Tamika selects two different numbers at random from the set $\{5, 6, 7\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 4, 8\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


Eight 6-sided dice are rolled. What is the probability that exactly four of these dice show the number 3?


The time right now is 3 o'clock. On a 12-hour clock, what time will it be 2517 hours from now?


Let $b_1, b_2, \ldots$ be a sequence defined by:

(i) $b_1 = 1$, and

(ii) $b_{3n}=n^2\cdot b_n$ for any positive integer $n$.

What is the value of $b_{3^{100}}$?


Eric is now building a larger pyramid with a height of 15 inches and a square base measuring 12 inches on each side. For extra safety, he desires a clearance of at least 3 inches above the pyramid when placed in a cube-shaped box. What is the minimum volume of the cube-shaped box required to hold the pyramid with the additional clearance?


What is the sum of the digits of the base-2 expression for $315_{10}$?


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 $SR = 11$, then what is the volume of pyramid $SPQR$?

Solution: \boxed{264 \text{ cubic units}}


If $(4, 20)$ and $(12, -6)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $y$-coordinates of the other two vertices?


When $5000^{150}$ is expanded out, the result is what digit followed by how many zeros?


From a standard deck of 52 cards, three cards are drawn consecutively without replacement. What is the probability that the first card is a spade, the second card is a heart, and the third card is a diamond?


In the six-digit integer $4B4,\!832$, what is the largest digit $B$ so that the six-digit integer will be divisible by both 3 and 4?


In a press conference before a major league game, ten players from four different teams will be taking questions. Three are from the Cubs, three are from the Red Sox, two are from the Yankees, and two are from the Dodgers. If teammates insist on sitting together, how many ways can the ten players be seated in a row?


A function \(g\) is defined on the complex numbers by \(g(z) = (c+di)z,\) where \(c\) and \(d\) are real numbers. This function has the property that for each complex number \(z\), \(g(z)\) is equidistant from both \(z\) and the origin. Given that \(|c+di| = 7\), find \(d^2.\)


Find the radius of the circle with equation $x^2 + 8x + y^2 - 10y + 41 = 0$.


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


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


Consider the first $30$ rows of Pascal's Triangle, from the $0$th row to the $29$th row. Calculate the sum of the binomial coefficients in the $10$th row.


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


The Big Eighteen Soccer League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other two divisions twice. How many league games are scheduled?


What is the smallest whole number $b$ such that 120 can be expressed in base $b$ using only three digits?


The exact amount of fencing that enclosed six congruent equilateral triangular corrals is reused to form one large equilateral triangular corral. Given that the side length of each small corral is 10 units, calculate the ratio of the total area of the six small corrals to the area of the new large corral. Express your answer as a common fraction.


In triangle $ABC$, $AC = \sqrt{17}$ and $AB = 4$. What is $\tan A$?

[asy]
pair A,B,C;
B = (0,0);
A = (0,4);
C = (1,0);
draw(A--B--C--A);
draw(rightanglemark(A,B,C,7));
label("$B$",B,SW);
label("$A$",A,NW);
label("$C$",C,SE);
label("$4$",(A+B)/2,NW);
label("$\sqrt{17}$",(A+C)/2,NE);
[/asy]


If $9:x^2 = x:25$, what is the value of $x$?


The Bank of New Town offers a Special Interest Account that compounds annually at a rate of two percent. If Mark invests $1500 in this account, how much interest will he earn after eight years?


The projection of $\begin{pmatrix} 5 \\ c \end{pmatrix}$ onto $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ is
\[5\begin{pmatrix} -3 \\ 2 \end{pmatrix}.\] Find $c.$


For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$.


Two rectangles have integer dimensions, and both have a perimeter of 200 cm. One dimension of one of the rectangles is fixed at 30 cm. What is the smallest possible area of this rectangle?

Solution: \boxed{2100 \text{ square cm}}


Let \( \mathbf{B} = \begin{pmatrix} a & 2 \\ -3 & d \end{pmatrix} \) for some real numbers \( a \) and \( d \). If
\[ \mathbf{B} + \mathbf{B}^{-1} = \mathbf{0}, \] determine \( \det \mathbf{B} \).


Let $A=(0,15)$ and $B=(0,18)$. Points $A'$ and $B'$ are on the line $y=x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C=(4,10)$. What is the length of $\overline{A'B'}$?


Given the points (2, 15), (9, 30), (15, 50), (21, 55), and (25, 60), what is the sum of the $x$-coordinates of the points that are situated above the line $y = 3x + 5$ in the coordinate plane?


The value of $3x + 15$ is one third of the value of $6x + 45$. After finding $x$, subtract 5 from the result. What is the final value?


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.


What is the digit in the ten-thousandths place of the decimal equivalent of $\frac{5}{32}$?


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $675$?


Let $\bold{v} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$ and $\bold{w} = \begin{pmatrix} 8 \\ -1 \end{pmatrix}$. Find the area of the parallelogram with vertices $\bold{0}$, $\bold{v}$, $2\bold{w}$, and $\bold{v} + 2\bold{w}$.


What is the remainder when the sum $1^2 + 2^2 + 3^2 + \dots + 15^2$ is divided by 16?


Find the positive value of $x$ that satisfies $cd = x + 6i$ given $|c|=3$ and $|d|=5$.


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.


Find the sum: $(-51) + (-49) + \cdots + (1)$. Additionally, identify the middle term of the series.


Consider the polynomial equation
\[ z^6 - z^4 + z^2 - 1 = 0. \]
Among all the roots of this polynomial, find the root with the maximum imaginary part, expressed as $\sin \theta$, where $-90^\circ \leq \theta \leq 90^\circ$. Determine $\theta$.


Find the smallest positive integer $n$ such that
\[\left(\begin{pmatrix} \cos 120^\circ & -\sin 120^\circ \\ \sin 120^\circ & \cos 120^\circ \end{pmatrix}^2\right)^n = \mathbf{I}.\]


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


Tamika selects two different numbers at random from the set $\{7, 8, 10\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 4, 6\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.

Solution: \boxed{\frac{6}{9} = \frac{2}{3}}


What is the greatest integer value of $b$ so that $5$ is not in the range of the function $y = x^2 + bx + 20$?


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


Let
\[
\mathbf{B} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{B}^{93}$.

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


In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac45$ and $\sin C = \frac12$, then what is $DC$?

[asy]
pair A,B,C,D;
A = (0,0);
B = (10*sqrt(5),24);
D = (10*sqrt(5),0);
C = (10*sqrt(5) + 24*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]


A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals?


A high school basketball team consists of 18 players, including a set of twins: Bob and Bill. In how many ways can we select 8 players for a game if both twins have to be in the lineup?


Six couples are at a gathering. Each person shakes hands with everyone else except for their spouse and the first new person they meet. How many handshakes are exchanged?


The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal asymptote is such that $y = 1/A < 1$. Find $A + B + C$.


Suppose I randomly pick an integer \( p \) between 1 and 20 inclusive. What is the probability that I choose a \( p \) such that there exists an integer \( q \) so that \( p \) and \( q \) satisfy the equation \( pq - 6p - 3q = 3 \)? Express your answer as a common fraction.

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


A ball bounces back up $\frac{3}{4}$ of the height from which it falls. If the ball is dropped from a height of $400$ cm, after how many bounces does the ball first rise less than $50$ cm?


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


Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{d}$ be vectors such that $\|\mathbf{a}\| = 2$, $\|\mathbf{b}\| = 4$, $\|\mathbf{d}\| = 6$, and
\[
\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{d}.
\]
If $\phi$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, find $\sin \phi$.


Compute
$$\sum_{k=1}^{500} k(\lceil \log_{3}{k}\rceil- \lfloor\log_{3}{k} \rfloor).$$


In the seven-term sequence $P, Q, R, S, T, U, V$, the value of $S$ is $7$ and the sum of any three consecutive terms is $21$. What is $P+V$?


Determine the range of the function $g(x) = \cos^4 x - \sin x \cos x + \sin^4 x$.

Solution: \boxed{\left[0, \frac{9}{8}\right]}


Let $g$ be defined by
\[g(x) = \left\{
\begin{array}{cl}
3-x & \text{ if } x \leq 2, \\
3x-x^2 & \text{ if } x>2.
\end{array}
\right.\]
Calculate $g^{-1}(-4)+g^{-1}(0)+g^{-1}(5)$.


An ellipse has its foci at $(1, 2)$ and $(4, 2)$. Given that it passes through the point $(-1, 5)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $h+k.$


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$ such that the graph of $y = g(x)$ consists of just three points. These three points form a triangle of area $45$. What is the area of the triangle formed by the points on the graph of $y = 3g(3x)$?


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math.


How many positive integers $N$ less than $2000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


In a bag, there are 6 blue chips, 5 red chips, and 4 yellow chips. One chip is drawn from the bag, then put back in, and a second chip is drawn. Determine the probability that the two selected chips are of different colors. Express your answer as a common fraction.


Thirty tiles are numbered from 5 through 34 and are placed into box $A$. Another thirty tiles numbered 15 to 44 are placed into box $B$. One tile is drawn randomly from each box. What is the probability that the tile from box $A$ is less than 20 and the tile from box $B$ is either odd or greater than 40? Express your answer as a common fraction.


In a certain sequence the first term is \( a_1 = 2010 \) and the second term is \( a_2 = 2011 \). Further, the values of the remaining terms are chosen so that \( 2(a_n + a_{n+1} + a_{n+2}) = 2n + 3 \) for all \( n \geq 1 \). Determine \( a_{1000} \).


Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 5x^2 - 3x)(3x^2 - 5x + 1)$.


Let $T$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2024}$. What is $\log_2(T)$?


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


Circle $P$ is on the coordinate plane with its center at $(1, 2)$. One endpoint of a diameter is at $(4, 6)$. What are the coordinates of the other endpoint of this diameter? Express your answer as an ordered pair.

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


In the World Series of baseball, the Astros are playing against the Dodgers. To win the World Series, a team must win 4 games before the other team does. If the Dodgers win each game with a probability of $\dfrac{3}{4}$ and there are no ties, what is the probability that the Astros will win the World Series but that the contest will need all seven games to be decided?


The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle?


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


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these numbers $abc = 8$.


How many two-digit numbers are composed of two distinct digits such that one digit is twice the other?


There are positive integers $a$ and $b$ that satisfy the system of equations\begin{align*}
\log_{10} a + 2 \log_{10} (\text{gcd}(a,b)) &= 12\\
\log_{10} b + 2 \log_{10} (\text{lcm}(a,b)) &= 42.
\end{align*}
Let $p$ be the number of (not necessarily distinct) prime factors in the prime factorization of $a$, and let $q$ be the number of (not necessarily distinct) prime factors in the prime factorization of $b$. Find $3p+2q$.


Let $a, b, c$ be non-negative integers less than $5$ satisfying the equations:
\begin{align*}
a + 2b + 3c &\equiv 3 \pmod{5}, \\
2a + 3b + c &\equiv 2 \pmod{5}, \\
3a + b + 2c &\equiv 1 \pmod{5}.
\end{align*}
Find the remainder when $abc$ is divided by $5$.


Three of the following test scores are Jake's and the other three are Linda's: 80, 86, 90, 92, 95, 97. Jake's mean score is 89. What is Linda's mean score?


Nalia needs to travel from point $X$ to $Y$, then from $Y$ to $Z$, and then from $Z$ to $X$. Each of these segments is either by road or rail. The cities form a right-angled triangle with $Z$ located 4000 km from $X$ and $Y$ located 5000 km from $X$. Compute the total distance Nalia travels on her journey.


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


Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/3 chance that she will toss it to Bob and a 2/3 chance that she will keep it. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after three turns?


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


The real number $x$ satisfies $x^2 - 4x + 3 < 0.$ Find all possible values of $x^2 + 4x + 3.$

Solution: \boxed{(8, 24)}


Given a function \( f(x) \) defined for all real numbers \( x \) such that for all non-zero values \( x \),
\[ 2f(x) + f\left(\frac{1}{x}\right) = 6x + 3 \]
Let \( T \) denote the sum of all values of \( x \) for which \( f(x) = 2023 \). Compute the integer nearest to \( T \).


How many factors of \(5400\) are perfect squares?


When a polynomial is divided by $3x^3 - 5x^2 + 3x - 8,$ what are the possible degrees of the remainder?  Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$.


When the base-12 integer $2625_{12}$ is divided by $10$, what is the remainder?


I have 6 books, three of which are identical copies of a science book, and the other three books are different. In how many ways can I arrange them on a shelf?


What is the sum of all integers in the set $\{1,2,3,4,5,6,7,8,9,10\}$ that are primitive roots $\pmod {11}$?


Find the sum of the squares of the solutions to the equation
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\]


Find the projection of the vector $\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$ onto the plane given by the equation $2x + 3y - z = 0$.

Solution: \boxed{\begin{pmatrix} \frac{17}{7} \\ \frac{-5}{14} \\ \frac{53}{14} \end{pmatrix}}


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


Determine the radius of the circle with equation \(x^2 + 8x + y^2 - 4y + 20 = 0\).


Let \( m = \underbrace{33333333}_{8 \text{ digits}} \) and \( n = \underbrace{666666666}_{9 \text{ digits}} \).

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


If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 72$ and $\gcd(a,c) = 240$, what is the smallest possible value of $\gcd(b,c)$?


At the conclusion of a match, each of the six members of a basketball team shakes hands with each of the six members of the opposite team, and all of the players shake hands with three referees. How many handshakes occur?


What is the greatest positive integer that must divide the sum of the first twelve terms of any arithmetic sequence whose terms are positive integers?


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


Simplify $\left((9 \times 10^8) \times 2^2\right) \div \left(3 \times 2^3 \times 10^3\right)$.


How many numbers less than 30 are abundant numbers?


I have four distinct mystery novels, four distinct fantasy novels, and four distinct biographies. I want to take exactly two books from different genres with me on a vacation. How many possible combinations of books can I choose?


A circle has a radius of five inches. The distance from the center of the circle to chord $PQ$ is four inches. How many inches long is chord $PQ$, and what is the area of the circle? Express the length of $PQ$ in simplest radical form and the area in terms of $\pi$.


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


Determine the sum of the rational roots of the polynomial $h(x) = x^3 - 7x^2 + 8x + 6$.


A binary operation $\diamondsuit$ is defined such that $a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c$ and $a \diamondsuit a = 1$ for all nonzero $a, b, c$. Solve the equation $504 \diamondsuit (3 \diamondsuit x) = 50$.

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


Find an ordered pair $(x, y)$ that solves the system: \begin{align*}
7x &= -10 - 3y, \\
4x &= 5y - 35.
\end{align*}

Solution: \boxed{\left(-\frac{155}{47}, \frac{205}{47}\right)}


In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?

[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy]


Let $A$ be the number of four-digit even numbers, and let $B$ be the number of four-digit numbers that are multiples of both 5 and 3. Calculate $A + B$.


A parallelogram has adjacent sides of lengths \( s \) units and \( 3s \) units, forming a 60-degree angle. What is the area of the parallelogram?


The points $(2, 9), (14, 18)$ and $(6, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?


For a real number \(x,\) find the maximum value of
\[
\frac{x^4}{x^8 + 2x^6 + 4x^4 + 8x^2 + 16}.
\]


Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend?


Find the remainder when the polynomial \( x^{100} \) is divided by the polynomial \( (x^2 + 1)(x - 1) \).


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


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.


In a certain sequence, the first term is \(a_1 = 1010\) and the second term is \(a_2 = 1011\). The values of the remaining terms are chosen so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \(n \geq 1\). Determine \(a_{1000}\).


Suppose a drawer contains 3 shirts, 6 pairs of shorts, and 7 pairs of socks. If I randomly select four articles of clothing, what is the probability that I get one shirt, two pairs of shorts, and one pair of socks?


Rationalize the denominator of $\frac{7}{\sqrt{175} - \sqrt{75}}$.

Solution: \boxed{\frac{7(\sqrt{7} + \sqrt{3})}{20}}


Find the sum of the first seven terms in the geometric sequence $\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots$.


How many integers between $200$ and $250$ have three different digits, all in increasing order?


For what real value of $t$ is $\frac{-15-\sqrt{145}}{6}$ a root of $3x^2 + 15x + t$?


The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 30, $x$ is three times $y$. Determine the value of $y$ when $x=-12$.


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


How many 3-letter words can we make from the letters A, B, C, D, and E, if we are allowed to repeat letters, and we must use the letters A and B at least once each in every word? (Here, a word is an arbitrary sequence of letters.)


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


Find the sum of the series $2_6 + 4_6 + 6_6 + \cdots + 100_6$ in base $6$.


Let \(p\) and \(q\) be the solutions to the equation \(3x^2 - 7x - 6 = 0\). Compute the value of \((5p^3 - 5q^3)(p - q)^{-1}\).


The pond in Jenny's garden grows algae such that the surface area covered by algae doubles each day. By day $24$, the pond is completely covered. Determine on which day the pond was only $12.5\%$ covered with algae.


What is the median number of moons per celestial body? (Include Pluto, Eris, and Ceres in the list.) \begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0\\
Venus & 0\\
Earth & 1\\
Mars & 2\\
Jupiter & 20\\
Saturn & 25\\
Uranus & 15\\
Neptune & 2\\
Pluto & 5\\
Eris & 1\\
Ceres & 0\\
\end{tabular}


How many ways are there to distribute 6 distinguishable balls into 4 indistinguishable boxes?


Calculate the greatest integer less than or equal to the expression \[\frac{4^{150} + 3^{150}}{4^{145} + 3^{145}}.\]


A sphere intersects the xy-plane in a circle centered at $(1,3,0)$ with a radius of 2. The sphere also intersects the yz-plane in a circle centered at $(0,3,-8),$ with radius $r.$ Find $r$ and verify the radius of the sphere.


Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one?


A hurricane in Melbourne, Australia, caused $\$$45 million in damage. Initially, the exchange rate was 1.8 Australian dollars for 1 British pound. Due to economic changes during the storm, the exchange rate shifted to 1.7 Australian dollars for 1 British pound. Calculate the damage in British pounds considering an average exchange rate for the period.


Billy and Bobbi each selected a positive integer less than 250. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction.


Let $c$ and $d$ be positive real numbers such that each of the equations $x^2 + cx + 3d = 0$ and $x^2 + 3dx + c = 0$ has real roots. Find the smallest possible value of $c + 3d$.


In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.


For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $15n$?


The polynomial $g(x) = x^4 + px^3 + qx^2 + rx + s$ has real coefficients, and $g(3i) = g(1+2i) = 0$. What is $p + q + r + s$?


