. The rectangle is then dilated with the dilation center at $(0,0)$ and a scale factor of 3. What are the coordinates of the vertex of the image of rectangle EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(21, -12)}


The Rockets are playing the Bulls in the NBA finals. To win the finals, a team must win 4 games. The probability that the Bulls win any given game is $\frac{3}{5}$. Assuming no ties, what is the probability that the Rockets will win the NBA finals and that the contest will need all seven games to be determined? Provide your answer as a fraction.


Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle.


Medians $\overline{AF}$ and $\overline{BD}$ of $\triangle ABC$ are perpendicular, and they intersect the third median $\overline{CE}$. If $AF = 12$ and $BD = 16$, compute the area of $\triangle ABC$.


What is the shortest distance from the origin to the circle defined by \( x^2 - 30x + y^2 - 8y + 325 = 0 \)?

Solution: \boxed{\sqrt{241} - 2\sqrt{21}}


Billy is hiking in Colorado. He starts by walking eastward seven miles, then turns 45 degrees northward and walks eight miles, and finally walks straight north for another four miles. How far is he from his starting point? Express your answer in simplest radical form.

Solution: \boxed{\sqrt{129 + 88\sqrt{2}}}


The coefficients of the polynomial
\[x^3 + bx^2 + cx + d = 0\]
are all integers. Let $m$ be the exact number of integer roots of this polynomial, counting multiplicity. For example, the polynomial $(x + 2)^2 (x - 3) = 0$ has three integer roots counting multiplicity, because the root $+2$ is counted twice, and $-3$ is counted once.

Enter all possible values of $m,$ separated by commas.

Solution: \boxed{0, 1, 2, 3}


Determine how many different positive integers will leave a remainder of 7 when divided into 61.


What is the fifteenth term in the geometric sequence $12, 4, \frac{4}{3}, \ldots$?


Suppose $g(x)$ is a function defined for all real $x$, and suppose $g$ is invertible. Consider the graphs of $y=g(x^3)$ and $y=g(x^6)$. How many points do these graphs intersect?


If the price of a stamp is 45 cents, and the price drops to 40 cents per stamp if more than 100 stamps are purchased, what is the maximum number of stamps that could be purchased with $50?


Convert $172_{10}$ to base 13. Represent $10$ as $A$, $11$ as $B$, and $12$ as $C$ if necessary.


A girl has the following eight coins in her pocket: $3$ pennies, $2$ nickels, $2$ dimes, and $1$ half-dollar. She takes out two coins, records the sum of their values, and then puts them back with the other coins. She continues to do this repeatedly. What is the maximum number of different sums she can record?


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


A polynomial of degree $17$ is divided by $d(x)$ to give a quotient of degree $7$ and a remainder of $5x^5 + 2x^2 - 3x + 8$. What is $\deg d$?


Solution: \boxed{60x^5(1 - 3x^4)}


Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$


What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by $17$ and $11$, and remainder $2$ when divided by $6$?


What is the value of the expression $\frac{x^2 + x - 12}{x - 4}$ when $x = 5$?


How many 3-letter words can we make from the letters A, B, C, D, and E, if we are allowed to repeat letters, and we must use the letter A at least twice?


In $\triangle ABC$, where $\angle A = 90^\circ$ and $\tan C = 2$, compute $\cos C$ and $\sin C$.


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


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


A library has ten identical copies of a specific book. At any given time, some of these copies are at the library and some are checked out. Each time, at least two books must be in the library and at least three must be checked out. How many different arrangements are possible?


Consider two solid spherical balls where the first ball is centered at $\left( 0, 0, 10 \right)$ with radius 5, and the second ball is centered at $(0, 0, 2)$ with radius 4. Calculate the volume of the intersection of these two spheres.


Lily is taking a 30-question, multiple-choice Biology quiz. Each question offers four possible answers. Lily guesses on the last six questions. What is the probability that she will get at least two of these last six questions wrong?


What is half of the absolute value of the difference of the squares of 20 and 15?


The first term of a sequence is 2, and each successive term is the sum of all previous terms of the sequence. Find the value of the first term which exceeds 10000.


