How many positive integers $n$ less than 150 have a corresponding integer $m$ divisible by 4 such that the roots of $x^2-nx+m=0$ are consecutive even integers?


Express $(7 - 3i) - 3(2 - 5i) + 4i(1 - 2i)$ as a complex number.

Solution: \boxed{9 + 16i}


Rationalize the denominator of $\frac{\sqrt{12}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}$.

Solution: \boxed{4 - \sqrt{6}}


Find the largest constant $D$ so that
\[2x^2 + 2y^2 + 3 \ge D(x + y)\] for all real numbers $x$ and $y.$


The mean (average) of 8, 12, and 24 is equal to the mean (average) of 16 and $x$. What is the value of $x$?


Calculate the expression $2w + 3 - 4w - 5 + 6w + 2^3 - 8w - 9$.

Solution: \boxed{-4w - 3}


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


What is the smallest number of whole 3-by-4 non-overlapping rectangles required to exactly cover a square region, without extra overhangs and without gaps?


Lucy goes to a café that serves blueberry muffins only on certain days. Over a span of 7 days, the café has a probability of \( \frac{3}{4} \) of serving blueberry muffins each day. What is the probability that the café serves blueberry muffins exactly 5 out of the 7 days Lucy visits?


Two tour guides are leading eight tourists. The guides decide to split up. Each tourist must choose one of the guides, and each guide must take at least one tourist. How many different groupings of guides and tourists are possible?


A line is described by the vector equation
\[\begin{pmatrix} 3 \\ -4 \end{pmatrix} \cdot \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} \right) = 0.\]
Express the equation of the line in the form $y = mx + b$ and provide the ordered pair $(m,b)$.

Solution: \boxed{\left(\frac{3}{4}, \frac{19}{2}\right)}


If $5x + 9 \equiv 3 \pmod{16}$, find the value of $3x + 8 \pmod{16}$.


A pyramid has an initial volume of 60 cubic inches and a square base. If each side of the base is tripled and the height is doubled, what is the new volume of the pyramid?

Solution: \boxed{1080 \text{ cubic inches}}


I have 8 shirts, 4 pairs of pants, and 8 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, plus white, yellow, red, and green. I refuse to wear an outfit in which all 3 items are the same color or where any item has stripes. If there are 2 striped shirts and 2 striped hats, how many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


What is the greatest common divisor of $12345$ and $6789$?


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


Let \( B, S, N, \) and \( K \) be positive real numbers such that
\begin{align*}
\log_{10} (BS) + \log_{10} (BN) &= 3, \\
\log_{10} (NK) + \log_{10} (NS) &= 4, \\
\log_{10} (KB) + \log_{10} (KS) &= 5.
\end{align*}
Compute the value of the product \( BSNK \).


If $g(x) = cx + d$ and $g^{-1}(x) = dx + c$ with $c$ and $d$ real, what is the value of $c+d$?


A convex polyhedron \( Q \) has \( 30 \) vertices, \( 70 \) edges, and \( 42 \) faces, \( 30 \) of which are triangular and \( 12 \) of which are quadrilaterals. Determine the number of space diagonals in \( Q \).


Factor completely: $x^8 - 4x^6 + 6x^4 - 4x^2 + 1$.


Let $\mathbf{v} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -1 \\ 5 \\ 2 \end{pmatrix}$. The columns of a matrix $A$ are $\mathbf{u}$, where $\mathbf{u}$ is any unit vector, $\mathbf{v}$, and $\mathbf{w}$. Find the largest possible determinant of matrix $A$.


In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 48. What is the perimeter of the shaded region? [asy]

defaultpen(1);

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draw(unitcircle);
draw(shift(2 * dir(-60)) * unitcircle);
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How many different positive three-digit integers can be formed using only the digits in the set $\{1, 3, 4, 4, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?


For each positive integer $n$, let $f(n) = n^4 - 400n^2 + 900$. Determine the sum of all values of $f(n)$ that are prime numbers.


Two positive integers \( m \) and \( n \) are chosen such that \( m \) is the second smallest positive integer with only two positive divisors and \( n \) is the largest integer less than \( 200 \) with exactly three positive divisors. What is \( m+n \)?


How many prime numbers have squares between 2500 and 4500?


Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 758^\circ.$


A ball is dropped from 800 feet high and always bounces back to three-quarters of the distance it just fell. After how many bounces will the ball first reach a height of less than 10 feet?


If $0 \le p \le 1$ and $0 \le q \le 1$, define a new function $H(p, q)$ by
\[
  H(p, q) = -3pq + 4p(1-q) + 2(1-p)q - 5(1-p)(1-q).
\]
Define $I(p)$ to be the maximum of $H(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $I(p)$?


Find the value of \( t \) if:
\[ \sqrt{3\sqrt{t-3}} = \sqrt[4]{10 - t} \]


Find \( x \) such that \( \lceil x \rceil \cdot x = 216 \). Express \( x \) as a decimal.


Given that $3 - 5\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx - 47 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Positive integers \(a\), \(b\), \(c\), and \(d\) satisfy \(a > b > c > d\), \(a + b + c + d = 2200\), and \(a^2 - b^2 + c^2 - d^2 = 2200\). Find the number of possible values of \(a\).


A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 20, and the axes of the cylinder and cone coincide. The height of the cylinder is three times its radius. Find the radius of the cylinder.


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


BoatWorks built 4 canoes in January of this year and then each subsequent calendar month they built three times the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of April of this year?


Find an ordered pair $(x,y)$ that solves the system:
\begin{align*}
4x &= -10 - 3y,\\
6x &= 5y - 32
\end{align*}

Solution: \boxed{\left(-\frac{73}{19}, \frac{34}{19}\right)}


Ice-cream-o-rama has now expanded its flavors to include a fourth one: mint. They still create new flavors by blending different proportions of their flavors into a mixture. If they make new flavors by taking five scoops of ice cream consisting of the flavors: chocolate, vanilla, strawberry, and mint, how many total flavors can they create if at least one scoop must be of mint?


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


Define $f(x)$ as the function that returns the remainder when $x^2$ is divided by $13$. Suppose we want to find the order of $3$ with respect to this function. What is the smallest positive integer $m$ for which $f^{(m)}(3)$ equals $3$?


Consider $b$ as a nonzero constant such that the quadratic equation $bx^2 + 15x + 4 = 0$ has exactly one solution. Determine the value of this solution.


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $720$?


The digits $1, 2, 3, 4, 5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers are both digits $1$ to the left of digit $2$ and digit $1$ to the left of digit $3$? (The digits $1$, $2$, and $3$ do not have to be next to each other.)


Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*}
x + y + z &= 5, \\
x^2 + y^2 + z^2 &= 11, \\
x^3 + y^3 + z^3 &= 35.
\end{align*}
Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$


Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 9x^2 + 42x + a$$ form an arithmetic progression and are not all real.


Let $x$, $y$, and $z$ be real numbers such that
\[\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = \cos x + \sin y + \cos z = 0.\]
Find the sum of all possible values of $\cos 2x + \cos 2y + \cos 2z.$


A new lattice pattern starts exactly as the previous one but has 7 columns per row instead of 5. The sequence is continued through 9 rows. What is the fifth number in the 9th row?


Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 3 \\ -1 \\ -5 \end{pmatrix} + t \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} -1 \\ 6 \\ 2 \end{pmatrix}.\]


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


$PQRS$ is a rectangle whose area is 24 square units. How many square units are contained in the area of trapezoid $TURS$?

[asy]
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for(int i=0; i < 4; ++i){
for(int k=0; k < 6; ++k){
draw((0,i)--(6,i));
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label("$Q$",(6,0),SE);
label("$R$",(6,4),NE);
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label("$T$",(1,4),N);
label("$U$",(5,4),N);
[/asy]


Simplify $\sqrt{48x} \cdot \sqrt{27x} \cdot \sqrt{32x}$. Express your answer in simplest radical form in terms of $x$.


Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[36, m] = 180$ and $\mathop{\text{lcm}}[m, 50] = 300$, what is $m$?


In triangle $ABC$, $AB = 7$, $AC = 15$, and $BC = 20$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $BC$. Find $GP$.


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a common fraction.


What is the least four-digit positive integer, with all different digits, at least one of which is a prime number, that is divisible by each of its digits?


Solution: \boxed{3x^{13} + 17x^{12} + x^{11} + 14x^9 + 8x^7 + 4x^4 + 6x^2 + 9}


If the consecutive integers from $60$ to $1$ were written as $$6059585756...,$$ what would be the $85^{\text{th}}$ digit to be written?


Each page number of a 589-page book is printed one time in the book. The first page is page 1 and the last page is page 589. When printing all of the page numbers, how many more 4's are printed than 8's?


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


What is the value of $m$ such that $10^m = 10^{2}\times \sqrt{\frac{10^{90}}{0.0001}}$?


A large rectangle has side lengths of $(2x + 14)$ and $(2x + 10)$. Within this rectangle, there is a rectangular hole with side lengths of $(4x - 6)$ and $(2x - 4)$. Additionally, there is a square-shaped addition to the area of side length $(x + 3)$ on the side of the rectangle, outside the hole. What is the area of the large rectangle excluding the hole but including the added square?

Solution: \boxed{-3x^2 + 82x + 125}


Linda thrice chooses a random integer between 1 and 60, inclusive. What is the probability that at least one of the numbers Linda chooses is a multiple of 4?


Find the product of all values of \( t \) such that \( t^2 - 10t + 25 = 0 \).


Suppose that $a$ and $b$ are digits, and the repeating decimal $0.\overline{ab}$, where $a$ and $b$ are not both zero, is expressed as a fraction in lowest terms. How many different denominators are possible, assuming $a$ and $b$ could be nine?


Write $0.\overline{156}$ as a simplified fraction.


How many three-digit integers less than 800 have all distinct digits?


You have a certain quantity of gold coins that you want to distribute equally among your 7 best friends. After dividing your gold coins into 7 equal piles, you observe that if you allocate all the coins, 2 friends would receive one extra gold coin each. The total number of gold coins you have is more than 50 but less than 150. What is the largest number of gold coins you could have in this scenario?