Find all integer values of $n$ such that $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 5$.


Expanding $(1+0.1)^{2000}$ by the binomial theorem yields
\[ \sum_{k=0}^{2000} \binom{2000}{k} (0.1)^k = A_0 + A_1 + \dots + A_{2000}, \]
where $A_k = \binom{2000}{k} (0.1)^k$ for $k = 0, 1, 2, \ldots, 2000$. Determine the value of $k$ for which $A_k$ is the largest.


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


Construct a quadratic function that has zeroes at $x=1$ and $x=5$, and that takes the value $10$ when $x=3$.

Solution: \boxed{-\frac{5}{2}x^2 + 15x - \frac{25}{2}}


There are two positive integers \( c \) for which the equation \[ 3x^2 + 7x + c = 0 \] has rational solutions. What is the product of those two values of \( c \)?


Let $G$ and $H$ denote the centroid and orthocenter of triangle $ABC$, respectively. If $AG = AH$, find all possible values of angle $A$ (in degrees).


A curve is described parametrically by:
\[
(x,y) = (3 \cos t + \sin t, 3 \sin t).
\]
The graph of the curve can be expressed in the form:
\[
ax^2 + bxy + cy^2 = 1.
\]
Determine the ordered triple $(a, b, c)$.

Solution: \boxed{\left( \frac{1}{9}, -\frac{2}{27}, \frac{10}{81} \right)}


Let $\mathbf{u}_0$ be a vector. The vector $\mathbf{u}_0$ is projected onto $\begin{pmatrix} 4 \\ 2 \end{pmatrix},$ resulting in the vector $\mathbf{u}_1.$ The vector $\mathbf{u}_1$ is then projected onto $\begin{pmatrix} 2 \\ 2 \end{pmatrix},$ resulting in the vector $\mathbf{u}_2.$ Find the matrix that takes $\mathbf{u}_0$ to $\mathbf{u}_2.$

Solution: \boxed{\begin{pmatrix} \frac{3}{5} & \frac{3}{10} \\ \frac{3}{5} & \frac{3}{10} \end{pmatrix}}


Given that $x+\cos y=3000$ and $x+3000 \sin y=2999$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$.


In triangle $ABC$, $AB = 25$ and $BC = 20$. Find the largest possible value of $\tan A$ and $\sin A$.


A regular octahedron is formed by joining the centers of adjoining faces of a cube with a side length of 2 units. Calculate the ratio of the volume of the octahedron to the volume of the cube.


How much greater, in square inches, is the area of a circle with a radius of 12 inches and a surrounding ring of 2 inches thickness than a circle of radius 10 inches? Express your answer in terms of $\pi$.


What is the largest prime factor of $1739$?


Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $n$ be the number of ordered pairs of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.


Points $A$, $B$, $C$, and $P$ are in space such that each of $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ is perpendicular to the other two. If $PA = PB = 12$ and $PC = 7$, what is the volume of pyramid $PABC$?

Solution: \boxed{168 \text{ cubic units}}


Evaluate the expression $\sqrt{80} - 4\sqrt{5} + 3\sqrt{\frac{180}{3}}$ and express it as $\sqrt{M}$, where $M$ is an integer.


A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing?


Let $F_n$ be the Fibonacci sequence, defined by $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute:
\[
\sum_{n=0}^{\infty} \frac{F_n}{5^n}.
\]


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


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

Solution: \boxed{\frac{3ab}{\sqrt{a^4 + 7a^2b^2 + b^4}}}


Commander Krel from Planet Verx has received 30 resumes for four specialized roles on his starship: Chief Science Officer, Head Engineer, Navigation Expert, and Communications Officer. Unfortunately, Commander Krel finds only 10 of these candidates appropriate for any role. Additionally, for the position of Chief Science Officer, only 4 of the suitable candidates have the specific scientific expertise required. In how many ways can Commander Krel staff his starship, considering these constraints?


Both roots of the quadratic equation $x^2 - 107x + k = 0$ are prime numbers. Find the number of possible values of $k.$


Bob's password now consists of a non-negative single-digit number followed by any letter (uppercase or lowercase) and another non-negative single-digit number. What is the probability that Bob's password consists of an odd single-digit number followed by an uppercase letter and a positive single-digit number?


Simplify the expression $(3x-2)\cdot(5x^{12} + 3x^{11} + 7x^9 + 3x^8)$ and express your answer as a polynomial in descending order of degrees.

Solution: \boxed{15x^{13} - x^{12} - 6x^{11} + 21x^{10} - 5x^9 - 6x^8}


The average of five different positive whole numbers is $5.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the sum of the two smallest numbers after setting aside the smallest and largest numbers?


In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $BC = \sqrt{75}$. What is $\tan A$?


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


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


Alex has 12 different kinds of lunch meat and 11 different kinds of cheese. He can make a sandwich with one kind of meat and up to two kinds of cheese (it does not matter in which order he chooses the cheese). Additionally, there are 3 different kinds of bread he can choose from. How many different sandwiches could Alex make?


Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-15$, the remainder is $8$, and when $Q(x)$ is divided by $x-19$, the remainder is $10$. What is the remainder when $Q(x)$ is divided by $(x-15)(x-19)$?

Solution: \boxed{\frac{1}{2}x + \frac{1}{2}}


Four marbles are randomly selected, without replacement, from a bag containing three red, three blue, and two green marbles. What is the probability that exactly one marble of each color is selected, with an additional red marble?


Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\] is true if and only if either $x < -1$ or $|x-10| \le 2.$ Given that $a < b$, find the value of $a + 2b + 3c.$


The first few rows of a new sequence are given as follows:
- Row 1: $3$
- Row 2: $6, 6, 6, 6$
- Row 3: $9, 9, 9, 9, 9, 9$
- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$

What is the value of the $40^{\mathrm{th}}$ number if this arrangement were continued?


Find the largest integral value of \( x \) which solves: \( \frac{2}{5} < \frac{x}{7} < \frac{8}{11} \)

Solution: \boxed{x = 5}


A certain ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(-3 + \sqrt{5}, 2)$ and $(-3 - \sqrt{5}, 2).$ Find the length of the major axis.


If the system of equations
\[
4x - 2y = c, \quad
6y - 12x = d
\]
has a solution, find $\frac{c}{d},$ assuming $d \neq 0.$

Solution: \boxed{\frac{c}{d} = -\frac{1}{3}}


What is the largest possible median for the five-number set \(\{x, 2x, 6, 4, 7\}\) if \(x\) can be any integer?


In $\triangle ABC$, point D is on side BC such that BD = 4 and DC = 14. Calculate the ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$. Additionally, find what fraction of side BC does point D divide the side into.


What is the value of $b$ for which $\frac{1}{\log_2 b} + \frac{1}{\log_3 b} + \frac{1}{\log_5 b} + \frac{1}{\log_6 b} = 1$?


The area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is 12. Find the area of the parallelogram generated by the vectors $3\mathbf{a} + 4\mathbf{b}$ and $2\mathbf{a} - 6\mathbf{b}$.


A deck of 104 cards, consisting of two standard 52-card decks, is shuffled and dealt out in a circle. What is the expected number of pairs of adjacent cards which are both red?


A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time?


Determine the leading coefficient in the polynomial $5(x^5 - 2x^3 + x^2) - 8(x^5 + x^4 + 3) + 6(3x^5 - x^3 - 2)$ after it is simplified.


A rectangular dining table arrangement requires placing $48$ identical plates such that each row and each column contains the same number of plates, and each row and column has at least two plates. Determine how many different arrangements (configurations of rows and columns) are possible if we assume all plates are used and the arrangement respects the condition mentioned. Note that configurations are considered distinct based on the orientation (i.e., swapping rows and columns counts as different).


Let $(x, y)$ be a solution to the system of equations
\[
\begin{aligned}
\lfloor x \rfloor - \{y\} &= 3.7, \\
\{x\} + \lfloor y \rfloor &= 6.2.
\end{aligned}
\]
Compute $|x - y|$.


Each of $a_1,$ $a_2,$ $\dots,$ $a_{50}$ is equal to $1$ or $-1.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} a_i a_j.\]


Jamie made a complete list of the prime numbers between 1 and 50. What is the sum of the second smallest prime number and the second largest prime number on her list?


Let $m$ be the integer such that $0 \le m < 29$ and $4m \equiv 1 \pmod{29}$. What is $\left(5^m\right)^4 - 3 \pmod{29}$?


The terms $250, b, \frac{81}{50}$ are the first, second, and third terms, respectively, of a geometric sequence. If $b$ is positive, what is the value of $b$?


Determine the largest interval including the point $x=2$ where the function $g(x) = 3x^2 - 9x + 4$ is invertible.

Solution: \boxed{[\frac{3}{2}, \infty)}


The product of three times the circumference of a circle and two is equal to twice the circle's area. What is the length of the radius of the circle, in inches?


A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$.


For $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 7 \\ 3.5 \end{pmatrix}.\]Find $y$.


Find the least common multiple (LCM) and the greatest common divisor (GCD) of 12 and 18.


A block of measurements 9-inches by 7-inches by 12-inches is filled with as many solid 4-inch cubes as possible. What percentage of the volume of the block is occupied by these cubes?


