I had $\$60$ in allowance money and spent it as indicated in the pie graph shown. How much did I spend on video games?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.966,.259);
C=(-.707,-.707);
D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Books", W, N);
label("$\frac{1}{4}$", W, S);
X=(-.55, 0);
label("Video Games", X, S);
Y=(-.17,-.7);
label("Snacks", Y, N);
label("$\frac{1}{6}$", Y, S);
Z=(.5, -.15);
label("Toys", Z, N);
label("$\frac{2}{5}$", Z, S);
[/asy]


Eleven stones are arranged in a straight line. They are counted from left to right as $1, 2, 3, \ldots, 11$, and then from right to left, so that the stone previously counted as 10 is counted as 12. The pattern is continued to the left until the stone previously counted as 1 is counted as 21. The pattern then reverses so that the stone originally counted as 2 is counted as 22, 3 as 23, and so on. The counting continues in this manner. Which of the original stones is counted as 137?


In a hexagon, the measures of five angles are \(130^\circ\), \(95^\circ\), \(115^\circ\), \(120^\circ\), and \(110^\circ\). What is the measure of the sixth angle, \(Q\)?


A triangular array of squares has one square in the first row, two in the second, and in general, \( k \) squares in the \( k \)th row for \( 1 \leq k \leq 9 \). Each square, except those in the bottom row, contains a number that is the sum of the numbers in the two squares directly below it. In each square of the ninth row, a 0 or a 1 is placed. Determine for how many initial distributions of 0's and 1's in the bottom row, the number in the top square is a multiple of 2.


Ray climbs a flight of stairs in two ways. When he takes $6$ steps at a time, there are $4$ steps left at the top. When he takes $7$ steps at a time, there are $3$ steps left. Calculate the smallest number of steps in the staircase, $n$, that is greater than $30$.


In square \( ABCD \), point \( P \) is the midpoint of side \( AB \), point \( Q \) is a point on side \( BC \) such that \( BQ = \frac{2}{3}BC \). What is the ratio of the area of triangle \( APQ \) to the area of square \( ABCD \)?


Eight students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Aaron ate the most apples and Zeb ate the fewest. How many more apples than Zeb did Aaron eat?

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[11];
for( int i = 0 ; i < 11 ; ++i ){

yaxis[i] = (0,i);

draw(yaxis[i]--yaxis[i]+(16,0));
}
draw((0,0)--(0,10));
draw((16,10)--(16,0));

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


William is participating in a 30-question multiple choice exam. Each question has five answer choices. He randomly guesses the answers for the last six questions. What is the probability that he will get at least two of these six questions correct?


Find the product of the divisors of \(72\).


Let $n$ equal the number of sides in a regular polygon. For $4 \leq n < 12$, how many values of $n$ result in a regular polygon where the common degree measure of both the interior and exterior angles is not an integer?


The Cookie Monster finds another peculiar cookie represented by the boundary equation \(x^2 + y^2 + 36 = 6x + 24y\). He needs to determine if this cookie is large enough for his voracious appetite. What is the radius of this cookie?


Find the largest value of $n$ such that $3x^2 + nx + 108$ can be factored as the product of two linear factors with integer coefficients.


The noon temperatures for nine consecutive days were $82^\circ$, $78^\circ$, $84^\circ$, $86^\circ$, $88^\circ$, $90^\circ$, $92^\circ$, $84^\circ$, and $85^\circ$ Fahrenheit. Calculate the mean noon temperature, in degrees Fahrenheit, for these days, excluding any temperatures below $80^\circ$.


Evaluate the determinant of the given matrix:
\[
\begin{vmatrix} \sin \theta \sin \phi & \sin \theta \cos \phi & \cos \theta \\ \cos \phi & -\sin \phi & 0 \\ -\cos \theta \sin \phi & -\cos \theta \cos \phi & \sin \theta \end{vmatrix}
\]


Let $x$, $y$, $z$, and $w$ be real numbers such that $|x-y|=1$, $|y-z|=2$, and $|z-w|=3$. Find the sum of all possible values of $|x-w|$.


$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 2, CD = 3,$ and $DE = 8$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?


Find the remainder when $3 \times 7 \times 13 \times 17 \times 23 \times 27 \times \ldots \times 193 \times 197$ is divided by $6$.


Janine is creating a design using a large square and inside it are placed six small circles. Each circle is tangent to one side of the square and the three circles adjacent to it. The side length of the square in Janine's design is 30 inches. How many square inches will be shaded in her design?

[asy]
size(120);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);

fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,grey);

draw(circle((1,1),1));
fill(circle((1,1),1),white);
draw(circle((5,1),1));
fill(circle((5,1),1),white);
draw(circle((1,5),1));
fill(circle((1,5),1),white);
draw(circle((5,5),1));
fill(circle((5,5),1),white);

draw(circle((3,2),1));
fill(circle((3,2),1),white);
draw(circle((3,4),1));
fill(circle((3,4),1),white);
[/asy]

Solution: \boxed{900 - 150\pi \text{ square inches}}


Find the quotient when $x^4 - 22x^3 + 12x^2 - 13x + 9$ is divided by $x - 3$.

Solution: \boxed{x^3 - 19x^2 - 45x - 148}


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


How many distinct sequences of six letters can be made from the letters in "EXEMPLARY" if each sequence must begin with "E", end with "Y", and no letter can appear in a sequence more than once?


Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 30x^2 + 65x - 42 = 0$. Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}$.


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


What is the value of $x^2 + y^2 + z^2 + 2xy - 2yz$ when $x=2$, $y=-3$, and $z=6$?


Evaluate $\sec 225^\circ$ and $\csc 225^\circ$.

Solution: \boxed{\sec 225^\circ = -\sqrt{2}, \quad \csc 225^\circ = -\sqrt{2}}


A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. However, the side length of the smaller square inscribed in the first circle is \(3r\) instead of \(2r\) where \(r\) is the radius of the smaller circle. What is the ratio of the area of the smaller circle to the area of the larger square?


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{13}$?


The time right now is 3:15 p.m. What time will it be in 1850 minutes?

Solution: \boxed{10\!:\!05 \text{ p.m.}}


Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \[
n^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx + 2x + 2y + 2z - 12.
\]


A projectile is launched with an initial velocity of $u$ at an angle of $\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:
\[
x = ut \cos \phi,
\]
\[
y = ut \sin \phi - \frac{1}{2} gt^2,
\]
where $t$ is time and $g$ is the acceleration due to gravity. Suppose $u$ is constant but $\phi$ varies from $0^\circ$ to $180^\circ$. As $\phi$ changes, the highest points of the trajectories trace a closed curve. The area enclosed by this curve can be expressed as $d \cdot \frac{u^4}{g^2}$. Find the value of $d$.


A club has 15 members, 6 boys and 9 girls. Three of the members are chosen at random. What is the probability that all three are girls?


Suppose that each of the numbers \[2 - \sqrt{3}, \; -2-\sqrt{3}, \; 1+\sqrt{5}, \; 1-\sqrt{5}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


Simplify: $\frac{3^{n+3} - 3(3^n)}{3(3^{n+2})}$ and express your answer as a common fraction.


A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such composite numbers?


The graph of $xy = 4$ is a hyperbola. Identify the position of the foci and calculate the distance between them.


A beach ball must have a maximum circumference of 33 inches. Calculate the surface area on the beach ball excluding the valve area. Assume the valve has a circular shape with a diameter of 1 inch. Express your answer as a common fraction in terms of $\pi$.

Solution: \boxed{\frac{4356 - \pi^2}{4\pi}}


What is the largest four-digit number whose digits add up to 23?


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 $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $D, E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ is parallel to $\overline{AC}$ and $\overline{EF}$ is parallel to $\overline{AB}$. What is the perimeter of parallelogram $ADEF$?


Calculate $(18y^3) \cdot (9y^2) \cdot \left(\frac{1}{(6y)^2}\right)$.


Find the product of all positive integral values of $n$ such that $n^2 - 41n + 420 = q$ for some prime number $q$. Note that there is at least one such $n$.


Find the area of the parallelogram generated by vectors $\begin{pmatrix} 2 \\ 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}$.


What is the coefficient of \(x^2\) when \(3x^3 + 4x^2 + 2x - 3\) is multiplied by \(2x^2 + 7x - 4\) and the like terms are combined?


Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55.


Bonnie constructs the frame of a larger cube using 12 pieces of wire each 8 inches long. Concurrently, Roark uses 0.5-inch-long pieces of wire to create a collection of smaller cube frames. The total volume of Roark's smaller cubes combined equals the volume of Bonnie's larger cube. Compute the ratio of the total length of Bonnie's wire to the total length of Roark's wire.


Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $10$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $35$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors?


Let \( y \) be a very large number given by \( 3^44^55^66^77^88^99^{10} \). What is the smallest positive integer that, when multiplied by \( y \), results in a product that is a perfect square?


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 17. What is the greatest possible perimeter of the triangle?


Automobile license plates for a region consist of four letters followed by a dash and three single digits. How many different license plate combinations are possible if exactly two letters are repeated once each, and digits increase consecutively?


Simplify the expression:
\[
\sin^2 x + \sin^2 (x + y) - 2 \sin x \sin y \sin (x + y).
\]

Solution: \boxed{\sin^2 y}


How many nonzero terms are in the expansion of $$(x^2 + 2)(3x^3 + 5x^2 + 2) - 2(x^4 - 3x^3 + 2x^2)?$$


If \( a \) is a constant such that \( 9x^2 + 27x + a \) is the square of a binomial, then what is \( a \)?


In triangle $ABC$, $AD=DC$ where $D$ is a point on $\overline{AC}$, and $BD$ is a point on $\overline{BC}$ such that $BD = BC$. If $\overline{BD}$ bisects angle $ABC$, what is the measure, in degrees, of angle $B$?


