The club now has 30 members, 12 of whom are left-handed. 20 members like jazz music, and 5 are right-handed who dislike jazz music and do not play basketball. 10 members play basketball, including 3 left-handed who also like jazz music. How many club members are left-handed, like jazz music, but do not play basketball?


Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbb{R}^3$, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Suppose $\mathbf{m} = \begin{pmatrix} 4 \\ 8 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 12.$ Additionally, $\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 0 \\ 0 \\ 6 \end{pmatrix}$. Find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$


Jack is considering a positive integer less than 150. It is two less than a multiple of 10 and four less than a multiple of 9. What is the greatest possible integer Jack could be considering?


Suppose that $a$ and $b$ are integers such that $$5b = 12 - 3a.$$ How many of the first four positive integers must be divisors of $3b + 15$, and ensure that $5$ is a divisor of $3b + 15$?


Both roots of the quadratic equation \( x^2 - 29x + k = 0 \) are prime numbers. How many possible values of \( k \) are there?


Suppose that $3x^2 - 6x + m = 0$ is a quadratic equation with one solution for $x$. Express $m$ as a common fraction.


How many numbers are in the list starting from $-48$, increasing by $7$ each time, up to and including $119$?


A standard die is rolled eight times. What is the probability that the product of all eight rolls is a multiple of 3? Express your answer as a common fraction.


During the first year, DEF's stock price starts at $ \$120 $ and increases by 120%. During the second year, its stock price decreases by 30% from its end price at the first year. What is the price of the stock, in dollars, at the end of the second year?


The prime factorization of 8820 is $2^2 \times 3^2 \times 5 \times 7^2$. How many of its positive integer factors are perfect cubes?


Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $31x + y = 2017$. Find the number of such distinct triangles whose area is a positive integer.


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


Circle $Q$ has a circumference of $18\pi$ meters, and segment $AB$ is a diameter. If the measure of angle $QAC$ is $45^{\circ}$, what is the length, in meters, of segment $AC$?

Solution: \boxed{9\sqrt{2} \text{ meters}}


In the diagram, $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Point $F$ is located such that $\overline{DF}$ is perpendicular to $\overline{BC}$, and $F$ lies on $\overline{BC}$. Given $A(0,10)$, $B(0,0)$, and $C(10,0)$, calculate the area of quadrilateral $DBCF$.


What is the coefficient of $x^5$ in the expansion of $$(x + 3\sqrt{2})^9?$$


Find integers $x$ and $y$ such that $x > y > 0$ and $x + y + xy = 99$. What is the integer value of $x$?


A line contains the points $(7, -3)$ and $(3, 1)$. Where does this line intersect the $x$-axis?

Solution: \boxed{(4, 0)}


Point $H$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BH$ intersects diagonal $AC$ at $J$ and side $DC$ at $K$. If $JK = 12$ and $KH = 36$, then $BJ = ?$
[asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), H = (5.7, 9.5); pair K = intersectionpoints(B--H, D--C)[0]; pair J = intersectionpoints(A--C, B--H)[0]; draw(A--D--C--B--cycle); draw(A--C); draw(D--H--B); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$H$", H, N); label("$K$", K, NE); label("$J$", J, SE); //Credit to MSTang for the asymptote[/asy]


One of the six faces of a rectangular prism will be used as the base of a new pyramid. Calculate the maximum value of the sum of the numbers of exterior faces, vertices, and edges of the resulting shape after fusing the rectangular prism and the pyramid.


Calculate $\sqrt{15p^3} \cdot \sqrt{20p} \cdot \sqrt{8p^5}$ and express your answer in simplest radical form in terms of $p$.

Solution: \boxed{20p^4 \sqrt{6p}}


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


A function $f$ is defined by $f(z) = (3 - 2i) z^3 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma are complex numbers and $i^2 = -1$. Suppose that $f(1)$ and $f(-1)$ are both real. What is the smallest possible value of $|\alpha| + |\gamma|$?


The sum of 64 consecutive integers is \( 8^4 \). What is their median?


A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed?


Given that $\log_{10} \sin x + \log_{10} \cos x = -2$ and that $\log_{10} (\sin x + \cos x)^2 = \log_{10} n + 1,$ find $n.$


You have $6$ red shirts, $4$ green shirts, $5$ blue shirts, $7$ pairs of pants, $9$ red hats, $7$ green hats, and $6$ blue hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?


Find the product of all real values of $r$ for which $\frac{1}{3x} = \frac{r-2x}{10}$ has exactly one real solution.


How many ways are there to put 6 balls into 3 boxes if the balls are not distinguishable and neither are the boxes?


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

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


The sum of the first and the fourth of four consecutive even integers is 160. What is the sum of the second and third integers?


Determine the greatest common divisor (GCD) of 13680, 20400, and 47600.


Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{3}{5}$. Jackie flips the three coins, and then Phil flips the same three coins. Let $\frac {p}{q}$ be the probability that Jackie gets the same number of heads as Phil, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.


When $1 + 3 + 3^2 + \cdots + 3^{1001}$ is divided by $500$, determine the remainder.


What integer $n$ satisfies $0\le n<127$ and $$126n\equiv 103\pmod {127}~?$$


Given the six digits 1, 2, 3, 7, 5, and 8, if these digits are randomly arranged to form a six-digit integer, what is the probability that this integer is divisible by 15? Express your answer as a common fraction.


How many of the integers between 1 and 500, inclusive, can be expressed as the difference of the squares of two nonnegative integers?


The sum of the areas of two squares is 130, while the difference in their areas is 42. Find the sum of their perimeters.

Solution: \boxed{4(\sqrt{86} + 2\sqrt{11})}


Find all values of $k$ such that the function
\[b(x) = \frac{kx^2 + 3x - 4}{3x^2 - 4x + k}\]
has its domain as the set of all real numbers.

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


Two standard 8-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is either a prime number or a perfect square? Express your answer as a common fraction.


How many integers between 700 and 900 have the sum of their digits equal to 20?


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes "$AB$" and "$CD$", where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle{ABP} = 30^{\circ}$. Given that $AP = 1$ and $CP = 3$, compute the area of $\triangle ABC$.


During a business lunch, I have to choose a meal at a restaurant that is known for its diverse menu. I am a vegetarian who also needs to avoid gluten. The restaurant I visited has an extensive menu, with five dishes that do not contain any meat. These dishes comprise one fifth of their total offerings. However, four out of these five vegetarian dishes contain gluten. What fraction of the total menu can I eat?


What is the remainder when the binary number $110111100101_2$ is divided by 8 (give your answer in base 10)?


Evaluate $\dbinom{n+2}{n} \cdot n!$ for any positive integer $n$.


Factor the following expression: $286x^2y + 143x$.

Solution: \boxed{143x(2xy + 1)}


If $a$ and $b$ are positive integers for which $ab - 4a + 3b = 504$, what is the minimal possible value of $|a - b|$?


Solution: \boxed{\frac{2}{2-\log_7 5}}


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


Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x + 17 \equiv 63 \pmod{38}$ have?


What is the ratio of the volume of a cube with edge length four inches to that of a cube with edge length two feet? Additionally, calculate the total surface area of the smaller cube.


Let $\mathcal{T}$ be the set $\{1, 2, 3, \ldots, 15\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Find the remainder obtained when $n$ is divided by $1000$.


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


What is the least natural number that can be added to 52,350 to create a palindrome?


A right circular cone has a base radius $r$ and height $h$. The cone is laid on its side on a flat table. As it rolls on the surface without slipping, the point where the cone's base meets the table traces a circular arc centered at the vertex touching the table. The cone first returns to its original position after making $23$ complete rotations. Express the ratio $h/r$ in the form $p\sqrt{q}$, where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p + q$.


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


If $5x + 9 \equiv 3 \pmod{19}$, then what integer between $0$ and $18$, inclusive, is $3x + 15$ congruent to modulo $19$?


Solution: \boxed{33x^5(2 - 5x^4 + 3y)}


The quadratic \( x^2 - 5x + 7 = x + 35 \) has two solutions. What is the positive difference between these solutions?


When $600_{10}$ is expressed in a certain base, it has four digits in the form ABBA, where A and B are different digits. What is this base?


Solve the following system of equations:
\[
\begin{align*}
4x - 3y &= -7, \\
5x + 6y &= -20.
\end{align*}
\]
Express your answer as an ordered pair \((x, y)\).

Solution: \boxed{\left(-\frac{34}{13}, -\frac{15}{13}\right)}


Let $a$ and $b$ be positive real numbers such that $6a + 5b = 45$. Find the maximum value of $ab$.


For how many values of the digit $A$ is it true that $54$ is divisible by $A$ and $273{,}1A4$ is divisible by $4$?


Given a rectangular array of chairs, where each row and column must contain at least three chairs and all chairs in the room must be used without any leftover, determine how many possible arrays can be formed in a classroom containing $48$ chairs.


Say that an integer $B$ is scrumptious if there exist several consecutive integers, including $B$, that add up to 2021. What is the smallest scrumptious integer?


In the town of Gearville, each bike license plate consists of three letters. The first letter is chosen from the set $\{B, F, G, T, Y\}$, the second from $\{E, U\}$, and the third from $\{K, S, W\}$. Gearville decided to increase the number of possible license plates by adding three new letters. These new letters can be added to one set or distributed among the sets. What is the largest possible number of ADDITIONAL license plates that can be created by optimally placing these three letters?


Express the following as a common fraction: $\sqrt[3]{9 \div 22.5}$.


In a special sequence, the first term is $a_1 = 1010$ and the second term is $a_2 = 1015$. The sequence is then defined such that for all $n \geq 1$,
\[a_n + a_{n + 1} + a_{n + 2} = 2n + 1.\]
Determine the value of $a_{1000}$.


Alex is at the candy store buying chocolates. He wants to buy at least 150 chocolates. He wishes to buy the least amount such that he would have exactly $17$ leftover after dividing the chocolates evenly among $19$ people. How many chocolates should Alex buy?