In a different hyperbola setting, the center is at $(1, 3),$ one focus is at $(1, 9),$ and one vertex is at $(1, 0).$ 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{7 + 3\sqrt{3}}


Let $x$ and $y$ be positive real numbers, and $a$ be a positive constant. Find the maximum value of
\[
\frac{(x+y+a)^2}{x^2+y^2+a^2}.
\]


If $\log_2 x^2 + \log_4 x = 6,$ compute $x.$


The Boston weatherman says there is a 75 percent chance of rain for each day of a five-day holiday weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny exactly two of those days for a historical reenactment. What is the probability they get the weather they want? Give your answer as a fraction.


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


The Estimable Mathematics Institute is organizing its freshman reception. The new freshman class has fewer than **700** people. When these students are arranged in columns of **20**, the last column contains **19** students. When they line up in columns of **25**, the last column has **24** students. When they line up in columns of **9**, there are **3** students in the last column. How many students are in the freshman class?


The lengths of the sides of a triangle with positive area are $\log_{10}20$, $\log_{10}45$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$.


What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect fifth power?


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


Triangle $DEF$ has sides of $7$ units, $10$ units, and $13$ units. A rectangle, whose area is equal to that of triangle $DEF$, has a length of $7$ units. What is the perimeter of this rectangle?

Solution: \boxed{14 + \frac{40\sqrt{3}}{7}}


Determine the smallest possible median for the five number set $\{x, 3x, 4, 1, 6\}$ if $x$ can be any integer.


It is currently $8\!:\!45\!:\!00 \text{ a.m.}$. What time will it be in $9876$ seconds?

Solution: \boxed{11\!:\!29\!:\!36 \text{ a.m.}}


The expression $512x^3 + 27$ can be decomposed into the form $(ax+b)(cx^2+dx+e)$. Find the value of $a + b + c + d + e$.


A number between $0000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?


The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.


The function $y=\frac{x^3+11x^2+38x+35}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?


Let **a** and **b** be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{2}{5} \end{pmatrix}}


In a hexagon $ABCDEF$, where angle $A$ is unknown, the following angles are given:
$\angle B = 134^\circ$, $\angle C = 98^\circ$, $\angle D = 120^\circ$, $\angle E = 139^\circ$, $\angle F = 109^\circ$. What is the measure of angle $A$?


Find the smallest composite number that has no prime factors less than 20.


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


Find the remainder when $8x^5 - 10x^4 + 3x^3 + 5x^2 - 7x - 35$ is divided by $x - 5$.


Determine the values of $x$ for which $\frac{\log{(5-2x)}}{\sqrt{2x-3}}$ is defined.

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


A teacher offers candy to her class of 50 students, with the mean number of pieces taken by each student being 7. If every student takes at least one candy but no more than 20 candies, what is the greatest number of pieces one student could have taken?


What is the smallest positive integer $n$ such that $n^2$ is divisible by 24, $n^3$ is divisible by 900, and $n^4$ is divisible by 1024?


Solution: \boxed{x = 40}


If $q(x) = x^4 - 4x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.


Given a quadrilateral $PQRS,$ side $\overline{PQ}$ is extended past $Q$ to $P'$ so that $P'Q = PQ.$ Points $Q'$, $R'$, and $S'$ are similarly constructed.

After this construction, points $P,$ $Q,$ $R,$ and $S$ are erased. You only know the locations of points $P',$ $Q',$ $R'$ and $S',$ and want to reconstruct quadrilateral $PQRS.$

There exist real numbers $u,$ $v,$ $w,$ and $x$ such that
\[\overrightarrow{P} = u \overrightarrow{P'} + v \overrightarrow{Q'} + w \overrightarrow{R'} + x \overrightarrow{S'}.\]
Find the ordered quadruple $(u,v,w,x).$

Solution: \boxed{\left( \frac{1}{15}, \frac{2}{15}, \frac{4}{15}, \frac{8}{15} \right)}


Emily is thinking of a number. She gives the following 3 clues. "My number has 216 as a factor. My number is a multiple of 45. My number is between 1000 and 3000." What is Emily's number?


Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.


Line \( m \) has the equation \( y = 2x + 8 \). Line \( n \) has the equation \( y = kx - 9 \). Lines \( m \) and \( n \) intersect at the point \((-4, 0)\). What is the value of \( k \)?

Solution: \boxed{k = -\frac{9}{4}}


Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=2x-4$. Find $g^{-1}(f(-3))$.


Find the smallest constant $n$, so that for any positive real numbers $a, b, c, d, e,$ we have
\[
\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > n.
\]


Two cross sections of a right pentagonal pyramid are obtained by cutting the pyramid with planes parallel to the pentagonal base. The areas of the cross sections are \(125\sqrt{3}\) square feet and \(500\sqrt{3}\) square feet. The two planes are \(12\) feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.


Find the equation of the oblique asymptote for the graph of the function $\frac{3x^2 + 8x + 12}{3x + 4}$.

Solution: \boxed{y = x + \frac{4}{3}}


What is the sum of the digits of the base $8$ representation of $888_{10}$?


If $x$ is an even number, then find the largest integer that always divides the expression \[(15x+3)(15x+9)(5x+10).\]


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

Solution: \boxed{FQ = 6}


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


Each of the numbers $b_1, b_2, \dots, b_{97}$ is $\pm 1$. Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 97} b_i b_j.\]


What is the smallest positive integer that can be added to 725 to make it a multiple of 5?


Define \( g \) by \( g(x)=3x-2 \). If \( g(x)=f^{-1}(x)-2 \) and \( f^{-1}(x) \) is the inverse of the function \( f(x)=ax+b \), find \( 3a+4b \).


Find all values of \( x > 6 \) which satisfy
\[
\sqrt{x - 6 \sqrt{x - 9}} + 3 = \sqrt{x + 6 \sqrt{x - 9}} - 3.
\]

Solution: \boxed{[18, \infty)}


A square has a side length of 8 cm, and each vertex of the square is the center of a circle. Each circle has a radius of 3 cm. Calculate the area of the shaded region formed in the square, assuming that the circles intersect similar to the original problem.

Solution: \boxed{64 - 9\pi \text{ cm}^2}


How many positive perfect square integers are factors of the product $\left(2^{14}\right)\left(3^{9}\right)\left(5^{20}\right)$?


Each of the first ten prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number greater than 10?


A cylindrical container of honey, which is 4 inches in diameter and 5 inches high, sells for $\$$0.80. At the same rate, what would be the price for a container that is 8 inches in diameter and 10 inches high?


What is the remainder when $1520 \cdot 1521 \cdot 1522$ is divided by 17?


Point \(Q\) lies on the line \(x= 1\) and is 8 units from the point \((-4, -3)\). Find the product of all possible \(y\)-coordinates that satisfy the given conditions.


Simplify the product \[\frac{10}{5} \cdot \frac{15}{10} \cdot \frac{20}{15} \dotsm \frac{5n + 5}{5n} \dotsm \frac{2520}{2515}.\]


Yan is somewhere between his home and the stadium. He can either walk directly to the stadium or walk back home and then ride his bicycle to the stadium. He rides 9 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?


Let $T$ be the set of 12-tuples $(b_0, b_1, \dots, b_{11})$, where each entry is 0 or 1, so $T$ contains $2^{12}$ 12-tuples. For each 12-tuple $t = (b_0, b_1, \dots, b_{11})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 11 such that
\[q_t(n) = b_n\]for $0 \le n \le 11.$
Find
\[\sum_{t \in T} q_t(12).\]


Simplify
\[
\frac{\sin{25^\circ}+\sin{35^\circ}}{\cos{25^\circ}+\cos{35^\circ}}.
\]
Enter your answer as a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.)

Solution: \boxed{\tan 30^\circ}


Let $A$ be the greatest common factor and let $B$ be the least common multiple of 12, 18, and 30. Find twice the value of $A$ added to $B$. What is the value of $2A + B$?


Let \(a\), \(b\), and \(c\) be angles such that
\[
\cos a = \tan b, \quad \cos b = \tan c, \quad \cos c = \tan a.
\]
Find the largest possible value of \(\sin a\).

Solution: \boxed{\frac{\sqrt{5} - 1}{2}}


Consider the arithmetic sequence $2$, $5$, $8$, $11$, $\ldots$. Find the $20^\text{th}$ term and the sum of the first $20$ terms.


What is the smallest five-digit positive integer that is divisible by 53?


In Mr. Kennedy's class, 8 of the 20 students received an A on the latest exam. If the same ratio of students received an A in Mr. Holden's exam where he teaches 30 students, how many students in Mr. Holden's class received an A? If Mr. Holden had an additional quiz where the ratio of students who did not receive an A was half as much as those in the exam, how many students did not receive an A in the quiz?


How many ways are there to choose 4 cards from a standard deck of 52 cards, if all four cards must be of different suits?


What is the smallest positive perfect square that is divisible by both 5 and 7?


At the end of a game, each of the six members of a basketball team shakes hands with each of the six members of the other team, each player shakes hands with each of his own team members, and all of the players shake hands with three referees. How many handshakes occur?


James now has 10 apples, 6 of which are red and 4 of which are green. If he chooses 3 apples at random, what is the probability that at least one of the apples he chooses is green?


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{9}{14} & -\frac{3}{14} & \frac{6}{14} \\ -\frac{3}{14} & \frac{1}{14} & -\frac{2}{14} \\ \frac{6}{14} & -\frac{2}{14} & \frac{4}{14} \end{pmatrix}}


A line segment begins at $(2, 2)$ and ends at the point $(x, 5)$, with $x > 0$. If the length of the line segment is 6 units, what is the value of $x$?