Find the matrix $\mathbf{N}$ that quadruples the first row and doubles the second row of a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. If no such matrix $\mathbf{N}$ exists, then use the zero matrix.

Solution: \boxed{\begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}}


For how many integer values of $m$ between 1 and 500 inclusive does the decimal representation of $\frac{m}{980}$ terminate?


Add $2_4 + 13_4 + 132_4 + 1320_4.$ Express your answer in base 4.


If 3400 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


The prime factorization of 4410 is $2 \times 3^2 \times 5 \times 7^2$. How many of its positive integer factors are perfect squares?


In the diagram, $CP$ and $CQ$ trisect $\angle ACB$. $CM$ bisects $\angle PCQ$. Find the ratio of the measure of $\angle MCQ$ to the measure of $\angle ACQ$.


What is the smallest positive integer \(n\) such that all the roots of the equation \(z^5 - z^3 + z = 0\) are \(n^\text{th}\) roots of unity?


A man labels the squares of a chessboard from $1$ to $64$. On each square $k$, he places $3^k$ grains of rice. How many more grains of rice are placed on the $12^{th}$ square than on the first $10$ squares combined?


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


A square pyramid has a base edge of 40 inches and an altitude of 18 inches. A smaller similar square pyramid, whose altitude is one-third of the original altitude, is cut away at the apex of the original pyramid. What fractional part of the volume of the original pyramid does the remaining frustum represent?


Find the value of \( k \) so that the line \( 4x + 6y + k = 0 \) is tangent to the parabola \( y^2 = 16x \).


A rectangle has a perimeter of 72 cm and each side has an integer length. How many non-congruent rectangles meet these criteria?


How many positive two-digit integers are multiples of 3 and of 7?


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


Let $h(x)$ be a function defined as \[h(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
3x-50 & x>0.
\end{array}
\right.\] If $b$ is negative, find $b$ such that $h(h(h(15)))=h(h(h(b)))$.


Determine the sum of the two solutions for the quadratic equation $81 - 18x - 3x^2 = 0$.


An equilateral triangle is inscribed in the ellipse whose equation is $x^2+3y^2=3$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


When Claire divides her cupcakes into groups of 6, she has 2 remaining, and when she divides her cupcakes into groups of 8, she has 6 remaining. If Claire has fewer than 100 cupcakes, what is the sum of all possible quantities of cupcakes that she could have?


What is the perimeter of hexagon $ABCDEF$ where $AB = 1$, $BC = 1.5$, $CD = 1.5$, $DE = 1.5$, $EF = \sqrt{3}$, and $FA = 2$?

Solution: \boxed{7.5 + \sqrt{3}}


We roll a fair 8-sided die 6 times. What is the probability that we get an odd number in exactly 5 of the 6 rolls?


The function
\[f(x) = \left\{ \begin{aligned} 2x - 1 & \quad \text{ if } x < 3 \\
x^2 & \quad \text{ if } x \ge 3 \end{aligned} \right.\]
has an inverse $f^{-1}.$ Find the value of $f^{-1}(-3) + f^{-1}(-2) + \dots + f^{-1}(8) + f^{-1}(9).$

Solution: \boxed{6 + 3 = 9}


Convert $153_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


Find the number of integers $x$ that satisfy
\[20 < x^2 < 120.\]


Given the air distance in miles between selected world cities in the table below, if two different cities are chosen at random, what is the probability that the distance between them is less than $7500$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London \\ \hline
Bangkok & & 6500 & 6800 & 6100 \\ \hline
Cape Town & 6500 & & 11,800 & 6200 \\ \hline
Honolulu & 6800 & 11,800 & & 7400 \\ \hline
London & 6100 & 6200 & 7400 & \\ \hline
\end{tabular}


Define the function
\[ f(n) =
\begin{cases}
n^2 + 1 & \text{if } n < 3, \\
3n + 2 & \text{if } 3 \leq n < 6, \\
2n - 1 & \text{if } n \geq 6.
\end{cases}
\]
Find $f(f(f(1)))$.