Find the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 12\]and $x,$ $y,$ $z \ge 0,$ with the additional constraint that $x + y \ge 2z$.


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 $p+q$?


If the six solutions of \(x^6=-729\) are written in the form \(a+bi\), where \(a\) and \(b\) are real numbers, find the product of those solutions for which \(a>0\).


Given that $g(x) = x^{-2} + \frac{x^{-2}}{1 - x^{-2}}$, calculate $g(g(2))$. Express your answer as a common fraction.


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and
\[ f(x^2 + y^2) = (x + y)(f(x) - f(y)) \]
for all real numbers $x$ and $y$. Determine $n$, the number of possible values of $f(2)$, and $s$, the sum of all possible values of $f(2)$. Finally, compute $n \times s$.


Piravena is planning a triangular journey starting from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally returning from $Z$ to $X$. The cities form a right-angled triangle, with $Z$ a distance of $4000\text{ km}$ from $X$ and city $Y$ a distance of $4500\text{ km}$ from $X$. Traveling by bus costs $\$0.20$ per kilometer. To fly, there is a $\$120$ booking fee plus $\$0.12$ per kilometer. Piravena begins her trip by flying from $X$ to $Y$. Determine the cost to fly from $X$ to $Y$.


Given the circle with equation $x^2 + 8x - 10y = 10 - y^2 + 6x$, find the center $(a,b)$ and the radius $r$ and compute $a+b+r$.


Chloe wants to buy a hoodie that costs \$45.50. She empties her wallet and finds she only has four \$10 bills, ten quarters, three $1 coins, and a pile of dimes. What is the minimum number of dimes that must be in her pile so she can pay for the hoodie?


Let $\omega$ be a nonreal root of $z^4 = 1.$ Find the number of ordered pairs $(a, b)$ of integers such that $|a \omega + b| = 1.$


For each positive integer $n$, let $h(n) = n^4 - 500n^2 + 625$. What is the sum of all values of $h(n)$ that are prime numbers?


The eighth, ninth, and tenth terms of an arithmetic sequence are 14, 18, and 22, respectively. What is the sum of the first six terms of the sequence?


Let
\[\bold{B} = \begin{pmatrix} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{pmatrix}.\]
Determine the constants $p$, $q$, and $r$ so 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. Enter the ordered triple $(p, q, r)$.

Solution: \boxed{(-3, -6, 8)}


Suppose a parabola has vertex \(\left(\frac{1}{3},-\frac{5}{9}\right)\) and follows the equation \(y = ax^2 + bx + c\), where \(a > 0\) and \(a + b + c\) is an integer. Find the smallest possible value of \(a\).


A right triangle has one leg of 15 meters and a hypotenuse of 17 meters. A second triangle, similar to the first, has a hypotenuse of 51 meters. What is the length of the other leg in the second triangle?

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


What is the smallest positive integer $n$ such that $\frac{n}{n+103}$ is equal to a terminating decimal?


If a 5'' by 5'' square is added at each successive stage, what will be the area of the rectangle at Stage 4, in square inches?


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


In an equilateral triangle, the area is one and a half times the perimeter. What is the radius of the inscribed circle?


Find the largest of five consecutive two-digit positive integers, all less than 40, which are not prime.


A binary operation \(\diamondsuit\) has properties that \(a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c\) and that \(a\,\diamondsuit \,a=1\) for all nonzero real numbers \(a, b,\) and \(c\). Find the solution to the equation \(504 \,\diamondsuit\, (3\,\diamondsuit\, x)=50.\)

Solution: \boxed{x = \frac{25}{84}}


What is the value of $x$ if it is a negative integer divisible by 3 and in the set $\{15, x, 50, 19, 37\}$, the median is four less than the mean?


How many positive four-digit integers with a $5$ in the units place are divisible by $15$?


What is the smallest prime whose digits sum to \(28\)?


The sum of 81 consecutive integers is $9^5$. What is their median?


The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years between the ages of any two consecutive children. What is the age of the oldest child?


Consider the sequence where the $n$-th term is given by $a_n = 120 + n^2$, for $n=1,2,3,\ldots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Determine the probability that $$(a-2)(b-1)(c-1) \neq 0 $$


The operation $\star$ is defined as $a \star b = a - \frac{a}{b}$. Calculate $18 \star 6$ and then add this result to 2 times $6$.


Find the minimum of the function
\[\frac{x^2y}{x^3 + y^3}\]
in the domain \(\frac{1}{3} \le x \le \frac{2}{3}\) and \(\frac{1}{4} \le y \le \frac{1}{2}.\)


Determine the product of the solutions to the quadratic equation $49 = -2x^2 - 8x$.


Determine the range of the function $g(x) = \arcsin x + \arccos x + \arctan x + 2\arcsin x$ for $-1 \leq x \leq 1.$

Solution: \boxed{\left[-\frac{3\pi}{4}, \frac{7\pi}{4}\right]}


Emily has two numbers, $c$ and $d$. When she divides $c$ by 60, she gets a remainder of 53. When she divides $d$ by 42, she gets a remainder of 35. What remainder does she get when she divides $c+d$ by 21?


If the seven-digit number $945n631$ is divisible by $11$, what is $n$?


Determine the number of angles between 0 and $2\pi$, other than integer multiples of $\frac{\pi}{4}$, such that $\sin \theta$, $\cos \theta$, and $\sec \theta$ form a geometric sequence in some order.


Julia has more than $\$2$ but less than $\$20$ worth of dimes. When she arranges her dimes in stacks of 6, she has 1 dime left over. When she puts them in stacks of 7, she has 1 dime left over. When she stacks them in 8, she again has 1 dime left over. How many dimes does Julia have?


Given two positive real numbers $a$ and $b$, the operation $a \, \star \, b$ is defined such that $(a \star b) \star b = a(b \star b)$ and $(a \star 1) \star a = a \star 1$ for all $a, b > 0$. Moreover, $1 \star 1 = 2$. Find $5 \star 10$.


Compute the smallest positive angle $y$, in degrees, such that
\[\tan 3y = \frac{\cos y - \sin y}{\cos y + \sin y}.\]

Solution: \boxed{y = 11.25^\circ}


Factor the expression $3y(y^2 - 4) + 5x(y^2 - 4)$.

Solution: \boxed{(3y + 5x)(y + 2)(y - 2)}


What is the greatest common divisor of $2197$ and $2209$?


In three-dimensional space, find the number of lattice points in the first octant that are a distance of 5 from the origin.

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


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

\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((a_i),\) \((b_i),\) \((c_i)\) be the sequence, from left to right, of elements in the 10th, 11th, and 12th rows, respectively, with the leftmost element occurring at \(i = 0.\) Compute
\[
\sum_{i = 0}^{10} \frac{b_i}{c_i} - \sum_{i = 0}^{9} \frac{a_i}{b_i}.
\]


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


$216_{10}$ in base $b'$ has exactly $5$ digits. What is the value of $b'$?


If the roots of the quadratic equation $\frac{5}{2}x^2 + 17x + c = 0$ are $x = \frac{-17 \pm \sqrt{23}}{5}$, then what is the value of $c$?


What is the units digit of the sum of the six terms of the sequence $1! + 3, \, 2! + 3, \, 3! + 3, \, 4! + 3, \, 5! + 3, \, 6! + 3$?


In rolling five standard six-sided dice, it is found that two of the dice form a pair, another die shows a different number distinct from the pair, and the other two dice each show different numbers as well. The last two dice which show different numbers are set aside for rerolling. What is the probability that, after rerolling these two dice, at least three of the five dice show the same value?


Determine the smallest positive integer $n$ such that $4n$ is a perfect square and $5n$ is a perfect cube.


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

Solution: \boxed{3^7 = 2187}


Both roots of the quadratic equation $x^2 - 71x + k = 0$ are prime numbers. How many possible values of $k$ are there?


If
\[\cot x = \frac{\sqrt{a^2 - b^2}}{2ab},\] where $a > b > 0$ and $0^\circ < x < 90^\circ$, then determine $\sin x$ in terms of $a$ and $b$.

Solution: \boxed{\frac{2ab}{\sqrt{4a^2b^2 + a^2 - b^2}}}


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


For how many integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 6?


What is the value of $n$ in the equation $n + (n + 1) + (n + 2) + (n + 3) = 20$?


A ball is dropped from 800 feet high and bounces back with each bounce reaching 40% of the height of the previous bounce. After how many bounces will the ball first reach a maximum height less than 5 feet?


The equation of the ellipse is given by
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{c^2} = 1.\]
An ellipse has a center at \((-3, 1)\), a horizontal semi-major axis of length 4, and a vertical semi-minor axis of length 2. Calculate \(h + k + a + c.\)


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


The surface area of a sphere with radius $r$ is $4\pi r^2$. Including the area of its circular base, what is the total surface area of a hemisphere with radius 8 cm? Additionally, calculate the surface area of a cylinder with the same height and radius as the hemisphere. Express your answer in terms of $\pi$.


If \(\frac{9}{x^2} = \frac{x}{25}\) and \(x\) is not equal to zero, what is the value of \(x\)?


Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 18, what is the smallest value of $n$ such that $g(n) > 18$?

Solution: \boxed{n = 342}


Determine how many integers \( n \) satisfy \( 200 < n < 300 \), with \( n \) having the same remainder when divided by \( 7 \) or \( 9 \), and \( n \) is even.


Starting with the number 150, Emily repeatedly divides her number by two and then takes the greatest integer less than or equal to that number. How many times must she do this before she reaches the number 1?


A frustum of a right circular cone is formed by cutting a small cone from the top of a larger cone. If this frustum has an altitude of \(20\) centimeters, the area of its lower base is \(324\pi\) sq cm, and the area of its upper base is \(36\pi\) sq cm, what is the altitude of the small cone that was cut off?