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


Starting with the number 250, Shaffiq repeatedly divides his number by three and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1?


Given $ab + ac + bd + cd = 40$ and $a+d = 6$, find $b+c$.


Determine the smallest integer $y$ such that $\frac{y}{4} + \frac{3}{7} > \frac{4}{7}$.


Solution: \boxed{89 + \frac{36}{89}}


Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is the length of $QS$?


A point is selected at random from the portion of the number line from 0 to 8. What is the probability that the point is closer to 6 than to 0? Express your answer as a decimal to the nearest tenth.


Ryosuke drives his friend from a cafe to a library. The odometer reads 85,340 when he picks his friend up at the cafe, and it reads 85,368 when he drops his friend off at the library. Ryosuke's car gets 32 miles per gallon and the price of one gallon of gas is $\$3.95$. Calculate the cost of the gas used for this journey.


Real numbers $a$ and $b$ are chosen with $2<a<b$ such that no triangle with positive area has side lengths $2, a,$ and $b$. Additionally, it is given that $a + b = 7$. What is the smallest possible value of $b$?


Suppose that \(x^2\) varies inversely with \(y^4\). If \(x = 10\) when \(y = 2\), find the value of \(x\) when \(y = \sqrt{8}\).


The sequence $\{b_n\}$ is defined such that $b_1 = 2$ and $3^{b_{n + 1} - b_n} - 1 = \frac{1}{n + \frac{1}{4}}$ for $n \geq 1$. Find the least integer $k$ greater than $1$ for which $b_k$ is an integer.


The fourth, fifth, and sixth terms of an arithmetic sequence are 3, 7, and 11, respectively. What is the sum of the first five terms of the sequence?


Given that \(\frac{1}{n} - \frac{1}{n+2} < \frac{1}{15}\), what is the least possible positive integer value of \( n \)?


Find the point on the line
\[ y = \frac{x + 5}{2} \]
that is closest to the point $(7,2)$.

Solution: \boxed{\left( \frac{27}{5}, \frac{26}{5} \right)}


An equilateral triangle has an area of $144\sqrt{3}$ cm². If each side of the triangle is decreased by 5 cm, by how many square centimeters is the area decreased?

Solution: \boxed{53.75\sqrt{3} \text{ cm}^2}


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?


Compute
$$\sum_{k=1}^{500} k(\lceil \log_{\sqrt{3}}{k}\rceil- \lfloor\log_{\sqrt{3}}{k} \rfloor).$$


Calculate the product: $\frac{8}{4} \times \frac{10}{5} \times \frac{21}{14} \times \frac{16}{8} \times \frac{45}{15} \times \frac{30}{10} \times \frac{49}{35} \times \frac{32}{16}$.


Allison, Derek, and Sophie each have a 6-sided cube. All the faces on Allison's cube have a 4. Derek's cube is numbered from 1 to 6 uniformly. On Sophie's cube, four of the faces have a 3, and two of the faces have a 5. All three cubes are rolled simultaneously. What is the probability that Allison's roll is greater than each of Derek's and Sophie's? Express your answer as a common fraction.


How many distinct triangles can be constructed by connecting three different vertices of a regular octahedron? (Two triangles are distinct if they have different locations in space.)


Compute the smallest base-10 positive integer greater than 15 that is a palindrome when written in both base 2 and 4.


Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and purple. She randomly draws five socks this time. What is the probability that she has exactly one pair of socks with the same color and the rest are all different colors?


What is the sum of the digits of the base-2 expression for $315_{10}$?


Find the area of the triangle with vertices $(2, 3),$ $(0, 7),$ and $(5, 0).$


A frustum of a right circular cone is formed by slicing off a smaller cone from a larger cone. Given that the frustum has a height of $30$ centimeters, the area of its lower base is $400\pi$ sq cm, and the area of its upper base is $100\pi$ sq cm, determine the height of the small cone that was removed.

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


The graph of the line $x-y=7$ is a perpendicular bisector of the line segment from $(2,4)$ to $(10,-6)$. What is the value of $b$ in the equation $x-y=b$?


How many three-digit whole numbers have no digits that are 0, 3, 5, 7, or 9?


A circle with center $O$ has radius $10$ units and circle $P$ has radius $4$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively.  What is the length of segment $OS$? Express your answer in simplest radical form.


What is the result when we compute the sum $$1^3 + 2^3 + 3^3 + 4^3 + \dots + 49^3+50^3$$ and the sum $$(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + \dots + (-49)^3+(-50)^3,$$ then multiply the result of these two sums by the cube of 25?


Twelve people sit down at random seats around a circular table. Five of them are computer scientists, four others are chemistry majors, and the remaining three are history majors. What is the probability that all five computer scientists sit in consecutive seats?


Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to construct this dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.)


What is $\log_{8}{1600}$ rounded to the nearest integer?


The expression $x^2 + 18x + 77$ can be written as $(x + d)(x + e)$, and the expression $x^2 - 19x + 88$ can be written as $(x - e)(x - f)$, where $d$, $e$, and $f$ are integers. What is the value of $d + e + f$?


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


Simplify $\tan \frac{\pi}{8} + \tan \frac{3\pi}{8}$.


Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new trailer homes were added five years ago?


A convex polyhedron \( Q \) has \( 30 \) vertices, \( 72 \) edges, and \( 44 \) faces, \( 30 \) of which are triangular and \( 14 \) of which are quadrilaterals. Determine how many space diagonals \( Q \) has.


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.


A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?


How many even integers between 8000 and 9000 have four different digits, where the last digit is also a prime number?


Determine how many trailing zeroes are at the end of the number $500!$, and additionally, find out how many times the number $500!$ is divisible by $3$.


For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$.


What are the rightmost three digits of $7^{1983} + 123$?


Let $f(x) = x^2 - x + 2008$. Compute the greatest common divisor of $f(102)$ and $f(103)$.


The function $f(x)$ satisfies
\[ f(xy) = \frac{f(x)}{y^2} \]
for all positive real numbers $x$ and $y$. If $f(40) = 50$, find $f(80)$.

Solution: \boxed{f(80) = 12.5}


Find the mean of all solutions for the equation $x^3 + 5x^2 - 2x - 8 = 0$.


David works at a widget factory. On Monday, David produces $w$ widgets every hour for $t$ hours, knowing that $w = 2t$. Feeling more motivated on Tuesday, he works for one fewer hour and increases his production rate by $5$ widgets per hour. Determine how many more widgets David produces on Monday than he does on Tuesday.

Solution: \boxed{-3t + 5}


Find the vector $\mathbf{u}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 2 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\]and
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 1 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.\]

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


Find the distance between the planes \(x + 2y - z = 3\) and \(2x + 4y - 2z = 6\).


Three dice, each with eight faces numbered 1 through 8, are tossed. What is the probability that the sum of the numbers shown on the top faces is even? Express your answer as a common fraction.


Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $10$, where $a, b, c, d$ are distinct positive integers less than $10$ and are invertible modulo $10$.


What is the median number of moons per celestial body? (Include Ceres, a dwarf planet, along with the traditional planets and Pluto.) Use the following data:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 20 \\
Saturn & 25 \\
Uranus & 17 \\
Neptune & 3 \\
Pluto & 5 \\
Ceres & 0 \\
\end{tabular}


Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$.

Solution: \boxed{\frac{9\pi}{2} - 9}


When two fair 9-sided dice are tossed, each with faces showing numbers from 1 to 9, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits) and both $a$ and $b$ are divisible by 4?


Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}^4 = \mathbf{0}$. Determine the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite".


What is the total number of digits used when the first 2500 positive even integers are written?


The value of $z$ varies inversely as $x^2$ and when $x=3$, $z=2$. What is $x$ when $z=8$?


Berengere and her American friend Emily are at a café in Paris trying to buy a pastry. The pastry costs 8 euros. Emily has seven American dollars with her. How many euros must Berengere contribute if the exchange rate is now 1 euro = 1.1 USD?

Solution: \boxed{1.64\text{ euros}}


Let
\[g(x) =
\begin{cases}
\frac{x}{3} &\quad \text{if } x \text{ is divisible by } 3, \\
x^2 + 2 &\quad \text{if } x \text{ is not divisible by } 3.
\end{cases}
\]
What is $g(g(g(g(3))))$?


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $25$. After increasing each vertex's $x$-coordinate by $10\%$, find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


Evaluate the polynomial
\[ x^3 - 3x^2 - 9x + 7, \]
where $x$ is the positive number such that $x^2 - 3x - 9 = 0$.


Jane has received the following scores on her quizzes: 98, 97, 92, 85, 93, 88, and 82. What is her mean score now?


A pyramid with a volume of 60 cubic inches has a square base. If the side length of the base is tripled and the height is decreased by $25\%$, what is the volume of the modified pyramid, in cubic inches?


Lark has forgotten her bike lock combination. It’s a sequence of four numbers, each in the range from 1 to 40, inclusive. She remembers the following: the first number is odd, the second number is a prime number less than 30, the third number is a multiple of 4, and the fourth number is a perfect square. How many combinations could possibly be Lark's?


Calculate the value of the expression $(45 + 18)^2 - (45^2 + 18^2 + 10)$.


What is the sum of all two-digit primes that are greater than 20 but less than 99, and are still prime when their two digits are interchanged?