How many solutions does the equation $\sin x = \sin (\sin x)$ have in the interval $0 \le x \le \sin^{-1} 0.99$?


If a school bus leaves school with 64 students on board, and one-third of the students get off the bus at each of the first four stops, how many students remain on the bus after the fourth stop?


Calculate the number of seven-digit palindromes.


Suppose that $n$ is a positive integer such that in base $5$, $n$ can be expressed as $\overline{XYZ}_5$, and in base $9$, $n$ can be expressed as $\overline{ZYX}_9$. Find the largest possible value of $n$ in base $10$.


$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 5$ and $AE = 2.$ Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$


The measure of the angles of a hexagon are in the ratio of 3:3:3:4:5:6. What is the number of degrees in the measure of the largest angle?


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x + y) + xg(y) - 2xy - x + 2 \]
for all real numbers \( x \) and \( y \). Determine the number of possible values of \( g(1) \), denoted as \( m \), and the sum of all possible values of \( g(1) \), denoted as \( t \). Find \( m \times t \).


An arithmetic sequence with the 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?


$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 3$ and $AE = 2.$ Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$


The four digits $a$, $b$, $c$ and $d$ are such that $a = a$, $b = a+2$, $c = a+4$, and $d = a+6$. Using these digits, the four-digit numbers $abcd$ and $dcba$ are formed. Determine the greatest common divisor (GCD) of all numbers of the form $abcd + dcba$ where $a, b, c, d$ follow the stated relationships.


Let $g(n)$ be a function that, given an integer $n$, returns an integer $j$, where $j$ is the smallest possible integer such that $j!$ is divisible by $n$. Given that $n$ is a multiple of 21, what is the smallest value of $n$ such that $g(n) > 21$?

Solution: \boxed{n = 483}


How many digits are located to the right of the decimal point when $\frac{5^7}{10^5 \cdot 125}$ is expressed as a decimal?


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(1000,0),(1000,1005),$ and $(0,1005)$. What is the probability that $x > 5y$? Express your answer as a common fraction.


Solution: \boxed{120 \sqrt{10}}


Maria needs to build a circular fence around a garden. Based on city regulations, the garden's diameter needs to be close to 30 meters, with an allowable error of up to $10\%$. After building, the fence turned out to have a diameter of 33 meters. Calculate the area she thought she was enclosing and the actual area enclosed. What is the percent difference between these two areas?


A grocer makes a display with cans arranged in rows. The top row has three cans and each subsequent row has three more cans than the row above it. How many rows are needed to have at least 150 cans in the display?


Compute the following expression:
\[
\left( 1 - \frac{1}{\cos 37^\circ} \right) \left( 1 + \frac{1}{\sin 53^\circ} \right) \left( 1 - \frac{1}{\sin 37^\circ} \right) \left( 1 + \frac{1}{\cos 53^\circ} \right).
\]


If the pattern observed in the first 4 rows continues, with each row \( n \) containing \( 2n \) occurrences of the number \( 2n \), determine the value of the \( 100^{\mathrm{th}} \) number in this arrangement.


Suppose $a$ and $b$ are positive integers such that $(a+bi)^3 = 2 + 11i$. Determine $a+bi$.


Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$


A truncated pyramid is formed by slicing the top off of a larger pyramid. If this truncated pyramid has a square lower base with a side length of 10 inches, a square upper base with a side length of 5 inches, and a height of 6 inches, what is its lateral surface area?

Solution: \boxed{195 \text{ square inches}}


How many non-negative integers $n$ less than 120 have a corresponding integer $m$ divisible by 4 such that the roots of $x^2 - nx + m = 0$ are consecutive non-negative integers?


Provide an example of a quadratic function that has zeroes at \(x=1\) and \(x=5\), and that takes the value \(8\) when \(x=3\).

Solution: \boxed{-2x^2 + 12x - 10}


Find the positive value of $x$ such that $x \cdot \lfloor x \rfloor = 50$. Express your answer as a decimal.


Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2023} \right| = \frac{1}{2023}.\]


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q - p$?