Let $A,$ $B,$ and $C$ be points on a circle of radius $12.$ If $\angle ACB = 90^\circ,$ what is the circumference of the minor arc ${AB}$? Express your answer in terms of $\pi.$


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = 9$, $SQ = 8$, and $SR = 7$, what is the volume of pyramid $SPQR$?


I had planned to work for 25 hours a week for 15 weeks during the summer to save \$4500 for a study abroad program. Unfortunately, after working for three weeks, I had to reduce my work schedule for another two weeks to 10 hours per week due to a family commitment. How many hours per week will I have to work for the remaining weeks of the summer to still reach my financial goal?


What is the area, in square units, of a trapezoid bounded by the lines $y = x + 2$, $y = 12$, $y = 6$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.


Find the largest value of $n$ such that $5x^2 + nx + 80$ can be factored as the product of two linear factors with integer coefficients, including at least one prime number as a factor.

Solution: \boxed{n = 401}


Both roots of the quadratic equation $x^2 - 59x + k = 0$ are prime numbers. How many possible values of $k$ are there?


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

Solution: \boxed{-1524 + 6\sqrt{3}}


The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?


Arthur walks 8 blocks south and then 10 blocks west. Each block is one-fourth of a mile. How many miles did Arthur walk in total?


The pages of a book are numbered from 1 through $n$. One of the page numbers was mistakenly added twice, resulting in a total page number sum of 2378. Determine the page number that was added twice, given that $n$ lies in the range of 60 to 70.


A truncated cone has horizontal bases with radii 12 and 3. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. Calculate the radius of the sphere.


You have five shirts and six pairs of pants. However, one specific shirt cannot be paired with two specific pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make?


Determine the total count of numbers in the sequence $$ 2, 5, 8, \ldots, 2005, 2008. $$


\[
\text{There are } 30 \text{ different complex numbers } z \text{ such that } z^{30}=1. \text{ For how many of these is } z^5 \text{ a real number?}
\]


What is the midpoint of the segment with endpoints \( (5.5, -3) \) and \( (-1.5, 7) \)?

Solution: \boxed{(2, 2)}


What is the constant term in the expansion of $(x^4 + x^2 + 7)(x^6 + x^3 + 3)(2x^2 + 9)$?


What is the smallest prime whose digits sum to $22$?


If $16^8 = 8^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.


Simplify $5 \cdot \frac{21}{8} \cdot \frac{32}{-63}$.


Determine the coefficient of $x$ when the expression $5(x - 6) + 6(9 - 3x^2 + 3x) - 9(5x - 4)$ is simplified.


A paperboy delivers newspapers to 12 houses along a street. He does not always deliver to every house, but to avoid trouble he never misses four consecutive houses. Compute the number of ways the paperboy could deliver papers under these conditions.


If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^8$ are odd?


A perfectly round pizza with a diameter of $16\text{ cm}$ is cut into four equal-sized sector-shaped slices. Calculate $m^2$, where $m$ is the number of centimeters in the length of the longest line segment that can be drawn in one of these slices.


A cube with an edge length of 5 units has the same volume as a square-based pyramid with base edge lengths of 10 units. What is the value of height \( h \) of the pyramid?

Solution: \boxed{h = 3.75}


Find the product of $237_9 \cdot 17_9$. Express your answer in base 9.


Juan needs to calculate the area of a circle but, due to a measurement error, he might estimate the diameter wrong by up to 30%. If the true diameter of the circle is 30 cm, what is the maximum possible percent error, in percent, in Juan's computed area of the circle?


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


The sum of the lengths of the twelve edges of a rectangular box is $180$, and the distance from one corner of the box to the farthest corner is $25$. Additionally, one of the dimensions of the box is known to be $10$. Calculate the total surface area of the box.


Solution: \boxed{x = 4}


A sequence consists of $1500$ terms. Each term after the first is 1 larger than the previous term. The total sum of the $1500$ terms is $3000$. When every third term is summed, starting with the first term, what is the sum?


Determine the value of $y$ in the equation $8^5 + 8^5 + 2 \cdot 8^5 = 2^y$.


A bus, when its brakes are applied, travels 6 feet less in each second than it did the previous second until it comes to a complete stop. The bus travels 30 feet in the first second after the brakes are applied. How many feet does the bus travel from the time the brakes are applied until it stops?


Joyce now counts the pennies in her bank differently. When she counts them by sevens, she has three left over. When she counts them by fives, there are four left over. When she counts them by threes, there remains two left over. What is the least number of pennies in Joyce's bank?


A conservatory houses five pairs of different animals, one male and one female of each type. The feeder must alternate between feeding a male and a female each time. If the feeder begins by feeding a female hippopotamus, how many ways can the feeder complete feeding all the animals?


How many ways are there to distribute 5 distinguishable balls into 3 indistinguishable boxes?


How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable but the boxes are not?


I have 12 distinguishable socks in my drawer: 3 white, 3 brown, 3 blue, and 3 black. In how many ways can I choose a pair of socks, provided that I get two socks of different colors?


Spinners $C$ and $D$ are spun. Spinner $C$ has numbers 1 through 5, and Spinner $D$ has numbers 1 through 4. Each number is equally likely to be selected. What is the probability that the product of the two spinners' numbers is even? Express your answer as a common fraction.


Find the positive integer $n$ such that
\[
\tan \left( \frac{\pi}{2n} \right) + \sin \left (\frac{\pi}{2n} \right) = \frac{n}{3}.
\]


At Lakeview Book Club, the average number of books read in a year by regular members is 5 and by premium members is 12. The average number of books read by all members combined is 8. What is the ratio of regular members to premium members? Express your answer as a common fraction.


The digits of a three-digit number $ABC$ are reversed to form a second three-digit number $CBA$. If $A \neq C$, what is the smallest prime number that must be a factor of the difference $ABC - CBA$?


When the greatest common divisor and least common multiple of two integers are multiplied, their product is 180. How many different values could be the greatest common divisor of the two integers?


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


Determine the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 3027 - 3028 \operatorname{sign}(y + z + 1), \\
y &= 3027 - 3028 \operatorname{sign}(x + z + 1), \\
z &= 3027 - 3028 \operatorname{sign}(x + y + 1).
\end{align*}
Note: For a real number $a,$
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]


What is the remainder when $2011 \cdot 2012 \cdot 2013 \cdot 2014 \cdot 2015$ is divided by 17?


If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^8$ are odd?


In triangle $PQR$, $\cos(Q - 2P) + \sin(P + Q) = 2$ and $PQ = 6$. What is $QR$?


Let $\triangle PQR$ be a right triangle with angle $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $192$ and $PR = 32$, what is the length of $QS$?


Suppose we have three numbers using which, when added two at a time, the sums are 37, 58, and 72. After finding the sum of all three numbers, subtract 10 to get the final result. What is the final result?


What is the sum of the different prime factors of $390390$?


John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green?


How many positive four-digit integers of the form $\_\_90$ are divisible by 90?


If \( x^2 - x + 6 = y + 2 \) and \( x = -8 \), what is the value of \( y \)?


A club consists of 30 members, 18 boys and 12 girls. In how many ways can we choose a president, a vice-president, and a treasurer if all selected officers must be of the same gender and no one can hold more than one office?


Solution: \boxed{b = \frac{1}{125}}


In $\triangle PQR$, point $S$ is the midpoint of side $PQ$. Point $T$ is on $QR$ such that $QT:TR = 1:3$. Point $U$ is on $PS$ such that $PU:US = 2:1$. If the area of $\triangle STU$ is 12, determine the area of $\triangle PQR$.


Samantha's Sandwich Shop offers custom sandwiches with these 10 toppings: mustard, ketchup, mayo, lettuce, tomato, onions, pickles, olives, cheese, and peppers. A customer can choose a sandwich with 1, 2, or 3 meat slices or opt for a veggie slice instead. With any combination of the toppings, how many different kinds of sandwiches can be made?


In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is one-third of the way from $B$ towards $C$ on side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction.


In a map using a scale where 1 cm equals 1 mile, a similar plot of land is shaped like a trapezoid with a bottom base of 20 cm, top base of 25 cm, and height of 15 cm. One square mile equals 640 acres. Calculate the total area of the actual plot of land in acres.

Solution: \boxed{216000 \text{ acres}}


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


How many distinct four-digit positive integers are there such that the product of their digits equals 8?


A garden in the shape of a square has an area of 784 square meters. After measuring the garden, an additional 10 meters of fencing is required on each side to enhance protection. Calculate the total length of fencing needed.

Solution: \boxed{152 \text{ meters}}


Circle $Q$ has a circumference of $16\pi$ feet. If segment $AB$ is a diameter and the measure of angle $AQB$ is $120^\circ$, what is the length, in feet, of segment $QB$?

Solution: \boxed{8 \text{ feet}}


Completely factor the following expression: \[(10x^3 + 50x^2 - 4) - (3x^3 - 5x^2 + 2).\]

Solution: \boxed{7x^3 + 55x^2 - 6}


Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac{1}{2}$ the measure of $\angle 2$. On line $k$, $\angle 1$ and $\angle 3$ are supplementary, as are $\angle 2$ and $\angle 4$. What is the degree measure of $\angle 5$?


Find the greatest possible value of $k$ for which the roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{85}$.


The polynomial $P(x) = x^3 + ax^2 + bx + c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. Given that the $y$-intercept of the graph of $y = P(x)$ is 3, find $b$.


Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
\[
f(x) f(-x) = f(x^3).
\]


Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=60^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=p+\sqrt{q},$ where $p$ and $q$ are integers, find $p+q.$


In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 75 square units, and the area of triangle $ADE$ is 30 square units, what is the area of trapezoid $ABCD$?

Solution: \boxed{147 \text{ square units}}


Given positive integers $x$ and $y$ such that $x \neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, what is the smallest possible value for $x + y$?


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2008} \times 7$?