**How many numbers less than 50 are abundant numbers?**


Triangle DEF has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle UVWX has vertex U on $\overline{DE}$, vertex V on $\overline{DF}$, and vertices W and X on $\overline{EF}$. In terms of the side length $UV = \omega$, the area of UVWX can be expressed as the quadratic polynomial \[ \text{Area}(UVWX) = \alpha \omega - \beta \omega^2. \]
Then the coefficient $\beta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


The diagonals of a rhombus are $12$ inches and $30$ inches. What is the perimeter of the rhombus, in inches?


How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable, the boxes are indistinguishable, and each box must contain at least one ball?


A park in the shape of a rectangle covers an area of 600 square meters. If its length is decreased by 20% and the width is increased by 30%, what would be the new area of the park?

Solution: \boxed{624 \text{ square meters}}


Two congruent cylinders each have a radius of 5 inches and a height of 4 inches. The radius of one cylinder is increased by 2 inches, and the height of the other cylinder is increased by a different non-zero number of inches, denoted as \( y \). If the resulting volumes of both cylinders are equal, how many inches is the increase \( y \)?


What is the remainder when \( 10! \) is divided by 13?


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

[asy]
unitsize(0.4 cm);

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

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

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

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

draw(L--R, blue);

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


A square is divided into four congruent rectangles. If the perimeter of each of these rectangles is 40 inches, what is the perimeter of the square?

Solution: \boxed{64 \text{ inches}}


In a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books each as possible?


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathbf{c}) + r(\mathbf{c} \times \mathbf{a}),
\]
and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2$, find the value of $p + q + r$.


A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,2)$, and $(0,2)$. What is the probability that $x < 2y$?


On modulo $9$ graph paper, we want to find points in common between the graphs of $$y \equiv 3x + 5 \pmod{9}$$ and $$y \equiv 7x + 3 \pmod{9}.$$
What is the sum of the $x$-coordinates of those points?


A grocer arranges a display of cans where the top row has three cans, and each lower row has two more cans than the row above it. The total number of cans in the display is 169. How many rows does the display contain?


Compute \[ \left\lfloor \dfrac {1005^3}{1003 \cdot 1004} - \dfrac {1003^3}{1004 \cdot 1005} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


If two distinct numbers are selected at random from the first ten prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.


How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?


Find the minimum value of
\[ 2 \sqrt{2x} + \frac{4}{x} \]
for \( x > 0 \).


In right triangle $PQR$, where $PQ=8$, $QR=15$, and $\angle Q = 90^\circ$. Points $M$ and $N$ are midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively; $\overline{QN}$ and $\overline{MR}$ intersect at point $Z$. Compute the ratio of the area of triangle $PZN$ to the area of quadrilateral $QZMR$.


Find the last two digits of the following sum: $$3! \times 2 + 6! \times 2 + 9! \times 2 + \cdots + 99! \times 2$$


Allison, Brian, and Noah each have a 6-sided cube. Three of the faces on Allison's cube have a 3 and three of the faces have a 4. The faces on Brian's cube are numbered 0, 1, 2, 3, 4 and 5. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction.


A bag contains 4 tan, 3 pink, and 5 violet chips. If the 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn such that the 4 tan chips are drawn consecutively, the 3 pink chips are drawn consecutively, and the 5 violet chips are drawn consecutively? Assume the order of consecutively colored blocks can vary.


What is the coefficient of \( x^6 \) in the expansion of \( (3x + 4)^8 \)?


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?

Solution: \boxed{22.5 \text{ daps}}


The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{225} - \frac{y^2}{144} = \frac{1}{36}\]
coincide. Find \(b^2\).


What is the greatest common divisor of $7654321$ and $6543210$?


In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.]


What is the degree measure of the smaller angle formed by the hands of a clock at 3:30?


The terms $280, b, \frac{90}{56}$ are the first, second, and third terms, respectively, of a geometric sequence. If $b$ is positive, what is the value of $b$?


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


Real numbers $x$ and $y$ satisfy the equations $2^x=16^{y+1}$ and $27^y=3^{x-2}$. What is $xy$?


Billy and Bobbi each selected a positive integer less than 300. Billy's number is a multiple of 36, and Bobbi's number is a multiple of 48. What is the probability that they selected the same number? Express your answer as a common fraction.


In an international mathematics conference in 2024, a puzzle competition involves finding distinct positive integers $A$, $B$, and $C$ such that the product $A\cdot B\cdot C = 2401$. Determine the largest possible value of the sum $A+B+C$.


If $547\,932\,BC4$ is divisible by $12$, where $B$ and $C$ represent digits, what is the sum of all unique values of $B + C$?


How many positive factors of 1800 are both perfect cubes and perfect squares?


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^7 \cdot 5^3}$?


A number $x$ is equal to $5 \cdot 16 \cdot 27$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0 \leq x, y < m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence.

Consider the congruence $$5x \equiv 2y + 3 \pmod{29}.$$
This graph has a single $x$-intercept $(x_0,0)$ and a single $y$-intercept $(0,y_0)$, where $0 \leq x_0, y_0 < 29$.

What is the value of $x_0 + y_0$?


How many of the 729 smallest positive integers written in base 9 use either 7 or 8 (or both) as a digit?


Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $17x + y = 144$. Find the number of such distinct triangles whose area is a positive integer multiple of $\frac{1}{2}$.


A framed artwork consists of a painting surrounded by a uniform frame. The outer dimensions of the framed artwork are 90 cm by 120 cm, and the width of the frame is 15 cm. What is the area of the painting inside the frame?

Solution: \boxed{5400 \text{ cm}^2}


How many positive factors of 180 are perfect squares?


At a school cafeteria, Thomas wants to buy a lunch combo consisting of one main dish, one side, one drink, and one dessert. The table below lists Thomas's available choices in the cafeteria. How many distinct possible lunch combos can Thomas buy from these options?

\begin{tabular}{ |c | c | c | c| }
\hline \textbf{Main Dishes} & \textbf{Sides} & \textbf{Drinks} & \textbf{Desserts} \\ \hline
Burger & Fries & Soda & Apple Pie \\ \hline
Taco & Onion Rings & Iced Tea & Brownie \\ \hline
Pasta & & & \\ \hline
\end{tabular}


Four of the following test scores are Cyprian's and the other four are Margaret's: 82, 85, 88, 90, 92, 95, 97, 99. Cyprian's mean score is 88.5. The highest and lowest scores belong to different students. What is Margaret's mean score?


Let $g(x) = x^4 + 8x^3 + 18x^2 + 8x + 1$. Let $w_1, w_2, w_3, w_4$ be the four roots of $g$. Find the smallest possible value of $|w_a w_b + w_c w_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.


The third and the eleventh terms of an arithmetic sequence are 12 and 32, respectively. What is the thirtieth term?


Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by the letter 'A' and ends with an even single-digit number?


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


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


The arithmetic mean of 15 scores is 90. When the highest and two lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the sum of the two lowest scores?


Rectangle $EFGH$ has sides $\overline {EF}$ of length 6 and $\overline {FG}$ of length 8. Divide $\overline {EF}$ into 200 congruent segments with points $E=P_0, P_1, \ldots, P_{200}=F$, and divide $\overline {FG}$ into 200 congruent segments with points $F=Q_0, Q_1, \ldots, Q_{200}=G$. For $1 \le k \le 199$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {EH}$ and $\overline {GH}$, and then draw the diagonal $\overline {EG}$. Find the sum of the lengths of the 399 parallel segments drawn.


Robin and Alex each bought an ice cream cone. Robin chose a cone with five different flavors: vanilla, chocolate, strawberry, cherry, and mint. Alex chose a cone with four different flavors: banana, grape, peach, and lime. In how many different ways can the five scoops be stacked on Robin's cone and the four scoops be stacked on Alex's cone?


Write the expression $\frac{5 + 2d}{8} + 3$ as a single fraction.

Solution: \boxed{\frac{29 + 2d}{8}}


How many three-digit whole numbers have at least one 5 or at least one 8 as digits?


What numeral is in the 250th decimal place in the decimal representation of $\frac{13}{17}$?


Determine the measure of angle \( Q \) in a hexagon where the other angles measure \( 134^\circ, 108^\circ, 122^\circ, 99^\circ, \) and \( 87^\circ \).


How many positive integers less than $151$ are multiples of either $6$ or $8$, but not both at once?


Contractor Anna agreed to complete a project in 40 days. After 10 days, she observed that the 10 people working had completed 40% of the project. If each worker maintains the same productivity, what is the minimum number of workers she must retain to ensure the project is completed by the deadline?


The isosceles triangle and the rectangle shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the length $l$ and width $w$ of the rectangle?


Find the number of ordered triples $(a,b,c)$ of nonnegative real numbers such that:
\begin{align*}
a^2 + b^2 + c^2 &= 3, \\
(a + b + c)(a^2b + b^2c + c^2a) &= 9.
\end{align*}


Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 50^\circ$, and $\angle ADC = 120^\circ$. What is the degree measure of $\angle BAD$?


In a regular hexagon with side length 12 units, find the length of the diagonal connecting two opposite vertices. Express your answer in simplest radical form.


What is the expected value of a roll of a fair dodecahedral die, where each face is numbered from 1 to 12, and scores are doubled for numbers 1 to 6? Express your answer as a decimal.


In a pentagon $ABCDE$, two interior angles $A$ and $B$ are $70^\circ$ and $120^\circ$ respectively. The angles $C$ and $D$ are equal, and the fifth angle $E$ is $30^\circ$ less than three times $C$. Determine the measure of the largest angle.


The sixth and ninth terms of a geometric sequence of real numbers are $9!$ and $10!$ respectively. What is the first term?

Solution: \boxed{a = \frac{9!}{10^{5/3}}}


What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors?