The 2022 AMC 12 awards 7.5 points for each correct response, deducts 2 points for each incorrect response, and grants 2 points for each problem left unanswered. Olivia decides to attempt the first 15 problems and leave the last 10 unanswered. How many of the first 15 problems must she solve correctly to score at least 120 points?


For $1 \leq n \leq 200$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal?


Solution: \boxed{k = 8}


The diagonals of rectangle $ABCD$ intersect at point $O$. If $AB = 15$ and $BC = 20$, then what is $\cos \angle AOB$?


To repair a damaged part at the center $C$ of a circular garden, workers positioned a 20-foot plank from point $A$ to point $B$ and then a 12-foot plank from point $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Determine the area of the circular garden. Express your answer in terms of $\pi$.


How many different 8-digit positive integers exist if the digits from the second to the eighth can only be 0, 1, 2, 3, or 4?


Chewbacca has 30 pieces of cherry gum and 40 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of cherry gum, then the ratio of the number of pieces of cherry gum he has to the number of pieces of grape gum will be exactly the same as if he instead finds 4 packs of grape gum. Find $y$.


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1089$, then what is the sum of the numbers written on the faces?


At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink, and one dessert. The table below lists Crystal's options:

\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks} & \textbf{Desserts} \\ \hline
Pizza & Lemonade & Frozen Yogurt \\ \hline
Chicken Teriyaki & Ice Tea & Chocolate Chip Cookie \\ \hline
Burger & Root Beer & Apple Pie \\ \hline
\end{tabular}

How many distinct possible meals can she buy using these options?


Modern widescreen television screens are rectangles and are typically measured by the length of their diagonals. The ratio of the horizontal length to the height in these widescreen televisions is $16:9$. Calculate the horizontal length (in inches) of a ``50-inch'' widescreen television screen.


A parabola with the equation $y=ax^2+bx+c$ is graphed below:

[asy]
xaxis(-3,7);
yaxis(-5,32);
real g(real x)
{
return 3(x-3)^2-9;
}
draw(graph(g,-1,5));
dot((3,-9));
label("Vertex: $(3,-9)$", (3,-9), SE);
dot((5,6));
label("$(5,6)$", (5,6), E);
[/asy]

The zeros of the quadratic $ax^2 + bx + c$ are at $x=p$ and $x=q$, where $p>q$. What is $p-q$?


Let \(x\) and \(y\) be real numbers such that \(4x + 3y \leq 12\) and \(3x + 6y \leq 9.\) Find the largest possible value of \(x + 2y.\)


At the Pawful Talent Training Center, cats can learn to perform three tricks: jump, fetch, and meow on command. Of the cats at the center:
- 60 cats can jump
- 35 cats can fetch
- 40 cats can meow on command
- 20 cats can jump and fetch
- 15 cats can fetch and meow
- 25 cats can jump and meow
- 11 cats can do all three
- 10 cats can do none of the tricks

How many cats are in the training center?


Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[f(a^2 + 2b^2) = [f(a)]^2 + 2[f(b)]^2\] for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $f(34),$ and let $s$ be the sum of the possible values of $f(34)$. Find $n \times s.$


Triangle $PQR$ is an isosceles right triangle with the measure of angle $P$ equal to 90 degrees. The length of segment $PR$ is 8 cm. What is the area of triangle $PQR$, in square centimeters?


If $n$ is $3$ less than a multiple of $60$, then what is the remainder when $n^3 + 2n^2 + 3n + 4$ is divided by $60$?


How many ways can you arrange 4 different mathematics books and 4 different history books on a shelf if all the mathematics books must stay together, and all the history books must stay together, with the additional requirement that a specific mathematics book (let's call it M1) must be the first book in the mathematics group?


A box contains tiles, numbered 1, 2, 3, ..., 49, 50, 51, 52. Only tiles which are marked with a number congruent to $3 \pmod{4}$ are red. One tile is chosen randomly from the box. What is the probability that the tile is red?


Let \(a, b, c, d\) be nonzero integers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.
\]
Find the smallest possible value of \(|a| + |b| + |c| + |d|\).


