Find the vector \(\mathbf{v}\) such that
\[
\operatorname{proj}_{\begin{pmatrix} 3 \\ 2 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 6 \\ 4 \end{pmatrix}
\]
and
\[
\operatorname{proj}_{\begin{pmatrix} 1 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 2 \\ 8 \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 3.6 \\ 7.6 \end{pmatrix}}


By partial fractions,
\[
\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 5)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 5}
\]
for some constants \( A, B, C, D, \) and \( E \). Find \( A + B + C + D + E \).


A regular dodecahedron is constructed using twelve congruent regular pentagonal faces, each of a different color. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.)

Solution: \boxed{\frac{39916800}{5} = 7983360}


Let $P$ equal the number of four-digit prime numbers. Let $M$ equal the number of four-digit multiples of 3. Find $P + M$.


In the coordinate plane, the curve \( xy = 2 \) intersects a circle at four points, three of which are \( (3, \frac{2}{3}), (-4, -\frac{1}{2}), \) and \( (\frac{1}{2}, 4) \). Find the fourth point of intersection.

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


What is the total number of digits used when the first 1500 positive even integers are written?


If $a$ and $b$ are positive integers such that $\gcd(a,b) = 10$, what is the smallest possible value of $\gcd(12a, 20b)$?


Let $x$ be a multiple of $7200$. Determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+5)(4x+7)$ and $x$.


Calculate how many nonnegative integers can be represented in the form
\[
a_8 \cdot 3^8 + a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0,
\]
where each coefficient $a_i \in \{-1, 0, 1\}$ for $0 \leq i \leq 8$?


A square has sides of length 8 units, and a circle centered at one of its vertices has a radius of 8 units. Find the area of the union of the regions enclosed by the square and the circle. Express your answer in terms of $\pi$.

Solution: \boxed{64 + 48\pi}


How many three-digit whole numbers have no 5's, no 7's, and no 9's as digits?


Let \(a,\) \(b,\) and \(c\) be real numbers such that \(a + 3b + c = 6.\) Find the maximum value of
\[ab + ac + bc.\]


Solution: \boxed{x = \frac{89}{23}}


Jessica completed six assignments with scores of 85, 90, 93, 87, 92, and 95. To calculate her final score, her instructor decides to drop the lowest score and then calculate the arithmetic mean of the remaining scores. What is Jessica's adjusted mean score?


Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a rectangular field. The rectangle has three times as many posts on its longer side as on the shorter side. What is the perimeter, in feet, of the fence?

Solution: \boxed{236 \text{ feet}}


Billy and Bobbi each selected a positive integer less than 150. Billy's number is a multiple of 15, and Bobbi's number is a multiple of 20. What is the probability that they selected the same number, given that at least one of their numbers is a square? Express your answer as a common fraction.


The complete graph of \( y=g(x) \), which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is \( 1 \)).

What is the sum of the \( x \)-coordinates of all points where \( g(x) = x+2 \)?


Calculate the area of trapezoid EFGH with vertices E(0, 0), F(0, -3), G(5, 0), and H(5, 8).


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


Evaluate $\lfloor -6 - 0.5 \rfloor \cdot \lceil 6 + 0.5 \rceil \cdot \lfloor -5 - 0.5 \rfloor \cdot \lceil 5 + 0.5 \rceil \cdot \dots \cdot \lfloor -0.5 \rfloor \cdot \lceil 0.5 \rceil$.


For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 3?


Marius is entering a wildlife photo contest, and wishes to arrange his eight different height Jaguars in a line. He wants to ensure that the tallest and shortest Jaguars are placed at the ends of the line. How many ways can he arrange the Jaguars?

Solution: \boxed{1440 \text{ ways}}


Determine how many non-zero digits are to the right of the decimal point when $\frac{120}{2^4 \cdot 5^9}$ is expressed as a decimal.


Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[x^2 + y^2 + \frac{4}{(x + y)^2}.\]


Suppose $x-2$ and $y+2$ are multiples of $7$.

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

Solution: \boxed{n = 3}


A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Determine the radius of the sphere if it can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)?


Let $a$, $b$, $c$, $d$, $w$, $x$, $y$, and $z$ be real numbers satisfying:
\begin{align*}
  17w + bx + cy + dz &= 0 \\
  aw + 29x + cy + dz &= 0 \\
  aw + bx + 37y + dz &= 0 \\
  aw + bx + cy + 53z &= 0.
\end{align*}
Suppose that $a \neq 17$, $b \neq 29$, $c \neq 37$, and $w$, $x$, $y$ do not all equal zero. Determine the value of
\[
\frac{a}{a-17} + \frac{b}{b-29} + \frac{c}{c-37} + \frac{d}{d-53}.
\]


For some constants $a$ and $b,$ let \[f(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 3, \\
9 - 2x & \text{if } x \ge 3.
\end{array}
\right.\] The function $f$ has the property that $f(f(x)) = x$ for all $x.$ Additionally, $f(x)$ is continuous at $x = 3.$ What is $a + b?$


A book containing 63 pages, numbered 1 to 63, has each page renumbered in reverse, from 63 to 1. Determine how many pages have the same units digit for both the old page number and the new page number.


Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, ..., 10\}$. What is the probability that the positive difference between the two numbers is $3$ or greater? Express your answer as a common fraction.


There are integers $b,c$ for which both roots of the polynomial $x^2-2x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$.


What is the largest four-digit negative integer congruent to $2 \pmod{25}$?


In triangle $ABC$, $BC = 10$. The lengths of medians $AD$ and $BE$ are 6 and 7 respectively. Let $N$ be the largest possible value of $AB^2 + AC^2 + BC^2$, and let $n$ be the smallest possible value. Find $N - n$.


An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (4,0)$. The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection?

Solution: \boxed{\left(\frac{24}{5}, 0\right)}


Consider a polynomial of the form
\[ x^m + x^{m-1} + x^{m-2} + \ldots + x^2 + x + 1 = 0, \]
where $m = 2n+1$ and $n$ is a non-negative integer. Determine the maximum number of real roots of this polynomial.


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


Find the quadratic polynomial $q(x)$ such that $q(-1) = 4,$ $q(2) = 1,$ and $q(4) = 20.$

Solution: \boxed{2.1x^2 - 3.1x - 1.2}


What is the least positive integer $n$ such that $2700$ is a factor of $n!$?


The graph of the equation \[\sqrt{x^2+y^2} + |y-2| = 4\] consists of portions of two different parabolas. Compute the distance between the vertices of these parabolas.


A heptagon is formed by connecting sequentially the points \( (0,1), (1,3), (3,3), (4,2), (4,1), (3,0), (1,0), (0,1) \). Compute the perimeter of the heptagon, assuming it can be expressed in the form \( a + b\sqrt{2} + c\sqrt{3} + d\sqrt{5} + e\sqrt{10} \), where \( a, b, c, d, \) and \( e \) are integers. Find the sum \( a+b+c+d+e \).


What is the units digit of the product of all odd positive integers between 10 and 150?


Consider a polynomial
\[ b_6 x^6 + b_5 x^5 + b_4 x^4 + \dots + b_1 x + b_0 = 0 \]
with integer coefficients, where all roots \( s_1, s_2, \dots, s_6 \) are integers. Moreover, the roots of the polynomial
\[ b_0 x^6 + b_1 x^5 + b_2 x^4 + \dots + b_4 x^2 + b_5 x + b_6 = 0 \]
are also \( s_1, s_2, \dots, s_6 \). Determine the number of possible multisets \( T = \{s_1, s_2, \dots, s_6\} \).


Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?


A large number \( y \) is equal to \( 2^{10}3^{15}4^{20}5^{25}6^{30}7^{35}8^{40}9^{45} \). What is the smallest positive integer that, when multiplied with \( y \), results in a product that is a perfect square?


How many perfect squares less than 2000 have a ones digit of 4, 5, or 6?


What is the total distance on a Cartesian coordinate plane starting at $(2, -3)$, going to $(8, 9)$, and then moving to $(3, 2)?$

Solution: \boxed{6\sqrt{5} + \sqrt{74}}


In a small book club, 11 out of 15 members must agree on a book to be chosen for the next reading. What is the ratio of required agreements to the total members, rounded to the nearest tenth?


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


For how many integer values of $n$ between 1 and 569 inclusive does the decimal representation of $\frac{n}{570}$ terminate?


Find the number of odd digits in the base-4 representation of $157_{10}$.


How many three-digit whole numbers have no 5's and no 8's as digits?


Chandra has five bowls of different colors: red, blue, yellow, green, and purple. She also has matching glasses for each bowl except for the purple bowl, for which she has no glass. If she chooses a bowl and a glass from the cupboard, excluding the possibility of choosing the purple bowl unless she chooses no glass, how many valid pairings are possible?


Let $x$ and $y$ be positive real numbers such that $6x + 8y = 72$. Find the maximum value of $xy$ given that $x = 2y$.


In right triangle $GHI$, we have $\angle G = 30^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth.


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


Define the function \( g(n) \) by:
\[
g(n) =
\begin{cases}
n^2 + 1 & \text{if } n < 3, \\
2n + 3 & \text{if } 3 \leq n < 6, \\
4n - 2 & \text{if } n \geq 6.
\end{cases}
\]
Find \( g(g(g(1))) \).


Two reals \(a\) and \(b\) are such that \(a + b = 10\) and \(a^3 + b^3 = 172\). Compute \(ab\).


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$?


Pizzas are sized by diameter. Calculate the overall percent increase in area if Jenna's pizza goes from a 10-inch pizza to a 12-inch pizza, then from a 12-inch pizza to a 15-inch pizza.


For a nonnegative integer $n$, let $r_8(n)$ denote the remainder when $n$ is divided by $8$. What is the $15^\text{th}$ entry in an ordered list of all nonnegative integers $n$ that satisfy
$$ r_8(7n) \leq 3 $$


What is the smallest positive value of $x$ such that $x + 1234$ results in a palindrome?


Suppose that the number \( b \) satisfies the equation \( 3 = b + b^{-1} \). What is the value of \( b^6 + b^{-6} \)?


Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 72$. Express $x$ as a decimal.


In triangle $DOG$, $\angle DGO = \angle DOG$ and $\angle DOG = 50^\circ$. Segment $OP$ bisects $\angle DOG$. Determine the measure of $\angle DPO$.