A line contains the points $(3,5)$, $(-1, k)$, and $(-7, 2)$. What is the value of $k$?


There are 60 chips in a box. Each chip is either small or large. If the number of small chips exceeds the number of large chips by a composite number of chips, what is the greatest possible number of large chips?


From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$.


Solution: \boxed{x^4 - 3x^2 + 9}


In 1950, there were 600,000 cases of a certain disease reported in a country. In 2000, there were 300 cases reported. How many cases of the disease would have been reported in 1975 if the number of cases reported from 1950 to 2000 decreased linearly?


In the expression \( e \cdot f^g - h \), the values of \( e \), \( f \), \( g \), and \( h \) are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the result?


A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 6 meters from its base. When Matthew walks 4 meters from the base of the pole toward the point where the cable is attached to the ground, his head just touches the cable. Matthew is 1.8 meters tall. How many meters tall is the pole?

Solution: \boxed{5.4\text{ meters}}


The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+1)^2 = 1$ are tangent. Compute $m$.


Find an ordered pair $(x,y)$ that solves the system:
\begin{align*}
7x &= -5 - 3y, \\
4x &= 5y - 34
\end{align*}

Solution: \boxed{\left(-\frac{127}{47}, \frac{218}{47}\right)}


Let \(x, y, z\) be non-zero complex numbers such that:
\[ x + y + z = 18 \]
and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz\]
Find the value of \(\frac{x^3 + y^3 + z^3}{xyz}\).


If $3x - 2y = 7$ and $x + 3y = 8$, what is the value of $x$?


For all positive integers $n$, let $g(n)=\log_{3003} n^2$. Calculate $g(7) + g(11) + g(13)$.


Two spinners are divided into quarters and fifths, respectively. The first spinner (quarters) has numbers 2, 4, 6, and 8. The second spinner (fifths) has numbers 1, 3, 5, 7, and 9. If each spinner is spun once, what is the probability that the product of the two spin results will be an even number? Express your answer as a common fraction.


Find the $2 \times 2$ matrix $\mathbf{N}$ such that $\mathbf{N} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ -7 \end{pmatrix}$ and $\mathbf{N} \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 10 \\ 1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{25}{7} & \frac{15}{7} \\ -\frac{20}{7} & \frac{9}{7} \end{pmatrix}}


A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base is $45^\circ$. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side?


Each of the first two digits must be 1, 2, 4, or 5, the last two digits must be 5, 6, or 7, and the last two digits must not be the same. Calculate the number of 4-digit positive integers satisfying these conditions.


Let $ T $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ T $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?


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 $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$ Enter the ordered pair $(t,u).$

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


Ms. Rivera's high school basketball league wants to organize a championship with eighteen teams in a knockout format. However, the two top-ranked teams are given a bye in the first round. How many games will be played to determine the winner?


Determine how many lattice points lie on the graph of the equation $x^2 - y^2 = 61$ in the $x,y$-plane.


Calculate $\left(\frac{3}{5}\right)^{10} \cdot \left(\frac{2}{3}\right)^{-4}$.


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


Quadrilateral $ABCD$ is a parallelogram. Its area is $48$ square units. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. What is the area of triangle $ACF?$


Solution: \boxed{-7 - 24i}


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


What is the smallest positive multiple of $17$ that is $6$ more than a multiple of $73$?


What is the largest two-digit whole number, the product of whose digits is 12?


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=10$. Find the volume of the tetrahedron.


Martha distributes 540 apples equally into $m$ baskets, ensuring that each basket contains more than one apple and there are more than one basket. How many values of $m$ can this be done?


In triangle $ABC$, the sides are tangent to a circle with center $D$ as shown. Given that $\angle ABC = 75^\circ$ and $\angle BCD = 40^\circ$, find $\angle BAC$, in degrees.


The side length of square $C$ is 12.5 cm. The side length of square $D$ is 18.5 cm. What is the ratio of the area of square $C$ to the area of square $D$? Express your answer as a common fraction.


Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 6-dog group?


Below is the graph of \( y = a \csc bx \) for some positive constants \( a \) and \( b \). Assume \( a = 3 \). The vertical dashed lines are located at \( x = -2\pi \) and \( x = 2\pi \). Find \( b \).

\[
\text{[asy]}
import TrigMacros;

size(300);
real g(real x)
{
    return 3*csc(x/2);
}

draw(graph(g, -4\pi + 0.01, -2\pi - 0.01), red);
draw(graph(g, -2\pi + 0.01, 0.01), red);
draw(graph(g, 0.01, 2\pi - 0.01), red);
draw(graph(g, 2\pi + 0.01, 4\pi - 0.01), red);
limits((-4\pi, -5), (4\pi, 5), Crop);
draw((-2\pi, -5) -- (-2\pi, 5), dashed);
draw((2\pi, -5) -- (2\pi, 5), dashed);
trig_axes(-4\pi, 4\pi, -5, 5, \pi/2, 1);
layer();
rm_trig_labels(-11, 11, 2);

label("$1$", (0, 1), E);
label("$3$", (0, 3), E);
label("$-1$", (0, -1), E);
label("$-3$", (0, -3), E);
\text{[/asy]}
\]

Solution: \boxed{b = \frac{1}{2}}


Define $\$N$ by the formula $\$N = 0.3N + 2$. Calculate $\$(\$(\$60))$.


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


Given that the values of \( a, b, c, \) and \( d \) are 5, 6, 7, and 8, but not necessarily in that order, what is the largest possible value of the sum of the four products \( ab, bc, cd, \) and \( da \)?


In triangle $PQR$, points $M$ and $N$ are midpoints of sides $\overline{PQ}$ and $\overline{PR}$, respectively. If the area of $\triangle PQR$ is 40 square units, find the area of $\triangle QMN$.


If the least common multiple of $A$ and $B$ is $180$, and the ratio of $A$ to $B$ is $2:3$, then what is their greatest common divisor?


A quadratic equation is given by $x^2 + bx + \frac{1}{5}$, where $b$ is a specific negative number. Thomas rewrites this quadratic by completing the square and results in the form $(x+n)^2 + \frac{1}{20}$. Determine $b$.


For how many positive integers \( n \) does \( 1+2+\cdots+n \) evenly divide \( 30n \)?


Let $T$ be the sum of all positive integers $n$ such that $n^2 + 12n - 1981$ is a perfect square. Find the remainder when $T$ is divided by $1000$.


The digits $1,2,3,4,5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of both the digits $2$ and $3$?


Two circles are drawn in a 15-inch by 20-inch rectangle. Each circle has a diameter of 8 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance between the centers of the two circles?

Solution: \boxed{\sqrt{193} \text{ inches}}


What is the value of $3^2 \times 5^2 \times 7 \times 11^2$?


A right circular cylinder with radius 3 is inscribed in a sphere with radius 7 so that its bases are parallel to the base of the sphere. What is the height of this cylinder?


Two scientists were both born in the last 300 years. Each lives (or will live) to be 80 years old, then dies. Each scientist is equally likely to be born at any point during those 300 years. What is the probability that they were contemporaries for any length of time?


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


How many of the divisors of $10!$ are larger than $9!$?


A right triangle has sides 15 cm and 20 cm. This triangle is similar to another triangle which has one of its sides as twice the length of a rectangle's shorter side. If the rectangle's shorter side is 30 cm and the longer side is 60 cm, what is the perimeter of the larger triangle?

Solution: \boxed{240 \text{ cm}}


Suppose $198 \cdot 864 \equiv m \pmod{75}$, where $0 \leq m < 75$.


Joel chose an acute angle $y$ (strictly between $0$ and $90$ degrees) and calculated $\sin y$, $\cos y$, and $\tan y$. He wrote these values on three different cards, handing them out to Sofia, Liam, and Nina, each receiving one card. After sharing their values, they found that only Sofia could confidently determine her card's associated trigonometric function based solely on the relational size (greater than, less than) of her value compared to others. Identify the trigonometric function on Sofia's card and compute the value for angle $y$ given she is the only one able to deduce her function entirely.

Solution: \boxed{\text{Function on Sofia's card is } \tan, \text{ and } y = 45^\circ}


Find the sum of all real solutions to the equation \[\frac{x-3}{x^2+3x+2} = \frac{x-7}{x^2-7x+12}.\]


Consider a geometric sequence where the sum of the first 1500 terms is 300, and the sum of the first 3000 terms is 570. Determine the sum of the first 4500 terms of this sequence.


Given that $1+\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx + 6 = 0\] and that $a$ and $b$ are rational numbers, compute $b.$


What is the radius of the circle inscribed in triangle $PQR$ if $PQ = 30$, $PR=26$, and $QR=28$? Express your answer in simplest radical form.


Suppose that \( a^2 \) varies inversely with \( b^4 \). If \( a = 7 \) when \( b = 2 \), find the value of \( a^2 \) when \( b = 4 \).


Monica tosses a fair 8-sided die. If the roll is a prime number, she wins that amount of dollars. If the roll is composite, she wins half that amount (rounded down to the nearest whole dollar). If she rolls a 1, she loses 3 dollars. What is the expected value of her winnings on one die toss? Express your answer as a dollar value to the nearest cent.


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


The two lines $y = 5x - 20$ and $3x + y = 110$ intersect. What is the value of $x$ at the point of intersection?


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


If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{a} + \mathbf{a} \times \mathbf{b},$ and $\mathbf{b}.$


A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle such that the length of the curb between the stripes is 20 feet. If each stripe is 75 feet long, find the distance, in feet, between the stripes.


Determine the sum of all single-digit replacements for $z$ such that the number ${35{,}z91}$ is divisible by 9.


Mary is interested in collecting stickers from a coffee shop that gives out stickers each day with a 3/4 chance of being a gold sticker. She plans to visit the coffee shop for 7 consecutive days. What is the probability that she receives exactly 5 gold stickers during her visits?