If 2 cards from a standard deck are selected randomly, what is the probability that either two Queens or at least 1 Ace occurs? (There are 4 Aces, 4 Queens, and 52 total cards in a standard deck.)


Find the modular inverse of $28$, modulo $29$.

Express your answer as an integer from $0$ to $28$, inclusive.


Compute the product of \( 0.\overline{1357} \) and 7, and express your result as a fraction in simplified form.


What is the greatest common factor of all three-digit palindromes? (A palindrome is a number that reads the same forwards as backwards.)


Consider a triangle $ABC$ where a circle with center $O$ is tangent to each side of the triangle. Given that $\angle BAC = 50^\circ$ and $\angle BCO = 20^\circ$, find the measure of $\angle ACB$, in degrees.


Suppose we have 15 dogs and we need to divide them into three groups: one with 4 dogs, one with 6 dogs, and one with 5 dogs. How many ways can we form these groups such that Duke is in the 4-dog group and Bella is in the 6-dog group?


What is the least possible five-digit palindrome that is divisible by 5?


Find the number of permutations \((b_1, b_2, b_3, b_4)\) of \((1,2,3,4)\) that satisfy
\[
\left(\frac{b_1^2 + 1}{2}\right) \cdot \left(\frac{b_2^2 + 2}{2}\right) \cdot \left(\frac{b_3^2 + 3}{2}\right) \cdot \left(\frac{b_4^2 + 4}{2}\right) \geq 4!.
\]


Given a rectangular box whose volume is 360 cubic units and where $a, b,$ and $c$ are integers with $a > b > c > 2$, what is the largest possible value of $b$?


Let \( F_1 = (-4, 2 - \sqrt{3}) \) and \( F_2 = (-4, 2 + \sqrt{3}) \). Then the set of points \( P \) such that
\[ |PF_1 - PF_2| = 2 \]
form a hyperbola. The equation of this hyperbola can be written as
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1, \]
where \( a, b > 0 \). Find \( h + k + a + b \).

Solution: \boxed{-1 + \sqrt{2}}


Compute the square of 1033 without a calculator.


The ratio of irises to roses in Nikki's garden is 3:7. Initially, she has 35 roses but gives away 15 roses. After this, she plans to add 25 more roses and wants to plant enough new irises to maintain the same ratio. How many irises will she need to have in total after these adjustments?


In a new sequence, the first term is $b_1 = 3010$ and the second term is $b_2 = 3011$. Additionally, the terms are designed such that $b_n + b_{n+1} + b_{n+2} = n + 4$ for all $n \ge 1$. Determine $b_{1000}$.


Determine the curve described by the equation $r = 6 \tan \theta \sec \theta.$

Solution: \boxed{x^2 = 6y}


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


Marty wants to paint a box. He can choose to use either blue, green, yellow, black, or white paint. However, the white paint cannot be applied with a roller. He can style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose?


Simplify $\sqrt{5 \cdot 3} \cdot \sqrt{3^5 \cdot 5^2}$.


Cory made a complete list of the prime numbers between 30 and 60. What is the sum of the smallest prime number, the largest prime number, and the median prime on his list?


Round 6437.5054 to the nearest even number.


The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:

(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left( \frac{2x}{5} \right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$

Find $g\left( \frac{3}{5} \right).$


Compute $\cos \left( \arcsin \frac{3}{5} \right)$ and $\tan \left( \arcsin \frac{3}{5} \right)$.


Find the sum of all angles \( x \in [0^\circ, 360^\circ] \) that satisfy
\[
\sin^3 x - \cos^3 x = \frac{1}{\cos x} - \frac{1}{\sin x}.
\]


Given that $x$ is a multiple of $34567$, what is the greatest common divisor of $g(x) = (3x+4)(8x+3)(15x+11)(x+15)$ and $x$?


A line on a Cartesian plane passes through points $\left(-3, 1\right)$ and $\left(2, 5\right)$. This line is parameterized so that the direction vector is of the form $\begin{pmatrix} 1 \\ b \end{pmatrix}.$ Find the value of $b$.