What is the smallest positive integer $n$ such that $3n$ is a perfect square and $5n$ is a perfect cube?


A rectangle is divided into three congruent rectangles, as shown in an imaginary diagram. Each smaller rectangle has a perimeter of 40 inches. If the width of the initial rectangle is half of its length, what is the perimeter of the original rectangle?

Solution: \boxed{72 \text{ inches}}


What is the remainder when \(5x^6 - 3x^5 + 6x^4 - 7x^3 + 3x^2 + 5x - 14\) is divided by \(3x - 6\)?


At time $t=0$, a ball is thrown downward at 30 feet per second from a height of 200 feet above the ground. The equation $h = -16t^2 - 30t + 200$ describes the height (in feet) of the ball. Determine how many seconds will elapse before the ball hits the ground.


Consider a quadratic polynomial \[x^2 - sx + q,\] with roots \(a\) and \(b\), where it is given that \[a+b = a^2+b^2 = a^3+b^3 = \cdots = a^{2008}+b^{2008}.\] Determine the maximum possible value of \[\dfrac{1}{a^{2009}}+\dfrac{1}{b^{2009}}.\]


Compute without using a calculator: $10! - \frac{9!}{3!}$


In a bag containing 9 green marbles and 6 purple marbles, Phil randomly draws a marble, notes its color, returns it, and repeats this process until he has drawn 8 marbles. What is the probability that exactly five of the marbles Phil draws are green? Express your answer as a decimal rounded to the nearest thousandth.


Determine the value of $k$ such that $\frac{-25-\sqrt{369}}{12}$ is a root of the quadratic equation $6x^2 + 25x + k = 0$.


What is the remainder when $2022 \cdot 2023 \cdot 2024 \cdot 2025$ is divided by 17?


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


The sequence starts from 1,000,000, and each term is obtained by repeatedly dividing the previous term by 3. What is the last integer in this sequence?


What is the total area of two overlapping circles where circle A has center at point $A(2, -1)$ and passes through point $B(5, 4)$, and circle B has center at point $C(3, 3)$ and passes through point $D(5, 8)$? Express your answer in terms of $\pi$.


Let $r$ and $s$ be the solutions to the quadratic equation $3x^2 - 5x - 8 = 0$. Compute the value of $(9r^2 - 9s^2)(r - s)^{-1}$.


Let function $t(x)$ have a domain $\{-2, -1, 0, 1\}$ and range $\{-1, 1, 3, 5\}$. Function $u(x)$ has a domain $\{0, 1, 2, 3\}$ and is defined by $u(x) = x + 2$. Determine the sum of all possible values of $u(t(x))$.


Consider a 4x4 square quilt made from sixteen unit squares. Some unit squares are divided into two right-angled triangles, and some are kept whole. The top two rows have alternating squares shaded (starting with the first square shaded) and the bottom two rows have the second and fourth squares in each row fully divided into two triangles, both of which are shaded. Calculate the fraction of the entire quilt that is shaded.


An airplane features three sections: First Class (30 seats), Business Class ($20\%$ of the total number of seats), and Economy ($\frac{3}{4}$ of the total number of seats). Calculate the total number of seats on the airplane.


Let $(x, y)$ be a solution to the system of equations \[\begin{aligned} \lfloor x \rfloor + \{y\} &= 3.7, \\ \{x\} + \lfloor y \rfloor &= 4.2. \end{aligned} \] Compute $|x - 2y|.$


How many different primes are in the prime factorization of $101 \cdot 103 \cdot 105 \cdot 107$?


How many two-digit numbers have digits whose sum is either a perfect square or a perfect cube?


In the updated diagram, there are additional triangles formed. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.

[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
draw((0,0)--(.5,0)--(.5,.5)--(0,0)--cycle,linewidth(1));
draw((.5,0)--(1,0)--(1,.5)--(.5,.5)--cycle, linewidth(1)); // Added new triangle
draw((1,0)--(1,1)--(.5,.5)--cycle, linewidth(1)); // Added new triangle

label("A",(0,1),NW);
label("B",(.5,.5),NE);
label("C",(1,0),SE);
label("D",(.5,0),S);
label("E",(0,0),SW);
label("F",(1,.5),SE);  // New point labelled
label("G",(1,1),NE);  // New point labelled

filldraw((.5,0)--(1,0)--(.5,.5)--(.5,0)--cycle,gray,black);
filldraw((1,0)--(1,.5)--(.5,.5)--cycle, gray, black);  // Shading added to new triangle
[/asy]


In the diagram, $DE = 15 \mbox{ cm},$ $DF = 36 \mbox{ cm},$ and $\angle D = 90^\circ.$ What is the area of triangle $DEF?$ Also, determine if triangle $DEF$ is isosceles.


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


Arc $AD$ is a semicircle with center $B$. The shaded region $ABD$ is "rolled" along a straight board $PQ$ until part $AD$ is aligned in a reversed orientation, with point $B$ landing at point $B'$ for the first time. If $BD$ is \( \frac{4}{\pi} \) cm, compute the length of the path that point $B$ travels.

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


Compute
$$\sum_{k=1}^{1000} k(\lceil \log_{\sqrt{3}}{k}\rceil- \lfloor\log_{\sqrt{3}}{k} \rfloor).$$


Find the value of $m + n$ where $m$ and $n$ are integers such that the positive difference between the roots of the equation $4x^2 - 12x - 9 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, with $m$ not divisible by the square of any prime number.


A girl has the following six coins in her purse: $3$ pennies, $1$ nickel, $1$ dime, and $1$ quarter. She draws two coins, records the sum of their values, and then returns them to her purse. This process is repeated. Determine the maximum number of different sums she can record.


Calculate the result of $203_4 + 321_4 - 112_4$ and express your answer in base 4.


I take variable $a$, double it, and add six. I subtract $3a$ from this new expression, and divide the resulting difference by two. What is my final expression in the simplest form?

Solution: \boxed{-\frac{a}{2} + 3}


The base of a triangular piece of paper $ABC$ is $20\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base $AB$. The area of the triangle that projects below the base is $81\%$ of the area of the triangle $ABC$. What is the length of $DE$, in cm?

Solution: \boxed{18 \text{ cm}}


The circle centered at $(3, -2)$ and with radius $5$ intersects the circle centered at $(3, 4)$ and with radius $\sqrt{17}$. Find $(AB)^2$, where $A$ and $B$ are the points of intersection.


How many different prime values of \( x \) will make this statement true: there are exactly \( 3 \) positive two-digit multiples of \( x \)?


Seventy percent of a train's passengers are women, and fifteen percent of those women are in the business class compartment. What is the number of women in the business class compartment if the train is carrying 300 passengers?


Calculate the value of $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \cdot \frac{1}{243} \cdot \frac{729}{1}$.


The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, etc. How many of the first 15 such sums are also prime?


In writing the integers from 10 through 149 inclusive, how many times is the digit 7 written?


Find the focus of the parabola \( y = 4x^2 + 8x - 5 \).

Solution: \boxed{(-1, -\frac{143}{16})}


For how many bases between two and nine inclusive does the representation of $385_{10}$ have a final digit of 1?


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


Find the smallest odd deficient number. A number is deficient if the sum of its proper divisors is less than the number itself.


Solution: \boxed{2022 - \frac{16}{2023}}


The shaded region shown consists of 16 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is now rotated about the $y$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.

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draw((0,1)--(5,1));
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draw((2,0)--(2,1));
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draw((4,0)--(4,1));

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draw((0,4)--(0,5),EndArrow(4));
draw((0,0)--(0,-1),EndArrow(4));
label("$y$", (0,5), N);
[/asy]


What is the smallest positive four-digit number divisible by 9 which has two even digits and two odd digits, starting with the digit '2'?


A sequence has $2500$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2500$ terms is $7000$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?


If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
  F(p, q) = -4pq + 5p(1-q) + 2(1-p)q - 3(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?


The Greenhill Soccer Club has 25 players, including 4 goalies. During an upcoming practice, the team plans to have a competition in which each goalie will try to stop penalty kicks from every other player, including the other goalies. How many penalty kicks are required for every player to have a chance to kick against each goalie?


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


How many times does the digit 8 appear in the list of all integers from 1 to 1000?


Milton spilled some ink on his algebra homework again, obscuring another coefficient. This time, he can only see part of his quadratic equation: $x^2 + bx + 48 = 0$. He recalls that the equation has two distinct negative integer solutions. What is the sum of all possible integers that could be under the ink stain where $b$ is?


A square has sides of length 12, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.

Solution: \boxed{144 + 108\pi}


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


How many positive $3$-digit numbers are multiples of $15$, but not of $45$?


Julia is taking a 20-question, multiple choice test where each question has four answer choices. She guesses the answers for the last five questions. What is the probability that she will get at least two of these last five questions correct?


How many integers between 1 and 2000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?


Solution: \boxed{\log_3 \left(\frac{43}{3}\right)}


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


Two spinners are divided into thirds and quarters, labeled as follows:
- Spinner A: 3, 5, 7
- Spinner B: 2, 6, 8, 10
Both spinners are spun twice. What is the probability that at least one of the products from the two spins will be an even number?


Suppose $x$ and $y$ are positive integers such that $\gcd(x, y)$ is divisible by exactly $9$ distinct primes and $\mathop{\text{lcm}}[x, y]$ is divisible by exactly $36$ distinct primes.

If $x$ has fewer distinct prime factors than $y$, then $x$ has at most how many distinct prime factors?


The equation
\[(x - \sqrt[3]{20})(x - \sqrt[3]{70})(x - \sqrt[3]{170}) = \frac{1}{2}\] has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$


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


A triangle is divided into four smaller triangles of equal area. The top triangle is then divided into four smaller triangles of equal area and the pattern continues indefinitely. The top triangle of each subsequent division is shaded. What fractional part of the triangle is shaded?


Compute the value of \[ M = 150^2 - 147^2 + 144^2 - 141^2 + \cdots - 6^2 + 3^2, \] where the additions and subtractions alternate every two terms, beginning with a subtraction.