A point is chosen at random on the number line between 0 and 2, and the point is colored blue. Then, another point is chosen at random on the number line between 0 and 2, and this point is colored red. What is the probability that the number of the red point is greater than the number of the blue point, but less than three times the number of the blue point?


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


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


Define a new operation $a \nabla b = 3 + b^a$. What is the value of $(2 \nabla 3) \nabla 2$?


Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 2 by 3 by 6 units. Given that the volume of this set is $\displaystyle \frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$.

Solution: \boxed{m+n+p = 324 + 37 + 3 = 364}


Charlie has 57 cherry lollipops, 128 wintergreen lollipops, 14 grape lollipops, and 246 shrimp cocktail lollipops, and decides to introduce 12 raspberry lollipops. He plans to distribute these equally among his 13 best friends. Every friend receives the same number of lollipops. How many lollipops does Charlie keep for himself?


Points $D(2, -3), E(2, 0)$, and $F(5, -3)$ are the vertices of $\triangle DEF$. What will be the coordinates of the image of point D if $\triangle DEF$ is rotated 180 degrees about the point $(3, -2)$?

Solution: \boxed{(4, -1)}


How many integers satisfy $-3 \leq 3x - 2 \leq 8$?


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


A spiral staircase turns \(315^\circ\) as it rises 12 feet. The radius of the staircase is 4 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.


By restricting the domain of the function $g(x) = 3x^2 + 6x - 2$, to an interval, can this function be made invertible? What is the largest such interval that includes the point $x=-2$?

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


Side $AB$ of regular hexagon $ABCDEF$ is extended past $B$ to point $X$ such that $AX = 4AB$. Given that each side of the hexagon is $2$ units long, what is the length of segment $FX$? Express your answer in simplest radical form.


Find the number of real solutions to the equation:
\[(x^{12} + 1)(x^{10} + x^8 + x^6 + x^4 + x^2 + 1) = 12x^{11}.\]


A corporation plans to expand its sustainability team to include specialists in three areas: energy efficiency, waste management, and water conservation. The company needs 95 employees to specialize in energy efficiency, 80 in waste management, and 110 in water conservation. It is known that 30 employees will specialize in both energy efficiency and waste management, 35 in both waste management and water conservation, and 25 in both energy efficiency and water conservation. Additionally, 15 employees will specialize in all three areas. How many specialists does the company need to hire at minimum?


Marty wants to paint a room and he has decided to either paint the ceiling or the walls, but not both. He can choose from five different colors: blue, green, yellow, black, or white. He also has options to paint using a brush, roller, or spray gun. How many different combinations of choices can Marty choose considering the part of the room and painting method?


What integer $n$ satisfies $0 \le n < 17$ and $$-250 \equiv n \pmod{17}~?$$


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


In a 6 by 6 grid, each of the 36 small squares measures 1.5 cm by 1.5 cm. The grid is fully shaded in grey. Six unshaded shapes are then added: one medium hexagon, four small circles placed symmetrically, and one larger circle in the center. The diameter of the small circles equals the side of a small square (1.5 cm), and the larger circle has a diameter of 3 cm; the side length of the hexagon equals 1.5 cm. Assume all circles and the hexagon are placed such that no parts of these shapes overlap. The area of the visible shaded region can be written in the form $A-B\pi - C\sqrt{3}$. What is the value $A+B+C$?


In triangle $QCB$, formed by three tangents to circle $D$, and $\angle CQB = 60^\circ$, determine $\angle COD$.


How many distinct sequences of four letters can be made from the letters in "EXAMPLE" if each letter can be used only once and each sequence must begin with E and not end with X?


Lucy enters a cafeteria where she has to select one type of fruit, two different salads, and one dessert. How many different meal combinations can Lucy choose?

$\bullet$ Fruit: apple, banana, orange

$\bullet$ Salads: lettuce, spinach, kale, cabbage

$\bullet$ Dessert: cookie, pie, cake, pudding, gelato


The Dunkers are coached by Coach Loudvoice. The Dunkers have 15 players, but three of them, Alex, Ben, and Cam, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loudvoice create, ensuring that no two among Alex, Ben, and Cam are in the same lineup?


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.


Let $r_1,$ $r_2,$ and $r_3$ be the roots of the polynomial
\[x^3 - 4x^2 + x + 6 = 0.\]
Find the monic polynomial, in $x,$ whose roots are $3r_1,$ $3r_2,$ and $3r_3.$

Solution: \boxed{x^3 - 12x^2 + 9x + 162}


An $\textit{annulus}$ is formed by two concentric circles with radii $r$ and $s$ ($r > s$). Let $\overline{OP}$ be a radius of the outer circle, and $\overline{PQ}$ be tangent to the inner circle at point $Q$. Let $\overline{OR}$ be the radius of the outer circle that intersects $Q$. Define $k = PQ$, $m = QR$, and $n = PR$. Determine the area of the annulus in terms of $\pi$ and at most one of the variables $k, r, s, m, n$.

Solution: \boxed{\pi k^2}


The sum of three different numbers is 100. The two larger numbers differ by 8 and the two smaller numbers differ by 5. What is the value of the largest number?


Determine the product of all positive integer values of $k$ such that the quadratic equation $5x^2 + 20x + k = 0$ has two distinct real roots.


If $x(x+y) = 49$ and $y(x+y) = 63$, what is the value of $(x+y)^3$?


How many natural numbers greater than 8 but less than 100 are relatively prime to 35?


The square root of $3x$ is greater than 4 and less than 5. How many integer values of $x$ satisfy this condition?


Find the value of $x$ such that $\sqrt{5x + 13} = 15$.

Solution: \boxed{x = 42.4}


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

Solution: \boxed{x^2 + 6x + 12}


Thirteen is the sixth prime number. What is the fifteenth prime number?


Find the remainder when $3^{2003}$ is divided by $13$.


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


In an isosceles triangle, two sides are each 12 cm, and the base is 18 cm. If a similar triangle has its longest side as 54 cm, find the perimeter of this larger triangle.

Solution: \boxed{126 \text{ cm}}


The average of the numbers $1, 2, 3, \dots, 149,$ and $x$ is $150x$. What is $x$?


Consider a $10\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 25 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $BC$ and $AD$, respectively. What is the area of quadrilateral $EFCD$ in square units?


Each day, Barry Sotter uses his magic to increase the length of an object as follows: On the first day, he increases the length by $\frac{1}{2}$; on the second day by $\frac{1}{3}$; on the third day by $\frac{1}{4}$; and so on. On the $n^{\text{th}}$ day, he increases the day's starting length by $\frac{1}{n+1}$. Barry wants the object’s length on the $n^{\text{th}}$ day to be exactly 50 times its original length. What is the value of $n$?


The graph of \(y = h(x)\) is a piecewise function given as follows:
- \(h(x) = -x - 2\) for \(-4 \leq x \leq 0\),
- \(h(x) = \sqrt{9 - (x - 3)^2} - 3\) for \(0 < x \leq 3\),
- \(h(x) = 3(x - 3)\) for \(3 < x \leq 4\).

For certain constants \(p\), \(q\), and \(r\), the function \(j(x)\) is defined by:
\[j(x) = ph(qx) + r.\]
The graph of \(y = j(x)\) is obtained by stretching the graph of \(y = h(x)\) horizontally by a factor of 3 and shifting it downward by 5 units.

Determine the ordered triple \((p, q, r)\).

Solution: \boxed{(1, \frac{1}{3}, -5)}


How many even perfect square factors does $2^6 \cdot 7^{10}$ have?


Rectangle $ABCD$ is inscribed in triangle $PQR$ such that side $AD$ of the rectangle is on side $PR$ of the triangle. The triangle’s altitude from vertex $Q$ to side $PR$ is 8 inches, and $PR = 12 \text{ inches}$. The length of segment $AB$ is equal to a third the length of segment $AD$. What is the area of rectangle $ABCD$?

Solution: \boxed{\frac{64}{3}\text{ square inches}}


Find the product of the greatest common divisor and the least common multiple of $90$ and $135.$


Two scientists were both born in the last 300 years. Each lives to be 80 years old. What is the probability that they lived at the same time for any length of time?


In a sequence of positive integers, each term after the first is $\frac{1}{3}$ of the sum of the term that precedes it and the term that follows it. What is the 6th term of this sequence if the 2nd term is 7 and the 3rd term is 20?


Let $\mathbf{p}$ and $\mathbf{q}$ be vectors, and let $\mathbf{n}$ be the midpoint of $\mathbf{p}$ and $\mathbf{q}$. Given $\mathbf{n} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ and $\mathbf{p} \cdot \mathbf{q} = 12$, find $\|\mathbf{p}\|^2 + \|\mathbf{q}\|^2$.


Abby, Bart, Cindy, and Damon weigh themselves in pairs. Together Abby and Bart weigh 280 pounds, Bart and Cindy weigh 230 pounds, and Cindy and Damon weigh 260 pounds. How many pounds do Abby and Damon weigh together?


A cylindrical jar of honey is 4 inches in diameter and 5 inches high and sells for $\$0.75$. At a different rate, what would be the price for a jar that is 8 inches in diameter and 10 inches high if the price per unit volume increases by 10% for the larger jar?


On a balance scale, $4$ green balls balance $8$ blue balls, $3$ yellow balls balance $7.5$ blue balls, and $5$ blue balls balance $3.5$ white balls. How many blue balls are needed to balance $5$ green, $4$ yellow and $3$ white balls?

Solution: \boxed{\frac{170}{7} \text{ blue balls}}


Define a set of integers as "spacy" if it includes no more than one out of any three consecutive integers. Determine the count of all "spacy" subsets of the set $\{1, 2, \dots, 15\}$, including the empty set.


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


Crestview’s school celebration is coming up, and new colors have been added while designing a flag using three solid-colored horizontal stripes. The school's primary colors have been updated to purple, gold, and silver. When designing the flag, the middle stripe must be a different color from the ones directly above and below it. How many different flag designs are possible under these conditions?


In a right-angled triangle, the sum of the squares of the three side lengths is 2500. What is the length of the hypotenuse of this triangle?


How many positive integers less than 1000 are both a square, a cube, and a fourth power?