The function $\lceil x \rceil$ is defined as the smallest integer greater than or equal to $x$. For example, $\lceil 5.67 \rceil = 6$, $\lceil -\tfrac 14 \rceil = 0$, and $\lceil 8 \rceil = 8$.

What is the range of the function $$g(x) = \lceil x \rceil - x~?$$ Express your answer in interval notation.

Solution: \boxed{[0, 1)}


Given that $11^{-1} \equiv 31 \pmod{53}$, find $42^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)


How many license plates are possible if the plate consists of 3 letters followed by 3 digits, such that one digit is odd, another is even, and one is any number from 0 to 9?


On another episode of the game show $\text{\emph{Wheel of Fraction}}$, the spinner has been updated. The spinner now includes the following amounts: "$\text{Bankrupt}$", "$\$2000$", "$\$600$", "$\$3500$", "$\$800$". Each region has equal area. What is the probability that you earn exactly $\$3400$ in your first three spins? Assume you avoid bankruptcy in all spins.

Solution: \boxed{\frac{6}{64} = \frac{3}{32}}


The product of two 2-digit numbers is 2210. What is the smaller of the two numbers?


An isosceles trapezoid is inscribed around a circle. The longer base of the trapezoid is 18, and one of the base angles is arccos(0.6). Find the area of the trapezoid.


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


Bill is sent to a donut shop to purchase exactly eight donuts. The shop has five kinds of donuts (types A, B, C, D, E). Bill is required to get at least two donuts of type A and at least one of each of the remaining four types. How many combinations will satisfy Bill’s order requirements?


Find the minimum value of
\[(15 - x)(13 - x)(15 + x)(13 + x).\]


Given a function \( f(x) \) from positive real numbers to real numbers such that
\[ xf(y) - yf(x) = f(xy) \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( f(100) \).


The sequence in the lattice is altered such that each row now increments by 4 starting from 2, and each row contains 6 elements. The sequence is as follows:
\[
\text{Row 1: } 2, 6, 10, 14, 18, 22
\text{Row 2: } 26, 30, 34, 38, 42, 46
\text{Row 3: } 50, 54, 58, 62, 66, 70
\text{Row 4: } 74, 78, 82, 86, 90, 94
\]
What will be the fourth number in the $9$th row?


On the 6 by 6 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. Determine the product of the area of rectangle $EFGH$ (in cm$^2$) and the perimeter of rectangle $EFGH$ (in cm). Express your answer in simplest radical form.

[asy]
unitsize(1cm);
defaultpen(linewidth(0.7));
for(int i = 0; i <= 5; ++i) {
    for(int j = 0; j <= 5; ++j) {
        dot((i,j));
    }
}
draw((1,4)--(4,5)--(5,2)--(2,1)--cycle);
label("$E$",(1,4),W);
label("$F$",(4,5),N);
label("$G$",(5,2),E);
label("$H$",(2,1),S);
[/asy]


Compute \[ \left\lfloor \dfrac {2015^2}{2013 \cdot 2014} - \dfrac {2013^2}{2014 \cdot 2015} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


The median of the set of numbers $\{15, 43, 48, x, 17\}$ is six less than the mean. If $x$ is a negative integer, what is the value of $x$?


A point \((x, y)\) is randomly picked from inside the rectangle with vertices \((0, 0)\), \((6, 0)\), \((6, 2)\), and \((0, 2)\). What is the probability that \(x < 3y\)?


Determine the range of the function $f(x) = (x^2 - 2)^2 + 1$ for the domain $[0,\infty)$.

Solution: \boxed{[1, \infty)}


The coefficients of the polynomial
\[ b_8 x^8 + b_7 x^7 + b_6 x^6 + \dots + b_1 x + b_0 = 0 \]
are all integers, and its roots \( s_1, s_2, \dots, s_8 \) are all integers. Additionally, the roots of the polynomial
\[ b_0 x^8 + b_1 x^7 + b_2 x^6 + \dots + b_6 x^2 + b_7 x + b_8 = 0 \]
are also \( s_1, s_2, \dots, s_8 \). Determine the number of possible multisets \( T = \{s_1, s_2, \dots, s_8\} \), given that \( b_8 \neq 0 \) and \( b_0 \neq 0 \).