The Westlake Youth Soccer Club has 24 players on the team, including reserves. Of these, four are goalies. During a special training session, each goalie will face penalty kicks from every other player on the team, including the other goalies. How many penalty kicks must be taken to ensure that every goalie has faced a shot from each player?


Two triangles, $\triangle ABC$ and $\triangle DEF$, are drawn such that $\triangle ABC \sim \triangle DEF$. The lengths of segments $BC = 8$ cm and $AB = 10$ cm in $\triangle ABC$, and $DE = 24$ cm in $\triangle DEF$. If angle $BAC = 90^\circ$ and angle $EDF = 90^\circ$, what is the length of segment $EF$?

Solution: \boxed{19.2 \text{ cm}}


A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis.


In a large square, points $P$, $Q$, $R$, and $S$ are midpoints of the sides. Inside the square, a triangle is formed by connecting point $P$ to the center of the square and point $Q$. If the area of the larger square is 80, what are the areas of the smaller square and the triangle formed?


Let \( c \) be a complex number. Suppose there exist non-zero, distinct complex numbers \( r, s, t \) forming the vertices of an equilateral triangle in the complex plane such that for every complex number \( z \),
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of \( c \).


How many three-digit whole numbers have no 6's, 7's, or 9's as digits?


What is the greatest 3-digit base 12 positive integer that is divisible by 7? (Express your answer in base 12.)


Line segment $\overline{AB}$ is a diameter of a circle with $AB = 30$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?


In the diagram shown, $\overrightarrow{OA}\perp\overrightarrow{OB}$ and $\overrightarrow{OC}\perp\overrightarrow{OD}$. If $\angle{BOD}$ is 4 times $\angle{AOC}$, what is $\angle{BOD}$? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;

pair O=(0,0), A=dir(0), B=dir(90), C=dir(50), D=dir(140);
pair[] dots={O,A,B,C,D};

dot(dots);
draw(O--1.2*D,EndArrow(4));
draw(O--1.2*C,EndArrow(4));
draw(O--1.2*B,EndArrow(4));
draw(O--1.2*A,EndArrow(4));

label("$D$",D,SW);
label("$C$",C,E);
label("$B$",B,N);
label("$A$",A,E);
label("$O$",O,S);
[/asy]


Let $g(x) = \left\lceil\dfrac{1}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{1}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


Each outcome on the spinner described below has equal probability. If you spin the spinner four times and form a four-digit number from these outcomes, where each spin result corresponds to a successive digit from left to right, what is the probability that you will end up with a four-digit number that is divisible by 5? Express your answer as a common fraction.

[asy]
draw(Circle((0,0),10));
draw((0,0)--(8.7,-5));
draw((0,0)--(-8.7,-5));
draw((0,0)--(0,10));
label("1",(7,7),SW);
label("3",(-7,7),SE);
label("2",(0,-2),S);
draw((0,0)--(3,5),Arrow);
[/asy]


Find the matrix \(\mathbf{N}\) with real entries such that:
\[
\mathbf{N}^2 - 3\mathbf{N} + 2\mathbf{N} = \begin{pmatrix} 6 & 12 \\ 3 & 6 \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}}


Rectangle $PQRS$ has area $2014$. An ellipse with area $2014\pi$ passes through points $P$ and $R$ and has foci at $Q$ and $S$. Determine the perimeter of the rectangle.


Regular hexagon \(ABCDEF\) has its center at \(G\). Each of the vertices and the center are to be associated with one of the digits \(1\) through \(7\), with each digit used once, in such a way that the sums of the numbers on the lines \(AGC\), \(BGD\), and \(CGE\) are all equal. How many ways can this be done?


Margo walks to her friend's house in 12 minutes. Using the same route, it takes Margo 24 minutes to get back home due to carrying a heavy backpack. If her average walking rate for the entire trip is 3 miles per hour, how many total miles did she walk?

Solution: \boxed{1.8 \text{ miles}}


What is the sum of the odd integers from 15 through 55, inclusive, where each next odd number increases by 4 from the previous one?


In a different hyperbola configuration, the center is at $(1,2)$, one focus is at $(1,9)$, and one vertex is at $(1,-1)$. The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]
Find $h + k + a + b.$

Solution: \boxed{6 + 2\sqrt{10}}


What is the domain of the function $f(x) = \log_2(\log_3(\log_6(\log_7 x)))$?

Solution: \boxed{(117649, \infty)}


The vertices of a parallelogram in a coordinate plane are at $(2, 1)$, $(6, 1)$, $(9, 5)$, and $(5, 5)$. Determine the perimeter and area of the parallelogram and find the sum of the perimeter and area.


Melinda will roll two standard six-sided dice and make a two-digit number with the two numbers she rolls. For example, if she rolls a 4 and a 2, she can either form 42 or 24. What is the probability that she will be able to make an integer between 20 and 30, inclusive? Express your answer as a common fraction.


Find the smallest integer in a set of consecutive odd integers where the median is 150, and the greatest integer is 167.


In convex quadrilateral $PQRS, \angle P \cong \angle R, PQ = RS = 200,$ and $PR \neq QS.$ The perimeter of $PQRS$ is $680$. Find $\lfloor 1000 \cos P \rfloor.$


A complex number $-8 - 4i$ is first scaled by 2 and then rotated $270^\circ$ clockwise around the origin. What is the resulting complex number?

Solution: \boxed{8 - 16i}


In a modified cross country meet between 2 teams of 4 runners each, a runner who finishes in the \( n \)th position contributes \( n \) points to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?


The sequence starts at 8640 and each subsequent term is obtained by repeatedly dividing by 2. How many integers are in this sequence?


Determine the diameter, in inches, of a sphere with three times the volume of a sphere of radius 12 inches. Express the diameter in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.


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


In the standard coordinate system, trapezoid \(EFGH\) has vertices \(E(2, -3)\), \(F(2, 2)\), \(G(6, 8)\), and \(H(6, 2)\). What is the area of trapezoid \(EFGH\)?


A sequence consists of $1500$ terms. Each term after the first is 2 smaller than the previous term. The sum of the $1500$ terms is $3000$. When every second term is added up, starting with the first term, what is the sum?


One of the asymptotes of a hyperbola has equation \( y = 2x \). The foci of the hyperbola have the same \( x \)-coordinate, which is \( 4 \). Find the equation of the other asymptote of the hyperbola, giving your answer in the form \( y = mx + b \).

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


The members of a choir are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the choir if the membership is between 200 and 300?


Find $113^{-1} \pmod{114}$, as a residue modulo 114. (Give an answer between 0 and 113, inclusive.)


In a sequence of natural numbers listed in ascending order, identify the smallest prime number that appears after a sequence of seven consecutive integers, all of which are nonprime.


What is the least common multiple of 153, 180, and 560?


How many three-digit integer palindromes are there between 200 and 600 where the middle digit is an even number?


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


A circle with center $C$ is tangent to the positive $x$-axis and positive $y$-axis and externally tangent to the circle centered at $(5,0)$ with radius $2$. What is the sum of all possible radii of the circle with center $C$?


Find the number of positive divisors of 2744.


Suppose $x-4$ and $y+4$ are multiples of $7$.

What is the smallest positive integer, $n$, for which $x^2 + xy + y^2 + n$ is a multiple of $7$?


How many 12-digit positive integers have all digits either 1 or 2, and have either two consecutive 1's or two consecutive 2's?


Let \( a\), \( b\), \( c\), \( x\), \( y\), and \( z\) be real numbers that satisfy the three equations
\[
   17x +  by +  cz = 0, \\
   ax + 29y +  cz = 0, \\
   ax +  by + 53z = 0.
\]
Suppose that \( a \ne 17\) and \( x \ne 0\). What is the value of
\[
   \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53} \, ?
\]


How many positive divisors does 8! have?


Determine the number of angles between 0 and $2\pi$, other than integer multiples of $\frac{\pi}{3}$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order.


What is the value of the following expression: $1 - 4 + 7 - 10 + 13 - \cdots + 91 - 94 + 97$?


What is the remainder when $1488 \cdot 1977$ is divided by $500$?


If eight people decide to come to a concert, but four of them are only 1/3 sure that they will stay for the entire time (the other four are certain they'll stay the whole time), what is the probability that at least 7 people stayed the entire time?


Anna, Abe, Bobby, Charles, Devin, Edwin, and Fiona are participants in a race. How many different 1st-2nd-3rd place outcomes are possible if Bobby must be one of the top three finishers and there are no ties?


What is the base 4 representation of the base 2 number \(10101110_2\)?


Triangle $DEF$ is similar to triangle $ABC$. If $DE = 9$, $EF = 18$, $DF = 12$, and $BC = 27$ units, what is the length of segment $AC$?


Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 4:1$. Then $\overrightarrow{P}$ can be expressed as $t \overrightarrow{A} + u \overrightarrow{B}$ for some constants $t$ and $u$. Find the ordered pair $(t,u)$.

[asy]
unitsize(1 cm);

pair A, B, P;

A = (0,0);
B = (5,0);
P = interp(A,B,4/5);

draw(A--B);

dot("$A$", A, W);
dot("$B$", B, E);
dot("$P$", P, N);
[/asy]

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


A marine biologist wants to estimate the number of dolphins in a coastal area. On January 1st, she captures 100 dolphins, tags them, and releases them back. On June 1st, she captures another sample of 90 dolphins and finds that 4 of them are tagged. She hypothesizes that only 20% of these dolphins might have left the coastal area by June 1st (due to migration), and that 50% of the dolphins present on June 1st were not in the area on January 1st (due to new arrivals). What is her estimate for the number of dolphins in the coastal area on January 1st?


Simplify the following expression:
\[
\frac{\sin^3 30^\circ + \cos^3 30^\circ}{\sin 30^\circ + \cos 30^\circ}.
\]

Solution: \boxed{1 - \frac{\sqrt{3}}{4}}


What is the base ten equivalent of $65432_7$?