I won a trip for six to a luxury cruise. I can bring 5 of my friends. I have 10 friends including my best friend. In how many ways can I form my travel group, if I must include my best friend?


You have a certain number of gold coins that you intended to divide equally among your 15 best friends. However, after attempting to distribute them equally, you discovered that 3 of your friends would receive one extra gold coin each compared to the others. You have fewer than 150 gold coins. What is the largest number of gold coins you could have for this distribution to occur?


Calculate the vector $\begin{pmatrix} -3 \\ 6 \\ 2 \end{pmatrix}$ projected onto the line given by the vector equation $\frac{x}{4} = \frac{y}{-1} = \frac{z}{3}$.

Solution: \boxed{\begin{pmatrix} -\frac{24}{13} \\ \frac{6}{13} \\ -\frac{18}{13} \end{pmatrix}}


A pet store has 12 puppies, 10 kittens, 5 hamsters, and 3 birds. Alice, Bob, Charlie, and David each want to buy a pet. For variety, they each want a different kind of pet. In what ways can Alice, Bob, Charlie, and David buy pets and leave the store satisfied?


A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts?


Determine the distance between the vertices of the parabolas represented by the graph of the equation:
\[
\sqrt{x^2+y^2} + |y-2| + 1 = 5
\]


In a different ellipse, the center is located at $(2, 3)$, one focus is at $(2, 1)$, and one endpoint of a semi-major axis is at $(0, 3)$. Determine the length of the semi-minor axis of this ellipse.


How many of the 150 smallest positive integers are congruent to 3 (mod 7)?


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


Three cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart, the second card is a diamond, and the third card is a club?


In triangle $ABC$, the orthocenter $H$ divides altitude $\overline{CF}$ such that $HF = 8$ and $HC = 24$. Calculate $\tan A \tan B$.


What is the largest value of $y$, if $\frac{y}{7} + \frac{2}{3y} = 3$?

Solution: \boxed{\frac{63 + \sqrt{3801}}{6}}


Solution: \boxed{\frac{\log_2 5}{6}}


Convert the point $(\rho,\theta,\phi) = \left( 15, \frac{5 \pi}{4}, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates.

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


Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2016} \right| = \frac{1}{2016}.\]


Simplify $\tan \frac{\pi}{12} + \tan \frac{5\pi}{12}.$


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2017,0),(2017,2018),$ and $(0,2018)$. What is the probability that $x > 9y$? Express your answer as a common fraction.


Determine the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix} + t \begin{pmatrix} 5 \\ 3 \\ 2 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}.\]


What is the greatest number of Mondays that can occur in the first 45 days of a year if the year starts on a Monday?


Simplify
\[
\frac{\sin^2 x - \cos^2 x}{2 \sin x \cos x}.
\]

Solution: \boxed{-\cot 2x}


Consider the function $h(x)$ defined by the equation
\[
(x^{2^5-1}-1)h(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^4}+1) - 1
\]
Find $h(2)$.


Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{01}$ is $\frac{1}{99}$.


Find \(89^{-1} \pmod{90}\), as a residue modulo 90. (Answer should be between 0 and 89, inclusive.)


The bottoms of two vertical poles are 18 feet apart on a region of flat ground. One pole is 6 feet tall, and the other is 20 feet tall. The base of the shorter pole is raised 3 feet above the ground. How long, in feet, is a wire stretched from the top of one pole to the top of the other pole?


In triangle $XYZ$, you know that $\angle X = 90^\circ$, the hypotenuse $YZ = 15$, and $\tan Z = 3\sin Z$. What is the length of side $XZ$?


Let $P(x) = 3\sqrt{x}$, and $Q(x) = x^3$. Consider functions $R(x) = x + 2$, and $S(x) = 2x$. Calculate $P(Q(R(S(P(Q(2))))))$.


When $\frac{18}{50000}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


A bookstore is evaluating its pricing strategy for a bestselling novel. The store's analysis shows that if the novel is priced at $p$ dollars (where $p \le 30$), the number of novels it can sell per month is given by $150 - 4p$. Determine the price $p$ that maximizes the bookstore's monthly revenue from this novel.


Let $a$ and $b$ be the roots of $mx^2 - (3m - 2) x + 7 = 0.$ Let $m_1$ and $m_2$ be the values of $m$ for which $a$ and $b$ satisfy
\[
\frac{a}{b} + \frac{b}{a} = \frac{3}{2}.
\]
Find
\[
m_1 + m_2.
\]


In a local pond, algae coverage doubled each day starting significantly earlier in the month. By day $25$ of the month, the pond was completely covered in algae. On what day of that month was the pond $87.5\%$ algae-free?


A square has sides of length 8, and a circle centered at one of its vertices has a radius of 8. What is 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}


The number of gallons of coffee a teacher drinks on any given day is inversely proportional to the hours of sleep she gets plus the number of hours she spends preparing for classes the previous night. On Wednesday, she got 8 hours of sleep and spent 2 hours preparing, drinking 3 gallons of coffee. On Thursday, she got 5 hours of sleep and spent 3 hours preparing. How many gallons of coffee did she drink?


How many paths are there from point $A$ to point $B$ in a grid of size 7x8, if movement is restricted to upward or rightward steps only?


Determine the coefficient of $x^4$ after simplifying the expression $2(x^4 - 2x^3) + 3(2x^2 - 3x^4 + x^6) - (5x^6 - 2x^4)$.


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 280^\circ.$


Determine the sum of all integers $k$ that can be written in the form:
\[k = (a_2a_1a_0)_{-2+i}\]
where $a_2 \neq 0$, $k$ is a real number, and $a_i$ are integers in $\{0, 1, 2, \ldots, 4\}$.


Let $a$ and $b$ be real numbers such that
\[\frac{a}{b} + \frac{a}{b^2} + \frac{a}{b^3} + \dots = 2.\]
Find
\[\frac{a}{2a + b} + \frac{a}{(2a + b)^2} + \frac{a}{(2a + b)^3} + \dots.\]


A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (3,0), (3,2), (0,2)$. What is the probability that $P$ is closer to the point $(1,1)$ than it is to the point $(4,1)$?


How many two-digit positive integers are congruent to 1 (mod 5)?


In triangle $ABC$, let $\cos(3A-B) + \sin(A+B) = 2$ and $AB = 4$. Determine the length of side $BC$.

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


Find the distance between the foci of the ellipse defined by the equation
\[\sqrt{(x-4)^2 + (y-5)^2} + \sqrt{(x+6)^2 + (y-9)^2} = 24.\]


Tracy originally had a bag of candies. She ate $\frac{1}{5}$ of them and then gave $\frac{1}{3}$ of the remaining candies to her friend Sam. Tracy and her dad then each ate 10 candies from what was left. Lastly, Tracy's sister took anywhere from two to six candies, leaving Tracy with five candies. How many candies did Tracy start with?


A valid license plate in Xanadu now consists of three letters followed by two digits, and then one more letter at the end. How many valid license plates are possible under this new format?


Given that the roots of the polynomial $x^2 - x - 1$ are also roots of a polynomial $x^6 - bx - c$, find the product $bc$.


Solution: \boxed{\sqrt{985 - 56\sqrt{197}}}


If $x$ and $y$ are positive real numbers such that $7x^2 + 21xy = x^3 + 3x^2 y^2,$ find the value of $x$.


A girl has the following seven coins in her pocket: $2$ pennies, $2$ nickels, $1$ dime, $1$ quarter, and $1$ half-dollar. She takes out two coins, records the sum of their values, and then puts them back with the other coins. She continues to take out two coins, record the sum of their values, and put them back. How many different sums can she record at most?


Determine how many three-digit positive integers exist such that the sum of the digits equals $7$ and the last digit is even.


The entry fee at a science fair is $ \$30$ per adult and $ \$15$ per child. On a particular day, the science fair collected a total of $ \$2250$ in entry fees, with at least one adult and one child attending. What is the ratio of adults to children on that day that is closest to 2?


Solution: \boxed{\frac{-1 + \sqrt{3 + 2\sqrt{3}}}{2}}


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


What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{31}{198}$?


What is the expected value of the roll of a biased 6-sided die where the probabilities of rolling 1, 2, 3, 4, 5, and 6 are $\frac{1}{10}$, $\frac{1}{10}$, $\frac{1}{10}$, $\frac{1}{10}$, $\frac{1}{10}$, and $\frac{1}{2}$ respectively?


How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 4, 5, 5, 6\}$ if no digit may be used more times than it appears in the given set of available digits?


How many odd numbers are greater than 150 and less than 350?


A right pyramid has a square base with each side measuring 8 cm. Its peak is 10 cm above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges, rounding your answer to the nearest whole number.


If the consecutive integers from $60$ to $1$ were written as $$6069585756...,$$ what would be the $85^{\text{th}}$ digit to be written?


Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.


Solution: \boxed{16 \sqrt{2}}


Simplify: $$\dfrac{\sqrt{450}}{\sqrt{200}}+\dfrac{\sqrt{242}}{\sqrt{121}}.$$
Express your answer as a common fraction.

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


The operation $\oplus$ is defined by
\[a \oplus b = \frac{a + b}{1 + ab}.\] Compute
\[1 \oplus (2 \oplus (3 \oplus (\dotsb \oplus (999 \oplus 1000) \dotsb))).\]


In an isosceles triangle, two angles are the same and the third angle is $40^\circ$. What is the degree measure of each of the larger angles?


How many even multiples of 12 are there between 50 and 250?


The sequence $3, 10, 17, x, y, 31, z$ is arithmetic. What is the value of $x + y + z$?


A line with slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. Both lines intersect at $E(3,3)$. What is the area of the shaded quadrilateral $OBEC$?


Solve for $z$ in the equation $4 - 3iz = 1 + 5iz$. Express your answer in standard form.

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


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $42$ faces, $30$ of which are triangular and $12$ of which are quadrilaterals. Calculate the number of space diagonals in $Q$.