Graphs of $$y \equiv 8x + 3 \pmod{20}$$ and $$y \equiv 14x + 15 \pmod{20}$$ on modulo $20$ graph paper intersect at some points. Determine the sum of all distinct $x$-coordinates of these points.


In Pascal's Triangle, if the sum of the interior numbers in the sixth row is 30, what is the sum of the interior numbers in the eighth row?


Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 15x^2 + 26x - 8 = 0.$ Find the value of $(1+r)(1+s)(1+t).$


If $m$ and $n$ are positive 3-digit integers such that $\gcd(m,n)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[m,n]$?


What number must we add to $5 - 3i$ after multiplying it by 2, to get $4 + 11i$?

Solution: \boxed{-6 + 17i}


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


What is the smallest number of whole 3-by-4 rectangles needed to completely cover a square region exactly, with no overlaps and without any portion of the square uncovered?


A bubble assumes a semi-spherical shape when it rests on a wet, horizontal table. The volume of the hemispherical bubble is the same as the original spherical bubble. If the hemisphere's radius is \( 5\sqrt[3]{2} \) cm after it settles, calculate the radius of the original spherical bubble.


What is the greatest common divisor (gcd) of $9125$, $4257$, and $2349$?


A curious historian is hosting a puzzle where participants must guess the year a certain artifact was discovered. The year is made up of the six digits 1, 1, 1, 5, 8, and 9, and the year must start with an even digit. How many different possibilities are there for the artifact's discovery year?


Let there exist vectors $\mathbf{u}$ and $\mathbf{v}$, and a scalar $d$ such that
\[\mathbf{i} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{i}) + \mathbf{j} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{j}) + \mathbf{k} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{k}) = d (\mathbf{u} + \mathbf{v}).\]
Determine the value of $d.$ Assume $\mathbf{u}$ is a constant vector and $\mathbf{v}$ can be any vector.


Find the matrix $\mathbf{Q}$ such that:
\[\mathbf{Q} \mathbf{v} = -3 \mathbf{v}\]
for all vectors $\mathbf{v}.$

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


Po is challenged with a new equation: $$64x^2+80x-72 = 0.$$ He aims to rewrite the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. Determine the value of $a + b + c$.


Jenny places a total of 20 red Easter eggs in several green baskets and a total of 30 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket?


How many positive factors of 120 that are perfect cubes?


Suppose I have 8 shirts, 5 pairs of pants, 4 ties, and 2 hats. An outfit requires one shirt and one pair of pants. Additionally, the outfitter can choose to wear one of the ties or no tie at all, and can also select whether to wear one of the hats or not. How many different outfits can I create?


There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?


Maria borrowed money from her friend to buy a bicycle. Her friend has agreed to let her work off her debt by washing cars under the following conditions: her first hour of work is worth $\$2$, the second hour worth $\$4$, the third hour $\$6$, the fourth hour $\$8$, the fifth hour $\$10$, the sixth hour $\$12$, the seventh hour $\$2$, the eighth hour $\$4$, etc. If she repays her debt by working for 48 hours, how many dollars did she borrow?


Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 5)^{x^2 - 8x + 12} = 1.\]


Solution: \boxed{27x^{12} - 64z^9}


A $180\times 360\times 450$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?


Let $x$ and $y$ be two positive real numbers such that $x + y = 50.$ Find the ordered pair $(x,y)$ for which $x^6 y^3$ is maximized.

Solution: \boxed{\left(\frac{100}{3}, \frac{50}{3}\right)}


If $\sqrt{8 + x} + \sqrt{25 - x} = 8$, what is the value of $(8 + x)(25 - x)$?


Let $p$ be the smallest positive, four-digit integer congruent to 3 (mod 13). Let $q$ be the smallest positive, four-digit integer congruent to 3 (mod 7). Compute the absolute difference $|p - q|$.


How many positive four-digit integers of the form $\_\_25$ are divisible by 25?


In a tennis best-of-five series, a player needs to win 3 matches to claim the series. Assume Player A wins each match with a probability of $\frac{3}{4}$, and there are no draws. What is the probability that Player B will win the series, but the series will go to the full five matches?


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


Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$.

Solution: \boxed{\frac{9\pi}{2} - 9}


Roberto has four pairs of trousers, eight shirts, three jackets, and five pairs of shoes. He can wear any jacket with any shirt but has specific preferences for jackets with trousers: each jacket can only be worn with two specific pairs of trousers. How many different outfits can Roberto put together if an outfit consists of a pair of trousers, a shirt, a jacket, and a pair of shoes?


The polynomial $h(x) = x^3 - 2x^2 + 4x - 1$ has three roots. Let $j(x) = x^3 + px^2 + qx + r$ be a cubic polynomial with a leading coefficient of $1$ such that the roots of $j(x)$ are two less than the roots of $h(x)$. Find the ordered triple $(p, q, r)$.

Solution: \boxed{(4, 8, 7)}


The Maplewood Youth Soccer Team consists of 25 players, including four who are goalies. During a special drill, each goalie takes a turn in the net while the other 24 players (including the remaining goalies) attempt penalty kicks.
Calculate the total number of penalty kicks that must be taken so each player has the opportunity to take a penalty kick against each goalie.


Define a $\emph{brilliant integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 10. What fraction of all brilliant integers is divisible by 15? Express your answer as a common fraction.


A whole number is called ''12-heavy'' if the remainder when the number is divided by 12 is greater than 8. What is the least three-digit 12-heavy whole number?


Sheila has been invited to a picnic tomorrow. The picnic will occur, rain or shine. If it rains, there is a 30% probability that Sheila will decide to go, but if it is sunny, there is a 70% probability that she will decide to go. Additionally, Sheila can only go if her friend agrees to drive. There is a 50% chance her friend will agree to drive. The forecast for tomorrow states that there is a 50% chance of rain. What is the probability that Sheila will attend the picnic? Express your answer as a percent.


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


I have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?


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

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


The ''roundness'' of an integer greater than 1 is defined as the sum of the exponents in the prime factorization of the number. Determine the roundness of 1728.


Genevieve decides to create a new design for her kite display. In her revised design, the small kite has vertices at points on a grid where the distance between each point is one inch. The vertices are at the following coordinates on this new grid:
- (0,6), (4,8), (8,6), (4,1).

For her larger kite, Genevieve doubles the height and quadruples the width of the grid for the entire kite compared to the small kite.

Calculate the number of square inches in the area of the newly designed small kite.

Solution: \boxed{28 \text{ square inches}}


Find the sum of $753_8$ and $326_8$ in base $8$.


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


Amanda took a loan of $\$x$ from her friend to purchase a laptop. Her friend agreed to let Amanda pay off the debt by pet sitting. The payment structure for pet sitting is cyclic every 4 hours: the first hour she earns $\$2$, the second hour $\$3$, the third hour $\$2$, the fourth hour $\$3$, and then the cycle repeats. If Amanda works for 50 hours to clear her debt, how much did she borrow?


Jackie and Phil each has one fair coin and one biased coin. The biased coin shows heads with a probability of $\frac{2}{5}$. They each flip both coins once. Find the probability that both Jackie and Phil get the same number of heads, and if $\frac{p}{q}$ is this simplified probability (where p and q are coprime integers), compute $p + q$.


Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$


In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 12, sides $\overline{CD}$ and $\overline{DA}$ both have length 20, and the measure of angle $ADC$ is $45^\circ$. What is the length of diagonal $\overline{AC}$?

Solution: \boxed{20\sqrt{2 - \sqrt{2}}}


The expression \(x^2 + 15x + 36\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 7x - 60\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?


A circular pie has a diameter of 18 cm and is cut into four equal-sized sector-shaped pieces. Let \( l \) be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is \( l^2 \)?


Three faces of a right rectangular prism have areas of 56, 63, and 72 square units. The area of the face with an area of 72 square units is represented as twice a product of its dimensions. What is the volume of the prism, in cubic units?


A square with sides 8 inches is shown. If $P$ is a point such that the segment $\overline{PA}$, $\overline{PB}$, $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{FD}$, what is the area, in square inches, of triangle $APB$? [asy]
pair A, B, C, D, F, P;
A = (0,0); B= (4,0); C = (2,4); D = (4,4); F = (0,4); P = (2,2);
draw(A--B--D--F--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$P$",P,NW);label("$F$",F,NW);
label("$8''$",(2,0),S);

[/asy]


Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number?


A bridge is constructed by linking a wooden plank between two triangular supports, as shown in the diagram below:

```
[Drawing not to scale]
       A         D
       /\        /\
      /  \      /  \
     /    \    /    \
    B------C  E------F
```
Both triangles are isosceles with equal heights, such that AB = AC and DE = DF. The apex angles of the triangles are $\angle BAC = 25^\circ$ and $\angle EDF = 35^\circ$. Determine the sum $\angle DAC + \angle ADE$.


A map of a region is depicted by the Cartesian plane. John is located at $(3, -15)$ and Linda is located at $(-2, 20)$. They decide to meet at the closest point equidistant from both their starting points, and then walk upwards together to the location of their friend Maria at $\left(0.5, 5\right)$. How many units do John and Linda walk upwards together to get to Maria's location?


Rationalize the denominator of \(\frac{1}{\sqrt[3]{5} - \sqrt[3]{3}}\) and express your answer in the form \(\frac{\sqrt[3]{X} + \sqrt[3]{Y} + \sqrt[3]{Z}}{W}\), where the fraction is in lowest terms. What is \(X + Y + Z + W\)?