Using only the digits 5, 6, 7, 8, and 9, how many positive nine-digit integers can be made that are palindromes?


Let \( M \) be the largest integer whose square has exactly 4 digits when written in base 7.

What is \( M \), expressed in base 7?


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


Find the largest value of $n$ such that $6x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.


Jill is studying a bacteria population that initially has 50 bacteria at 10:00 a.m. The population doubles every 4 minutes. Assuming 10% of the bacteria die every 4 minutes, how many bacteria are present at 10:20 a.m. the same day?


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


Determine the smallest positive real number \(x\) such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 7.\]


If altitude $CD$ is $\sqrt{2}$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?

[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(2)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$45^{\circ}$",A-(0.05,0.5),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,SW);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(225)--D + sqrt(2)*.1*dir(180)--D+.1*dir(135));
[/asy]


Compute \[ \left\lfloor \dfrac {2025^3}{2023 \cdot 2024} - \dfrac {2023^3}{2024 \cdot 2025} \right\rfloor,\]where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


Simplify $8 \cdot \frac{18}{5} \cdot \frac{-40}{27}$.


Consider one positive even integer and one positive odd integer less than $16$, where the even integer is a multiple of $3$. Compute how many different possible values result when the sum of these two numbers is added to their product.


Laura is training for a triathlon, but she doesn't feel like swimming. She bikes 25 miles at \(3x + 2\) miles per hour, then spends 10 minutes parking her bike, changing to her running shoes, and catching a drink of water before running 6 miles at \(x\) miles per hour. Her total workout lasts 140 minutes. How fast did Laura run, to the nearest hundredth of a mile per hour?

Solution: \boxed{6.24 \text{ mph}}


How many positive integers \( n \) less than 200 have a corresponding integer \( m \) divisible by 4 and a perfect square such that the roots of \( x^2 - nx + m = 0 \) are consecutive positive integers?


The deli now offers five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich still consists of one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, swiss cheese, and rye bread are each offered at the deli. If 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 could Al order?


A line segment with endpoints $A(-3, 7)$ and $B(9, -1)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates of point $C$? Express your answer as an ordered pair.

Solution: \boxed{(15, -5)}


An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint three faces of a die red so that the numbers on the red faces don't all add up to $12$?


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 12 & -6 \\ 9 & -3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$

Solution: \boxed{\begin{pmatrix} 12 & -6 \\ 9 & -3 \end{pmatrix}}


Let $g(x) = \frac{2x + 8}{x}$. The sequence $(g_n)$ of functions is defined by $g_1 = g$ and
\[g_n = g \circ g_{n - 1}\]
for all $n \ge 2.$ For example,
\[g_2(x) = g(g(x)) = \frac{\frac{2x + 8}{x} + 8}{\frac{2x + 8}{x}} = \frac{10x + 8}{2x + 8}\]
and
\[g_3(x) = g(g_2(x)) = \frac{\frac{10x + 8}{2x + 8} + 8}{\frac{10x + 8}{2x + 8}} = \frac{18x + 72}{10x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[g_n(x) = x\]
for some positive integer $n.$ Find the number of elements in $T.$


Let $\theta$ be the angle between the planes $3x - 2y + z - 1 = 0$ and $9x - 6y - 4z + 3 = 0$. Find $\cos \theta$.


What is the sum of all integer solutions to $4 < (x-3)^2 < 36$?


Three different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these three numbers is one? Express your answer as a common fraction.


Five concentric circles are drawn with radii of 2, 6, 10, 14, and 18. The innermost circle is painted black, the first ring around it is white, the next ring is black, the following ring is white, and the outermost ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


The first term of a new sequence is $3079$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $3079^{\text{th}}$ term of this sequence?


How many non-empty subsets \( S \) of \( \{1, 2, 3, \ldots, 12\} \) have the following two properties?
1. No two consecutive integers belong to \( S \).
2. If \( S \) contains \( k \) elements, then \( S \) contains no number less than \( k \).


A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?


The sum of four positive integers that form an arithmetic sequence is 50. Of all such possible sequences, what is the greatest possible third term?


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


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


In a public park, a circular arrangement is designed as follows: at the center, there is a circular water fountain surrounded by a ring-shaped flower garden. Outside the garden, there is a walking path and then another inner walking path adjacent to the garden but inside it. The water fountain has a diameter of 12 feet. The flower garden ring is 9 feet wide, the outer walking path is 7 feet wide, and the inner walking path is 3 feet wide. Calculate the diameter of the outermost boundary of the outer walking path.


Find the domain of the function
\[ f(x) = \sqrt{2 - \sqrt{3 - \sqrt{4x - 12}}}.\]

Solution: \boxed{\left[3, \frac{21}{4}\right]}


What is the smallest positive integer $n$ such that $725n \equiv 1275n \pmod{35}?$


How many divisors of $10!$ are exactly 10 times greater than $9!$?


Starting with $10^6$, Sally creates a sequence by first multiplying by 3, then dividing by 2, and repeats these steps. Write the last integer of this 12-step sequence in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as possible.


What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $4$:$50$?


A rectangular box has a volume of 216 cubic feet. Convert this volume into cubic meters knowing that 1 cubic yard equals 0.764 cubic meters.


What is the remainder when the sum of the seven consecutive odd integers starting from $11075$ to $11087$ is divided by $14$?


Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will roll her die?


The first and fifteenth terms of an arithmetic sequence are 7 and 35, respectively. Calculate the sixtieth term and the sum of the first sixty terms of the sequence.


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


What is the sum of the $x$-values that satisfy the equation $10 = \frac{x^3 - 5x^2 - 10x}{x+2}$?


If the product $(4x^2 - 6x + 5)(8 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $9a + 3b + c + d$.


Calculate $(-7)^7\div7^4 + 2^6 - 8^2$ as an integer.


How many positive five-digit integers of the form $\_\_325$ are divisible by 25?


Find the solution to \( x|x| = 3x - 2 \) which has the smallest value.

Solution: \boxed{\frac{-3 - \sqrt{17}}{2}}


In a survey done by the community library, the table below shows a partially filled record of people who prefer physical books over e-books. How many of the seniors surveyed prefer physical books?

\begin{tabular}{| l | c | r |}
\hline
& prefer physical books & prefer e-books \\
\hline
adults & 80 & \\ \hline
seniors & & 130 \\ \hline
total & 180 & 200 \\
\hline
\end{tabular}


Let $n$ be a positive integer. If $b\equiv (5^{2n}+6)^{-1}\pmod{11}$, what is the remainder when $b$ is divided by 11?


Let $c$ and $d$ be real numbers such that the roots of
\[z^2 + (15 + ci) z + (35 + di) = 0\] are complex conjugates. Determine the ordered pair $(c,d)$.


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


Each week, Tim goes to the supermarket and buys the following: $8$ apples at $\$1$ each, $4$ bottles of milk at $\$3$ each, $3$ pineapples at $\$4$ each, $3$ bags of flour at $\$6$ each, and a $\$10$ box of chocolates. This week the store has a special where pineapples are at half price and milk is discounted by $\$1$ per bottle. Tim also has a coupon for $\$10$ off any order of $\$50$ or over. Calculate the total expenditure for Tim on this trip.


Find the integer $n,$ $-90 \le n \le 90,$ such that $\cos n^\circ = \sin 712^\circ.$


In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ Determine the area of $\triangle ABC.$

[asy]
draw((0,0)--(2,5)--(11,5)--(18,0)--cycle);
draw((11,5)--(0,0));
label("$D$",(0,0),W);
label("$A$",(2,5),NW);
label("$B$",(11,5),NE);
label("$C$",(18,0),E);
[/asy]


I have seven apples and twelve oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)


A wire is cut into two pieces, one of length $x$ and the other of length $y$. The piece of length $x$ is bent to form a square, and the piece of length $y$ is bent to form a regular pentagon. The square and the pentagon have equal perimeter. What is $\frac{x}{y}$?


Let $a,$ $b,$ $c$ be distinct real numbers such that
\[\frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} = 0.\]Find all possible values for
\[\frac{a}{(b - c)^3} + \frac{b}{(c - a)^3} + \frac{c}{(a - b)^3}.\]


A rectangular floor that measures 15 feet by 20 feet is to be tiled with 3-inch-by-9-inch tiles. Additionally, there is a 1-foot wide border of different tiles along all edges of the floor. How many 3-inch-by-9-inch tiles are needed to cover the main area of the floor?


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


One root of the equation \(4x^2 + kx + 2 = 0\) is -0.5. What are \( k \) and the other root?


Equilateral triangle $ABC$ and rectangle $BCDE$ are coplanar, with $BC = DE = 2 \times CD$. What is the number of degrees in the measure of angle $CAD$?

Solution: \boxed{\angle CAD = 15^\circ}


Solve for $x$ if $3x^2 + 7 = 6x - 4.$ Express any complex solutions in the form $x = a \pm b i$, and find the value of $a + b^2$.


A right triangle has legs measuring 12 cm and 12 cm. The hypotenuse of a larger, similar triangle is 30 cm. Determine the perimeter of the larger triangle.

Solution: \boxed{30\sqrt{2} + 30 \text{ cm}}


Lisa places a total of 30 blue Easter eggs in several yellow baskets and a total of 45 purple Easter eggs in some red baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Lisa put in each basket?


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 4x^2 - 6x + 8 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a - 3,$ $b - 3,$ and $c - 3.$

Solution: \boxed{x^3 + 5x^2 - 3x - 19}


Define the operation $x \star y = xy - 2x + y^2$. What is the value of $(7 \star 4) - (4 \star 7)$?


Two horizontal sections of a right octagonal pyramid are obtained by slicing the pyramid with planes parallel to the octagonal base. The areas of these sections are $360$ square units and $810$ square units respectively. The two planes are $10$ units apart. Determine the distance from the apex to the larger section.