A rectangle has an area of 576 $\text{cm}^2$. If the length of the rectangle is twice the width, what is its perimeter?

Solution: \boxed{72\sqrt{2} \ \text{cm}}


If $n$ is $1$ less than a multiple of $100$, what is the remainder when $n^2 + 2n + 3 + n^3$ is divided by $100$?


Suppose the graph of a function $y=g(x)$ has the property that if it is shifted $30$ units to the right, then the resulting graph is identical to the original graph of $y=g(x)$.

What is the smallest positive $b$ such that if the graph of $y=g\left(\frac{x}{3}\right)$ is shifted $b$ units to the right, then we know that the resulting graph is identical to the original graph of $y=g\left(\frac{x}{3}\right)$?


Three of the following test scores belong to John and the other three to Lisa: 82, 88, 91, 95, 96, 97. John's mean score is 91. What is Lisa's mean score?


What is the domain of the function $f(x) = \log_2(\log_5(\log_3 x))$?

Solution: \boxed{(3, \infty)}


The median of a set of consecutive odd integers is 148. If the greatest integer in the set is 159, what is the smallest integer in the set?


What is the first year after 2010 for which the sum of the digits is 10?


An eight-sided die is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?


The five partners in a business decide to split the profits of their company in the ratio 2:3:4:4:6. If the profit one year is $\$38,000$, what is the largest number of dollars received by any of the five partners?


Suppose $w$ is a complex number such that $w^3 = 64 - 48i$. Find $|w|$.


Given that $x + \sin y = 2021$ and $x + 2021 \cos y = 2020$, where $\frac{\pi}{2} \leq y \leq \pi$, find the value of $x + y$.

Solution: \boxed{2020 + \frac{\pi}{2}}


For all real numbers $r$ and $s$, define the mathematical operation $\#$ such that: $r \# 0 = r$, $r \# s = s \# r$, and $(r + 1) \# s = (r \# s) + s + 2$. Compute the value of $12 \# 6$.


A point has rectangular coordinates \((3, -4, 12)\) and spherical coordinates \((\rho, \theta, \phi)\). Find the rectangular coordinates of the point with spherical coordinates \((\rho, -\theta, \phi)\).

Solution: \boxed{(3, 4, 12)}


In pentagon $PQRST$, angles $P$ and $Q$ are $90^\circ$ and $70^\circ$ respectively. Angles $R$ and $S$ are equal, and the fifth angle $T$ is $20^\circ$ more than twice $R$. Find the measure of the largest angle.


For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?


Seven couples were at a gathering. If each person shook hands exactly once with everyone else except for their spouse and one specific individual (not the same for everyone), how many handshakes were exchanged? Assume that each of these selected individuals did not shake hands with one person who is also avoiding shaking hands with them.


A repair crew uses a 20-foot plank to span from point $A$ to point $B$ across a circular fountain, and then a 12-foot plank from point $D$ to the center $C$ of the fountain, where $D$ is the midpoint of $\overline{AB}$. Determine the area of the circular fountain in terms of $\pi$.

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


How many perfect squares less than 5000 have a ones digit of 4, 5, or 6?


Bill can buy jags, jigs, and jogs for $\$3$, $\$2$, and $\$8$ each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and he has exactly $\$60$ to spend?


The vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}.$ There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} 1 \\ 4 \\ -6 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).\] Find $r.$


Let $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} \sqrt{3} \\ 5 \\ -2 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 11 \\ -6 \\ 23 \end{pmatrix}$. Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$, in degrees.


Consider ten dice, where five are standard 6-sided dice and five are special 6-sided dice displaying only even numbers (2, 4, 6). What is the probability that exactly one of the standard dice shows a 1 and exactly one of the special dice shows a 4? Express your answer as a decimal rounded to the nearest thousandth.


A line segment with endpoints \(A(-3, 1)\) and \(B(9, 7)\) is extended from \(B\) to a point \(C\) such that \(BC = \frac{1}{2} \cdot AB\). What are the coordinates of point \(C\)?

Solution: \boxed{(15, 10)}


Given that \(5^{-1} \equiv 39 \pmod{79}\), find \(125^{-1} \pmod{79}\), and provide the answer as a residue modulo 79.


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 $C$, the second letter must be $D$, one of the remaining letters must be $B$, and no letter can be used more than once in the arrangement?


Alex picks three different cards without replacement from a standard 52-card deck. What is the probability that all three cards are of different suits?


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


Find $b$ if the remainder is constant when $8x^3 - 9x^2 + bx + 5$ is divided by $3x^2 - 2x + 1$.


Suppose California and Texas have different license plate formats. California plates have the format where the first two slots and the last three slots are letters, and the third and fourth slots are digits. Texas plates, by contrast, use a format where the first two slots are digits followed by four letters. Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can California issue than Texas?


How many positive three-digit integers less than 300 have at least two digits that are the same?


The sum of the areas of two squares is 113, while the difference in their areas is 47. The perimeter of the larger square is tripled before adding to the perimeter of the smaller square to get the total. Find the resulting sum of the perimeters.

Solution: \boxed{48\sqrt{5} + 4\sqrt{33}}


If $x + \frac{1}{x} = \sqrt{2}$, then find $x^8$.


Determine the number of intersection points between the graphs of the following equations:
\[
y = |3x + 4|
\]
\[
y = -|4x - 1|
\]


The letters of the alphabet are given numeric values based on the two conditions below.

$\bullet$ Only the numeric values of $-3,$ $-2,$ $-1,$ $0,$ $1,$ $2,$ and $3$ are used.

$\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern:
$$
2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1,\ldots
$$

The pattern is now extended to length 10. Calculate the sum of the numeric values of the letters in the word "algebra".


In right triangle $\triangle ABC$ with a right angle at $C$ and $\angle A = 45^\circ$, if side $AC = x$, find the radius of the incircle of triangle $\triangle ABC$.

Solution: \boxed{x(1 - \frac{\sqrt{2}}{2})}


Let $\mathbf{S}$ be the matrix for rotating about the origin counter-clockwise by an angle of $45^\circ$ and then scaling by a factor of 2 along both axes. Find $\det \mathbf{S}$.


A space agency discovers 13 habitable planets: 7 Earth-like and 6 Mars-like. Each Earth-like planet requires 3 units of colonization, and each Mars-like requires 1 unit. If 18 total units of colonies are available, in how many different ways can the agency choose a combination of planets to colonize, assuming all planets are unique and different from each other, with the condition that at least one Earth-like planet must be colonized?


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 9$. Find the minimum value of
\[\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.\]


A circle is tangent to the lines $3x - 4y = 24$ and $3x - 4y = -12.$ The center of the circle lies on the line $3x + 2y = 0.$ Find the center of the circle.

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


The volume of a sphere is $72\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bottom) of the cylinder.

Solution: \boxed{90\pi \sqrt[3]{4}}


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


A toy store manager received a shipment of DinoDolls just before a big sale. The manager decides to place $30\%$ of the DinoDolls on the shelves, leaving the other 140 DinoDolls in storage. How many DinoDolls were in the shipment?


A total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?


The square quilt block shown is made from sixteen unit squares arranged in a 4x4 grid. Some of these unit squares are divided diagonally into two triangles, and half of these triangles are shaded. If exactly two whole squares are fully shaded, and four squares are divided into two shaded triangles such that they form another square, what fraction of the quilt is shaded? Express your answer as a common fraction.


Calculate $\sqrt[3]{\sqrt{0.000008}}$. Express your answer as a decimal to the nearest thousandth.


How many integers between $300$ and $700$ contain the digits $3$, $4$, and $5$?


Find the point on the line
\[ y = \frac{x + 3}{2} \]
that is closest to the point $(8,3)$.

Solution: \boxed{(7, 5)}


From the following infinite list of numbers, how many are integers? $$\sqrt{2401},\sqrt[3]{2401},\sqrt[4]{2401},\sqrt[5]{2401},\sqrt[6]{2401},\ldots$$


In triangle $XYZ$, we have $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$.


A right-angled triangle has an inscribed circle with a radius of 4 cm. If the hypotenuse makes an angle of 45 degrees with one of the legs, what is the length of the hypotenuse? Express your answer in simplest radical form.

Solution: \boxed{8 + 8\sqrt{2}}


Let $S$ be the set of positive integer divisors of $18^7$. Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1, a_2,$ and $a_3$ in the order they are chosen. The probability that $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m$.


Let p and q be any two odd numbers, with q less than p. Determine the largest integer which divides all possible numbers of the form p^2 - 3q^2.


Tara ran 3 miles in half the time it took Alex to run 4.5 miles. If it took Alex 27 minutes to run 4.5 miles, using the same rates, how many minutes would it take Tara to run 6 miles?

Solution: \boxed{27 \text{ minutes}}


Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 2$ and $QS = 9$, find the length of $RS$.


Find the sum of the first eight terms in the geometric sequence $\frac13, \frac19, \frac1{27}, \dots$.


A photo is surrounded by a frame with a two-inch-wide wooden border. The entire frame (not including the photo) occupies 48 square inches, and one of the outer edges of the frame measures 10 inches. What is the perimeter of the photo?


Compute the exact value of the expression
\[ |2\pi - |2\pi - 9||. \]
Write your answer using only integers and \(\pi\), without any absolute value signs.

Solution: \boxed{4\pi - 9}


How many positive, three-digit integers contain at least one $7$ but do not contain the digit $4$ or $6$?


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?


An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction.


Calculate the number of distinct ways 8 distinct beads can be placed on a bracelet, considering that the bracelet can be rotated or flipped.


My friend Emma likes numbers divisible by 8. How many different ones digits are possible in numbers that Emma likes?


Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-3 - i \sqrt{8}$ as a root.