Let \(A=(0,6)\) and \(B=(0,10)\). Points \(A'\) and \(B'\) are on the line \(y=x\), and lines \(\overline{AA'}\) and \(\overline{BB'}\) intersect at point \(C=(3,7)\). What is the length of \(\overline{A'B'}\)?


Of the numbers $\frac{7}{8}, \frac{9}{10}$, $\frac{4}{5}$, and $\frac{17}{20}$, which number is the arithmetic mean of any two of the other numbers?


At Lincoln High School, there are 30 players on the soccer team. All players are participating in at least one of the three science courses: biology, chemistry, or physics. There are 15 players taking biology, 10 players taking chemistry, 5 players taking physics, and 2 players are taking all three sciences. How many unique players are taking chemistry, considering some might be taking more than one science subject?


What is the smallest positive integer $n$ such that $23n \equiv 5678 \pmod{11}?$


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $10 \leq y \leq 99$, what is the largest possible value of $a+b+c$?


Let $g$ be a function defined on the positive integers such that:

(i) $g$ is strictly increasing (i.e., $g(n+1) > g(n)$ for all positive integers $n$).
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(18)$.


If $g(n)$ is defined as the sum of all the positive divisors of integer $n$, how many integers $j$ exist such that $1 \le j \le 3025$ and $g(j) = 1 + \sqrt{j} + j$?


When the base-10 integers 500 and 2500 are expressed in base 2, how many more digits does 2500 have than 500 (after being converted)?


Jane has a bag containing 9 blue marbles and 6 red marbles. She randomly draws a marble, records its color, returns it to the bag, and then repeats this process 8 times. What is the probability that she draws exactly 5 blue marbles? Express your answer as a decimal rounded to the nearest thousandth.


Evaluate the expression \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{20}}{x^2\cdot x^4 \cdot x^6 \cdots x^{30}}\] when $x = 3$.


Five horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed, but no selection includes the first and fifth horizontal lines together?


Find $n$ if $n\cdot n! - n! = 5040 - n!$, where $n! = n\cdot (n-1)\cdot (n-2)\cdots 2\cdot 1$.


Suppose all four numbers \[2 - \sqrt{5}, \; -2-\sqrt{5}, \; 3+\sqrt{8}, \; 3-\sqrt{8}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Calculate the product $\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{2012}{2016}\right)$. Express your answer as a common fraction.


A solid rectangular block consists of $N$ congruent 1-cm cubes glued together face to face. When positioned so that three faces are visible, precisely $252$ of the 1-cm cubes are not visible. Determine the smallest possible value of $N.$


It is currently $10\!:\!45\!:\!00 \text{ a.m.}$. What time will it be in $4500$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)


The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, and $z$ are $10$, $6$, and $4$, respectively. What is the value of $x^2 + y^2 + z^2$?


In a recent survey conducted by Mary, she found that $72.4\%$ of participants believed that rats are typically blind. Among those who held this belief, $38.5\%$ mistakenly thought that all rats are albino, which is not generally true. Mary noted that 25 people had this specific misconception. Determine how many total people Mary surveyed.


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


At the Supernatural Gathering, there are 25 gremlins and 20 imps. This year, half of the imps are open to shaking hands amongst each other. All gremlins are friendly and shake hands with each other and with all imps. No participant shakes hands more than once with the same counterpart. How many total handshakes occurred at the event?


In a bike race, Alberto, Bjorn, and Charlie are participating. Their speed is plotted on a graph over a period of 6 hours. Alberto travels 10 miles per hour, Bjorn travels 8 miles per hour, and Charlie travels 9 miles per hour. After 6 hours, how many more miles has Alberto traveled compared to Bjorn and Charlie?


Jane shoots basketballs and makes 60% of her initial 30 shots. After taking an additional 10 shots, her shooting percentage increases to 62%. How many of these additional 10 shots did she make?


A rectangle and two circles, each with a radius of 3 cm, are arranged as shown. What is the area, in square centimeters, of the rectangle? The circles are placed such that each touches two sides of the rectangle and each other.

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


Luna writes down all six-digit numbers that contain each of the digits 1, 2, 3, 4, 5, and 6 exactly once. What is the smallest number in Luna's list that is divisible by 6?


What is the greatest number of Mondays that can occur in the first 45 days of a year?


What is $4512_6 - 2324_6 + 1432_6$? Express your answer in base 6.


How many positive $3$-digit numbers are multiples of $35$, but not multiples of $70$?


Find the numerical value of
\[
\frac{\sin 15^\circ \cos 10^\circ + \cos 165^\circ \cos 105^\circ}{\sin 25^\circ \cos 5^\circ + \cos 155^\circ \cos 95^\circ}.
\]


For each positive integer \( n \), let \( n! \) denote the product \( 1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n \).

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


Find the complex number \( z \) such that
\[ |z - 2| = |z + 4| = |z - 2i|. \]

Solution: \boxed{-1 - i}


Three tour guides are leading eight tourists. The guides decide to split up such that each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?


Suppose functions $g$ and $f$ have the properties that $g(x) = 2f^{-1}(x)$ and $f(x) = \frac{5}{x^2 + 1}$. For what value of $x$ does $g(x) = 9$?


In the complex plane, $z,$ $z^2,$ $z^3$ represent, in some order, three vertices of a non-degenerate equilateral triangle. Determine all possible perimeters of the triangle.


Let \(a, b, c\) be positive real numbers such that \(a + b + c = 4.\) Find the minimum value of
\[
\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.
\]


In the diagram, square $ABCD$ has sides of length $6,$ and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x.$ Determine the measure of angle $BPC.$


A positive real number $x$ is such that \[
\sqrt[3]{1-x^4} + \sqrt[3]{1+x^4} = 1.
\]Find $x^8.$


Two sides of an isosceles triangle are 15 inches and 30 inches. If the shortest side of a similar triangle is 45 inches, what is the perimeter of the larger triangle?

Solution: \boxed{225\text{ inches}}


What is the largest integer \( n \) for which \( \binom{10}{4} + \binom{10}{5} = \binom{11}{n} \)?


Find the smallest positive integer $b$ for which $x^2 + bx + 2007$ factors into a product of two polynomials, each having integer coefficients.


What is the $62$nd digit after the decimal point when $\frac{51}{777}$ is expressed as a decimal?


Let $D'$ be the determinant of the matrix whose column vectors are **$\mathbf{u}, \mathbf{v}, \mathbf{w}$**. Find the determinant of the matrix whose column vectors are **$\mathbf{u} - \mathbf{v}, \mathbf{v} - \mathbf{w}, \mathbf{w} - \mathbf{u}$**, in terms of $D'$.


Calculate the sum of the first eight terms of the geometric sequence starting with \( \frac{1}{3} \) and where each term is one third of the previous term.


Bag X has 5 red marbles and 3 green marbles. Bag Y has 7 orange marbles and 5 purple marbles. Bag Z has 4 orange marbles and 6 purple marbles. A marble is drawn at random from Bag X. If it is red, a marble is drawn at random from Bag Y, otherwise, if it is green, a marble is drawn at random from Bag Z. What is the probability that the second marble drawn is orange?


If
\[\frac{\cos^6 \alpha}{\cos^3 \beta} + \frac{\sin^6 \alpha}{\sin^3 \beta} = 1,\]
then find the sum of all possible values of
\[\frac{\sin^6 \beta}{\sin^3 \alpha} + \frac{\cos^6 \beta}{\cos^3 \alpha}.\]


The function \( g \) is defined on the set of integers and satisfies
\[
g(n) = \begin{cases}
n-2 & \mbox{if } n \ge 500 \\
g(g(n+5)) & \mbox{if } n < 500
\end{cases}
\]
Find \( g(50) \).


Fully factor the expression: $3y^2 - 75z^2$.

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


Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $2 + i$. Find $a + b$.


Lucas, Jane, and Ellie plan to meet at a park. Lucas starts from the location \( (2, 5) \) and Jane starts from \( (10, 1) \). If they decide to meet at a point which divides the line segment joining them in the ratio 1:3 starting from Lucas' point, at what coordinates should Ellie go to meet them?

Solution: \boxed{(4, 4)}


In kite $EFGH$, the measure of angle $EFG$ is twice the measure of angle $FGH$. What is the measure of angle $EHG$?


A five-digit integer is selected at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number? Express your answer as a common fraction.


Given vectors $\mathbf{u}$ and $\mathbf{z}$ with magnitudes $\|\mathbf{u}\| = 5$, $\|\mathbf{z}\| = 8$, and the dot product $\mathbf{u} \cdot \mathbf{z} = 20$, calculate $\|\operatorname{proj}_{\mathbf{z}} \mathbf{u}\|$.


Simplify
\[\frac{\sin{40^\circ}+\sin{80^\circ}}{\cos{40^\circ}+\cos{80^\circ}}.\]
Enter your answer as a trigonometric function evaluated at an integer angle, such as "sin 7".

Solution: \boxed{\tan 60^\circ}


The coefficients of the polynomial
\[ x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 \]
are all integers. Let $n$ be the exact number of integer roots of this polynomial, counting multiplicity. Enter all possible values of $n$, separated by commas.

Solution: \boxed{0, 1, 2, 3, 5}


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be either less than 3 or equal to 7? Express your answer as a common fraction.


In right triangle $ABC$, we have $\angle A = 40^\circ$, $\angle B = 90^\circ$, and $BC = 7$. Find $AB$ to the nearest tenth. You may use a calculator for this problem.


Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[
\{-8, -6, -4, -1, 3, 5, 7, 10\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


There are two **positive** integers \( c \) for which the equation
\[ 7x^2 + 15x + c = 0 \]
has rational solutions. What is the product of those two values of \( c \)?


A bookstore is deciding what price it should charge for a novel. They discovered that if the book's price is $p$ dollars (where $p \le 30$), then the number of books sold per month can be modeled by the formula $150-6p$. Furthermore, the bookstore incurs a fixed cost of $200 per month for storing these books. What price should the store charge to maximize its profit?


Define $h(x) = 7x - 6$. If $h(x) = f^{-1}(x) - 2$ and $f^{-1}(x)$ is the inverse of the function $f(x) = cx + d$, find $7c + 7d$.


Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$  Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\]


Kelly is attempting to unlock her electronic device with a four-digit password. She remembers that she only used digits from 1 to 6, each digit possibly being repeated, and that each odd digit must be followed by an even digit, with no specific rule for the sequences following even digits. How many combinations might Kelly need to consider?


What is the value of $x$ for which $(10-2x)^2 = 4x^2 + 16$?


Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?


A ball is dropped from 30 feet high and always bounces back up one-third the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


6 12-sided dice are rolled. What is the probability that exactly three dice show a two-digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.)


Find the positive real value of $t$ that satisfies $|t+3i\sqrt{2}| |8-3i| = 40$.

Solution: \boxed{t = \sqrt{\frac{286}{73}}}


Let $a,$ $b,$ and $c$ be angles such that:

\[
\sin a = \cot b, \quad \sin b = \cot c, \quad \sin c = \cot a.
\]

Find the largest possible value of $\cos a.$

Solution: \boxed{\sqrt{\frac{3 - \sqrt{5}}{2}}}


A Senate committee has 7 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions?


What is the remainder when $2005 \cdot 2006 \cdot 2007 \cdot 2008 \cdot 2009$ is divided by 17?


John has $\$50$. He already spent 180 cents on a bus ticket. If the price of each stamp is 45 cents, how many stamps can he buy with his remaining money?


When the vectors $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ are both projected onto the same vector $\mathbf{u},$ the result is $\mathbf{q}$ in both cases.  Find $\mathbf{q}.$

Solution: \boxed{\begin{pmatrix} \frac{33}{10} \\ \frac{11}{10} \end{pmatrix}}


A tank needs to be filled with water. When the tank is one-eighth full, 150 gallons of water are added, making the tank two-thirds full. What is the total capacity of the tank when full?


How many prime numbers are between 50 and 70 that are also of the form \(6k + 1\) or \(6k - 1\)?


What is the smallest possible median for the six number set $\{x, 2x, 3, 2, 5, 4x\}$ if $x$ can be any positive integer?


A QR-like code consists of a $5 \times 5$ grid with either black or white squares. Every valid code must contain at least one square of each color. A code is termed $\textit{symmetric}$ if it looks the same when rotated by multiples of $90^\circ$ around the grid center, or when reflected across a line joining opposite corners or midline of opposite sides. Determine the total number of possible symmetric codes.


A number in the set $\{40, 41, 42, 43, ... , 990\}$ is randomly selected. What is the probability that it is a number that ends with the digit 5? Express your answer as a common fraction.


A cylindrical can of tomato sauce has a diameter of 4 inches and a height of 5 inches, and it is priced at $\$$0.80. Using the same pricing rate per volume, what would be the cost of a larger cylindrical can that is 8 inches in diameter and 10 inches high?


An infinite geometric series has a first term of \( 400 \) and a sum of \( 2500 \). What is its common ratio?


Let \(y\) be a real number such that \(y + \frac{1}{y} = 5\). Let \(T_m = y^m + \frac{1}{y^m}\). Calculate \(T_6\).


Alice needs to paint 10 cylindrical columns, including the top faces but not the bottom faces. Each column has a height of 14 feet and a diameter of 8 feet. One gallon of paint covers 400 square feet. How many gallons of paint does Alice need to buy if paint is sold only in full gallons?


Suppose $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1$, $x + \frac{1}{z} = 4$, and $y + \frac{1}{x} = 30$. Find $z + \frac{1}{y}$.


What integer $n$ satisfies $0 \leq n < 31$ and $$-250 \equiv n \pmod{31}~?$$


How many different positive values of \( x \) will make this statement true: there are exactly \( 3 \) positive two-digit multiples of \( x \).


If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 12, given that no two dice show the same number? Express your answer as a common fraction.


In a school cafeteria, Julia wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Julia's available options in the cafeteria:

\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Hamburger & Cola & Apple Pie \\ \hline
Pasta & Lemonade & Fruit Salad \\ \hline
Salad &  &  \\ \hline
Taco &  &  \\ \hline
\end{tabular}

The beverages Cola and Lemonade are only available if Julia chooses either Hamburger or Pasta as her main course. How many distinct possible lunch combinations can Julia buy?


Find the last two digits of the following sum: $$3! + 5! + 10! + 15! + \cdots + 95! + 100!.$$


In the diagram, $\triangle ABC$ is right-angled at $A,$ with $AB=45$ and $AC=60.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $BD.$


If \( 5x \equiv 10 \pmod{25} \) and \( 4x \equiv 20 \pmod{25} \), then what is the remainder when \( x^2 \) is divided by 25?


In triangle $PQR$, $PQ = 8$, $PR = 17$, and the length of median $PM$ is 12. Additionally, the angle $\angle QPR = 60^\circ$. Find the area of triangle $PQR$.


The projection of $\begin{pmatrix} -3 \\ c \end{pmatrix}$ onto $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ is
\[-4\begin{pmatrix} 3 \\ 2 \end{pmatrix}.\]Find $c.$


For a specific square in a Cartesian plane, two opposite vertices lie on the parabola $y = x^2$, and the other two opposite vertices lie on the parabola $y = -x^2 + 4$. Determine the smallest possible area of the square.


Determine the smallest positive integer \(m\) such that \(7^m \equiv m^7 \pmod{5}\).


Simplify the expression $2x - 3(2-x) + 4(2+x) - 5(1-3x)$.

Solution: \boxed{24x - 3}


In a basket, there are 49 chips, each either red or blue. If the number of red chips exceeds the number of blue chips by a prime number, what is the smallest possible number of blue chips?


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


A street has parallel curbs 30 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 10 feet and each stripe is 60 feet long. Find the distance, in feet, between the stripes.


Determine the values of $a$, $b$, and $r$ of the circle given by the equation $x^2 + 14y + 65 = -y^2 - 8x$. Let $a$ and $b$ be the coordinates of the center of the circle, and $r$ be its radius. What is the sum $a + b + r$?


Consider polynomials \(Q(x)\) of degree at most 3, each of whose coefficients is an element of \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). How many such polynomials satisfy \(Q(1) = 10\)?


Samantha is ten years older than Ravi. Twelve years from now, Samantha will be three times as old as Ravi was five years ago. What is the sum of their current ages plus the current age of Kim, who is half as old as Ravi?


Solution: \boxed{(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}


Let $a$, $b$, and $c$ be the roots of
\[ x^3 - 5x^2 + 7 = 0. \]
Find the monic polynomial, in $x$, whose roots are $a+3$, $b+3$, and $c+3$.

Solution: \boxed{x^3 - 14x^2 + 57x - 65}


What is the smallest odd number with four different prime factors, including the number $13$?


Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[P(1) = P(3) = a,\]
\[P(2) = P(4) = P(6) = P(8) = -a.\]
What is the smallest possible value of $a$?


What is the sum of the odd integers that are both greater than 5 and less than 20?


A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one book is in the library and at least one is with students? (The books should be considered indistinguishable.)


The domain of the function \( p(x) = (x^3 + 3)^2 \) is \( [-1, \infty) \). What is the range?

Solution: \boxed{[4, \infty)}


A biased coin flips heads with a probability of \( \frac{1}{4} \) and tails with a probability of \( \frac{3}{4} \). If you win \$4 for flipping heads and lose \$3 for flipping tails, what is the expected value, in dollars, of your winnings after one flip?


Find the maximum value of
\[
\begin{vmatrix}
1 & 1 & 1 \\
1 & 1 + \sin 2\theta & 1 \\
1 & 1 & 1 + \cos 2\theta
\end{vmatrix},
\]
as $\theta$ ranges over all angles.


A digit is written to the right of the units digit of $824$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?


Given that $-1 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 48 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$


Let $a$, $b$, $c$, and $d$ be four digits such that $a = d-3$, $b = d-2$, and $c = d-1$, forming the numbers $abcd$ and $dcba$. What is the greatest common divisor of all numbers of the form $abcd+dcba$, where $d$ is between 3 and 9, inclusive?


For all complex numbers $z$, define the function $f(z)$ as follows:
\[f(z) = \left\{
\begin{array}{cl}
2z^{2}&\text{ if }z\text{ is not real}, \\
-3z^2 &\text{ if }z\text{ is real}.
\end{array}
\right.\]
Find $f(f(f(f(1+i))))$.


The Bulls are playing the Knicks in a playoff series. To progress to the next round, a team must win 5 games out of a maximum of 9 games. If the Knicks win each game with a probability of $\frac{3}{4}$ and there are no ties, what is the probability that the Bulls will win the series in exactly 9 games?


If two sides of a triangle are 8 and 15 units, and the angle between them is 30 degrees, what is the length of the third side?

Solution: \boxed{\sqrt{289 - 120\sqrt{3}}}


Let $a$ and $b$ be the roots of the equation $x^2 - mx + 3 = 0.$ Suppose that $a + c\cdot(1/b)$ and $b + c\cdot(1/a)$ are the roots of the equation $x^2 - px + q = 0$. Determine $q$, where $c$ is a real number constant.

Solution: \boxed{3 + 2c + \frac{c^2}{3}}


In a triangle with sides of lengths \(a\), \(b\), and \(c\), given that
\[(a + b + c)(a + b - c) = 2ab + a^2.\]
Find the angle opposite the side of length \(c\).


Write the following expression as a polynomial: $$(3x^2+4x+8)(x-1)-(x-1)(x^2+5x-72)+(4x-15)(x-1)(x+6).$$

Solution: \boxed{6x^3 + 2x^2 - 18x + 10}


Real numbers $a$ and $b$ are chosen with $2 < a < b$ such that no triangle with positive area has side lengths $2, a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $\tfrac{1}{2}$. Determine the smallest possible value of $b$.

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


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

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


Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? Express your answer in simplest radical form.

Solution: \boxed{\sqrt{34 + 15\sqrt{2}}}


The function $f(x)$ satisfies
\[f(4^x) + xf(4^{-x}) = 3\]
for all real numbers $x.$ Find $f(4)$.


Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 325.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane form a right triangle with right angle at $q.$ Find the square of the hypotenuse, $h^2,$ of this triangle.


15 balls numbered 1 through 15 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and at least one of the drawn balls is even-numbered?


Write the expression $\frac{6+5c}{9}+3$ as a single fraction.


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $38$ faces, $20$ of which are triangular and $18$ of which are quadrilaterals. How many space diagonals does $Q$ have?


How many triangles with positive area can be formed such that all their vertices are at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $4$, inclusive?


Suppose $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 3^2 \pmod{2^4}, \\
3+x &\equiv 2^3 \pmod{3^4}, \\
4+x &\equiv 3^3 \pmod{2^3}.
\end{align*}
What is the remainder when $x$ is divided by $24$?


Jen has misplaced a friend’s number, recalling only the initial three digits that are either 324, 327, or 328. The subsequent digits are distinct and consist of 0, 2, 5, 8, and 9, though the precise sequence is forgotten. Jen decides to call a number at random that fits these stipulations. What is the probability that Jen will accurately dial her friend’s number? Express your result as a common fraction.


For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[8+8r+8r^2+8r^3+\cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=1152$. Find $T(b)+T(-b)$.


Tracy had a bag of candies. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. Tracy and her dad then each ate 10 candies from what Tracy had left. Finally, Tracy's sister took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the start?


What are the last two digits of the sum $1! + 2! + 3! + 4! + \cdots + 20!$?


Two circles are drawn in a 10-inch by 15-inch rectangle. Each circle has a diameter of 5 inches. Calculate the greatest possible distance between the centers of the two circles, assuming the circles do not extend beyond the rectangular region.

Solution: \boxed{5\sqrt{5} \text{ inches}}


The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$?


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


Two writers attend a library every day. They independently arrive at the library at a random time between 1 p.m. and 3 p.m., and each stays for exactly $n$ minutes. The probability that they are both in the library at the same time is $50\%,$ and $n = x - y\sqrt {z},$ where $x, y,$ and $z$ are positive integers, and $z$ is not divisible by the square of any prime. Find $x + y + z.$


Let set $\mathcal{A}$ be an 80-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.


Find the value of the first term in the geometric sequence $a, b, c, d, e, f, 128, 256$.


Solution: \boxed{74 + 28\sqrt{2}}


Determine the value of $n$ such that:
\[
(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^n.
\]


What is the base-10 integer 300 when expressed in base 6?


In a group of 30 people, 14 are left-handed, 20 enjoy rock music, and 5 are right-handed and dislike rock music. How many members are left-handed and like rock music, assuming people are either left-handed or right-handed?


The American Mathematics College is holding its orientation for incoming freshmen. The incoming freshman class contains fewer than $600$ people. When the freshmen are told to line up in columns of $17$, $16$ people are in the last column. When the freshmen are told to line up in columns of $19$, $18$ people are in the last column. Determine the number of people in the incoming freshman class.


Let \( a \) and \( b \) be positive real numbers such that \( a + 3b = 2 \). Find the minimum value of
\[\frac{1}{a} + \frac{3}{b}.\]


In rectangle $ABCD$, $AB = 150$. Let $E$ be the midpoint of $\overline{AD}$. Let $F$ be a point on $AD$ such that $\overline{BF}$ is perpendicular to $\overline{AD}$. Given that line $AC$ is also perpendicular to line $BE$, find the greatest integer less than $AD$.


Marius is entering a wildlife photo contest, and wishes to arrange his nine snow leopards of different heights in a row. If the shortest three leopards have inferiority complexes and demand to be placed at the ends of the row, with the shortest at one specific end, how many ways can he line up the leopards?

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


What is the smallest possible median for the five number set $\{x, 2x, 4, 3, 6\}$ if $x$ can be any integer?


Let $n$ equal the number of sides in a regular polygon. For $3 \leq n \leq 15$, how many values of $n$ result in a regular polygon where the common degree measure of the exterior angles is not an integer?


How many total days were there from the year 2004 through 2008?


How many non-similar triangles have angles whose degree measures are distinct positive integers in an arithmetic progression with the middle angle being a multiple of 10?


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


Consider a fictional language with ten letters in its alphabet: A, B, C, D, F, G, H, J, L, M. Suppose license plates of six letters utilize only letters from this alphabet. How many license plates of six letters are possible that begin with either B or D, end with J, cannot contain any vowels (A), and have no letters that repeat?


Find the product of the digits of the largest integer such that:
- The sum of the squares of its digits is $85$.
- Each digit is larger than the one on its left.


A four-inch wooden cube has all its faces painted blue. This cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least two faces?


If \( 2a + 5b = 40 \) and \( 4a + 3b = 41 \), what is the product \( ab \)?


Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a square. His leash is exactly long enough to reach the midpoint of each side of the square. If the fraction of the area of Fido's yard that he can reach while on his leash is expressed as $\frac{\sqrt{a}}{b}\pi$, find the value of the product $ab$.


If
\[\sin x = \frac{2ab}{\sqrt{4a^2b^2 + (a^2 + b^2)^2}},\] where $a > b > 0$ and $0^\circ < x < 90^\circ,$ find $\tan x$ in terms of $a$ and $b.$

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


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


Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?


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


There are two stages involved in digging the foundation for a duplex. In the first stage, it takes 10 men working steadily 6 days to dig half of the foundation. In the second stage, it takes 15 men working another number of days to complete the other half. How many days would it take 20 men to dig the second half, if they work at the same rate as before? Express your answer as a decimal to the nearest tenth.


Define \( E(a, b, c) = a \times b^c \). What positive value of \( r \) is the solution to the equation \( E(r, r, 5) = 1024 \)?


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


For the quadrilateral in the diagram, determine how many different whole numbers could be the length of the diagonal represented by the dashed line.

[asy]
draw((0,0)--(4,6)--(12,1)--(7,-7)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("6",(2,3),NW);
label("10",(8,3.5),NE);
label("14",(9.5, -3),SE);
label("12",(3.5,-4),SW);
[/asy]


In triangle $ABC$, $AB=12$, $BC=20$, and $CA=16$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy \(p+q+r=1\) and \(p^2+q^2+r^2=3/5\). The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Given $x^3 - \frac{1}{x^3} = C,$ where $C > 0$, find the minimum possible numerical value for $\frac{C^2 + 9}{C}$.


Eight friends plan to attend a baseball game. However, five of them are only 1/3 sure they will stay for the entire game (the other three will definitely stay for the whole time). What is the probability that at least 7 friends stay for the entire game?


Determine the sum of the positive integers $a$, $b$, and $c$ if the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ reduces to $\cos ax \cos bx \cos cx = 0$ for some positive integers $a$, $b$, $c$.


Solution: \boxed{\left(\frac{15}{16}\right)^{\frac{1}{7}}}


Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 4$ and $PT = AB - 2PA,$ then what is $PB$?

Solution: \boxed{14 + 2\sqrt{13}}


A bag contains 4 tan, 3 pink, and 5 violet chips. If all 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 4 tan chips are drawn consecutively, the 3 pink chips are drawn consecutively, and the 5 violet chips are drawn consecutively, and specifically in the tan-pink-violet order?


A 12-foot by 15-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles of radius 1 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total area of the floor that is shaded.

Solution: \boxed{180 - 45\pi}


Find the smallest value of $x$ that satisfies the equation $|5x - 3| = 45$.


What is the remainder when $4x^3 - 10x^2 + 15x - 17$ is divided by $4x - 8$?


To measure the distance of his cycling route, John cycled a total of $3618_9$ kilometers. How many kilometers (in base ten) did John cycle?


A square has sides of length 12, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.

Solution: \boxed{144 + 108\pi}


In how many different ways can 4 men and 4 women be placed into two groups of two people and one group of four people, with the condition that there must be at least one man and one woman in the groups of two, and at least two men and two women in the group of four? Note that identically sized groups are indistinguishable.


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


What is the sum of all the even integers between $400$ and $700$?


Let $f(x)=\frac{x+3}{x^2+ax+b}$, and $f(x)$ has vertical asymptotes at $x=2$ and $x=-3$. Additionally, the function has a horizontal asymptote at $y=0$. Find the values of $a$ and $b$, and the sum $a+b$.


Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.


How many distinct arrangements of the letters in the word "balloon" are there, given that it contains repeating letters?


If $f(x) = f(3-x)$ for all $x$, what line is necessarily an axis of symmetry for the graph of $y=f(x)$?

Solution: \boxed{x = 1.5}


Let $g$ be defined by  \[g(x) = \left\{
\begin{array}{cl}
x+3 & \text{ if } x \leq 2, \\
x^2 - 4x + 5 & \text{ if } x > 2.
\end{array}
\right.\]Calculate $g^{-1}(1)+g^{-1}(6)+g^{-1}(11)$.

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


How many non-congruent triangles with only integer side lengths have a perimeter of 18 units, where no side is shorter than 3 units?


Determine the units digit of the sum of the first twelve terms in the sequence: \(1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 11! + 11, \, 12! + 12\).


The current time is 6:00 a.m. What time will it be in 1876 minutes?

Solution: \boxed{1\!:\!16 \text{ p.m.}}


Calculate how many five-digit palindromes exist if the second digit cannot be 0 or 1.


An equilateral triangle has an area of $100\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is first decreased by 6 cm and then by an additional 2 cm, find the new area and by how many square centimeters the area is decreased.


Let $ABC$ be a triangle where $A=(0,0)$, $B=(24,10)$, and $C=(p,q)$ where both $p$ and $q$ are integers. What is the minimum area that $\triangle ABC$ can have?


Let \( A \), \( M \), and \( C \) be nonnegative integers such that \( A+M+C = 24 \). What is the maximum value of
\[ A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A? \]


How many positive integer values of \( x \) exist such that there are exactly \( 3 \) positive multiples of \( x \) that are three-digit numbers?


Solution: \boxed{\frac{4k^2 - 27k}{4}}


Find the point in the plane \( 4x - 3y + 6z = 24 \) that is closest to the point \( (2,1,4) \).