In a library, there are $1573$ boxes, each containing $42$ books. An assistant is tasked to repack all the books into boxes that must contain $45$ books each. After repacking as many boxes as possible, how many books will the assistant have left over?


Two positive integers differ by 5 and their product is 88. What is the larger integer?


Find the value of \( t \) for which the vector
\[
\bold{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 3 \\ -4 \\ 1 \end{pmatrix}
\]
is closest to
\[
\bold{a} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.
\]


Let $m$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers such that $\sum_{i = 1}^4 x_i = 84.$ Find $\frac m{100}.$


Compute the sum of the series given by:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \]


On a "prime date," both the month and the day are prime numbers. Calculate how many prime dates occurred in 2008, keeping in mind that it was a leap year.


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 5,$ $\|\mathbf{b}\| = 7,$ and $\|\mathbf{c}\| = 8,$ and
\[\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.\] Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$


In triangle $XYZ$, $XY = 12$, $YZ = 16$, and $XZ = 20$, with $ZD$ as the angle bisector. Find the length of $ZD$.


Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 40$, $b_1 = 70$, $b_n = 0$, and $$ b_{k+1} = b_{k-1} - \frac{2}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$.


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


Simplify $(a^4 \cdot a^3) \cdot (b^2 \cdot b^5)$.


Solution: \boxed{65 \frac{4}{5}}


Determine $p(x)$ if the graph of $\frac{x^3 - 3x^2 - 4x + 12}{p(x)}$ has vertical asymptotes at $3$ and $-1$, no horizontal asymptote, and $p(2) = 10$.

Solution: \boxed{p(x) = -\frac{10}{3}x^2 + \frac{20}{3}x + 10}


A semicircular arc AB with center O rolls along a straight line without slipping, until it completes half a rotation, returning its center O to point O'. If BO = OB' = 3 cm, calculate the length of the path that point O travels during this motion.

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


A triangle has sides of length 10 cm, 13 cm, and 7 cm. Find the perimeter of the triangle and also calculate its area using Heron's formula.

Solution: \boxed{20\sqrt{3} \text{ cm}^2}


If 500 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Determine \(p(x)\) if the graph of \(\frac{x^3 - 3x^2 - 4x + 12}{p(x)}\) has vertical asymptotes at \(3\) and \(-1\), no horizontal asymptote, and \(p(2) = 10\).

Solution: \boxed{-\frac{10}{3}x^2 + \frac{20}{3}x + 10}


If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Consider the quadratic equation $2x^2 - 6x + c = 0$. If this equation has two real solutions, and the discriminant is 12, find the average of these two solutions.


Consider the function $$g(x) = \frac{1}{2+\frac{1}{3+\frac 1x}}.$$ Determine the sum of all real numbers $x$ that are not in the domain of $g(x)$.


In a 3D space, the points \( E = (2, -5, 1) \), \( F = (4, -9, 4) \), \( G = (3, -4, -1) \), and \( H = (5, -8, 2) \) are given. Determine if these points form a parallelogram and if so, find the area of this parallelogram.


In a triangle $XYZ$, angle $\angle ZXY$ is $72^\circ$. It is given that side $XY$ is twice as long as side $YZ$. What is the measure of angle $\angle XYZ$?

Solution: \boxed{\angle XYZ = 36^\circ}


A line segment begins at $(4, 3)$. It is 15 units long and ends at the point $(x, 9)$ where $x < 0$. What is the value of $x$?

Solution: \boxed{x = 4 - \sqrt{189}}


$ABCDEFGH$ shown below is a cube with volume 8. Find the volume of pyramid $AEFG$.


What is the remainder when the sum $1^2 + 2^2 + 3^2 + \dots + 15^2$ is divided by 13?


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


A point $P$ is randomly chosen from the rectangular region with vertices $(0,0), (3,0), (3,1), (0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(4,1)$?


A 6x6x6 cube is made up of 216 unit cubes. On each face of the cube, only the central 2x2 square is painted. Determine how many of the 216 unit cubes remain without any paint.


Mr. Wong now has 12 grandchildren. Assuming that the gender of each grandchild is determined independently and with equal likelihood of being male or female, what is the probability that Mr. Wong has more grandsons than granddaughters or more granddaughters than grandsons?


Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 17.\] Find the largest possible value of $d.$

Solution: \boxed{\frac{5 + \sqrt{123}}{2}}


If the three points $(2,a,b),$ $(a,3,b),$ $(a,b,4)$ are collinear, what is the value of $a + b$?


A right triangle and a rectangle have the same area. The base of the right triangle is equal to the width of the rectangle, and both are 5 units. If the area of the rectangle is 45 square units, find the height \( h \) of the right triangle.


In rectangle $ABCD$, side $AB$ measures $8$ units and side $BC$ measures $4$ units. Points $F$ and $G$ are on side $CD$ such that segment $DF$ measures $2$ units and segment $GC$ measures $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$?


The difference between two perfect squares is 143. What is the smallest possible sum of the two perfect squares?


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


Three different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. What is the probability that the positive difference between the highest and lowest numbers is $5$ or greater? Express your answer as a common fraction.


Find the smallest negative integer $x$ which satisfies the congruence $45x + 8 \equiv 5 \pmod {24}$.


In triangle \( ABC \), where \( AB = 12 \), \( BC = 16 \), and \( CA = 20 \). Point \( D \) is on \( \overline{AB} \), \( E \) is on \( \overline{BC} \), and \( F \) is on \( \overline{CA} \). Let \( AD = p \cdot AB \), \( BE = q \cdot BC \), and \( CF = r \cdot CA \), where \( p \), \( q \), and \( r \) are positive and satisfy \( p + q + r = 3/4 \) and \( p^2 + q^2 + r^2 = 1/2 \). The ratio of the area of triangle \( DEF \) to the area of triangle \( ABC \) can be written in the form \( m/n \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \).


When the binary number $1110101101101_2$ is divided by 8, what is the remainder (give your answer in base 10)?


Each outcome on the spinner below has equal probability. The spinner is labeled with numbers 1, 2, and 5. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit, and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 5? Express your answer as a common fraction.


Mark needs to specify his height in centimeters for a health survey. If there are $12$ inches in a foot, $0.3048$ meters in a foot, and $100$ centimeters in a meter, what height should Mark specify, in centimeters, if his height is $70$ inches?


How many two-digit numbers are there in which the tens digit is exactly two times the ones digit?


A right triangle has one leg of 15 inches and the hypotenuse of 34 inches. A second triangle, which is similar to the first, has a hypotenuse that is twice the length of the first triangle’s hypotenuse. What is the length of the second triangle's shortest leg, if the longer leg of the second triangle is 33% larger than the corresponding leg of the first triangle?

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


In triangle $ABC,$ $AB = 8,$ $BC = 13,$ and $AC = 15.$  If $D$ and $E$ are chosen on $\overline{AB}$ and $\overline{AC}$ so that $AD = 3$ and $AE = 11,$ then find the area of triangle $ADE.$

Solution: \boxed{\frac{33 \sqrt{3}}{4}}


A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?


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


Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term.


Find the ordered pair $(k,a)$ such that when the polynomial $x^3 - 4x^2 + 9x - 6$ is divided by $x^2 - x + k$, the remainder is $2x + a$.

Solution: \boxed{(4, 6)}


The sum of the areas of two squares is 85, while the difference in their areas is 45. Find the sum of their perimeters.

Solution: \boxed{4(\sqrt{65} + 2\sqrt{5})}


Let $\mathbf{B} = \begin{pmatrix} x & 2 \\ -3 & y \end{pmatrix}$ for some real numbers $x$ and $y$. If
\[\mathbf{B} + \mathbf{B}^{-1} = \mathbf{0},\] then find $\det \mathbf{B}$.


For how many integers $n$ between 1 and 200 is the greatest common divisor of 24 and $n$ equal to 4?


A circle having center $(0,k)$, with $k>8$, is tangent to the lines $y=x$, $y=-x$ and $y=8$. What is the radius of this circle?

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


In triangle $ABC$, where $ABC$ is a right triangle at $B$, consider $AB = 4$ and $BC = \sqrt{17}$. Compute $\tan A$.


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


As $t$ varies over all real numbers, consider the curve defined by the equations:
\[
x = t^2 - 3,
\]
\[
y = t^3 - 6t + 4.
\]
Determine the coordinates $(x,y)$ where this curve intersects itself.

Solution: \boxed{(3, 4)}


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


In a new configuration, a large circle has a radius of 10 meters. Six smaller circles of equal size are arranged such that each smaller circle touches the larger circle internally at exactly one point, forming a regular hexagonal pattern inside. If the centers of the smaller circles form a regular hexagon, what is the radius of one of the smaller circles?


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


Let $Q_1$ be a regular $x$-gon and $Q_2$ be a regular $y$-gon $(x \geq y \geq 3)$ such that each interior angle of $Q_1$ is $\frac{29}{28}$ as large as each interior angle of $Q_2$. What is the largest possible value of $y$?


Suppose $g(x)$ is a rational function such that $4g\left(\dfrac{1}{x}\right)+\dfrac{3g(x)}{x}=2x^3$ for $x\neq 0$. Find $g(-1)$.


The sum of the lengths of all the edges of a rectangular prism is 88 cm. The prism's length is twice its width, and its height is half of its width. Find the volume of the rectangular prism.

Solution: \boxed{V = \frac{85184}{343} \text{ cm}^3}


The expression \( y^2 + 12y + 44 \) can be written as a combination of a square of a binomial and an integer. Find the integer.


Let $O$ be the origin. Determine the value of scalar $k$ for which any points $A$, $B$, $C$, and $D$ satisfying:
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 7 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0}\]
are guaranteed to be coplanar.


Given that
\[\tan x = \frac{3ab}{a^2 + 3b^2},\] where $a > b > 0$ and $0^\circ < x < 90^\circ,$ find $\sin x$ in terms of $a$ and $b.$

Solution: \boxed{\sin x = \frac{3ab}{\sqrt{a^4 + 15a^2b^2 + 9b^4}}}


Three tour guides are leading eight tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that at least one guide must take no tourists. How many different groupings of guides and tourists are possible where each tourist chooses a guide and each of the other two guides must have at least one tourist?