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


If 5 knicks = 3 knacks and 4 knacks = 6 knocks, how many knicks are equal to 36 knocks?

Solution: \boxed{40\text{ knicks}}


Find the quadratic polynomial, with real coefficients, which has $-3 - 4i$ as a root, and where the coefficient of $x$ is $-6$.

Solution: \boxed{-x^2 - 6x - 25}


After traveling 80 miles by taxi, Ann is charged a fare of $\$200$. The taxi has a basic charge of $20, and beyond that, the fare is directly proportional to the distance traveled. Given this scenario, how much would Ann be charged (in dollars) if she traveled 100 miles?

Solution: \boxed{245 \text{ dollars}}


How many integer palindromes are between 100 and 1000 where the sum of the digits is divisible by 4?


Express the quotient $2431_5 \div 23_5$ in base $5$.


Two vectors, \( \begin{pmatrix} -7 \\ 2 \end{pmatrix} \) and \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \), are projected onto the same vector \( \mathbf{v} \), resulting in the same projection \( \mathbf{p} \) for both. Find \( \mathbf{p} \).

Solution: \boxed{\begin{pmatrix} -\frac{17}{26} \\ \frac{85}{26} \end{pmatrix}}


A lattice point is a point with integer coordinates. How many lattice points lie on the boundary or inside the region bounded by \( y = 2|x| \) and \( y = -x^2 + 8 \)?


Circle $C$ has a circumference of $18\pi$ cm, and segment $AB$ is a diameter. Another point $D$ lies on the circle such that the measure of angle $ADB$ is $45^{\circ}$. What is the length of segment $AD$?

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


The real number $x$ satisfies $x^2 - 5x + 4 < 0.$ Find all possible values of $x^2 + 4x + 5.$

Solution: \boxed{(10, 37)}


What is the smallest positive integer $n$ for which $n^2$ is divisible by 24 and $n^3$ is divisible by 450?


In the diagram below, lines $m$ and $n$ are parallel. A transversal cuts lines $m$ and $n$ at points A and B respectively, forming a $40^\circ$ angle with line $m$. A second transversal intersects $m$ at C and $n$ at D, not at right angles. Find the measure of angle $y$ which is the angle between the second transversal and line $n$ at D. Assume the angle formed between the first transversal and the second transversal on the same side of the parallel lines is $60^\circ$.


Two hundred people were surveyed. Of these, 150 indicated they liked Beethoven, and 120 indicated they liked Chopin. Additionally, it is known that of those who liked both Beethoven and Chopin, 80 people also indicated they liked Vivaldi. What is the minimum number of people surveyed who could have said they liked both Beethoven and Chopin?


Factor the expression $x^2(x+3) + 2x(x+3) + (x+3)$.


A standard deck of cards has 52 cards divided into four suits of 13 cards each. Two of the suits are red (hearts and diamonds), and two are black (spades and clubs). The cards are shuffled into a random order. What is the probability that the first two cards drawn are red and the third card is a spade?


You have $7$ red shirts, $7$ green shirts, $9$ pairs of pants (4 blue and 5 black), $10$ green hats, and $10$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, under the condition that the shirt and hat must not be the same color, and the pants must be blue if the shirt is red?


Solution: \boxed{x = 2^{\frac{54}{11}}}


The Cookery club has 24 members: 14 boys and 10 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 2 girls?


Let \(y\) be a real number such that \(y + \frac{1}{y} = 5\). Define \(T_m = y^m + \frac{1}{y^m}\). Determine the value of \(T_5\).


What is the greatest integer less than 150 for which the greatest common factor of that integer and 30 is 5?


A rectangular array of tables is an arrangement of the tables in rows and columns such that each row contains the same number of tables as every other row and each column contains the same number of tables as every other column. If there must be at least two tables in every row and column and all of the tables in the room must be included, how many arrays are possible in a room containing $49$ tables? Note that $7$ rows of $7$ tables is different from $49$ rows of $1$ table (which is not allowed).


Let \( O \) be the origin. Determine the scalar \( k \) such that for any points \( A, \) \( B, \) \( C, \) and \( D \) where
\[ 4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0}, \]
the points \( A, \) \( B, \) \( C, \) and \( D \) are coplanar.


A projection takes $\begin{pmatrix} 3 \\ 3 \end{pmatrix}$ to $\begin{pmatrix} \frac{45}{13} \\ \frac{9}{13} \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ to?

Solution: \boxed{\begin{pmatrix} \frac{10}{13} \\ \frac{2}{13} \end{pmatrix}}


How many degrees are in the measure of the smaller angle formed by the hour and minute hands of a clock when the time is 3:15 a.m.?


Determine the minimum value of $w$ if $w = 3x^2 + 5y^2 + 8x - 10y + 34$.


The eighth, ninth, and tenth terms of an arithmetic sequence are 21, 17, and 13, respectively. What is the sum of the first four terms of the sequence?


Compute $\cos 315^\circ$ and $\sin 315^\circ$.

Solution: \boxed{\cos 315^\circ = \frac{\sqrt{2}}{2}, \sin 315^\circ = -\frac{\sqrt{2}}{2}}


Let \(\mathbf{r}\) be the projection of \(\mathbf{u}\) onto \(\mathbf{s},\) and let \(\mathbf{t}\) be the projection of \(\mathbf{r}\) onto \(\mathbf{u}.\) If \(\frac{\|\mathbf{r}\|}{\|\mathbf{u}\|} = \frac{3}{4},\) find \(\frac{\|\mathbf{t}\|}{\|\mathbf{u}\|}.\)


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


The first three stages of a pattern are shown below, where each line segment represents a toothpick. If the pattern continues such that at each successive stage, four toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 100th stage?


How many integers $n \neq 0$ satisfy the inequality $\frac{1}{|n|}\geq \frac{1}{5}$?


Calculate the modulo $11$ residue of $312 - 3 \cdot 52 + 9 \cdot 165 + 6 \cdot 22$.


Let $A$ equal the number of four-digit odd numbers. Let $B$ equal the number of four-digit numbers that are multiples of 3. Find $A+B$.


Alice and Bob are playing the Smallest Positive Integer Game again. This time, Alice says, "My number is 36." Bob replies, "Every prime factor of your number is also a prime factor of my number, but with possibly different exponents."

What is the smallest possible number that Bob could have?


Find the distance between the vertices of the hyperbola \(4x^2 + 24x - 4y^2 + 8y + 44 = 0.\)


If $a$, $b$ are nonzero real numbers such that $a^2+b^2=6ab$, find the value of $\left|\frac{a+b}{a-b}\right|$.


A piece of string fits exactly once around the perimeter of a rectangle whose area is 200. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?


In a survey conducted in Nevada, 620 people were asked about their favorite type of dessert. The pie chart results indicate that the central angle for "Ice Cream" is $198^\circ$. How many people chose "Ice Cream"? Provide your answer as a whole number.


If the price of one piece of gum is 2 cents, what is the total cost for buying 5000 pieces of gum? If a 5% discount is applied to purchases exceeding 4000 pieces, what is the final cost in dollars?


Determine the last two digits in the sum of factorials from 1! to 15!.


Tim decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 3/8 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers?


The points $(1,6)$ and $(5,2)$ are adjacent vertices of a square. What is the area of the square?


Consider the quadratic equation $3x^2 - 8x - 5 = 0$. The roots of this equation can also be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. Find the value of $n$.


Find the center of the hyperbola \(9x^2 - 45x - 16y^2 + 128y + 207 = 0.\)

Solution: \boxed{\left(\frac{5}{2}, 4\right)}


A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Find the number of values of $z$ such that $|z| = 3$ and $f(z) = z$.


The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks.

How many penalty kicks are needed to ensure that every player has had a chance to shoot against each of the goalies?


Let \(x\) and \(y\) be real numbers such that \(\frac{\sin x}{\sin y} = 4\) and \(\frac{\cos x}{\cos y} = \frac{1}{3}\). Find the value of
\[
\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.
\]


Quadrilateral $ABCD$ is a parallelogram. Suppose $\angle DCB = 75^\circ$. What is the degree measure of angle $A$?


Determine the maximum number of real solutions to the polynomial equation
\[x^n + yx^{n-1} + y^2x^{n-2} + \dots + y^{n-1}x + y^n = 0,\]
where \(n\) is a positive integer and \(y\) is a real parameter.


What is the sum of the two solutions to the equation $81 - 27x - 3x^2 = 0$?


In triangle $XYZ,$ the midpoint of $\overline{YZ}$ is $(2,3,1),$ the midpoint of $\overline{XZ}$ is $(1,2,3),$ and the midpoint of $\overline{XY}$ is $(3,1,4).$ Find the coordinates of vertex $X.$

Solution: \boxed{(2, 0, 6)}


Two circles, one with a radius of 7 inches and the other with a radius of 3 inches, are tangent at point P. Two ants start crawling at the same time from point P, the first one crawling along the larger circle at $5\pi$ inches per minute, while the other crawls along the smaller circle at $4\pi$ inches per minute. Calculate how many minutes pass before their next meeting at point P.


The graph of the equation \[\sqrt{x^2+y^2} + |y-2| = 4\] consists of portions of two different parabolas. Compute the distance between the vertices of these parabolas.


Bonnie constructs a frame of a rectangular prism using 8 pieces of wire, each 10 inches long. The prism has dimensions 10 inches, 5 inches, and 2 inches. Meanwhile, Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames. The total volume of Roark's cubes is the same as the volume of Bonnie's rectangular prism. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire?


A fair, twenty-faced die has $18$ of its faces numbered from $1$ through $18$ and two blank faces. Another fair, twenty-faced die has $19$ of its faces numbered from $2$ through $20$ and has one blank face. When the two dice are rolled, what is the probability that the sum of the two numbers facing up will be $25$? Express your answer as a common fraction.


In a 6 by 6 grid, each of the 36 small squares measures 3 cm by 3 cm and is shaded. Six unshaded circles are then placed on top of the grid. Five of these circles have a diameter equal to the side of a small square, and one larger circle has a diameter equal to the length of four small squares. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. Calculate the value $A+B$.


Determine the sum of the coefficients in the polynomial $2(4x^{6} + 9x^3 - 5) + 8(x^4 - 8x + 6)$ when fully simplified.


Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal vectors of lengths 2, 3, and 4 respectively, such that
\[\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a})\]
for some scalars $p, q,$ and $r,$ and the scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 24.$ Find $p + q + r.$


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le7$, given that $0.abc = \frac{abc}{900}$, what is the largest possible value of $a+b+c$?


Determine the domain of the function $\frac{\sqrt{x+3}}{x^2+4x+3}$.

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


Solution: \boxed{28a + 49b}


There are 3 math teams in the area, with 6, 9, and 10 students respectively. The first team has 3 co-captains, the second team has 2 co-captains, and the third team has 4 co-captains. If I randomly select a team, and then randomly select two members of that team to give a copy of $\emph{Introduction to Geometry}$, what is the probability that both of the people who receive books are co-captains?


Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term.


For some positive integers $c$ and $d$, the product \[\log_c(c+2) \cdot \log_{c+2} (c+4) \dotsm \log_{d-4} (d-2)  \cdot\log_{d-2} d\] contains exactly $435$ terms, and its value is $2$. Compute $c+d$.


Calculate the value of $a$ for which the expression $\log_{y^8}{x^2}\cdot\log_{x^4}{y^3}\cdot\log_{y^5}{x^4}\cdot\log_{x^2}{y^5}\cdot\log_{y^3}{x^4}$ can be simplified to $a\log_y{x}$.


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

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.5,xmax=4.5,ymin=-0.99,ymax=6.5;

pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return sqrt(abs(x-1))+(9/pi)*atan(sqrt(abs(x)));}
draw(graph(f2,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=F(x)$",(xmax+0.25,f2(xmax)),E);
[/asy]

The value of \( F(-2) \) is an integer. What is it?


How many positive integers less than 300 are divisible by 2, 3, and 5?


A rectangular prism has 8 vertices and 12 edges like a cube. However, it has three distinct dimensions: length, width, and height. Find the total number of diagonals in a rectangular prism. Assume that the diagonals can be either on the same face or spanning across the space inside the prism.


Given that $f(x) = x^2 - 2x + 5$ and $g(x) = x + 3$, and introducing a new function $h(x) = 2x - 1$, what is the value of $f(g(h(5))) - g(f(h(5)))$?


Given the equations $3x - 2y = 7$ and $x + 3y = 6$, find the value of $x$.


For which positive integer values of $k$ does $kx^2 + 24x + k = 0$ have rational solutions? Provide your answer as a list of integers separated by commas, in increasing order.


Find the number of positive solutions to
\[\sin^2 (\arccos(\sqrt{\tan^2 (\arcsin x)})) = x^2.\]


The diagram shows the miles traveled by cyclists Daniel and Evan. After five hours, how many more miles has Daniel biked than Evan? Daniel bikes at a constant rate and travels 65 miles in 5 hours. Evan bikes at a rate that took him 40 miles in the first three hours, then he took a half-hour break and continued at the same initial rate for the remaining time. How far behind Daniel is Evan after five hours?


In triangle $DEF$, $\sin D = \frac{4}{5}$ and $\cos E = \frac{12}{13}.$ Find $\cos F.$

Solution: \boxed{\cos F = -\frac{16}{65}}


Find the integer $n,$ $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 612^\circ.$


The graph of the rational function \(\frac{1}{q(x)}\) has vertical asymptotes at \(x = -2\) and \(x = 2\). Given that \(q(x)\) is a quadratic polynomial and \(q(3) = 9\), determine \(q(x)\).

Solution: \boxed{\frac{9}{5}x^2 - \frac{36}{5}}


A right triangle has legs $a$ and $b$ and hypotenuse $c$, which is also the diameter of the circle in which the triangle is inscribed. Identify the ratio $ \frac{a + b + c}{c} $ and find its maximum value.

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


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

Solution: \boxed{-\frac{6}{17} - \frac{10}{17}i}


Let \(x\) and \(y\) be positive real numbers. Find the minimum value of
\[
\frac{\sqrt{(x^2 + y^2)(4x^2 + y^2)}}{xy}.
\]


Find the number of even digits in the base-7 representation of $528_{10}$.


A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Determine the probability that the bug returns to its starting vertex on its eighth move and express this probability in lowest terms as $m/n$. Find $m+n$.


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

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


Convert the point in spherical coordinates $(\rho,\theta,\phi) = \left( 15, 225^\circ, 45^\circ \right)$ to rectangular coordinates, assuming $\theta$ and $\phi$ are in degrees.

Solution: \boxed{\left(-\frac{15}{2}, -\frac{15}{2}, \frac{15\sqrt{2}}{2}\right)}


A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $378$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$


Find the value of \(b\) if the remainder is constant when \(12x^3 - 9x^2 + bx + 8\) is divided by \(3x^2 - 4x + 2\).


A card is chosen at random from a deck of 54 cards (52 standard cards plus 2 jokers), and then it is replaced, and another card is chosen. What is the probability that at least one of the cards is a spade or an ace?


In how many ways can 8 distinct beads be placed on a bracelet? (Consider that two arrangements are the same if one can be rotated or reflected to produce the other.)


How many rows in Pascal's triangle, among the first 30 rows (from row 2 to row 30, exclusive of row 0 and 1), consist solely of even numbers, except for the $1$s at each end?


The length of a rectangle is increased by $30\%$, and the width is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased?


Yazmin buys pencils in packs of 5 and erasers in packs of 7. What is the smallest number of eraser packs she can buy in order to have an equal number of pencils and erasers?


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


Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 4x^2 + 5x - 3)(6x^2 - 5x + 1)$.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0 < y \le 12$, then what is the largest possible value of $a+b+c$?


A $6 \times 9$ rectangle can be rolled to form two different cylinders. Determine the ratio of the volume of the larger cylinder to the smaller cylinder when the rectangle is rolled along its length and width, respectively.


The formula for converting a Celsius temperature \( C \) to the corresponding Fahrenheit temperature \( F \) is \( F = \frac{9}{5}C + 32 \). An integer Celsius temperature \( C \) is converted to Fahrenheit, rounded to the nearest tenth, converted back to Celsius, and rounded to the nearest integer. Determine for how many integer Celsius temperatures between -10 and 50 inclusive does the original temperature equal the final temperature after these conversions?


Below is a portion of the graph of a function, $y=q(x)$:

[asy]
import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.5,xmax=3.5,ymin=-4.25,ymax=6.25;

pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return sgn(3x-3)*abs(3x-3)^(1/4)+3*sgn(3x-3)*abs(3x-3)^(1/6)+abs(3x-3)^(1/8);}
draw(graph(f2,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=q(x)$",(xmax+0.25,f2(xmax)),E);
[/asy]

The value of $q(2)$ is an integer. What is it?


Determine the number of pairs $(a,b)$ of real numbers such that $15, a, b, ab$ form an arithmetic progression.


The number $a+\sqrt{b}$ and its radical conjugate have a sum of $0$ and a product of $16$. Find $a+b$.


Annie lives at the point $(10, -15)$, Barbara lives at $(0, 20)$, and Charlie is at $(5, 12)$. They decide first to meet at a point equidistant from both their starting locations before going to Charlie. Calculate how many units they need to walk vertically from their meeting point to reach Charlie.


Let \(x, y, z\) be positive real numbers, and \(k \geq 2\) be a real number. Find the minimum value of
\[
\frac{(x^2 + kx + 1)(y^2 + ky + 1)(z^2 + kz + 1)}{xyz}.
\]


What is the least integer whose square is 12 more than three times the number?


Find \(AX\) in the diagram given \(CX\) bisects \(\angle ACB\).
\[ AC = 15, \ BC = 35, \ BX = 40. \]


Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $37x + y = 2223$. Determine the number of such distinct triangles whose area is a positive integer.


Two cylindrical tanks hold the same amount of liquid. One tank is twice as tall as the other. If the radius of the taller tank is 10 units, find the radius of the shorter tank in simplest radical form.


Berengere and her friend Liam, who is visiting from the US, are at a café in Paris that accepts both euros and dollars. They want to buy a pastry that costs 8 euros. Liam has a ten-dollar bill. How many euros does Berengere need to contribute to the cost of the pastry if 1 euro = 1.10 USD?

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


Three standard 6-sided dice are rolled. What is the probability that the sum rolled is a prime number?


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


Suppose $z$ and $w$ are complex numbers such that
\[|z| = |w| = 2, \quad z \overline{w} + \overline{z} w= 2.\]
Find the largest possible value of the real part of $z + w.$


Simplify $\dfrac{7+18i}{3-4i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary).

Solution: \boxed{-\frac{51}{25} + \frac{82}{25}i}


An infinite geometric series has a first term of $9$ and a second term of $3.$ A second infinite geometric series has the same first term of $9,$ a second term of $3+m,$ and a sum of three times that of the first series. Find the value of $m.$


Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $S$ is inside $\triangle PQR$, $\angle PQR = 50^\circ$, and $\angle PRS = 110^\circ$. What is the degree measure of $\angle SPC$?


Compute the following determinant:
\[
\begin{vmatrix} 2 & -1 & 4 \\ 3 & x & -2 \\ 1 & -3 & 0 \end{vmatrix}
\]

Solution: \boxed{-46 - 4x}


Triangle $PQR$ has side-lengths $PQ = 15, QR = 30,$ and $PR = 22.5$. 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 $\triangle PXY?$


Two circles each with a radius of 12 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord, expressed in simplest radical form.

Solution: \boxed{12\sqrt{3} \text{ cm}}


In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$, respectively, where $A$ is at $(0, 8)$, $B$ at $(0, 0)$, and $C$ at $(10, 0)$. Determine the sum of the $x$ and $y$ coordinates of $G$, the point of intersection of $\overline{AD}$ and $\overline{CE}$.


If the product $(4x^2 - 6x + 5)(8 - 3x)$ can be written in the form $px^3 + qx^2 + rx + s$, where $p, q, r, s$ are real numbers, then find $8p + 4q + 2r + s$.