Solution: \boxed{P = \left( \frac{102}{61}, \frac{76}{61}, \frac{214}{61} \right)}


Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ from left to right. For example, $a_3 = 123$ and $a_{12} = 123456789101112$. Compute the remainder when $a_{55}$ is divided by $45$.


Factor the expression $27x^6 - 512y^6$ completely and afterwards find the sum of all integer coefficients in the factorization if they can be written as products of linear and quadratic terms with integer coefficients.


Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of his fruit per day for a week such that he does not eat the banana on the first day, in how many orders can Cory eat the fruit? One such order is $AAAOOAB$.


For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers, starting with the largest number. For example, the alternating sum for $\{1, 2, 4, 7\}$ is $7-4+2-1=4$ and for $\{10\}$ it is simply $10$. Find the sum of all such alternating sums for $n=10$.


Let $\theta$ be the angle between the planes $3x - 2y - z + 4 = 0$ and $9x - 6y - 4z - 7 = 0.$  Find $\cos \theta.$


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)^(1/2)+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?


What is the smallest positive integer that can be written in the form $2010m + 44550n + 100p$, where $m$, $n$, and $p$ are integers?


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 130^{\circ}$ and $\angle AOB =135^{\circ}$, as shown. Determine the degree measure of $\angle ABC$.


A rectangular prism has dimensions 5 feet by 5 feet by 6 feet. A cylindrical section of radius 2.5 feet is removed from the solid, extending through the shortest dimension. What is the total remaining volume of the rectangular prism? Express your answer in cubic feet in terms of \(\pi\).

Solution: \boxed{150 - 31.25\pi \text{ cubic feet}}


In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$.


How many divisors of 75 are also divisors of 80 and divisors of 90? (Recall that divisors may be positive or negative.)


An arithmetic sequence starts with the term $3$ and has a common difference of $8$. A second sequence begins with $5$ and has a common difference of $9$. In the range of $1$ to $90$, what is the largest number that appears in both sequences?


Let $p$ and $q$ be the solutions to $3x^2 - 5x - 8 = 0.$ Compute the value of $(3p^2 - 3q^2)(p - q)^{-1}.$


A swimming pool can be filled by any of three hoses X, Y, or Z. Hoses X and Y together take 3 hours to fill the pool. Hoses X and Z together take 6 hours to fill the pool. Hoses Y and Z together take 5 hours to fill the pool. How many hours does it take hoses X, Y, and Z working together to fill the pool? Express your answer as a decimal to the nearest hundredth.


Alice cycles down a slope at a speed of $(x^2 - 5x - 14)$ miles per hour. Simultaneously, Bob cycled a total distance of $(x^2 + x - 20)$ miles in $(x-4)$ hours. If Alice and Bob cycled at the same speed, what is that speed, in miles per hour?

Solution: \boxed{8 + 2\sqrt{7}}


Find the positive real value of $t$ within the range $0 < t < 4$ that satisfies $|t+3i\sqrt{2}| |7-5i| = 35\sqrt{2}$.


Calculate the total number of days from the year 2010 through 2015, inclusive.


What is the greatest divisor of 630 that is smaller than 35 and also a factor of 90?


The graph of $y = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are quadratic, is known to have the horizontal asymptote $y = -3$ and the vertical asymptote $x = 3$. The graph also passes through the point $(2,0)$. Find $\frac{p(4)}{q(4)}$. Assume there is a hole at $x = -1$.


Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$?


How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 7?


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


Let $a$ and $b$ be positive real numbers such that $5a + 3b < 60.$ Find the maximum value of
\[ab (60 - 5a - 3b).\]


For what value of $n$ is $4 \times 6 \times 3 \times n = 8!$?


A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram.

Solution: \boxed{35 + 2\sqrt{34}}


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


Define $A\star B$ as $A\star B = \frac{(A+B)}{4}$. What is the value of $(3\star 12) \star 6$?


If $m+\frac{1}{m}=10$, then what is the value of $m^3 + \frac{1}{m^3} + 3$?


If $\tan x + \tan y = 30$ and $\cot x + \cot y = 40$, what is $\tan(x+y)$?


If the six digits 1, 3, 3, 4, 5, and 9 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 45 (which requires divisibility by both 9 and 5)? Express your answer as a common fraction.


A cylinder has a radius of 4 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?


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


Determine the smallest non-negative integer $n$ such that $0 \le n < 53$ and $$50n \equiv 47 \pmod{53}.$$


What is the sum of the last two digits of \(7^{25} + 13^{25}\)?


A positive real number \( y \) satisfies
\[
\sqrt{1-y^2} + \sqrt{1+y^2} = 2.
\]
Find \( y^4. \)


Determine the median salary from the following adjusted table of salaries and positions at a company with 63 employees:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$135,000$\\\hline
Senior Vice-President&4&$\$95,000$\\\hline
Manager&12&$\$80,000$\\\hline
Assistant Manager&8&$\$55,000$\\\hline
Clerk&38&$\$25,000$\\\hline
\end{tabular}


A zoo has six pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the female lion, how many ways can he feed all the animals?


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

Solution: \boxed{\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -2 \end{pmatrix}}


The first ten prime numbers are placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number, and their product is also a prime number?


Simplify $\sqrt{45y} \cdot \sqrt{20y} \cdot \sqrt{30y}$. Express your answer in simplest radical form in terms of $y$.

Solution: \boxed{30y \sqrt{30y}}


Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(3000,0),(3000,2000),$ and $(0,2000)$. What is the probability that $x > 5y$? Express your answer as a common fraction.


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

Solution: \boxed{n = 342}


Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ be vectors where $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 7,$ $\|\mathbf{c}\| = 6,$ \[\mathbf{a} \times (\mathbf{a} \times (\mathbf{b}+\mathbf{d})) = \mathbf{c}.\] If $\mathbf{d} = 2\mathbf{a}$ and $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$


Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen, ensuring that one side must be at least 5 feet longer than the other?


In right triangle $ABC$, $\tan{A}=\frac{3}{4}$, $AC=6$, and $\angle C = 90^\circ$. Find $AB$.


What is the product of the digits in the base 8 representation of $8670_{10}$?


A mother buys 4 yellow plates, 3 blue plates, 2 red plates, and 1 purple plate. How many ways are there to arrange these plates around a circular table if the 2 red plates must not be adjacent?


Let $\omega$ be a non-real root of $z^4 = 1$ (i.e., $\omega = i$ since the real root would be 1). Find the number of ordered pairs $(a, b)$ of integers such that $|a \omega + b| = \sqrt{2}$.


Determine the maximum number of real roots for the polynomial of the form
\[x^{2n} + x^{2n-1} + \dots + x + 1 = 0\]
where $n$ is a positive integer.


What is the value of the expression: $2 - 5 + 8 - 11 + 14 - \cdots - 47 + 50 - 53 + 56$?


Calculate the value of the sum
\[\binom{101}{0} - \binom{101}{2} + \binom{101}{4} - \dots - \binom{101}{100}.\]


In how many ways can I arrange 4 different math books and 4 different science books on my bookshelf, if I require there to be a science book on both ends?


Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^4 x - \cos^4 x = \frac{u-v}{uv}\]
where $u = \sin x$ and $v = \cos x$.


In a square, points $P$ and $R$ are midpoints of two adjacent sides. What fraction of the interior of the square is shaded if the triangle formed by the opposite vertex of $P$ and $R$ and point $P$ is unshaded? Express your answer as a common fraction.

[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((1,0)--(1,1)--(2,0)--(1,0)--cycle,white,linewidth(1));
label("P",(1,0),S);
label("R",(2,1),E);
[/asy]


A backhoe dropped off a load of gravel at a landscaping site, shaping it into a frustum of a cone. The diameters of the circular top and bottom of the frustum are $10$ feet and $14$ feet respectively, and the overall height of the frustum is $8$ feet. Calculate the volume of the gravel in terms of $\pi$.


Evaluate the sum of constants when the expression \[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)}\] is expressed in the form of partial fractions:
\[\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4} + \frac{F}{x + 5}.\]


Amelia, Bethany, and Claire are cousins. Amelia has 6 letters in her name, Bethany has 7 letters, and Claire has 6 letters. They decide to play a game using all 19 (6+7+6) letters from their names, placed on identical cards, one on each. Without replacement, two cards are selected at random from the 19 cards. What is the probability that each card comes from a different cousin's name?


Calculate the remainder when the sum of integers from $$88145$$ to $$88155$$ inclusive is divided by 11.


A right triangle has a hypotenuse of $8\sqrt{3}$ inches and one angle is 30°. What is the number of square inches in the area of the triangle?


Rationalize the denominator: $\sqrt{\frac{5}{2+\sqrt{3}}}.$

Solution: \boxed{\sqrt{10 - 5\sqrt{3}}}


In how many ways can 100 be written as the sum of three positive distinct perfect squares if the order of the three perfect squares does not matter?


Calculate the $30^{\text{th}}$ triangular number and find the sum of the first $30$ triangular numbers.


Six circles of diameter 2 are packed in the first quadrant of the coordinate plane. Let region $\mathcal{S}$ be the union of these six circular regions. Line $l,$ with slope 2, divides $\mathcal{S}$ into two regions of equal area. The equation of line $l$ can be expressed in the form $ax=by+c,$ where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1. Find $a^2+b^2+c^2.$


If $f(x) = \sqrt{x - 3}$ and $g(x) = \log_{2}(x - 10)$, what is the smallest real number $x$ in the domain of $f(g(f(x)))$?


Find the remainder when $x^{2040} - 1$ is divided by $x^9 - x^7 + x^5 - x^3 + 1.$


Let
\[ f(x) = \left\{
\begin{array}{cl}
2ax + 4, & \text{if } x > 2, \\
x^2-2 & \text{if } -2 \le x \le 2, \\
3x - c, & \text{if } x < -2.
\end{array}
\right. \]
Find $a+c$ if the piecewise function is continuous.