Two dice are rolled. What is the probability that the sum is greater than 6?


A biased 6-sided die has the probabilities of rolling each number as follows: $P(1) = \frac{1}{10}, P(2) = \frac{1}{10}, P(3) = \frac{2}{10}, P(4) = \frac{2}{10}, P(5) = \frac{2}{10}, P(6) = \frac{2}{10}$. What is the expected value of a roll of this die?


In a diagram, three angles are shown converging at a point. One of the angles measures $140^\circ$, and the other two angles are equal in measure, represented as $y^\circ$. What is the value of $y$?


Find the largest possible value of $k$ for which $3^{12}$ is expressible as the sum of $k$ consecutive positive integers.


The expression $x^2 - 14x + 40$ 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 $4d - c$?


Find the integer $n$, $12 \le n \le 18$, such that \[n \equiv 9001 \pmod{7}.\]


What is the least positive integer \( n \) such that \( 9450 \) is a factor of \( n! \)?


Simplify $\displaystyle\frac{2-i}{1+4i}$, where $i^2 = -1.$

Solution: \boxed{-\frac{2}{17} - \frac{9}{17}i}


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


The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 4x + 6.$ Find the leading coefficient of $g(x)$.


Define a **smart integer** as an even integer greater than 50, less than 200, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.


The positive integer divisors of 245, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 5?


In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 48. What is the perimeter of the shaded region created by the segments where each circle touches two others, given that each segment subtends an angle of $120^\circ$ at the center of its circle?


The mean of the set of numbers $\{92, 88, 86, 90, 91, y\}$ is 89.5. What is the median of the set of six numbers?


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

Solution: \boxed{k = 5}


In the arithmetic sequence $11, d, e, f, g, h, 61$, what is the value of $f$?


Suppose that \(S\), \(H\), and \(E\) are all distinct non-zero digits less than 6, and the following equation holds true in base 6:
$$\begin{array}{c@{}c@{}c@{}c}
  & S & H & E_6 \\
+ & E & S & H_6 \\
\hline
  & H & E & S_6 \\
\end{array}$$
Find the sum of the values \(S\), \(H\), and \(E\) expressed in base 6.


Find the greatest common divisor of $45121$ and $30340$.


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 $x.$


What is the base $2$ representation of $101_{10}$?


Calculate the sum of all integer solutions to $4 < (x-3)^2 < 36$.


Simplify $5 \cdot \frac{12}{7} \cdot \frac{49}{-60}$.


I had $\$45$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on video games?

[asy]
size(150);
pair A, B, C, D, 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("$\frac{2}{9}$", Y, S);
Z=(.5, -.15);
label("Toys", Z, N);
label("$\frac{1}{3}$", Z, S);
[/asy]


Determine the domain of the function $$k(x) = \frac{1}{x+5} + \frac{1}{x^2+5} + \frac{1}{x^3+5}.$$

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


The planet Zorion travels in an elliptical orbit around its sun, which is positioned at one focus of the ellipse. Zorion's closest approach to its sun is 3 astronomical units (AU), and its furthest point is 15 AU away. If the entire orbit of Zorion is tilted by 30 degrees from its original plane, calculate the distance from Zorion to its sun when Zorion is exactly halfway through its orbit from perigee to apogee.

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


Solve for $x$: \( x = \frac{\sqrt{8^2 + 15^2}}{\sqrt{25 + 16}} \). Express as a common fraction.


How many even integers are there between \(\frac{21}{5}\) and \(\frac{43}{3}\)?


Calculate the value of the expression $(25 + 15 + 8)^2 - (25^2 + 15^2 + 8^2)$.


How many ways can change be made for a half-dollar (50 cents) using only nickels, dimes, and quarters? (Don't count "2 quarters" as making change for a half-dollar.)


What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates 4:50?


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


Find $x$ such that $\log_x 8=\log_{81} 3$.


Let $(x, y)$ be a solution to the system of equations
\[\begin{aligned}
\lfloor x \rfloor + \{y\} &= 1.7, \\
\{x\} + \lfloor y \rfloor &= 3.6.
\end{aligned} \]
Compute the product $x \cdot y$.


For how many non-negative real values of $y$ is $\sqrt{169 - \sqrt[3]{y}}$ an integer?


A car's brakes are applied so that it travels 5 meters less in each second than it did in the previous second. If the car travels 35 meters in the first second after the brakes are applied, determine the total distance the car covers from the time the brakes are applied until the car comes to a complete stop.

Solution: \boxed{140 \text{ meters}}


Elmo makes $N$ sandwiches for a school event. For each sandwich, he uses $B$ dollops of peanut butter at $3$ cents each and $J$ spoonfuls of jelly at $7$ cents each. The total cost for the peanut butter and jelly to make all the sandwiches is $\$3.78$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the jelly Elmo uses to make the sandwiches?


A book has 688 pages, each page numbered from 1 to 688. How many more times does the digit '5' appear compared to the digit '2' when all the page numbers are printed?


Isosceles triangle $ABE$ of area 150 square inches is cut by $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 100 square inches. If the altitude of triangle $ABE$ from $A$ is 30 inches, what is the number of inches in the length of $\overline{CD}$?

Solution: \boxed{\frac{10\sqrt{3}}{3} \text{ inches}}


Given that $b$ is a multiple of $1428$, find the greatest common divisor of $b^2 + 11b + 30$ and $b + 6$.


A Senate subcommittee has 6 Democrats and 4 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table if no two Republicans are allowed to sit next to each other?


Solution: \boxed{s(g(4)) = \sqrt{18 - 12\sqrt{2}}}


Jessica has a 3 by 3 grid and two colors available to color it. No two adjacent squares (sharing a side) can be the same color. How many possible ways are there for Jessica to color the grid?


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?


If $f(x) = ax^2 + bx + c$ and $f^{-1}(x) = cx^2+bx+a$ for real numbers $a$, $b$, and $c$, what is the value of $a+b+c$ given that $f(f^{-1}(x))=x$?


Find the number of positive integers $n \le 1000$ that can be expressed in the form
\[
\lfloor x \rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor = n
\]
for some real number $x.$


A company that produces gadgets has a daily operational cost of $600 in maintenance fees. Each worker is paid $20 per hour. Every worker can manufacture 6 gadgets per hour, which are then sold at $4 each. Determine the minimum number of workers needed for the company to make a profit in a 10-hour workday.


Mr. Fat needs 15 minutes to eat a pound of cereal, while Mr. Thin needs 35 minutes, and Mr. Medium needs 25 minutes. If they all eat together, how long does it take for them to finish 5 pounds of cereal? Express your answer in minutes.


Solution: \boxed{-3x^{28} + 12x^{13} - 6x^9 + 24x^6 - 3x^3}


At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. Initially, hydrogen gas is placed in a 3.5-liter container where it exerts a pressure of 8 kPa. If this same gas is then transferred to a 7-liter container while maintaining the same temperature, what will the new pressure be in kPa?

Solution: \boxed{4 \text{ kPa}}


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


Calculate the following expression:
\[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2\]

Solution: \boxed{-x^2 + 23x - 3}


Consider the points $P(1, 2)$, $Q(4, 6)$, and $R(7, 2)$ arranged in the coordinate system to form a shape $\Delta PQR$. This shape is first rotated 180 degrees around the point $P$ and then translated three units to the right and four units downward. What is the midpoint of segment $Q'R'$ after these transformations?

Solution: \boxed{\left(-\frac{1}{2}, -4\right)}


You have 5 different shirts and 6 different pairs of pants. However, one special shirt cannot be paired with two specific pants. How many unique outfits can you make?


Let $T$ be the set of 12-tuples $(b_0, b_1, \dots, b_{11})$, where each $b_i$ is either 0 or 1. Thus, $T$ consists of $2^{12}$ 12-tuples. Define $q_t(x)$ for each 12-tuple $t = (b_0, b_1, \dots, b_{11}) \in T$ as a polynomial of degree at most 11 such that
\[q_t(n) = b_n\] for every integer $n$ from 0 to 11.
Find the sum:
\[\sum_{t \in T} q_t(12).\]


Find the $3 \times 3$ matrix $\mathbf{M}$ such that for any $3 \times 3$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ results in the first row of $\mathbf{N}$ becoming the negation of the second row of $\mathbf{N}$, the second row of $\mathbf{N}$ becoming the first row of $\mathbf{N}$, and the third row of $\mathbf{N}$ becoming tripled. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} -d & -e & -f \\ a & b & c \\ 3g & 3h & 3i \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 3 \end{pmatrix}}


Ana, Bob, and Cathy take turns rolling a six-sided die in their respective order. The first one to roll a 6 wins. What is the probability that Cathy will win? Express your answer as a common fraction.


Simplify $\sqrt{56x^3} \cdot \sqrt{10x^2} \cdot \sqrt{63x^4}$.


What is the product of all real numbers that are tripled when added to their reciprocals?


A rectangular tile measures 2 inches by 6 inches. What is the fewest number of these tiles needed to completely cover an L-shaped region composed of two rectangles, one measuring 3 feet by 4 feet and the other 2 feet by 1 foot, adjoining along one 3-foot side?


Determine the number of ways to arrange the letters of the word BANANAS such that no 'A' appears at the first position.


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

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


In a 7 by 7 grid, each of the 49 squares measures 1 cm by 1 cm and is shaded. Seven unshaded circles are placed on top of the grid as shown. The area of the visible shaded region can be written in the form $C-D\pi$ square cm. What is the value of $C+D$?