We define two sequences of vectors \((\mathbf{u}_n)\) and \((\mathbf{z}_n)\) as follows: First, \(\mathbf{u}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix},\) \(\mathbf{z}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.\) Then for all \(n \ge 1,\) \(\mathbf{u}_n\) is the projection of \(\mathbf{z}_{n - 1}\) onto \(\mathbf{u}_0,\) and \(\mathbf{z}_n\) is the projection of \(\mathbf{u}_n\) onto \(\mathbf{z}_0.\) Find
\[
\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dotsb.
\]

Solution: \boxed{\begin{pmatrix} 10 \\ 10 \end{pmatrix}}


Crestview's school has added a new color, silver, to its school colors, which were previously only purple and gold. The students are designing a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes must not be the same color?


How many distinct three-digit numbers can be written with the digits $1$, $2$, $3$, and $4$, if digits may be repeated, but no digit may appear more than twice in a three-digit number?


The value of $y$ varies inversely as $x^2$ and when $x = 3$, $y = 6$. What is the value of $x$ when $y = 2$?


Consider a diagram made up of seven contiguous hexagons arranged as follows: one hexagon at the center surrounded by six hexagons (each adjacent to the center hexagon). The central hexagon is pre-colored blue. The other hexagons are to be colored such that no two adjacent hexagons (including the center one) are colored the same. Apart from blue, you may use only red and yellow colors. In how many different ways can the surrounding hexagon be colored?


Terrell usually lifts two 25-pound weights 15 times. If he uses two 10-pound weights instead, how many times must Terrell lift them to lift the same total weight?


A school has between 140 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into seven distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


The line $y=-\frac{1}{2}x+6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


Determine the next term in the geometric sequence $$4, 12y^2, 36y^5, 108y^9, \ldots$$ Express your answer in terms of $y$.


Given the system of equations:
1. \( 3x - 2y = 8 \)
2. \( x + 3y = 7 \)

Find the value of \( x \).


The equation $x^2 - 4x = -2 + 2i$ has two complex solutions. Determine the product of their real parts.

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


The solution of the equation $9^{x+8} = 10^x$ can be expressed in the form $x = \log_b 9^8$. What is $b$?


Two friends decide to meet for a chess game every day. They arrive at the park independently, at random times between 3 p.m. and 4 p.m., and stay for exactly \( n \) minutes. The probability that either one arrives while the other is at the park is \( 25\% \), and \( n = d - e\sqrt{f} \), where \( d, e, \) and \( f \) are positive integers, and \( f \) is not divisible by the square of any prime. Find \( d + e + f \).


Six identical rectangles are arranged to form a larger rectangle $PQRS$, where the total area of $PQRS$ is $4500$. The width $w$ of each rectangle is $\frac{3}{4}$ of its length $x$. What is the length $x$ of each rectangle, rounded off to the nearest integer?


A hyperbola is centered at the origin with its axes oriented along the coordinate axes and passes through the points $(3,-4)$, $(0, 2)$, and $(t, -2)$. Find $t^2$.


A matrix projects onto a line $\ell$, passing through the origin, is given by
\[
\renewcommand{\arraystretch}{1.5}
\begin{pmatrix}
\frac{1}{18} & -\frac{1}{9} & \frac{1}{6} \\
-\frac{1}{9} & \frac{4}{18} & -\frac{1}{3} \\
\frac{1}{6} & -\frac{1}{3} & \frac{1}{2}
\end{pmatrix}
\renewcommand{\arraystretch}{1}
.\]
Find the direction vector of line $\ell$. Enter your answer as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a$, $b$, $c$ are integers, $a > 0$, and $\gcd(|a|,|b|,|c|) = 1$.

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


Find the real roots of the equation \[\sqrt{x} + \sqrt{x+6} = 12.\]


Suppose the least common multiple of two 5-digit integers is a 9-digit number. What is the maximum number of digits their greatest common divisor can have?


A function $g(x)$ is defined for all real numbers $x$. For all non-zero $x$, we have
\[3g\left(x\right) + 2g\left(\frac{1}{x}\right) = 7x + 6\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 2024$. Compute the integer nearest to $T$.


Cindy now wishes to arrange her coins into $X$ piles, each containing the same number of coins $Y$. Just as before, each pile will have more than one coin and no pile will have all the coins. If there are 19 possible values for $Y$ given all of the restrictions, what is the smallest number of coins Cindy could have now?


A biased coin is flipped six times. If the probability of getting heads exactly twice is equal to the probability of getting heads exactly three times in these flips, find the value of $i+j$ where $\frac{i}{j}$, in lowest terms, is the probability that the coin comes up heads exactly 4 out of 6 flips.


Solution: \boxed{6 - 5i}


Consider the function $f(x) = 3x^2 - 9x - 6$. Determine the largest interval containing $x = 0$ over which the function is invertible.

Solution: \boxed{(-\infty, \frac{3}{2}]}


Calculate \( \sqrt{15q} \cdot \sqrt{10q^3} \cdot \sqrt{8q^5} \).

Solution: \boxed{20q^4 \sqrt{3q}}


Thirteen people are standing in a circle. A ball is initially with the first person, Bob, who throws the ball clockwise but skips two people and throws it to the fourth person. This person then continues the pattern by skipping two people and throwing to the fourth person in sequence. How many total throws are necessary for the ball to return to Bob?


The circumference of a particular circle is 36 cm. Calculate both the area and the diameter of the circle. Express the area as a common fraction in terms of $\pi$ and the diameter in centimeters.


What is the largest five-digit number whose digits add up to 20?


Let \( f(x) = 4x^2 + 1 \) and \( g(f(x)) = x^2 - x + 1 \). Find the sum of all possible values of \( g(49) \).


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?


How many even natural-number factors does $n = 2^3 \cdot 3^2 \cdot 7^1 \cdot 5^1$ have?


Given a polynomial
\[ a_8 x^8 + a_7 x^7 + a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 \]
where \( a_0, a_8 \neq 0 \) and all coefficients are integers. Its roots \( r_1, r_2, \ldots, r_8 \) are integers. Additionally, the same roots \( r_1, r_2, \ldots, r_8 \) also satisfy the polynomial
\[ a_0 x^8 + a_1 x^7 + a_2 x^6 + a_3 x^5 + a_4 x^4 + a_5 x^3 + a_6 x^2 + a_7 x + a_8 = 0. \]
Find the number of possible multisets \( S = \{r_1, r_2, \ldots, r_8\} \).


Mr. Lee has 8 children. Assuming that the gender of each child is determined independently and with equal likelihood of being male or female, what is the probability that Mr. Lee has at least twice as many sons as daughters?


Solution: \boxed{\frac{\log_2 7}{8}}


A list of five positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?


Factor the following expression: $348b^2 + 87b + 261$.

Solution: \boxed{87(4b^2 + b + 3)}


On a "prime date," both the month and the day are prime numbers. How many prime dates occurred in 2004, a leap year?


What is the sum of all the solutions to the equation $x = \left|3x - |40-3x|\right|$?


Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) f(x - y) = (f(x) + f(y))^2 - 2x^2 y^2\]
for all real numbers $x$ and $y.$


The fifth and eighth terms of a geometric sequence are $-2$ and $-54$, respectively. What is the $11$th term of this progression?


Suppose we have 12 dogs and need to divide them into three groups: one with 4 dogs, one with 5 dogs, and one with 3 dogs. Determine how many ways the groups can be formed if Rocky, a notably aggressive dog, must be in the 4-dog group, and Bella must be in the 5-dog group.


How many perfect squares less than 2,500 can be represented as the difference of two consecutive perfect squares?


Define a function $g$ by $g(z) = -i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Determine how many values of $z$ satisfy both $|z| = 3$ and $g(z) = z$.


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


In right triangle $DEF$, $DE=15$, $DF=20$, and $EF=25$ units. What is the distance from $F$ to the midpoint of segment $DE$? Additionally, calculate the area of triangle $DEF$.


Find the square root of $\dfrac{9!}{126}$.


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 105, what is the lowest score?


John draws a regular six-pointed star (two overlapping equilateral triangles) in the sand, and at each of the 6 outward-pointing points and 6 inward-pointing points he places one of twelve different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?


Let $r$ and $s$ denote the two real roots of $x^2 - 2x \sqrt{6} + 3 = 0.$ Determine $r^8 + s^8.$


Given that $9^{-1} \equiv 79 \pmod{89}$, find $81^{-1} \pmod{89}$, as a residue modulo 89.


Let $m = 2^{12} \cdot 3^{16} \cdot 7^9$. How many of the natural-number factors of $m$ are multiples of 252?


In triangle $ABC$, where $\angle B = 90^\circ$ and $\angle A = \angle C$, the side $BC = 8$. What is the area of $\triangle ABC$?


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

[asy]
draw((0,0)--(1,3)--(10,3)--(15,0)--cycle);
draw((10,3)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,3),NW);
label("$B$",(10,3),NE);
label("$C$",(15,0),E);
[/asy]


Abby, Bart, Cindy, and Damon weigh themselves in pairs. Together Abby and Bart weigh 280 pounds, Bart and Cindy weigh 265 pounds, Cindy and Damon weigh 290 pounds, and Abby and Damon weigh together an unknown amount. If the total weight of Bart and Damon together is 275 pounds, how many pounds do Abby and Damon weigh together?


Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 12x - 8y - 44$. What is $xy$?


Consider the lines:
\[
y = 2x + 3, \quad 2y = 6x - 4, \quad 3y = 12x + 9, \quad 4y = -8x + 10, \quad 5y = 10x - 15
\]
Determine how many pairs of these lines are either parallel or perpendicular to each other.


An ellipse is tangent to both the $x$-axis and the $y$-axis, with its foci located at $(1 + \sqrt{3}, 2)$ and $(1 - \sqrt{3}, 2)$. Find the length of the major axis.


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


Simplify the sum: \(i^0 + i^1 + \cdots + i^{2011}.\)


Find the remainder when $7x^4 - 16x^3 + 3x^2 - 5x - 20$ is divided by $2x - 4.$


In the diagram, circle $O$ has a radius of 10 units. Chord $EF$ is 12 units long and parallel to segment $MB$. If $MA = 20$ units and points $M$, $A$, $O$, and $B$ are collinear, determine the area of triangle $MEF$. Represent the answer in simplest radical form.


Convert the point $(0, 4 \sqrt{3}, -4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left( 8, \frac{\pi}{2}, \frac{2\pi}{3} \right)}


**The Cookie Monster encounters another mysterious cookie, this time with the boundary given by the equation \(x^2 + y^2 + x - 5y = 10\). He wonders about the size of this cookie. What is the radius of this particular cookie?**


Determine how many distinct solutions satisfy both $|x-10| = |x+3|$ and $2x + 6 = 18$.