Solution: \boxed{14a + 30b}


Each row of a seating arrangement can seat either 6 or 7 people. Fifty-three people need to be seated, with every seat occupied. How many rows seat exactly 7 people?


Solution: \boxed{12\sqrt{2} + 2}


Suppose you flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads and that both the penny and half-dollar are heads?


In how many different ways can 3 men and 4 women be placed into two groups where one group consists of four people containing at least two men and two women, and the other group consists of three people with no further gender-specific conditions?


Find the remainder when \(x^5 - 2x^4 + 3x^3 + 4\) is divided by \(x^2 - 4x + 6\).

Solution: \boxed{2x - 44}


Consider the ellipse given in the equation
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]
The center and axes lengths are represented in the following graphical diagram:

[asy]
unitsize(0.3 cm);

int i, n = 10;

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

draw((0,-n)--(0,n));
draw((-n,0)--(n,0));

draw(shift((3,-1))*xscale(6)*yscale(4)*Circle((0,0),1),blue);

dot((3,-1));
[/asy]

Find the sum \( h + k + a + b \).


How many values of \( x \), \( -20 < x < 90 \), satisfy the equation \(\cos^2 x + 3\sin^2 x = 2\)? (Note: \( x \) is measured in radians.)


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

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


Find the minimum value of
\[ \sqrt{x^2 + (1 - x)^2} + \sqrt{(x - 1)^2 + (x - 1)^2} \]
over all real numbers $x$.


Let $m$ be the smallest positive, three-digit integer congruent to 6 (mod 13). Let $n$ be the smallest positive, four-digit integer congruent to 7 (mod 17). What is $n-m$?


How many perfect cubes less than 20,000 can be represented as the difference of two consecutive perfect squares?


Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=7x-4$.  Find $g^{-1}(f(2))$.


If $x^2 + 3x + 7$ is a factor of $x^4 + Rx^2 + S$, find $R + S$.


How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 6, 8, or 9?


What is the value of $\dfrac{5 \times 7}{10}?$


Solution: \boxed{\frac{3}{5}(1 + x - x^2)}


A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of twice the volume. Given the radius of the hemisphere is \( 6\sqrt[3]{4} \) cm, find the radius of the original bubble.

Solution: \boxed{R = 6 \text{ cm}}


A conservation group is planning to hire new staff so that 100 of these will work on forest conservation, 90 will work on wildlife conservation, and 40 of them will be involved in both areas. How many minimum employees should they hire?


Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $2x_1 + 3x_2 + 4x_3 = 120.$  Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2.\]


For how many bases between four and twelve inclusive does the representation of $1024_{10}$ have a final digit of 1?


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


What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome?


Determine how many more digits the base-3 representation of the number \(1357_{10}\) has than both the base-5 and base-8 representations combined of the same number.


If $\cos \theta = \frac{3}{5},$ then find both $\cos 3 \theta$ and $\sin 3 \theta$.

Solution: \boxed{\cos 3\theta = -\frac{117}{125}, \sin 3\theta = \frac{44}{125}}


If eight people decide to come to a basketball game, but four of them are only 1/3 sure that they will stay for the entire time (the other four are sure they'll stay the whole time), what is the probability that at least 4 people stayed the entire time?


In Pascal's Triangle, each entry is the sum of the two entries directly above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $4:5:6$?

(The top row of Pascal's Triangle has only a single $1$ and is the $0$th row.)


If the product $(4x^2 - 6x + 3)(8 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real numbers, then find $8a + 4b + 2c + d$.


For three points $(2, a+1, b+1)$, $(a+1, 3, b+1)$, and $(a+1, b+1, 4)$ to be collinear, determine the value of $a$ and $b$ and compute $a + b$.


Determine the values of $x$ for which $x^2 - 48x + 500 \leq 0$. Express your answer in interval notation.

Solution: \boxed{[24 - 2\sqrt{19}, 24 + 2\sqrt{19}]}