A bridge is constructed by placing a wooden plank between two triangular supports with equal heights, as shown in the diagram below:
[asy]
import olympiad;
import math;

// Draw triangles
pair A = (0, 1);
pair B = (-cos(1.3962), 0);
pair C = (cos(1.3962), 0);
pair D = (2, 1);
pair E = (2-cos(1.3089), 0);
pair F = (2+cos(1.3089), 0);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
draw(A--D);
label('$A$',A,N);
label('$B$',B,S);
label('$C$',C,S);
label('$D$',D,N);
label('$E$',E,S);
label('$F$',F,S);
[/asy]
If $AB = AC$ and $DE = DF,$ with $\angle BAC = 30^\circ$ and $\angle EDF = 40^\circ,$ determine $\angle DAC + \angle ADE$.


In the town of Gridlock, the streets are all $25$ feet wide, and the blocks are rectangles with lengths of $450$ feet and widths of $350$ feet. Sarah runs around a block on the $450$-foot side of the street, while Sam runs on the opposite side of the street. Moreover, Sarah must also run through a $25$ feet wide alleyway that cuts through the center of the $450$ foot side, extending her path. How many more feet than Sarah does Sam run for each lap around the block?


The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers?


Lucy has $21$ dollars and $45$ cents. She wants to buy as many notebooks as she can with her money. Each notebook costs $2$ dollars and $75$ cents. How many notebooks can she buy?


A right circular cone is inscribed in a right rectangular prism. The base of the prism measures $2a$ by $2b$ where $b < a$. The cone is tangent to the shorter sides of the rectangle and touches the midpoint of the longer sides. If the height of the cone is $h_c$ and the height of the prism is $h_p$, what is the ratio of the volume of the cone to the volume of the prism? Express your answer in terms of $\pi$.

Solution: \boxed{\frac{\pi b h_c}{12a h_p}}


A $120 \times 360 \times 400$ 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?


Mr. Lorenzo gives extra credits on tests to students whose test grades surpass the class average. There are 200 students in the class. What is the greatest number of students who can receive extra credits?


What is the smallest positive integer that can be written in the form $2024m + 48048n$, where $m$ and $n$ are integers?


Suppose we explore another number with similar properties, such as $999,999,999$. How many zeros are found in the square of $999,999,999$?


Let $g(x) : \mathbb{R} \to \mathbb{R}$ be a function such that
\[\frac{g(x) g(y) - g(xy)}{5} = x + y + 4\]for all $x,$ $y \in \mathbb{R}.$  Find $g(x).$

Solution: \boxed{x + 5}


A stage play has two specific male roles, two specific female roles, and three roles that can be played by either gender. Only a man can be cast in a male role and only a woman can be cast in a female role. If four men and five women audition, in how many ways can the seven roles be assigned?


What is the smallest positive value of $m$ so that the equation $12x^2 - mx + 360 = 0$ has integral solutions?


Triangle $XYZ$ has vertices $X(-1, 7)$, $Y(3, -1)$, and $Z(9, -1)$. A line through $Y$ cuts the area of $\triangle XYZ$ in half. Find the sum of the slope and the $y$-intercept of this line.


Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[
\sqrt{1 + \sqrt{45 + 20 \sqrt{5}}} = \sqrt{a} + \sqrt{b}.
\]


Andrew's father's age is eight times Andrew's age. Andrew's grandfather's age is three times Andrew's father's age. If Andrew's grandfather was 72 years old when Andrew was born, how many years old is Andrew now?


Juan, Carlos, Manu, and Ana take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Ana will win? Express your answer as a common fraction.


The sum of the first $n$ terms in the infinite geometric sequence $\{1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots \}$ is $\frac{85}{64}$. Find $n$.


Compute the multiplicative inverse of $217$ modulo $397$. Express your answer as an integer from $0$ to $396$.


What is the product of the digits in the base 6 representation of $7891_{10}$?


What common fraction (in lowest terms) is equivalent to $0.4\overline{36}$?


The operation $\#$ is redefined as $a \# b = a - \frac{b}{a}$. What is the value of $8 \# 4$?


What is the units digit of the sum of the squares of the first 1013 odd, positive integers?


Compute $2 \begin{pmatrix} 3 \\ -6 \end{pmatrix} + 4 \begin{pmatrix} -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 5 \\ -20 \end{pmatrix}$.

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


A square has sides of length 8, and a circle centered at the center of the square has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.


Find the projection of the vector $\begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix}$ onto the line defined by $\frac{x}{3} = \frac{y}{-2} = \frac{z}{1}$.

Solution: \boxed{\begin{pmatrix} \frac{51}{14} \\ -\frac{17}{7} \\ \frac{17}{14} \end{pmatrix}}


A right triangle is to be constructed in the coordinate plane with its legs parallel to the axes, and the medians to the midpoints of the legs are required to lie on the lines $y = 2x + 1$ and $y = mx + 3$. Determine the number of different constants $m$ for which such a triangle exists.


Points $A(-1,3)$ and $B(9,12)$ are given in a coordinate plane. Through calculation, determine the area of a circle where point $A$ is one endpoint of a diameter and point $B$ is displaced 4 units horizontally toward the right. How many square units are in the area of this new circle, expressed in terms of $\pi$?


Ella has designed a rectangular garden divided into four quadrants by paths, with a circular flower bed in each quadrant. Each flower bed is tangent to the paths and its adjacent flower beds. If the garden paths create a rectangle that is 30 inches long and 24 inches wide, how many square inches are not occupied by flower beds?

Solution: \boxed{720 - 144\pi \text{ square inches}}


Find constants \( A \), \( B \), and \( C \) such that
\[ \frac{x^2 - 2x + 5}{x^3 - x} = \frac{A}{x} + \frac{Bx + C}{x^2 - 1} \]
Enter your answer as the ordered triplet \( (A, B, C) \).

Solution: \boxed{(-5, 6, -2)}


The points \((2,10)\), \((14,19)\), and \((6,k)\), where \(k\) is an integer, are vertices of a triangle. What is the sum of the values of \(k\) for which the area of the triangle is a minimum?


Determine the value of \(\frac{c}{b}\) when the quadratic \(x^2 - 2100x - 8400\) is expressed in the form \((x+b)^2 + c\), where \(b\) and \(c\) are constants.


What is the arithmetic mean of the integers from -6 through 7, inclusive?


In a triangle \( \triangle ABC \), for \( \angle A \) to be the largest angle, it must be that \( m < x < n \). The side lengths of the triangle are given by \( AB = x+5 \), \( AC = 2x+3 \), and \( BC = x+10 \). Determine the smallest possible value of \( n-m \), expressed as a common fraction.


If $\frac{90}{2^4\cdot5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


Given $x, y,$ and $k$ are positive real numbers such that
\[4 = k^2\left(\frac{x^2}{y^2}+\frac{y^2}{x^2}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right),\]
find the maximum possible value of $k$.


The average age of the four Smith children is 9 years. If the three younger children are 6 years old, 9 years old, and 12 years old, how many years old is the oldest child?


Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$


What is the arithmetic mean of the integers from -5 through 6, inclusive?


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


A bag contains 5 red marbles, 7 white marbles, and 8 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that all three marbles drawn are the same color?


Find the remainder when $5x^4 - 13x^3 + 3x^2 - x + 15$ is divided by $3x - 9.$


What is the value of \[\frac{x^3\cdot x^5\cdot x^7\cdots x^{21}}{x^4\cdot x^8\cdot x^{12} \cdots x^{24}}\] if \( x=3 \)?


How many 3-digit squares are palindromes, considering only squares of numbers ending in $1$, $4$, or $9$?


Suppose $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-4,3]$ and the range of $g$ is $[-3,2]$. Determine the maximum possible value of $b$ if the range of $f(x) \cdot g(x)$ is $[a,b]$.


Given that \(b\) is a multiple of \(570\), find the greatest common divisor of \(5b^3 + 2b^2 + 5b + 95\) and \(b\).


In a shooting competition, ten clay targets are arranged in three hanging columns with four targets in the first column, three in the second, and three in the third column. A marksman must break all the targets following these rules:
1) The marksman first chooses a column.
2) The marksman must then break the lowest unbroken target in the chosen column.
Determine the number of different sequences in which all ten targets can be broken.


Calculate both the arithmetic and harmonic mean of the reciprocals of the first four prime numbers.


What is the value of the expression $\frac{x^3 - 2x^2 - 21x + 36}{x - 6}$ for $x = 3$? Express your answer in simplest form.


What is the length of the diagonal of a rectangle where the length is $100$ cm and the width is $100\sqrt{2}$ cm? Express your answer in simplest form.


Determine the domain of the function $$g(t) = \frac{1}{(t-2)^3 + (t+2)^3}.$$

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


A $6-8-10$ right triangle has vertices that are the centers of three mutually externally tangent circles. Furthermore, a smaller circle is inside the triangle, tangent to all three circles. What is the sum of the areas of the three larger circles?


Find the minimum value of the expression
\[x^2 + 2xy + 3y^2 + 2xz + 3z^2\]
over all real numbers \(x\), \(y\), and \(z\).


Add the numbers $254_{9}$, $367_{9}$, and $142_9$. Express your answer in base $9$.


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


Triangle $ABC$ has vertices $A(1,3)$, $B(3,7)$, and $C(5,3)$. The triangle is first reflected across the $y$-axis to form $A'B'C'$, and then $A'B'C'$ is reflected across the line $y = x - 2$. Find the coordinates of $C''$, the image of $C'$ after both reflections.

Solution: \boxed{(5, -7)}


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\beta,$ and the angle between $\mathbf{c}$ and $\mathbf{a} \times \mathbf{b}$ is $\beta.$ If $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{\sqrt{3}}{4},$ find the smallest possible value of $\beta,$ in degrees.


The numbers $a,$ $b,$ $c,$ $d$ are equal to -1, 1, 2, 3, in some order. Find the largest possible value of
\[
ab + bc + cd + da.
\]


Solve the inequality
\[\left| \frac{2x - 1}{x - 1} \right| > 3.\]

Solution: \boxed{\left( \frac{4}{5}, 1 \right) \cup (1, 2)}