Express $7.\overline{182}$ as a common fraction in lowest terms.


The arithmetic mean of an even number of consecutive even integers is $x$. Find the sum of the smallest and largest integers in terms of $x$.


Rectangle $ABCD$ is the base of pyramid $PABCD$ where $AB = 10$, $BC = 5$. The line segment $\overline{PA}$ is perpendicular to both $\overline{AD}$ and $\overline{AB}$, and $\overline{PC}$ is perpendicular to $\overline{CD}$. If $PB = 20$, determine the volume of pyramid $PABCD$.


How many positive four-digit integers are divisible by both 4 and 7?


Find the value of $b$ if the solution of the equation $9^{x+5} = 5^x$ can be expressed in the form $x = \log_b 9^5$.


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


In a drawer, there are 15 distinguishable socks: 5 white, 5 brown, 3 blue, and 2 green. How many ways can I choose a pair of socks, provided that I get two socks of different colors?


If 1000 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?


In the diagram below, a 4-inch by 4-inch square adjoins a 12-inch by 12-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction.
[asy]
unitsize(2mm);
defaultpen(linewidth(0.7pt)+fontsize(12pt));

pair H=(0,0), A=(0,12), B=(12,12), C=(12,4), G=(12,0), E=(16,4), F=(16,0);
pair D=extension(A,F,B,G);

draw(A--B--G--H--cycle);
fill(D--C--E--F--cycle, black);
draw(A--F);
draw(G--F--E--C);

label("$A$",A,NW);
label("$H$",H,SW);
label("$B$",B,NE);
label("$C$",C,E);
label("$G$",G,S);
label("$D$",D,WSW);
label("$E$",E,NE);
label("$F$",F,SE);
[/asy]

Solution: \boxed{10 \text{ square inches}}


In triangle $XYZ$, $XY = 5$, $XZ = 7$, and $YZ = 8$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$ respectively. Find $GP + GQ + GR$.


Assume a calendrical system in which leap years happen every three years without exception. In a 123-year period, what is the maximum possible number of leap years?


How many positive integers less than 500 are congruent to 3 (mod 7)?


Let \[f(x) =
\begin{cases}
x^3 + 3 &\text{if } x < n, \\
3x + 6 &\text{if } x \ge n.
\end{cases}
\]If the graph $y=f(x)$ is continuous at $x=n$, find the sum of all possible values of $n$.


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


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


The arithmetic mean, geometric mean, and harmonic mean of $a$, $b$, $c$ are $12$, $5$, and $4$ respectively. What is the value of $a^2+b^2+c^2$?


In the diagram below, we have \( \cos \angle VTX = \frac{4}{5} \). What is \( \cos \angle VTU \)?

[asy]
pair V,T,X,U;
U = (-2,0);
T = (0,0);
X = (2,0);
V = rotate(aCos(4/5))*(1.5,0);
dot("$U$",U,S);
dot("$X$",X,S);
dot("$V$",V,N);
dot("$T$",T,S);
draw(X--U);
draw(T--V);
[/asy]


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


Solution: \boxed{12z^6 + 16z^5 - 29z^4 - 12z^3 + 21z^2 + 8z - 10}


I had $\$50$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on video games?

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.866,.5);
C=(-.707,-.707);
D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Books", W, N);
label("$\frac{1}{2}$", W, S);
X=(-.55, 0);
label("Video Games", X, S);
Y=(-.17,-.6);
label("Snacks", Y, N);
label("$\frac{1}{10}$", Y, S);
Z=(.5, -.15);
label("Toys", Z, N);
label("$\frac{1}{4}$", Z, S);
[/asy]


In a bag, there are 6 green chips, 5 purple chips, and 4 orange chips. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two chips drawn are of different colors?


What is the constant term in the expansion of $\left(x^2+\frac{3}{x^2}\right)^{10}$?


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$.

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


The expression $\dfrac{\sqrt[6]{5}}{\sqrt[5]{5}}$ equals 5 raised to what power?


If 35% of a number is the same as 30% of 50, what is the number?


Convert the point \( (1, -1) \) in rectangular coordinates to polar coordinates. List your answer in the format \( (r, \theta) \), where \( r > 0 \) and \( 0 \leq \theta < 2\pi \).

Solution: \boxed{\left(\sqrt{2}, \frac{7\pi}{4}\right)}


The number $6\,28G\,409\,3H8$ is divisible by $72$. If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH$?


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


A standard deck of 52 cards has 13 ranks and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$). Suppose the deck is randomly shuffled. What is the probability that the top three cards include one from each suit ($\spadesuit$, $\heartsuit$, and $\diamondsuit$) irrespective of the order?


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


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{2}$, meaning that if the object's original length was $x$, then it is now $\frac{3}{2}x$. On the second day, he increased the object's longer length by $\frac{1}{2}$ again; on the third day, he increased the object's new length by $\frac{1}{2}$ again; and so on, such that on the $n^{\text{th}}$ day of performing this trick, Barry will make the object's length exactly 50 times its original length. What is the value of $n$?


Triangle $XYZ$ is equilateral. Let $\overline{KL}$, $\overline{MN}$, and $\overline{OP}$ be three lines parallel to $\overline{YZ}$, splitting the height of triangle $XYZ$ into four equal segments with points $K$ and $L$ being closest to vertex $X$. If $XK = KM = MN = NP$, calculate the ratio of the area of trapezoid $OPYZ$ to the area of triangle $XYZ$.


Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \pi h^3$ cubic units. Find the value of $M$.


At Frank's Fruit Market, 4 bananas cost as much as 3 apples, and 5 apples cost as much as 2 oranges. How many oranges cost as much as 24 bananas?


The year after 1998 during which the profit increased the most, compared to the previous year.


The states of Sunshine and Prairie have adopted new license plate configurations. Sunshine license plates are formatted "LDDLDL" where L denotes a letter and D denotes a digit. Prairie license plates are formatted "LDDLDD". Assuming all 10 digits and 26 letters are equally likely to appear in their respective positions, how many more license plates can Sunshine issue compared to Prairie?


Find the units digit of $8 \cdot 19 \cdot 1981 + 6^3 - 2^5$.


Each of \( b_1, b_2, \dots, b_{50} \) is equal to \( 2 \) or \( -2 \). Find the minimum positive value of
\[
\sum_{1 \le i < j \le 50} b_i b_j.
\]


Find the units digit of \(x\) given that \(ax = 14^8\) and \(a\) has a units digit of 9.


Calculate $7\cdot7! + 5\cdot5! + 3\cdot3! + 5!$.


There are 0.9072 kilograms in 2 pounds. To the nearest whole pound, how many pounds does an object that weighs 350 kg weigh?

Solution: \boxed{772\ \text{pounds}}


In the function machine shown, the input is 12. What is the output?

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{In}\mbox{put}$ = 12",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 3",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 20",(0,0));
draw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,"Divide by 2",(width1/2,height/2));
draw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,"Subtract 2",(width1/2,height/2));
draw(f,(0,0)--(widthE,0)--(widthE,0)+lengthE*dir(angleE)--lengthE*dir(angleE)--cycle);
label(f,"$\mbox{Out}\mbox{put}$ = ?",lengthE/2*dir(angleE) + (widthE/2,0));
add(shift(width1/2*left)*a); draw((0,0)--(0,-2),EndArrow(4));
add(shift(5*down + width2/2*left)*b);
add(shift((7+heightC/2)*down)*c); draw((0,-5)--(0,-7),EndArrow(4));
pair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);
draw("$\le 20$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 20?$",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));
rightpt = rightpt + 2.5E+2S;
leftpt = leftpt + 2.5W+2S;
add(shift(leftpt+height*down+.3*width3*left)*d);
add(shift(rightpt+height*down+.7*width4*left)*e);
rightpt = rightpt+.75height*down+.7*width4*left;
leftpt = leftpt+.75height*down+.7*width3*right;
draw(leftpt--rightpt);
pair midpt = (leftpt+rightpt)/2;
draw(midpt--(midpt+2down),EndArrow(4));
add(shift(midpt+.65widthE*left+(2+lengthE*Sin(angleE))*down)*f);
[/asy]


Solve for $y$: $\log_3 y + \log_9 y = 5$.


Charlie is planning to construct a boundary around a rectangular playground using exactly 380 feet of fencing. Regulations specify that the length of the boundary must be at least 100 feet and the width must be at least 60 feet. Charlie wants to maximize the area enclosed by the fence for more play equipment and sitting area. What are the optimal dimensions and the maximum possible area of the playground?


Determine an expression for the area of $\triangle QCA$ in terms of $p$. Your answer should be simplified as much as possible. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 15); pair b = (15, 0);
pair a = (3, 15); pair t = (3, 0); pair c = (0, 10);

draw((-2, 0)--(18, 0), Arrow);
draw((0, -2)--(0, 18), Arrow);
draw(q--a--b);
draw(a--c--b);

label("$Q(0, 15)$", q, W);
label("$A(3, 15)$", a, NE);
label("$B(15, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (18, 0), E);
label("$y$", (0, 18), N);
label("$C(0, p)$", c, W);
[/asy]

Solution: \boxed{\frac{45 - 3p}{2}}


Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be vectors such that \(\|\mathbf{a}\| = 2,\) \(\|\mathbf{b}\| = 5,\) and \(\|\mathbf{c}\| = 6,\) and
\[\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.\]
Compute \(\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.\)


In a modified version of a walking game, I play by different rules. On the first move, I stand still, but for each subsequent move $n$ where $2 \le n \le 25$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing all 25 moves, I must return to my original starting point. How many steps is my return journey?


The Grunters face the Screamers in a series of 6 games. The Grunters are the much better team with a 75% probability of winning each game. What is the probability that the Grunters will win all 6 games? Express your answer as a common fraction.


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


What is the value of $$\left(\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\right)^2?$$


Diane has one 1-cent stamp, two identical 2-cent stamps, three identical 3-cent stamps, four identical 4-cent stamps, and five identical 5-cent stamps. How many different arrangements can Diane paste exactly 15 cents worth of postage in a row across the top of an envelope, given that she must use at least one of each stamp from 1 to 5 cents?