Solution: \boxed{x^2 + 6x + 17}


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


Define a regular 12-pointed double-layered star where each vertex connects to two others, skipping some vertices between connections, and no three points of connections are collinear. Vertices are labeled $1, 2, \ldots, 12$. For each vertex like $P_k$, connect it to $P_{k+m \mod 12}$ and $P_{k+2m \mod 12}$ where $m$ is chosen such that $\gcd(m, 12) = 1$. Determine how many non-similar regular 12-pointed double-layered stars can be made.


What is the greatest three-digit number that is two more than a multiple of 11 and four more than a multiple of 7?


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

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


You have $6$ blue shirts, $4$ green shirts, $7$ pairs of pants, $9$ blue hats, and $7$ green hats, all distinct. How many outfits can you create consisting of one shirt, one pair of pants, and one hat, with the restriction that the shirt and hat must not be the same color?


Determine the sum of all positive even factors of the number $720$.


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


Solution: \boxed{(-\infty, 2) \cup (3, 4) \cup (5, 6) \cup (7, \infty)}


Consider the set \( T \) of all nonzero real numbers excluding \(-1\) and \(1\). Define a function \( g: T \to T \) that satisfies:

(i) \( g(x + y) = g(x) + g(y) \) for all \( x, y \in T \) such that \( x + y \in T \).

(ii) \( g\left(\frac{1}{\sin x}\right) + g\left(\frac{1}{\sin y}\right) = \sin x \sin y \cdot g\left(\frac{1}{\sin(x+y)}\right) \) for all \( x, y \in T \) such that \( \sin x + \sin y \neq 0 \).

Let \( p \) be the number of possible values of \( g(\frac{1}{\sin 1}) \), and let \( q \) be the sum of all possible values of \( g(\frac{1}{\sin 1}) \). Find \( p \times q \).


Let $x,$ $y,$ $z$ be nonnegative real numbers such that $x + y + z = 2.$ Find the maximum value of
\[\frac{xy}{x + y} + \frac{xz}{x + z} + \frac{yz}{y + z}.\]


Given an odd sequence of consecutive integers, which concludes with a single even integer, the arithmetic mean of these integers is $y$. If the number of integers (odd + 1 even) is also odd, find the sum of the smallest and largest integer of this sequence in terms of $y$.


A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(3, \frac{5\pi}{6}, \frac{\pi}{4}\right)$. Find the spherical coordinates of the point with rectangular coordinates $(x, -y, z)$. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2\pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left(3, \frac{7\pi}{6}, \frac{\pi}{4}\right)}


What is the smallest abundant number that is not a multiple of 4?


What is the smallest prime factor of 2023?


Container A holds 3 red balls and 5 green balls; Container B holds 5 red balls and 5 green balls; Container C holds 7 red balls and 3 green balls; and a new Container D holds 4 red balls and 6 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction.


A point $P$ is chosen at random in the interior of isosceles triangle $ABC$ with $AB = AC$. What is the probability that $\triangle ABP$ has a greater area than $\triangle ACP$ but less than $\triangle BCP$?


A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\frac{1}{4}$ and the probability of an immediate second chance (part of the LOSE sector) is $\frac{1}{2}$. What is the area, in sq cm, of the WIN and second chance sectors? Express your answer in terms of $\pi$.


If $\frac{80}{2^4 \cdot 5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


Let \(x\) and \(y\) be positive real numbers such that \(x + y = 2.\) Find the set of all possible values of \(\frac{1}{x} + \frac{1}{y}.\)

Solution: \boxed{[2, \infty)}


Let $a, b, c$ be positive real numbers. Find the set of all possible values of
\[ g(a,b,c) = \frac{a^2}{a^2 + b^2} + \frac{b^2}{b^2 + c^2} + \frac{c^2}{c^2 + a^2}. \]

Solution: \boxed{(1, 2)}


How many solutions does the equation $\tan x = \tan(\tan x + x)$ have on the interval $0 \leq x \leq \tan^{-1} 500$?


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


How many integers from 1 through 9999, inclusive, do not contain any of the digits 1, 3, 5, 7, or 9?


The sequence consists of all those positive integers which are either powers of 5 or sums of distinct powers of 5. Find the $50^\text{th}$ term of this sequence.


A rectangle has a perimeter of 80 inches and each side has an integer length. Additionally, one dimension must be at least twice as long as the other. How many non-congruent rectangles meet these criteria?


On a Cartesian coordinate plane, points $(1,2)$ and $(5,6)$ are adjacent points on a square. What is the area of the square?


The positive four-digit integers that use each of the four digits $5,$ $6,$ $7,$ and $8$ exactly once are ordered from least to greatest. What is the $10^{\text{th}}$ integer in the list?


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


The Happy Valley Zoo has 5 parrots, 3 dogs, and 4 cats. In how many ways can the 12 animals be placed in a row of 12 cages, such that all of the animals of each type are in adjacent cages, and the cats must be placed somewhere between the parrots and dogs?


What is the sum of all two-digit positive integers whose squares end with the digits 25?


Each of the $2500$ students at a university studies either Physics or Chemistry, and some study both. The number who study Physics is between $70\%$ and $75\%$ of the student population, and the number who study Chemistry is between $35\%$ and $45\%$. Let $m$ be the smallest number of students who could study both subjects, and let $M$ be the largest number of students who could study both subjects. Find $M-m$.


How many four-digit whole numbers have at least one digit as 8 or at least one digit as 3?


Consider the sequences \[C = 1 \times 2 + 3 \times 4 + \cdots + 41 \times 42 + 43\] and \[D = 1 + 2 \times 3 + \cdots + 40 \times 41 + 42 \times 43.\] Find the positive difference between integers $C$ and $D$.


The deli now has five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich is composed of one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, Swiss cheese, and rye bread are available at the deli. Al never orders a sandwich with a turkey/Swiss cheese combination nor a sandwich with a rye bread/roast beef combination. How many different sandwiches can Al order?


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

Solution: \boxed{9 - 2\pi}


Given the function $y = x^2 - 8x + 12$, find the vertex of the parabola and calculate the value of $y$ when $x = 3$.


A wonderful circular pie has a diameter of $16$ cm and is cut into four equal-sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that can be drawn in one of these slices. Find the value of $m^2$.


In the extended Number Wall, numbers are added from adjacent blocks directly below to form the sum in the block above. What number will be in the block labeled '$n$' in this configuration?
```plaintext
     __n__
   __   __
  __   __
 8  5  3  2
```


A regular polygon has a number of diagonals increased by 6, and this sum equals twice the number of its sides. Determine the number of sides of this polygon.

Solution: \boxed{n = 4}


What is the remainder when (103)(107) is divided by 13?


For how many different negative values of $x$ is $\sqrt{x +122}$ a positive integer?


Find the number of triples \((x, y, z)\) of real numbers that satisfy the system
\begin{align*}
x &= 1000 - 1001 \operatorname{sign}(y + z + 1), \\
y &= 1000 - 1001 \operatorname{sign}(x + z - 1), \\
z &= 1000 - 1001 \operatorname{sign}(x + y + 2).
\end{align*}


A sequence consists of $1500$ terms. Each term after the first is 2 larger than the previous term. The sum of the $1500$ terms is $7500$. What is the sum of the terms that are at odd positions in the sequence?


Solve the congruence $13m \equiv 9 \pmod{47}$, as a residue modulo 47. (Give an answer between 0 and 46.)


In a bag containing 15 balls—10 red and 5 blue—five balls are drawn randomly without replacement. What is the probability that exactly 3 of the drawn balls are red?


Two cards are chosen consecutively without replacement from a standard 52-card deck. What is the probability that the first card is a face card (Jack, Queen, or King) and the second card is a number card (2 through 10) with the two cards totaling to 15?


In $\triangle ABC$, we have $AC=BC=9$ and $AB=4$. Suppose that $D$ is a point on line $AB$ such that $A$ lies between $B$ and $D$ and $CD=10$. What is $BD$?

Solution: \boxed{2 + \sqrt{23}}


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


Find all real values of $y$ that satisfy $\frac{1}{y(y+2)} - \frac{1}{(y+2)(y+4)} < \frac{1}{4}$.

Solution: \boxed{(-\infty, -4) \cup (-2, 0) \cup (2, \infty)}


Consider the equation $x^3 - 12x^2 + 20x + 5 = 0$ with three real roots $a$, $b$, and $c$. Calculate the value of $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.


The line $y = b-x$ where $0 < b < 6$ intersects the $y$-axis at $P$ and the line $x=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:9, what is the value of $b$? Express the answer as a decimal to the nearest tenth.


How many positive integers less than $300$ can be written as the sum of two positive perfect cubes?


What is the remainder when 3572 is divided by 49?


Determine $x$ such that $\lfloor x \rfloor + x = \dfrac{17}{4}$. Express $x$ as a common fraction.


The number $n$ is a four-digit positive integer and is the product of the three distinct prime factors $x$, $y$ and $10y+x$, where $x$ and $y$ are each less than 10. What is the largest possible value of $n$?


How many even integers between 3000 and 7000 have four different digits?


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

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


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?


Consider the polynomial \( Q(x) = 3x^3 + dx^2 + ex + f \) which has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The \( y \)-intercept of the graph of \( y = Q(x) \) is 27. Determine the value of \( e \).


In triangle $ABC$, which is a right triangle with $\angle BAC = 45^\circ$ and $\angle ACB = 90^\circ$, altitude $CD$ from $C$ to the hypotenuse $AB$ measures $\sqrt{2}$ centimeters. Calculate the area of $\triangle ABC$.


Determine the values of $u$, $v$, and $s$ if the circle $C$ has the equation $x^2+8x+y^2-2y=-4$. Determine the value of $u+v+s$ where $(u,v)$ is the center of $C$ and $s$ is its radius.