[asy]
for(int i = 0; i < 7; ++i)
{
    for(int j = 0; j < 7; ++j)
    {
        filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,gray,linewidth(2));
    }
}

filldraw(circle((3.5,1),0.5),white,linewidth(2));
filldraw(circle((5.5,3.5),0.5),white,linewidth(2));
filldraw(circle((3.5,6),0.5),white,linewidth(2));
filldraw(circle((1,3.5),0.5),white,linewidth(2));
filldraw(circle((3.5,3.5),1.5),white,linewidth(2));
filldraw(circle((6.5,1),0.5),white,linewidth(2));
filldraw(circle((6.5,6),0.5),white,linewidth(2));
[/asy]


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


Given that $b$ is a multiple of $3150$, find the greatest common divisor of $b^2 + 9b + 54$ and $b + 4$.


John wants to buy two small pizzas, one large pizza, and two drinks. The drinks cost $q$ dollars each, the small pizza costs the same as one drink, and the large pizza costs four times as much as one drink. If John started with $50$ dollars, how much money would he have left after making all of his purchases? Give your answer as an expression of $q$.

Solution: \boxed{50 - 8q}


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


Let $a,$ $b,$ $c$ be three terms in an arithmetic series where all terms are positive and the common difference is $d$. If $abc = 125$, find the smallest possible value of $b$.


The following line passes through the points \(\begin{pmatrix} -3 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\), and its direction vector is of the form \(\begin{pmatrix} a \\ -2 \end{pmatrix}\). Find the scalar \(a\).

Solution: \boxed{a = 2}


Find the minimum value of
\[((\sin x + \csc x)^2 + (\cos x + \sec x)^2) \cdot e^{\sin x + \cos x}\]
for $0 < x < \frac{\pi}{2}$ and $\sin x + \cos x > 1$.


There is a prime number which is a factor of every sum of four consecutive integers. What is the prime number?


A pirate is searching for buried treasure on 8 islands. Each island has a $\frac{1}{4}$ probability of containing buried treasure and no traps, a $\frac{1}{12}$ probability of having traps but no treasure, and a $\frac{2}{3}$ probability of having neither traps nor treasure. What is the probability that, while searching all 8 islands, the pirate encounters exactly 5 islands with treasure and none with traps?


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


What is the modulo $13$ residue of $247+5 \cdot 40 + 7 \cdot 143 +4 \cdot (2^3 - 1)?$


In a drawer, there are 8 forks, 5 spoons, and 7 knives. If I reach in and randomly remove four pieces of silverware, what is the probability that I get two forks, one spoon, and one knife?


Michael's quiz scores so far are 84, 78, 95, 88, and 91. What score does he need to get on the sixth quiz to make the arithmetic mean of the six scores equal 90?


If $\sqrt{8 + x} + \sqrt{15 - x} = 6$, what is the value of $(8 + x)(15 - x)$?


How many positive multiples of 6 that are less than 150 have a units digit of 6?


Given $g(x) = \frac{2ax - b}{cx - 2d}$ where $abcd \neq 0$ and $g(g(x)) = x$ for all $x$ in the domain of $g$, find the value of $2a - 2d$.


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


If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Suppose the function $g(x)$ is defined on the domain $\{a, b, c\}$ such that the graph of $y=g(x)$ consists of just three points. These three points form a triangle of area $54$. The function $y=3g(1.5x)$ also consists of just three points. Determine the area of the triangle formed by these three points in the graph of the latter function.


What is the largest prime factor of $5! + 7!$?


Find the degree measure of the least positive angle \(\theta\) for which
\[\cos 10^\circ = \sin 20^\circ + \sin \theta.\]


Let $x$ and $z$ be the two real values for which \[\sqrt[3]{x} + \sqrt[3]{27 - x} = 3.\] The smaller of these two values can be expressed in the form $r - \sqrt{s}$ where $r$ and $s$ are integers. Compute $r+s$.


In a rectangular coordinate system, what is the distance from the point (6, -18) to the point (3, 9)?


For how many positive integers \( n \) does \( 1+2+\cdots+n \) evenly divide \( 10n \)?


The numbers \( x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3 \) are equal to the numbers \( 1, 2, 3, \ldots, 9 \) in some order. Find the smallest possible value of
\[ x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3. \]


What is the remainder when $2003 \cdot 2004 \cdot 2005 \cdot 2006 \cdot 2007$ is divided by 17?


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 10$, then what is the volume of pyramid $SPQR$?


The Metropolitan High School Basketball team consists of 18 players, including 2 who specialize as goalkeepers. During training, a practice session is organized where each goalkeeper takes turn guarding the hoop while the rest of the team (including the other goalkeeper) shoots basketballs from the free throw line.

How many free throws must be taken to ensure that everyone has a chance to shoot against each of the goalkeepers?


Determine the sum of the angles of the solutions to the equation $z^6 = 64i$ in degrees, where the solutions can be expressed as $z_k = r_k (\cos \theta_k + i \sin \theta_k)$, with $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ$ for $k=1, 2, ..., 6$.


Calculate the product of $101011_2$ and $1101_2$ and express your answer in base $2$.


In the following diagram, \( AB = 60 \). Find \( AX \).

[asy]
import markers;

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

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

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

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

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


What is the greatest common divisor of $2^{1100} - 1$ and $2^{1122} - 1$?

Solution: \boxed{2^{22} - 1}


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


Find the number of positive solutions to
\[\sin (\arccos( \cot (\arcsin x) ) ) = x.\]


A weighted dice has six faces. The probability of rolling a 2 is $\frac{1}{4}$, the probability of rolling a 5 is $\frac{1}{2}$, and the probability of rolling any other number is $\frac{1}{12}$ for each number. You win $\$4$ if you roll a 2, $\$6$ if you roll a 5, and lose $\$3$ for rolling any other number. What is the expected value, in dollars, of rolling this dice once?


Starting from 3, what is the maximum number of consecutive positive integers that can be added together before the sum exceeds 1000?


Expand $(1+0.1)^{500}$ by the binomial theorem and calculate each term as
\[{500 \choose k}(0.1)^k\] for $k = 0, 1, 2, \ldots, 500.$ Determine the value of $k$ for which the term is the largest.


Let $G$ be the centroid of triangle $ABC$ in a Cartesian coordinate system. If $GA^2 + GB^2 + GC^2 = 72$, and the coordinates of $A$, $B$, and $C$ are given by $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ respectively, find $AB^2 + AC^2 + BC^2$.


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


If three faucets can fill a 100-gallon tub in 6 minutes, how long will it take for four faucets to fill a 50-gallon tub? Express your answer in seconds.

Solution: \boxed{135 \text{ seconds}}


Every week, Ben goes to the supermarket and buys the following: $7$ apples at $\$2$ each, $4$ bottles of milk at $\$4$ each, $3$ loaves of bread at $\$3$ each, and $3$ bags of sugar at $\$6$ each. This week the store has a $25\%$ discount on all dairy products. Ben also has a coupon for $\$10$ off any order of $\$50$ or over. How much money does Ben spend on this shopping trip?


Let $x,$ $y,$ $z$ be integers such that
\[\mathbf{B} = \frac{1}{7} \begin{pmatrix} -5 & x \\ y & z \end{pmatrix}\]and $\mathbf{B}^2 = \mathbf{I}.$  Find the largest possible value of $x + y + z.$


Given that $16^{-1} \equiv 31 \pmod{101}$, find $256^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


What is the least five-digit positive integer which is congruent to 7 (mod 18)?


Determine the value of $x$ for which $10^x = 10^{-7} \times \sqrt{\frac{10^{105}}{0.0001}}$.


Find the matrix $\mathbf{M}$ that doubles both columns of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 2c & 2d \end{pmatrix}.\]

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


Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$.


Triangle $DEF$ has a perimeter of 1024 units. The sides have lengths which are all integer values and $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


What is the coefficient of \(x^2\) when \(3x^3 - 4x^2 + 5x - 2\) is multiplied by \(2x^2 + 3x + 4\) and the like terms are combined?


Suppose $f(x)$ is a rational function such that $4f\left(\dfrac{1}{x}\right)+\dfrac{3f(x)}{x}=x^3$ for $x\neq 0$. Find $f(-3)$.


What is the least integer whose square is 64 more than its double?


Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{7} \begin{pmatrix} -5 & a \\ b & c \end{pmatrix}\]and $\mathbf{A}^2 = 2\mathbf{I}.$  Find the largest possible value of $a + b + c.$


Let \[
f(x) =
\begin{cases}
3x + 4 &\text{if } x \leq 1, \\
7 - 3x^2&\text{if } x > 1.
\end{cases}
\]
Find $f(-2)$ and $f(3)$.


How many integers between 700 and 900 have the sum of their digits equal to 20?


Determine the range of the function
\[ g(x) = \frac{3(x + 5)(x - 4)}{x + 5}. \]

Solution: \boxed{(-\infty, -27) \cup (-27, \infty)}


Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$.


The dilation, centered at $1 + 2i$ with scale factor 2, takes $0 - 2i$ to which complex number?

Solution: \boxed{-1 - 6i}


In the six-digit integer $4B5,\!894$, what is the largest digit $B$ so that the six-digit integer will be divisible by 3?


What is the smallest positive integer that can be written in the form $3001m + 24567n$, where $m$ and $n$ are integers?


John draws a regular ten-pointed star in the sand, each with 10 outward-pointing and 10 inward-pointing points. He places one of twenty different sea shells at each of these 20 points. How many ways can he place the shells, if reflections and rotations of the arrangement are considered equivalent?


The arithmetic mean of an even number of consecutive even integers is \(z\). Find the sum of the smallest and largest of the integers in terms of \(z\).


Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 3 = 0.$ Find the values of $k$ for which $a$ and $b$ satisfy
\[
\frac{a}{b} + \frac{b}{a} = 2.
\]
Then, determine the sum of squares of these $k$ values.


What are the last two digits in the sum of factorials of the first 150 positive integers?


Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$?


Suppose there are two congruent triangles $\triangle ABC$ and $\triangle ACD$, such that $AB = AC = AD$. If $\angle BAC = 30^\circ,$ then what is $\angle BDC$?