How many integers $-10 \leq n \leq 12$ satisfy $(n-1)(n+3)(n+7) < 0$?


Let $\mathbf{u},$ $\mathbf{v},$ $\mathbf{w}$ be vectors in $\mathbb{R}^3$, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}.$ Find the determinant of the matrix whose column vectors are $\mathbf{u} \times \mathbf{v},$ $\mathbf{v} \times \mathbf{w},$ and $\mathbf{w} \times \mathbf{u}$ and express it in terms of $E$.


Two diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that they intersect inside the nonagon?


The angles of quadrilateral $EFGH$ satisfy $\angle E = 3\angle F = 2\angle G = 6\angle D$. What is the degree measure of $\angle E$, rounded to the nearest whole number?


Define the operation $a \oslash b = (\sqrt{3a - b})^4$. If $10 \oslash y = 81$, find the value of $y$.


When \(x\) is divided by each of \(3\), \(7\), and \(8\), remainders of \(2\), \(6\), and \(7\) (respectively) are obtained. What is the smallest possible positive integer value of \(x\)?


Determine the product $ABC$ for the partial fraction decomposition of
\[\frac{x^2 - 19}{x^3 - 3x^2 - 4x + 12}\]
into the form
\[\frac{A}{x - 1} + \frac{B}{x + 3} + \frac{C}{x - 4}.\]


A prism has 18 edges. How many faces does the prism have?


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

Solution: \boxed{z = -\frac{3i}{8}}


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


Evaluate the product \[ (a-5) \cdot (a-4) \cdot \cdot \ldots \cdot (a-1) \cdot a, \] where \( a = 7 \).


Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. [asy]
import markers;

real t=32/(32+35);
pair A=(-16.57,0);
pair B=(9.43,0);
pair C=(0,30.65);
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,NE);

label("$28$",.5*(B+A),S);
label("$32$",.5*(B+C),NE);
label("$35$",.5*(A+C),NW);
[/asy]


What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $7(5x-3) \equiv 14 \pmod{10}$?


For how many integer values of $n$ between 1 and 299 inclusive does the decimal representation of $\frac{n}{450}$ terminate?


In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB = 45^\circ$, $\angle BEC = 60^\circ$, and $\angle CED = 60^\circ$, and $AE=30$.
Find the length of $CE.$


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


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

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


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


What is the greatest possible number of digits in the product of a 5-digit whole number and a 3-digit whole number?


Circle $D$ has a radius of 10 cm. How many square centimeters are in the area of the largest possible inscribed equilateral triangle in circle $D$?


What is the sum of all integer solutions to $4 < (x - 3)^2 < 49$?


Determine the value of \( k \) such that the set of vectors \( \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ k \\ 6 \end{pmatrix} \right\} \) is linearly dependent.


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


A cylinder has a volume of $72\pi$ cm³ and its height is twice its radius. Calculate the volume of a cone that has the same height and radius as the cylinder.


If $x^5 - x^4 + x^3 - px^2 + qx + 9$ is divisible by $(x + 3)(x - 2),$ find the ordered pair $(p,q).$

Solution: \boxed{(-19.5, -55.5)}


Let
\[
\mathbf{B} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\]
Compute $\mathbf{B}^{98}$.

Solution: \boxed{\mathbf{B}^{98} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}}


Four vertices of a rectangle are given as $(2, 10)$, $(12, 10)$, $(12, -5)$, and $(2, -5)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 2)^2 + (y + 5)^2 = 16$?


Define a function \( f(x) \) for \( 0 \leq x \leq 1 \) with the following properties:

(i) \( f(0) = 0 \).
(ii) If \( 0 \leq x < y \leq 1 \), then \( f(x) \leq f(y) \).
(iii) \( f(1 - x) = \frac{3}{4} - \frac{f(x)}{2} \).
(iv) \( f \left( \frac{x}{3} \right) = \frac{f(x)}{3} \) for \( 0 \leq x \leq 1 \).

Determine the value of \( f \left( \frac{2}{9} \right) \).


Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them.


Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} - \mathbf{b}\| = 1,$ and
\[
\mathbf{c} - 2\mathbf{a} - 3\mathbf{b} = 4\mathbf{a} + 6\mathbf{b}.
\]
Find the value of $\mathbf{b} \cdot \mathbf{c}$.


A frustum of a right circular cone is created by cutting off the top of a larger cone. The frustum has a height of $18$ cm, the area of its larger base is $324\pi$ sq cm, and the area of its smaller base is $36\pi$ sq cm. Find the height of the smaller cone that was removed.

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


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 16 and an altitude of 24. Both cylinder and cone share the same central axis. Determine the radius of the inscribed cylinder. Express your answer as a common fraction.


Let \( z = e^{i\theta} \), where \( \theta \) is a real number, and \( |z| = 1 \). Find the real part of \( \frac{1}{2 - z} \).

Solution: \boxed{\frac{2 - \cos\theta}{5 - 4\cos\theta}}


Calculate the remainder of $11^{2023}$ when divided by 33.


How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


Let \(\mathbf{v}\) be a vector \((a, b)\) in \(\mathbb{R}^2\), and let \(\mathbf{u}(\theta)\) be a unit vector \((\cos\theta, \sin\theta)\). Find the minimum and maximum values of the dot product \(\mathbf{v} \cdot \mathbf{u}(\theta)\) as \(\theta\) varies from 0 to \(2\pi\).


Let \(c\) be a real number randomly selected from the interval \([-10,10]\). Then, \(p\) and \(q\) are two relatively prime positive integers such that \(p/q\) is the probability that the equation \(x^4 + 16c^2 = (3c^2 - 8c)x^2\) has \(\textit{at least}\) two distinct real solutions. Find the value of \(p+q\).


What is the area, in square units, of a trapezoid bounded by the lines $y = x + 2$, $y = 12$, $y = 3$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.


The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers.


In Townville, vehicle license plates each contain three letters. The first letter is chosen from the set $\{B, F, J, N, Z\}$, the second letter from $\{E, U, Y\}$, and the third letter from $\{K, Q, X, S\}$. Townville decided to expand their license plate capacity by adding four new letters to these sets. Each new letter could be added to any set, but at least one letter must be added to each of the second and third sets. What is the largest possible number of ADDITIONAL vehicle license plates that can be made by optimally placing these four new letters?


What is the least 3-digit base 8 positive integer that is divisible by 7?  (Express your answer in base 8.)


Evaluate the expression $\log_{y^3}{x^2}\cdot\log_{x^4}{y^3}\cdot\log_{y^5}{x^4}\cdot\log_{x^2}{y^5}$ and express it as $b\log_y{x}$ for some constant $b$.


The cards of a standard 40-card deck (20 black and 20 red) are dealt out in a circle. What is the expected number of pairs of adjacent cards that are both red? Express your answer as a common fraction.


Compute 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 \( a+b \).


In triangle $PQR$, the measure of $\angle P$ is $88^\circ$. The measure of $\angle Q$ is $18^\circ$ more than twice the measure of $\angle R$. Determine the measure, in degrees, of $\angle R$.

Solution: \boxed{\frac{74}{3} \text{ degrees}}


A point $(x, y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 7$. What is the probability that $x+y \leq 5$? Express your answer as a common fraction.


How many positive three-digit integers, where each digit is odd and greater than 4, are divisible by 6?


A line segment has endpoints at $A(3, 1)$ and $B(9, 7)$. The segment is extended through $B$ to a point $C$ such that $BC$ is half the length of $AB$. Determine the coordinates of point $C$.

Solution: \boxed{(12, 10)}


Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\sqrt{117}$ units and diagonals that differ by 8 units?


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}$.

Solution: \boxed{\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}}


For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 8\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 50x + c.\] What is \(f(1)\)?


Suppose functions $g$ and $f$ are such that $g(x) = 4f^{-1}(x)$ and $f(x) = \frac{30}{x+2}$. Determine the value of $x$ for which $g(x) = 20$.


Find the $2 \times 2$ matrix $\mathbf{M}$ such that for a $2 \times 2$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ results in swapping the first row and second row of $\mathbf{N},$ and tripling the second row of $\mathbf{N}.$ In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & d \\ 3a & 3b \end{pmatrix}.
\]

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


How many real numbers \( y \) are solutions to the equation \[ |y-2| = |y-1| + |y-4| \]?


The third and sixth terms of a geometric sequence of real numbers are $6!$ and $7!$ respectively. What is the first term?


In triangle $ABC$, the lengths of the sides are $AB = 4$, $AC = 7$, and $BC = 5$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.


Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{10} + \omega^{12} + \omega^{14} + \dots + \omega^{50}.\]


Find the matrix \(\mathbf{P}\) such that for any vector \(\mathbf{v},\) \(\mathbf{P} \mathbf{v}\) is the projection of \(\mathbf{v}\) onto the vector \(\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).

Solution: \boxed{\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \end{pmatrix}}


After finishing a meeting, Jay, Paul, and Sam start walking from the same point. Jay walks north at a rate of 1 mile every 20 minutes, Paul walks south at a rate of 3 miles every 40 minutes, and Sam walks north at a rate of 1.5 miles every 30 minutes. Calculate the total distance between Paul and the farthest among Jay and Sam after 2 hours.

Solution: \boxed{15 \text{ miles}}


Find the remainder when $3x^7 - 2x^5 + 5x^3 - 8$ is divided by $x^2 + 3x + 2$.