Determine the sum $A + B + C + D + E$ where $A, B, C, D, E$ are constants such that:
\[\frac{x+1}{(x+2)(x+3)(x+4)(x+5)(x+6)} = \frac{A}{x+2} + \frac{B}{x+3} + \frac{C}{x+4} + \frac{D}{x+5} + \frac{E}{x+6}\]


A sports club consists of two coaches and a certain number of players. Each year, the two existing coaches leave the club. Then, each player introduces one new player to join the club. After that, two new coaches are chosen from an external pool, and not from the current players. Initially, there are eight people in the club. Calculate the total number of people in the club after four years.


If $4^m = 4 \cdot 16^3 \cdot 64^2$, find the value of $m$.


The roots of
\[ z^7 = -\frac{i}{\sqrt{2}} - \frac{1}{\sqrt{2}} \]
are $\text{cis } \phi_1$, $\text{cis } \phi_2$, $\dots$, $\text{cis } \phi_7$, where $0^\circ \le \phi_k < 360^\circ$ for all $1 \le k \le 7$. Find $\phi_1 + \phi_2 + \dots + \phi_7$. Give your answer in degrees.


Find the product of the roots of the equation \[(3x^3 - 2x^2 + 9x - 15)(4x^4 + 3x^3 - 10x^2 + 7) = 0.\]


What is the sum of all of the positive even factors of $720$?


How many 4-digit numbers have the property that the units digit is at least twice the tens digit, and the thousands digit is odd?


The numbers $p,$ $q,$ $r,$ $s$ are equal to 1, 5, 3, 6, in some order. Find the largest possible value of
\[pq + qr + rs + sp.\]


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 317 hours, 58 minutes, and 30 seconds the time will be $A:B:C$. What is the value of $A + B + C$?


The volume of a certain rectangular solid is $512\text{ cm}^3$, its total surface area is $448\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.


In triangle $ABC$, $AB = 6$, $AC = 8$, and $BC = 10$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.


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

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


How many zeros are in the expansion of $999,999,999,995^2$?


Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD= 18$ and $BE = 24$, find the area of $\triangle ABC$.


Milton spilled some ink on his homework paper again. He can't read the coefficient of \(x^2\), but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?

[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,black);
label("$x+25=0$",(3,1.5),E);
label("$x^{2}+$",(0,1.5),W);
[/asy]


Two right triangles share a side as follows: $\triangle ABC$ and $\triangle ABD$ are right triangles such that $AB = 8$, $AC = 12$, and $BD = 8$. If $E$ is the point where $AD$ bisects $\angle CAB$, compute the area of $\triangle ACE$.

Solution: \boxed{[\triangle ACE] = \frac{144}{5}}


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


Find the matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = -7 \mathbf{v}\]
for all vectors $\mathbf{v}$ in $\mathbb{R}^3$.

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


A conference center is setting up chairs in rows for a seminar. Each row contains 14 chairs, and initially, there are 196 chairs in total. The center wants to minimize unused chairs while ensuring all rows are completely filled. If 120 participants are expected, how many chairs should be removed?


The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 8x + 6.$ Find the leading coefficient of $g(x)$.


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}}


Suppose $g(x) = f(\frac{x}{4})$ where $f(x)$ is a function with the property that $f(x) = f(x-24)$ for all $x$. Determine the smallest positive $b$ such that if the graph of $y=g(x)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y=g(x)$.


Determine the number of real solutions for the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -\frac{5}{3}.\]


We are allowed to remove exactly one integer from the list $$ -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 $$ and then we choose two distinct integers at random from the remaining list.  What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 9.$  Find the minimum value of
\[\frac{a^2 + b^2}{a + b} + \frac{a^2 + c^2}{a + c} + \frac{b^2 + c^2}{b + c}.\]


The first term of a sequence is $2007$. Each succeeding term is the sum of the squares of the digits of the previous term. Determine the $2007^{\text{th}}$ term of the sequence.


Calculate the product of the following sequence of matrices:
\[\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix} \dotsm \begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} 1 & 2550 \\ 0 & 1 \end{pmatrix}}


A chessboard has its squares labeled according to the rule $\frac{1}{c_i + r_j}$, where $c_i$ is the column number and $r_j$ is the row number. Eight squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of these eight chosen squares.


A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece?


In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$?


Find the range of the function \[f(x) = \frac{x}{x^2+2x+2},\] where $x$ can be any real number. (Give your answer in interval notation.)

Solution: \boxed{\left[\frac{-1 - \sqrt{2}}{2}, \frac{-1 + \sqrt{2}}{2}\right]}


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?


When a certain biased coin is flipped six times, the probability of getting heads exactly twice is equal to that of getting heads exactly three times. Let $\frac{i}{j}$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $6$ flips. Find $i+j$.


Suppose I have 8 shirts, 5 pairs of pants, 4 ties, and 2 belts. An outfit requires a shirt and pants, and can either have a tie or not have a tie, and can either have a belt or not have a belt. How many different outfits can I create?


Alexio has 200 cards numbered 1-200, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction.


The area of the parallelogram generated by vectors $\mathbf{a}$ and $\mathbf{b}$ is 10. Find the area of the parallelogram generated by the vectors $3\mathbf{a} - 2\mathbf{b}$ and $2\mathbf{a} + 4\mathbf{b}$.


Find all values of $r$ such that $\lfloor r \rfloor + 2r = 16$.


Determine all possible values of \( r \) so that for every \( s > 0 \), the inequality \( \frac{4(r s^2 + r^2 s + 4 s^2 + 4 r s)}{r+s} \leq 3 r^2 s \) holds true. Provide your answer in interval notation.

Solution: \boxed{\left[\frac{2+2\sqrt{13}}{3}, \infty\right)}


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{72}$ and $a^{-1}$ is also undefined $\pmod{90}$?


Compute \[ \left\lfloor \dfrac {2010^3}{2008 \cdot 2009} - \dfrac {2008^3}{2009 \cdot 2010} \right\rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.


Square $ABCD$ has an area of $144$. Point $E$ lies on side $\overline{BC}$ such that $BE:EC = 3:1$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $25$, find the area of triangle $GCD$.


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 150^{\circ}$ and $\angle AOB = 130^{\circ}$, as shown. What is the degree measure of $\angle ABC$?


In parallelogram $EFGH$, angle $F$ measures $125^\circ$. Determine the measure of angle $H$ if side $EF$ is twice the length of side $FG$.


A rectangular prism has twelve edges, eight vertices, and is formed by connecting points in 3D space such that its lengths are $a$, $b$, and $c$ where $a \neq b \neq c$. A segment $x$ which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. Calculate the total number of diagonals in this rectangular prism.


Determine the sum of the coefficients $A$, $B$, $C$, and the exception value $D$, for the simplified function $y = Ax^2 + Bx + C$, which originates from the function $y = \frac{x^3 + 5x^2 + 8x + 4}{x + 2}$.


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


Two concentric circles have radii of 23 and 37 units, respectively, and bound a shaded region. A third circle is to be drawn such that its area is equal to the area of the shaded region. Determine the radius of this third circle, expressing your answer in simplest radical form.

Solution: \boxed{r = 2\sqrt{210}}


What is the base-10 integer 789 when expressed in base 7?


Calculate the sum of the series given by $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5$.


Each of six, standard, six-sided dice is rolled once. What is the probability that exactly three pairs occur (that is, three different pairs of numbers are shown on the dice and no three dice show the same number)?


A pirate is searching for buried treasure on 8 islands. Each island has a $\frac{1}{3}$ chance that it has buried treasure without traps, a $\frac{1}{6}$ chance that it contains traps but no treasure, and a $\frac{1}{2}$ chance that it has neither traps nor treasure. What is the probability that while exploring all 8 islands, the pirate will find exactly 4 islands with treasure but none with traps?


Points $A$, $B$, $C$, $D$, and $E$ lie on a line, in that order. Given $AB=3$ units, $BC=4$ units, $CD=2$ units, and $DE=3$ units, calculate the ratio of $AC$ to $BD$.


How many positive $3$-digit numbers are multiples of $25$, but not of $45$?


Rodney uses the following clues to try to guess a secret number:
\begin{tabular}{ c }
It is a two-digit integer.\\
The tens digit is even.\\
The units digit is odd.\\
The number is greater than 75.
\end{tabular}
If Rodney guesses a number that has each of these properties, what is the probability that Rodney will guess the correct number? Express your answer as a common fraction.


Consider the geometric sequence $2$, $\dfrac{8}{3}$, $\dfrac{32}{9}$, $\dfrac{128}{27}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction.


Find the value of $k$ such that the equation $4x^2 + kx + 4 = 0$ has exactly one solution in $x$.

Solution: \boxed{k = \pm 8}


In the diagram, two circles intersect at points A and B. On circle 1, from point A to point B on the circumference, there are angles labeled $7x^\circ$ and $3x^\circ$. On circle 2, from point A to point B clockwise, an angle of $8x^\circ$ is marked. What is the value of $x$? The sum of angles around any point on the same circle must equal $360^\circ$. What is $x$ if the sum of the angles around point A and point B should each equal $360^\circ$?


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


Let \(x\) and \(y\) be positive real numbers such that \(4x + 3y < 60.\) Find the maximum value of
\[xy (60 - 4x - 3y).\]


Calculate the last two digits of the sum of the factorials of the first 15 integers minus the product of the factorials of 5, 10, and 15.


Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_1 + a_2 + \cdots + a_{50} = 50 \quad \text{and} \quad
a_{51} + a_{52} + \cdots + a_{100}=150.
$$
What is the value of $a_2 - a_1$?


Three positive integers have an arithmetic mean of 30 and the median of 28. If the median is 6 less than the largest number, what is the smallest number?


The sum of the first 500 terms of a geometric sequence is 400. The sum of the first 1000 terms is 720. Find the sum of the first 1500 terms.


The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first two terms are 1 and each subsequent term is the sum of the two preceding terms. What is the remainder when the $150^{\mathrm{th}}$ term of the sequence is divided by 7?