In a school, there are $1200$ notebooks distributed evenly among $30$ boxes. A teacher decides to redistribute these notebooks so that there are now $35$ notebooks in each box. How many notebooks will be left over after redistributing them into as many full boxes as possible?


Let $\omega$ be a complex number such that $\omega^9 = 1$ and $\omega \ne 1.$  Compute
\[\omega^{21} + \omega^{24} + \omega^{27} + \dots + \omega^{63}.\]


Consider the expression \[(x+y+z)^{2010} + (x-y-z)^{2010}.\] After expanding and combining like terms, how many terms are in the simplified expression?


In a circle with center $O$, the measure of $\angle RIP$ is $45^\circ$ and $OR=12$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$.


Given $J_k = 10\ldots 032$, where there are $k$ zeros between the $1$ and the $3$, and $M(k)$ be the number of factors of 2 in the prime factorization of $J_k$, find the maximum value of $M(k)$.


Humanity discovers a system with 11 habitable planets, 5 of which are "Earth-like" and 6 are "Mars-like." Earth-like planets require 2 units of colonization each, and Mars-like planets need 1 unit each. If humanity has 14 total units available for colonization, in how many different ways can they occupy these planets?


Rationalize the denominator of $\frac{4}{3\sqrt[4]{7}}$. The answer can be written in the form of $\frac{A\sqrt[4]{B}}{C}$, where $A$, $B$, and $C$ are integers, $C$ is positive, and $B$ is not divisible by the fourth power of any prime. Find $A+B+C$.


The United States Postal Service has updated its policy and now charges an extra $0.15 in postage if the length of an envelope, in inches, divided by its height, in inches, is less than $1.5$ or greater than $2.8.$ For how many of these four envelopes must the extra $0.15$ in postage be paid?
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
A &7 &5\\
B &10 &3\\
C &7 &7\\
D &12 &4
\end{tabular}


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(4) = 16$ and $g(16) = 1024$, find the largest possible value of $g(8)$.


In a circle with center $O$, the measure of $\angle TIQ$ is $45^\circ$ and the radius $OT$ is 12 cm. Find the number of centimeters in the length of arc $TQ$. Express your answer in terms of $\pi$.


How many two-digit primes have a ones digit of 3, and are greater than 50?


What is the area, in square units, of the interior region formed by the lines $y = 3x + 6$, $y = -4x + 20$ and the line $x = 5$?


A fair 8-sided die is rolled once. If I roll $n$, then I win $8 - n$ dollars. However, if $n$ is more than 4, I win an additional dollar. Calculate the expected value of my winnings.


What is the value of $x$ for which $(10-2x)^2 = 4x^2 + 20x$?


All the numbers 3, 4, 5, 6, 7, 8 are assigned to the six faces of a cube, one number for each face. Calculate the greatest possible value of the sum of the products for each of the eight vertices of the cube, where each product at a vertex is formed by multiplying the numbers assigned to the three faces meeting at that vertex.


There are $30n$ members in the Steinburg marching band, and when they line up in rows of 34, there are 6 band members left over. If $n$ is an integer and there are fewer than 1200 band members, what is the maximum number of people that could be in the Steinburg marching band?


At a local diner, there are five dishes that are vegan. These represent one fourth of the entire menu. Four out of these five vegan dishes contain soy, which I am allergic to. What fraction of the menu items can I eat, considering I am looking for vegan and soy-free options?


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


The graph of function $y = v(x)$ is given by the modified formula $y = x + 2\sin(x\pi/2)$. Determine the exact value of $v(-3.14) + v(-0.95) + v(0.95) + v(3.14)$.


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

Solution: \boxed{2 - 3i}


Find all angles $\theta,$ $0 \le \theta \le 2\pi,$ such that for all real numbers $x,$ $0 \le x \le 1,$
\[x^2 \sin \theta - x(1 - x) + (1 - x)^2 \cos \theta > 0.\]

Solution: \boxed{\left( \frac{\pi}{12}, \frac{5\pi}{12} \right)}


The product of the positive integer divisors of a positive perfect square integer $n$ is $1024$. Find $n$.


Find the phase shift, amplitude, and vertical shift of the graph of \( y = 2\sin(4x - \frac{\pi}{2}) + 1 \).


In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?


How many positive divisors does 8! have?


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


Two numbers $150$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?


A certain integer has $4$ digits when written in base $10$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?


The positive integer divisors of 147, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7?


A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students are left in the last row. When the students try to sit in rows of 6, 3 students remain in the last row. How many students are in this class?


How many three-digit numbers remain if we exclude all three-digit numbers in which there are exactly two identical digits that are adjacent and the third digit is different?


How many license plates consist of 2 letters followed by 2 digits, where one of the digits must be a prime number and the other must be a composite number?


Let
\[x^6 - x^3 - x^2 - x - 1 = q_1(x) q_2(x) \dotsm q_m(x),\]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \dots + q_m(2).$


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 6.$  Find $|\alpha|.$


Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair and exactly one triple (that is, three dice showing the same value and two other dice showing the same value)?


An equilateral triangle has two vertices at $(5, 0)$ and $(5, 8)$. Calculate the x-coordinate of the third vertex if it is located in the first quadrant.

Solution: \boxed{5 + 4\sqrt{3}}


The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ -1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{3}{2} \\ -\frac{1}{2} \end{pmatrix}\]
lie on a line. Enter the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = 3x - 5}


Two spinners are divided into quarters and fifths, respectively. The first spinner is labeled 2, 4, 6, 8, and the second spinner is labeled 3, 5, 7, 9, 11. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.


Bertha has 8 daughters and no sons. Some of her daughters have 4 daughters each, and the rest have none. Bertha has a total of 40 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


Let vectors $\mathbf{c}$ and $\mathbf{d}$ be such that
\[\mathbf{c} + \mathbf{d} = \begin{pmatrix} 8 \\ -4 \\ -8 \end{pmatrix},\]
where $\mathbf{c}$ is parallel to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ and $\mathbf{d}$ is orthogonal to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.$ Find $\mathbf{d}$.

Solution: \boxed{\begin{pmatrix} \frac{28}{3} \\ -\frac{8}{3} \\ -\frac{20}{3} \end{pmatrix}}


$\textbf{Anna's Vintage Postcards}$

Anna organizes the postcards in her collection by country and by the decade in which they were issued. The prices she paid for them at a collectibles shop were: Italy and Germany, $7$ cents each; Japan $5$ cents each; and India $6$ cents each. (Italy and Germany are European countries, and Japan and India are Asian countries.) In that collection:

- 60s: Italy (5), Germany (7), Japan (6), India (8)
- 70s: Italy (9), Germany (6), Japan (10), India (12)
- 80s: Italy (10), Germany (12), Japan (15), India (11)

In dollars and cents, how much did her Asian postcards issued in the 80s cost her?


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


Consider the polynomials
\[ f(x) = 2 - 15x + 4x^2 - 5x^3 + 6x^4 \]
and
\[ g(x) = 4 - 3x - 7x^3 + 12x^4. \]
Find $c$ such that the polynomial $f(x) + c g(x)$ has degree 3.


What is the largest four-digit palindrome that is divisible by 3? (A palindrome is a number that reads the same forwards and backwards.)


The cards of a modified 40-card deck (10 cards of each suit) are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both hearts?


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


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


What is the sum of all the solutions of the equation $x = \left|3x - |80 - 3x|\right|$?


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


In Mr. Gray's 20-student mathematics class, no two students have the same initials, and each student's first name and last name begin with the same letter. The initials only include letters from N to Z. If the letter "Y" is considered a vowel, what is the probability of randomly picking a student whose initials are vowels? Express your answer as a common fraction.


In triangle ABC, where AB = 24 and BC = 18, find the largest possible value of $\tan A$.


A woman labels the squares of a very large chessboard from $1$ to $64$. On each square $k$, she places $2^k$ grains of rice. How many more grains of rice are placed on the $15^{th}$ square than on the first $12$ squares combined?


Let $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ -4 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} \sqrt{3} \\ 5 \\ -2 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 8 \\ -7 \\ 19 \end{pmatrix}.$  Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} - (\mathbf{a} \cdot \mathbf{c}) \mathbf{b},$ in degrees.


Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\).


The length of a rectangular park exceeds four times its width by 15 feet, and the perimeter of the park is 780 feet. What is the area of the park in square feet?


A number in the set $\{30, 31, 32, 33, ... , 800\}$ is randomly selected. What is the probability that it is a three-digit number? Express your answer as a common fraction.


The four consecutive digits $p$, $q$, $r$, and $s$ are used to form the four-digit numbers $pqrs$ and $srqp$. What is the greatest common divisor of all numbers of the form $pqrs+srqp$?


Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2500$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$?


Suppose that $c$ and $d$ are positive integers such that $c-d=8$ and $\text{gcd}\left(\frac{c^3+d^3}{c+d}, cd\right) = 16$. Find the smallest possible value of $d$.


At a national competition, 30 Mathletes stay in the same hotel. Each Mathlete gets his/her own room with room numbers ranging from 1 to 30. However, the Mathletes assigned to rooms 15, 16, and 29 have not arrived yet. What is the median room number of the 27 Mathletes who have arrived?


Solve the following equation by completing the square: $$64x^2+80x-81 = 0.$$ Rewrite the equation in the form $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$?


In the six-digit integer $4B6,\!792$, what is the largest digit $B$ so that the six-digit integer will be divisible by 4?


A $240\times 360\times 400$ 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?


Given the function \(f(x) = x^{-1} + \frac{x^{-1}}{1+x^{-1}}\), calculate \(f(f(-3))\) and express your answer as a common fraction.


Let $a,$ $b,$ $c,$ $d,$ and $e$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} + \frac{1}{e + \omega} = \frac{3}{\omega^2},
\]
find
\[
\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1} + \frac{1}{d + 1} + \frac{1}{e + 1}.
\]


How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect cube factor other than one?


A library has eight identical copies of a particular book. At all times, a number of these copies are available at the library while others are checked out. Determine how many different configurations there are for some of the books to be in the library and the remainder to be checked out, assuming that at least one book is in the library and at least one is checked out. (The books should be considered indistinguishable.)