Find the matrix $\mathbf{N}$ that flips the rows of a matrix. That is, a matrix where:
\[
\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & d \\ a & b \end{pmatrix}.
\]
If no such matrix $\mathbf{N}$ exists, then provide the zero matrix.

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


Simplify $5 \cdot \frac{14}{3} \cdot \frac{9}{-42}$.


Circle \(D\) has a radius of 8 cm. Calculate the area of the largest possible inscribed triangle that has one side as a diameter of circle \(D\).

Solution: \boxed{64 \text{ square centimeters}}


Determine the smallest positive integer $n$ such that there exist positive real numbers $a$ and $b$ satisfying
\[(a + bi)^n = -(a - bi)^n.\]
If such $n$ exists, compute $\frac{b}{a}$.


Determine \( \frac{b}{a} \) if \( n \) is the smallest positive integer, now being \( n = 4 \), for which there exist positive real numbers \( a \) and \( b \) such that:
\[
(a + bi)^4 = (a - bi)^4
\]


A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form a square, and the piece of length $b$ is bent to form a regular octagon. The square and the octagon have equal perimeters. What is $\frac{a}{b}$?

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


Line segment \( \overline{AB} \) is extended past \( B \) to point \( Q \) such that \( AQ:QB = 7:2 \). Then \( \overrightarrow{Q} = x \overrightarrow{A} + y \overrightarrow{B} \) for some constants \( x \) and \( y \). Determine the ordered pair \( (x, y) \).

Solution: \boxed{\left( -\frac{2}{5}, \frac{7}{5} \right)}


Find the smallest integer $x$ such that $x$ has 24 positive factors. Factors of $x$ include 8 and 18. What is $x$?


Let $g(x)$ be a polynomial of degree 1004 with real coefficients, and let its roots be $s_1, s_2, \dots, s_{1004}$. There are exactly 502 distinct values among $|s_1|, |s_2|, \dots, |s_{1004}|$. What is the minimum number of real roots that $g(x)$ can have?


Let \( F_1 = \left( -4, 2 - \frac{\sqrt{8}}{2} \right) \) and \( F_2 = \left( -4, 2 + \frac{\sqrt{8}}{2} \right) \). The set of points \( P \) such that
\[ |PF_1 - PF_2| = 2 \]
forms a hyperbola. The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,\]
where \( a, b > 0 \). Find \( h + k + a + b \).


A school has between 100 and 250 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into seven distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Two quarters, a half dollar, and a one dollar coin are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent.


When $\sqrt[4]{2^9\cdot3^5}$ is fully simplified, the result is $a\sqrt[4]{b}$ where $a$ and $b$ are positive integers. What is $a+b$?


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


Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle.


A basketball player made the following number of free throws in 10 successive games: 8, 20, 16, 14, 20, 13, 20, 17, 15, and 18. What is the median and mean number of successful free throws?


From point $A$, Marie walks 30 yards north, 80 yards east, 15 yards south, and 30 yards west to point $C$, then moves 10 yards north to point $B$. What is the length, in yards, of $\overline{AB}$?


If $300\%$ of $x$ is equal to $75\%$ of $y$, and moreover $x + y = 30$, find the values of $x$ and $y$.


What is the smallest number of whole 3-by-4 rectangles needed to completely cover a rectangular region exactly, without overlapping and without gaps?


What is the sum of all the odd integers between -15.7 and 12.6?


In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than Anne does Bob run for every lap around the block?


The product of two positive integers plus their sum is 155. The integers are relatively prime, and each is less than 30. What is the sum of the two integers?


A four-inch sphere of silver weighs 8 pounds and is worth $\$500$. How much is a six-inch sphere of silver worth? Round your answer to the nearest dollar.


What non-zero, real value of $x$ satisfies $(4x)^5= (8x)^4$? Express your answer as a common fraction.


The area of a square garden is 675 square meters. Determine the perimeter of the square, and express your answer in simplest radical form.


The stem-and-leaf plot shows the ride duration on each of the $19$ top-rated roller coasters in a mix of minutes and seconds. In the stem-and-leaf plot, $3 \ 01$ represents $3$ minutes, $01$ second and $153$ represents $153$ seconds. Calculate the median ride time and express it in seconds.

\begin{tabular}{c|cccccc}
0&50&95&&&&\\
1&20&53&79&130&153&\\
2&10&45&60&&&\\
3&01&10&36&59&&\\
4&20&&&&&\\
\end{tabular}


In the diagram, circle $P$ has a radius of 10 units. Chord $EF$ has a length of 12 units and is parallel to segment $LM$. Another chord, $GH$, also parallel to segment $LM$, has a length of 16 units. Points $L$, $M$, $P$, and $N$ are collinear with $LP = 24$ units. Calculate the area of triangle $LGH$. [asy]
draw(Circle((0,0),10));
dot((0,0));
label("$P$",(0,0),S);
label("$M$",(-10,0),SW);
label("$N$",(10,0),SE);
label("$L$",(-24,0),W);
draw((-24,0)--(10,0));
label("$E$",(-6,sqrt(64)),NW);
label("$F$",(6,sqrt(64)),NE);
draw((-24,0)--(-6,sqrt(64)));
draw((-24,0)--(6,sqrt(64)));
draw((-6,sqrt(64))--(6,sqrt(64)));
label("$G$",(-8,sqrt(36)),NW);
label("$H$",(8,sqrt(36)),NE);
draw((-24,0)--(-8,sqrt(36)));
draw((-24,0)--(8,sqrt(36)));
draw((-8,sqrt(36))--(8,sqrt(36)));
[/asy]


A pile of gravel at a landscaping project is shaped like a cone. The diameter of the base of the cone is \(10\) feet and the height is \(80\%\) of its diameter. Calculate the volume of the gravel pile in terms of \(\pi\).

Solution: \boxed{\frac{200}{3} \pi \text{ cubic feet}}


Determine how many integer values of \( n \) satisfy \( -100 < n^3 < 100 \).


Let $P = (5, \phi_1)$ and $Q = (12, \phi_2)$ in polar coordinates. If $\phi_1 - \phi_2 = \frac{\pi}{3},$ find the distance $PQ.$


For how many values of the digit $A$ is it true that $45$ is divisible by $A$ and $273{,}1A5$ is divisible by $5$?


What is the matrix $\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$, where $A$ is rotated by $90^\circ$ counterclockwise and scaled by factor 2 relative to the origin to $A'$, and similarly for other points?

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


Given $b \neq 0,$ solve for $y$ in
\[\begin{vmatrix} y + b & y  & y \\ y & y + b & y \\ y & y & y + b \end{vmatrix} = 0.\]


Suppose that $(v_n)$ is a sequence of real numbers satisfying the recursive relation
\[v_{n+2}=2v_{n+1}+v_n.\]
Given that $v_2=7$ and $v_5=53$, find the value of $v_4$.


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


How many integers between 0 and 20 inclusive have an inverse modulo 17, and how many do not?


Let $M$ be the sum of the divisors of $300$. What is the largest prime factor of $M$?


In $\triangle ABC$, lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB} = \dfrac{4}{1}$ and $\dfrac{AE}{EB} = \dfrac{2}{3}$. Let $r = \dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Determine the value of $r$.


If the system of equations

\begin{align*}
8x - 6y &= c, \\
10y - 15x &= d.
\end{align*}

has a solution $(x,y)$ where $x$ and $y$ are both nonzero, find $\frac{c}{d},$ assuming $d$ is nonzero.


Compute the exact value of the expression $\left|\pi - \left| \pi - \frac{10}{3} \right| \right|$. Write your answer using only integers, $\pi$, fractions, and without any absolute value signs.

Solution: \boxed{2\pi - \frac{10}{3}}


Find the integer $n$, $0 \le n \le 12$, that satisfies \[n \equiv -3457 \pmod{13}.\]


Given $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 6,$ find
\[
\begin{vmatrix} x & 5x + 4y \\ z & 5z + 4w \end{vmatrix}.
\]


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


A teacher wants to arrange 4 copies of Introduction to Geometry and 5 copies of Introduction to Algebra on a bookshelf. Additionally, the first and last book on the shelf must be Introduction to Algebra. In how many ways can he do this?


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.


For what positive value of \( x \) does \( |6 + xi| = 15\sqrt{2} \)?


The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest point on the wall is 400 meters, and from B to the nearest point on the wall is 600 meters.


Solve the inequality:
\[
\frac{x}{x^2 + 3x + 2} \geq 0.
\]
Enter your answer using interval notation.

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


Given a matrix $\mathbf{M}$ that satisfies $\mathbf{M}(2\mathbf{i}) = \begin{pmatrix} 4 \\ 6 \\ -16 \end{pmatrix}$, $\mathbf{M}\mathbf{j} = \begin{pmatrix} 5 \\ 0 \\ -5 \end{pmatrix}$, and $\mathbf{M}(3\mathbf{k}) = \begin{pmatrix} 30 \\ -6 \\ 12 \end{pmatrix}$, find the matrix $\mathbf{M}$.

Solution: \boxed{\begin{pmatrix} 2 & 5 & 10 \\ 3 & 0 & -2 \\ -8 & -5 & 4 \end{pmatrix}}


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


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


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that it ends with two different digits?


Find the units digit of the following within the indicated number base: $65_7 - 12_7 + 24_7$


Let $x^2 + ax + b = 0$ be a quadratic equation whose roots are each three more than the roots of the quadratic $2x^2 - 3x - 8$. Determine the value of $b$.


A pet store has 12 puppies, 10 kittens, 8 hamsters, and 5 rabbits. Alice, Bob, Charlie, and David each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, Charlie, and David buy pets and leave the store satisfied?


If I choose five cards from a standard $52$-card deck, with replacement, what is the probability that I will end up with one card from each suit in the first four draws, and the fifth card is either a heart or a diamond?