If $p$, $q$, and $r$ are positive integers such that $\gcd(p,q) = 240$ and $\gcd(p,r) = 540$, what is the smallest possible value of $\gcd(q,r)$?


One of the asymptotes of a hyperbola has equation $y=2x+3$. The foci of the hyperbola have the same $x-$coordinate, which is $7$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = -2x + 31}


All the numbers 0, 1, 2, 3, 8, 9 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that meet at that vertex. What is the greatest possible value of the sum of these eight products?


How many even divisors of $8!$ are also divisible by $3$?


A right pyramid has a square base where each side measures 8 cm. Its peak is 15 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's edges and also find the volume of the pyramid.

Solution: \boxed{320 \text{ cm}^3}


The expression \(x^2 - 18x + 72\) can be written in the form \((x - c)(x - d)\), where \(c\) and \(d\) are both nonnegative integers and \(c > d\). What is the value of \(4d - c\)?


Each letter of the alphabet is assigned a random integer value, and $H=8$. The value of a word is the sum of its letters' values. If $MATH$ is 30 points, $TEAM$ is 36 points, and $MEET$ is 40 points, what is the value of $A$?


Find the remainder when $3x^5 + 2x^4 - 5x^3 + 6x - 8$ is divided by $x^2 + 3x + 2$.

Solution: \boxed{34x + 24}


A rectangular solid has dimensions $120 \times 350 \times 400$. It is composed of distinct types of $1 \times 1 \times 1$ cubes: type A and type B, alternately arranged in each dimension. An internal diagonal of this solid passes through the interiors of how many type A cubes?


Suppose 4 different integers are randomly chosen from between 10 and 59, inclusive. What is the probability that they each have a different tens digit?


A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element?


Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q| = 3$, $|q-r| = 4$, and $|r-s| = 5$. What is the sum of all possible values of $|p-s|$?


How many prime numbers are between 80 and 90, and what is their sum?


The quadratic equation \( x^2 + ax + b = 0 \) has roots that are three times those of \( x^2 + cx + a = 0 \), where \( a, b, \) and \( c \) are non-zero constants. What is the value of \( \frac{b}{c} \)?


Find \[\left|\left(2 - 3i\right)^5\right|\]


What is the value of $x^5 + 5x^3 + 2x$ when $x = 3$?


Calculate the value of \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{10}}{x^2\cdot x^4 \cdot x^6 \cdots x^{16}}\] if $x=3$.


The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where the first and second terms are 1, and each subsequent term is the sum of the previous two terms. What is the remainder when the $150^{\text{th}}$ term of the sequence is divided by 9?


Let $p$ and $q$ be angles such that $\cos p + \cos q = \frac{1}{3}$ and $\sin p + \sin q = \frac{4}{9}.$ Find
\[\tan \left( \frac{p + q}{2} \right).\]


Calculate the product $ABC$ for the partial fraction decomposition of
\[\frac{x^2 - 2x - 15}{x^3 - 3x^2 - x + 3}\]
into the form
\[\frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{x - 3}.\]


Let $h(x)$ be a function piecewise 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 $h(h(h(15)))=h(h(h(b)))$.

Solution: \boxed{b = -\frac{55}{3}}


What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,3), G(5,3), and H(5,-4)?


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


Find the point on the line
\[ y = \frac{3x - 1}{4} \]
that is closest to the point $(8,2)$.

Solution: \boxed{(6.2, 4.4)}


You are given a sequence of \( 50 \) terms; each term has the form \( Q+n \), where \( Q \) stands for the product \( 2 \times 3 \times 5 \times\ldots \times 53 \) of all prime numbers less than or equal to \( 53 \), and \( n \) takes, successively, the values \( 2, 3, 4, \ldots, 51 \). Let \( M \) be the number of primes appearing in this sequence. Then \( M \) is:


A four-inch cube of gold weighs 5 pounds and is worth $\$1200$. How much would a six-inch cube of the same material weigh and be worth if its density and price per pound are the same as the four-inch cube?


A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square?


What is the least positive integer $n$ such that $6375$ is a factor of $n!$?


Let triangle $PQR$ have vertices at $P=(-3, 5)$, $Q=(4, -2)$, and $R=(9, 2)$. You are to find the area of this triangle.


Twelve points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the twelve points as vertices?


The expression $x^2 + 17x + 72$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 8x - 63$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?


A bag contains 7 red, 8 green, 9 yellow, 10 blue, and 5 orange jelly beans. A jelly bean is selected at random. What is the probability that it is either blue or orange?


Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 1,$ $\|\mathbf{u} + \mathbf{v}\| = 2,$ and
\[\mathbf{w} - 2\mathbf{u} - 3\mathbf{v} = 2 (\mathbf{u} \times \mathbf{v}),\] then find $\mathbf{v} \cdot \mathbf{w}.$


Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.

Suppose that every edge of $DABC$ has length $20$ or $50$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$?


A line containing the points $(3, 27)$ and $(-7, -1)$ intersects the $y$-axis at what point?

Solution: \boxed{(0, 18.6)}


The Lucas numbers $L_n$ are defined by $L_0 = 2, L_1 = 1$, and $L_n = L_{n-1} + L_{n-2}$ for each $n \geq 2$. Calculate the units digit of $L_{L_9}$.


The polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
has coefficients that are all integers, and has roots $-3,$ $4,$ $8,$ and $-\frac{1}{4}.$ If \( e \) is a positive integer, find the smallest possible value of \( e \).


Find a positive integer that is divisible by 20 and whose cube root is a number between 8.2 and 8.3.


In the diagram, $O$ is the center of the larger circle with an area of $81\pi$. A smaller circle is inside the larger one and its topmost point touches the center of the larger circle. Each circle is divided into two equal areas. Calculate the total area of the shaded regions.


Two sides of a right triangle are 9 cm and 12 cm. What is the length of the hypotenuse? Also, find the area of the triangle.

Solution: \boxed{54 \text{ cm}^2}


Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $A$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 53^\circ$, find $\angle CED$, in degrees.


Compute the largest integer $k$ such that $2010^k$ divides $2010!$.


A ball is dropped from a height of 800 feet and rebounds to half of its previous height with every bounce. After how many bounces will the ball first reach a maximum height less than 2 feet?


Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses?


A rhombus has an area of 192 square units. The lengths of its diagonals have a ratio of 4 to 3. What is the length of the longest diagonal, in units?


Given two functions, \( p(x) = |x| - 3 \) and \( q(x) = -|x| \), evaluate \( q(p(x)) \) at \( x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \) and calculate the sum of these eleven values.


Determine the value of $p + q$ if the three points $(2,p,q)$, $(p,3,q)$, and $(p,q,4)$ are collinear.


Solution: \boxed{\sqrt{20 - 8\sqrt{5}}}


Determine how many perfect squares less than 20,000 can be represented as the difference of two consecutive perfect squares.


The function $f(x)$ satisfies
\[ f(xy) = \frac{f(x)}{y^2} \]
for all positive real numbers $x$ and $y.$ If $f(30) = 10$, find $f(40)$.


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


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

Solution: \boxed{x = \frac{\sqrt{30}}{100}}


A circle with center \( C \) is tangent to the positive \( x \)-axis and positive \( y \)-axis and externally tangent to the circle centered at \((4,0)\) with radius \( 2 \). What is the sum of all possible radii of the circle with center \( C \)?


The points \(A = (3, -5, 2)\), \(B = (5, -7, 5)\), \(C = (4, -5, 1)\), and \(D = (6, -7, 4)\) are given in space. Verify if these points form a flat quadrilateral and find the area of this quadrilateral if they do.


Compute the sum:
\[\sin^2 3^\circ + \sin^2 6^\circ + \sin^2 9^\circ + \dots + \sin^2 177^\circ.\]


All the numbers 1, 2, 3, 4, 5, 6 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, compute the product of the three numbers on the faces that meet at that vertex. What is the greatest possible value of the sum of these eight products?


John has $25$ dollars and $45$ cents. He wants to buy as many notebooks as possible, where each notebook costs $2$ dollars and $35$ cents. However, for every $5$ notebooks he buys, he gets a $15$ cents discount on each notebook. How many notebooks can John buy?


Let $f(x) = 4x^2 - 5$ and $g(f(x)) = 2x^2 - \cos(x) + 2$. Find the sum of all possible values of $g(51)$.

Solution: \boxed{30 - \cos(\sqrt{14})}


Calculate the sum of the first 6 terms of the geometric series \( 2 + 2\left(\frac{1}{3}\right) + 2\left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right)^3 + 2\left(\frac{1}{3}\right)^4 + 2\left(\frac{1}{3}\right)^5 \).


In a journey between three cities, $X, Y, Z$, forming a right-angled triangle, $Z$ is 4000 km from $X$, and $Y$ is 4500 km from $X$. The hypotenuse $YZ$ hence needs to be calculated. The cost to take a bus is $\$0.20$ per kilometer, and to take a flight, there is a $\$120$ booking fee plus $\$0.12$ per kilometer. Luna starts her travel by taking a bus from $X$ to $Y$. Calculate the cost for her to bus from $X$ to $Y$.


Find the units digit of the following sum in base 9: $45_9 + 76_9$.


How many distinct positive factors does 64 have, and what is their sum?


In a bag, there are 7 blue chips, 5 red chips, and 4 yellow chips. One chip is drawn from the bag and then placed back into the bag. A second chip is drawn with the condition that if the first chip was blue, the probability of drawing a blue chip again is reduced by half. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called `hearts' and `diamonds') are red, and the other two ($\spadesuit$ and $\clubsuit$, called `spades' and `clubs') are black. The cards in the deck are shuffled. What is the probability that the first three cards drawn from the deck are all red?


If the three lines $4y - 3x = 2$, $2x + 4y = 4$, and $6x - 8y = 3$ are drawn in the plane, how many points will lie at the intersection of at least two of the three lines?


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 9?