Solution: \boxed{354x + 340}


Find $n$ such that $2^6 \cdot 3^3 \cdot 5^1 \cdot n = 10!$.


What is the sum of all three-digit and four-digit positive integers up to 2000?


The area of the floor in a rectangular room is 320 square feet. The room is 16 feet long. The homeowners plan to cover the floor with tiles that are each 1 foot by 1 foot. How many tiles will be in each row?


Let $g(x) = \left\lceil\dfrac{2}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{2}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci.


Determine in how many different ways five students can stand in a line if three of those students insist on standing together.


Suppose that $a$, $b$, and $c$ are digits, and the repeating decimal $0.\overline{abc}$ is expressed as a fraction in lowest terms. Assume $a$, $b$, and $c$ are not all nine and not all zero. How many different denominators are possible?


How many different 8-digit positive integers are there?


Let $a,$ $b,$ and $c$ be the roots of
\[ x^3 - 5x + 7 = 0. \]
Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 9x^2 + 22x - 5}


Let $g(x)$ be a polynomial of degree 2011 with real coefficients, and let its roots be $s_1, s_2, \dots, s_{2011}$. There are exactly 1010 distinct values among
\[ |s_1|, |s_2|, \dots, |s_{2011}|. \]
What is the minimum number of real roots that $g(x)$ can have?


Let $\mathcal{T}$ be the set $\{1, 2, 3, \ldots, 12\}$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Find the remainder obtained when $m$ is divided by $1000$.


Let $a$, $b$, $c$ be the sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the respective angles opposite these sides. If $a^2 + b^2 = 2001c^2$, find the value of
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\]


What is the sum of all integer values $n$ such that $\binom{30}{15} + \binom{30}{n} = \binom{31}{16}$?


Calculate how many five-character license plates can be made if they consist of two consonants followed by two vowels and end with a digit? (Consider Y as a vowel for this problem.)


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


Find the quotient when $x^3 + 5x^2 + 3x + 9$ is divided by $x - 2$.

Solution: \boxed{x^2 + 7x + 17}


Find the units digit of the product, \( 1! \times 2! \times 3! \times \cdots \times 15! \).


Calculate the distance between the points $(2,6)$ and $(5,2)$.


What is the area enclosed by the graph of $|x| + |3y| = 9$?


Medians $\overline{AP}$ and $\overline{BQ}$ of $\triangle ABC$ are perpendicular. If $AP = 15$ and $BQ = 20$, what is ${AC}$?


I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not necessarily different)?


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


What is the sum of the squares of the coefficients of $3(x^4 + 2x^3 + 5x^2 + 2)$?


A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside this cylinder?

Solution: \boxed{10\sqrt{2} \text{ cm}}


Determine the last three digits of $7^{1987}$.


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


A right pyramid with a square base has a lateral face area of 120 square meters. If the slant height of this pyramid is 24 meters, find the length of a side of its base.


Determine the 50th number in the row of Pascal's triangle that contains 52 numbers.


Find the quotient when $x^6 - 4x^5 + 5x^4 - 27x^3 + 13x^2 - 16x + 12$ is divided by $x-3$.

Solution: \boxed{x^5 - x^4 + 2x^3 - 21x^2 - 50x - 166}


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


Simplify $\sqrt[3]{40a^5b^8c^{14}}$ and find the sum of the exponents of the variables that are outside the radical.


The number of liters of tea a writer drinks on any given day is inversely proportional to the number of hours she spends writing. On Sunday, she spent 8 hours writing and consumed 3 liters of tea. On Wednesday, she spent 4 hours writing. How many liters of tea did she drink on Wednesday? And if she drinks only 2 liters of tea on Thursday, how many hours does she spend writing?


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


How many ways are there to distribute 6 balls into 4 boxes if the balls are distinguishable but the boxes are not?


If $x$, $y$, and $z$ are positive integers satisfying $xy+z = yz+x = zx+y = 47$, what is the value of $x+y+z$?


Calculate the angle $\theta$ in degrees for the expression in the form $r \text{cis} \theta$ where
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 160^\circ.\]


Let $x, y, z$ be real numbers such that $x + y + z = 2$, and $x \ge -\frac{1}{2}$, $y \ge -2$, and $z \ge -3$. Find the maximum value of
\[
\sqrt{6x + 3} + \sqrt{6y + 12} + \sqrt{6z + 18}.
\]


What is the 42nd number in the row of Pascal's triangle that has 46 numbers?


In triangle $XYZ$, $YZ = 10$. The length of median $XM$ is 7. Let $N$ be the largest possible value of $XZ^2 + XY^2$, and let $n$ be the smallest possible value. Find $N - n$.


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


In a certain hyperbola, the center is at $(1, 0),$ one focus is at $(1 + \sqrt{41}, 0),$ and one vertex is at $(4, 0).$ The equation of this hyperbola can be written as
\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.\]
Find $h + k + a + b.$

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


Determine the next year after 2021 where the sum of its digits equals '5'.


Determine the number of terms in the simplified expression of \[(x+y+z)^{2010} + (x-y-z)^{2010}.\]


How many values of \(x\), \(0 < x < 3\pi\), satisfy \(3\cos^2 x + 2\sin^2 x = 2?\) (Note: \(x\) is measured in radians.)


Find the smallest value of $x$ such that \[\frac{x^2 - x - 72}{x-9} = \frac{3}{x+6}.\]


What is the smallest prime divisor of $3^{19} + 6^{21}$?


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 10?


Given that \( x \) is a multiple of \( 7263 \), determine the greatest common divisor (GCD) of \( g(x) = (3x+4)(9x+5)(17x+11)(x+17) \) and \( x \).


In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\sin \angle GAC$. (Consider this by extending the calculations needed for finding $\sin \angle HAC$)


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$

Solution: \boxed{\begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix}}


Find the quadratic polynomial $q(x)$ such that $q(-6) = 0,$ $q(3) = 0,$ and $q(4) = -45.$

Solution: \boxed{-4.5x^2 - 13.5x + 81}


In a right-angled triangle $PQR$ with angle $PQR = 90^{\circ}$, suppose $\cos Q = \frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.


A convex pentagon has interior angles with measures \(x+2\), \(2x+3\), \(3x-4\), \(4x+5\), and \(5x-6\) degrees. What is the measure of the largest angle?


John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions).


A company incurs a daily maintenance cost of \$600 and pays each worker \$20 per hour. Each worker manufactures 6 widgets per hour, which are then sold at \$3.50 each. Determine the minimum number of workers the company must employ to turn a profit during an 8-hour workday.


Find the remainder when the sum \(8930 + 8931 + 8932 + 8933 + 8934\) is divided by 13.


Consider $\mathcal{T}$ as the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are digits but can be the same (i.e., not necessarily distinct). Calculate the sum of the elements of $\mathcal{T}$.


What is the remainder when $3211$ is divided by $103$?


Find the greatest common divisor of 100 and 250.


The Chess Club at a high school has 25 members and needs to select 3 officers: president, treasurer, and secretary. Each member may hold at most one position. Two members, Alice and Bob, will only accept the positions if both are chosen as officers. Another member, Charlie, will only agree to be an officer if Alice is not chosen. In how many ways can the club select its officers?


The equation $y = -15t^2 + 75t$ represents the height (in feet) of a projectile launched from the ground with an initial velocity of 75 feet per second. Determine at what time $t$ will the projectile reach a height of 30 feet for the first time. Express your answer as a decimal rounded to the nearest tenth.


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 287 hours, 18 minutes and 53 seconds the time will be $X:Y:Z$. What is the value of $X + Y + Z$?


The function $g(x)$ satisfies $g(1) = 2$ and
\[g(x + y) = 4^y g(x) + 3^x g(y)\] for all real numbers $x$ and $y.$ Find the function $g(x).$

Solution: \boxed{2(4^x - 3^x)}


A box contains seven cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and one card is red on both sides. You pick a card uniformly at random from the box and see one side of the card. Given that the side you see is red, what is the probability that the other side is also red? Express your answer as a common fraction.

Solution: \boxed{\frac{2}{4} = \frac{1}{2}}


Determine the point on the plane $2x + 3y - z = 15$ that is closest to the point $(4,1,-2).$

Solution: \boxed{\left(\frac{30}{7}, \frac{10}{7}, -\frac{15}{7}\right)}


How many non-empty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ consist entirely of numbers that are odd or divisible by 3?


Triangle $ABC$ has side-lengths $AB = 12, BC = 26,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.

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


In the diagram, each circle is divided into unequal areas by a horizontal line, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi.$ The line divides the larger circle such that the shaded lower semi-circle is $\frac{2}{3}$ of its total area. The smaller circle’s center is located 2 units above $O$ and just touches the boundary of the larger circle. What is the total area of the shaded regions? [asy]
size(120);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,2),1,0,180)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((0,2),1));

dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,2)--(1,2));
[/asy]


Chandra now has six bowls of different colors (red, blue, yellow, green, orange, purple) and the same six colored glasses. She decides to choose a bowl and a glass wherein a pairing is valid if they are the same color or different colors. How many valid pairings are possible?


In the circle with center $O$, the measure of $\angle QIS$ is $45^\circ$ and $OQ=15$ cm. Find the number of centimeters in the length of arc $QS$. Express your answer in terms of $\pi$.

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


Given that $b$ is an odd multiple of $997$, find the greatest common divisor of $3b^2 + 17b + 31$ and $b + 7$.


What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 3?