There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 8 \\ -4 \\ -8 \end{pmatrix},\]
where $\mathbf{a}$ is parallel to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.$ Find $\mathbf{b}.$

Solution: \boxed{\mathbf{b} = \begin{pmatrix} \frac{28}{3} \\ -\frac{8}{3} \\ -\frac{20}{3} \end{pmatrix}}


Consider the calculation of
\[
\text{cis } 40^\circ + \text{cis } 48^\circ + \text{cis } 56^\circ + \dots + \text{cis } 128^\circ
\]
and express it in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


Solve for $x$ if $16^x \cdot 16^x \cdot 16^x = 64^3$.


Let \( p, q, r \) be three consecutive terms of a geometric series where all terms are positive, such that \( pqr = 216 \). Find the smallest possible value of \( q \).


The line \(y = \frac{3}{4}x + 2\) is parameterized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -6 \end{pmatrix}.
\]
Determine the ordered pair \((s, l)\).

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


A $10 \times 10$ checkerboard consists of nine horizontal and nine vertical lines forming $r$ rectangles, of which $s$ are squares. Calculate the fraction $s/r$ in the form of $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m + n$.


Let the line $q$ be the perpendicular bisector of $D = (15, 3)$ and $E = (6, 8).$ Given that $DE$ meets $q$ at $F = (x, y),$ what is $3x - 5y$?


In triangle $PQR$, the angle bisectors are $PL$, $QM$, and $RN$, which intersect at the incenter $I$. If $\angle PRQ = 52^\circ$, then find the measure of $\angle PIM$, in degrees.


In writing the integers from 100 through 999 inclusive, how many times is the digit 6 written?


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


What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A(-4, 0)$ and $B(0, 10)$?


In right triangle $DEF$ where $\angle D = 90^\circ$, the side lengths are $DE = 8$ and $EF = 17$. Find $\cos F$.


Let $p$ and $q$ be the solutions to the quadratic equation $3x^2 - 7x + 4 = 0$. Compute the value of $(5p^3 - 5q^3)(p - q)^{-1}$.


Find the number of even digits in the base-9 representation of $567_{10}$.


Five fair coins are to be flipped. What is the probability that either all five will be tails, or at least one coin will be heads? Express your answer as a common fraction.


What is the remainder when $8x^3 - 22x^2 + 30x - 45$ is divided by $4x - 8$?


Find the largest solution to \[\lfloor x \rfloor = 7 + 80 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


What is the least positive integer which when divided by 4 gives a remainder of 3, when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 11 gives a remainder of 10?


Consider the equation \(z^6 = -64i\). The solutions to this equation can be expressed in the form:
\[
z_k = r_k (\cos \theta_k + i \sin \theta_k),
\]
for \( k=1, 2, ..., 6 \), where \( r_k > 0 \) and \( 0^\circ \le \theta_k < 360^\circ\). Find the sum \( \theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6 \), in degrees.


In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the internal angle at $C$ intersects $AB$ at point $P$. Find the ratio $PA:PB$.


Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let $AB = a$, $AC = b$, and $AD = 2c$. Given that the areas of triangles $ABC$, $ACD$, and $ADB$ are denoted by $x$, $y$, and $z$ respectively, find the area of triangle $BCD$.

Solution: \boxed{\sqrt{x^2 + y^2 + z^2}}


What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c + \dfrac{1}{c}\right)^8$?


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that
\[\mathbf{a} + \mathbf{b} + 2 \mathbf{c} = \mathbf{0}.\]Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.


Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$?


Find the coefficient of $x^{60}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{10} - 10).\]


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


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 16. The axes of the cylinder and cone coincide. Determine the radius of the cylinder. Express your answer as a common fraction.


At another state contest, 25 Mathletes were accommodated in a hotel, each having their own room numbered consecutively from 1 to 25. If all Mathletes arrived except those in rooms 15 and 17, what is the median room number of the remaining Mathletes?


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


What is the remainder when the sum of the first 105 counting numbers is divided by 5270?


For certain real values of $a, b, c,$ and $d,$ the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has four non-real roots. The product of two of these roots is $8+3i$ and the sum of the other two roots is $2+5i$, where $i^2 = -1$. Find $b$.


Fully factor the following expression: $3y^3 - 75y$.


Given the function \( f(x) \) defined for \( -4 \leq x \leq 4 \) by the following piecewise segments: \( f(x) = -x-1 \) for \( -4 \leq x \leq -3 \), \( f(x) = -x \) for \( -3 < x \leq 3 \), and \( f(x) = -x+1 \) for \( 3 < x \leq 4 \), find the range of \( f(x) - 2x \).

Solution: \boxed{[-11, 11]}


What is the largest prime factor of 989?


A volleyball team has 18 members. I want to choose a starting lineup consisting of 8 players, one of whom will be the captain (the other seven positions are interchangeable). How many ways can I choose my starting lineup?


Let \( a, b, \) and \( c \) be distinct real numbers satisfying:
\[
\frac{a^3 + 12}{a} = \frac{b^3 + 12}{b} = \frac{c^3 + 12}{c}.
\]
Find \( a^3 + b^3 + c^3 \).


Ryan has 4 red lava lamps and 2 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on?


A conference is attended by 360 people. They are to be seated in a hall in such a manner that each row contains the same number of attendees, with at least 12 rows and at least 18 attendees in each row. Determine the sum of all possible numbers of attendees in each row.


What is the value of $a^3 - b^3$ given that $a+b=12$ and $ab=20$?


We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 12$, $BK = 9$, and $BC = 18$, then what is the area of $\triangle ABC$?


Six numbers are 1856, 1975, 2042, 2071, 2150, and 2203. Four of these numbers have a mean (average) of 2035. What is the mean (average) of the other two numbers?


Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pattern of increasing distance continues, what is the number of unit squares in the $50^{th}$ ring, if each ring’s distance from the center increases linearly (i.e., $n^{th}$ ring is at $n$ units from the center), and every second ring starting from the first is colored red?


The equation of another line shown can be written as $y=mx+b$. Find $mb$.
[asy]
size(100,0);
add(shift(-5,-5)*grid(10,10));
draw((-5,0)--(5,0),linewidth(2));
draw((0,-5)--(0,5),linewidth(2));
label("",(5,0),E);
label("",(0,5),N);
draw((1,-5) -- (-4,5),blue,Arrows);
[/asy]
Each grid square in the diagram shown is 1 unit by 1 unit.


Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$.


How many ways can 2210 be factored as a product of two two-digit numbers? (Two factorizations of the form $a \cdot b$ and $b \cdot a$ are considered the same).


Expand and simplify the following product: $$(x^2 - 3x + 3)(x^2 + 3x + 3).$$

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


John needs to order a custom belt online, where measurements need to be provided in centimeters. Given that there are $10$ inches in a foot and $25$ centimeters in a foot, calculate John's belt size in centimeters to the nearest whole number if his waist size is $42$ inches.

Solution: \boxed{105\ \text{cm}}


Two distinct positive integers $a$ and $b$ are factors of 100. If $a \cdot b$ is not a factor of 100, what is the smallest possible value of $a \cdot b$?


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

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


In triangle $PQR$, $PQ = 13$ and $PR = 15$. Let $S$ be the foot of the perpendicular from $P$ to $QR$. If $QS:SR = 3:4$, find $PS$.


The line $y=-\frac{2}{3}x+6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


Given the two functions $f(x) = x^3 + 1$ and $g(x) = 3x - 2$, find $f(g(f(g(2))))$.


For what values of $y$ is $\frac{\log{(5-y)}}{\sqrt{y-2}}$ defined?

Solution: \boxed{(2, 5)}


In $\Delta ABC$, $\overline{DE} \parallel \overline{AB}, CD = 6$ cm, $DA = 15$ cm, and $CE = 9$ cm. What is the number of centimeters in the length of $\overline{CB}$ if $\overline{DE}$ is $\frac{1}{3}$ the length of $\overline{AB}$?

Solution: \boxed{31.5\text{ cm}}


How many different positive, six-digit integers can be formed using the digits 1, 1, 3, 3, 3, and 6?


On the game show $\text{\emph{Wheel of Fortune Plus}}$, you see the following spinner. Given that each region is the same area, what is the probability that you will earn exactly $\$1900$ in your first three spins? Express your answer as a common fraction. [asy]
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8));
draw(unitcircle);
string[] labels = {"Bankrupt","$\$1000$","$\$500$","$\$4000$","$\$400$"};
for(int i = 0; i < 5; ++i){

draw(origin--dir(72*i));

label(labels[i],0.6*dir(36 + 72*i));
}
[/asy]


A region is bounded by semicircular arcs created on each side of an equilateral triangle, where each side measures \(1/\pi\). Calculate the total perimeter of this shaped region.


John started collecting coins on the first day he got 12 coins. Each subsequent day, he collected 10 more coins than the previous day. However, his collection was interrupted on the 7th day when he lost half of the coins he collected that day. If he collected coins for 8 consecutive days, what was the average number of coins collected per day?

Solution: \boxed{42.5 \text{ coins per day}}


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


In triangle $DEF$, the side lengths $d$, $e$, and $f$ satisfy $d + 3e + 3f = d^2$ and $d + 3e - 3f = -4$. Find the largest angle of the triangle, in degrees.


The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius using truncation (floor function), converted back to Fahrenheit using truncation, and again converted to Celsius using truncation.
For how many integer Fahrenheit temperatures between 30 and 1200 inclusive does the original temperature equal the final Fahrenheit temperature after two conversions?


Suzy is trying to solve the following equation by completing the square: $$25x^2+30x-24 = 0.$$ She aims to rewrite the given equation in the form: $$(dx + e)^2 = f,$$ where $d$, $e$, and $f$ are integers and $d > 0$. What is the value of $d + e + f$?


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


For how many integers $n$ with $1 \le n \le 2016$ is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 2 + e^{4 \pi i k / n} \right)^n - 1 \right)
\]
equal to zero?


An oreo shop now sells $6$ different flavors of oreos and $4$ different flavors of milk. Alpha and Beta decide to purchase some items. Alpha won't order more than 1 of the same flavor and refuses to buy chocolate flavor. Beta will only order oreos, and she is willing to have repeats of flavors. How many ways could they have left the store with a total of 3 products?


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