The chart below provides the air distance in miles between different cities. If two different cities are selected at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Tokyo & Sydney & New York & Paris \\ \hline
Tokyo & & 4800 & 6760 & 6037 \\ \hline
Sydney & 4800 & & 9954 & 10560 \\ \hline
New York & 6760 & 9954 & & 3624 \\ \hline
Paris & 6037 & 10560 & 3624 & \\ \hline
\end{tabular}


The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, and so on. Calculate how many of the first 15 such sums are both prime and divisible by 5.


Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?


A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


The quadratic $-3x^2 - 27x + 81$ can be rewritten in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is the value of $a+b+c$?


Emma initially received some coins from her grandparents. On her way to a friend's house, she lost exactly one-third of them. Later, by retracing her steps, she found exactly three-fourths of the coins she had lost. What fraction of the coins that she received from her grandparents were still missing after Emma retraced her steps? Express your answer as a common fraction.


A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$?


Determine all values of $x$ such that $$\frac{x}{4} \le 3+x < -3(1+x).$$ Express your answer in interval notation.

Solution: \boxed{\left[-4, -\frac{3}{2}\right)}


Let $m=\underbrace{3333333}_{\text{7 digits}}$ and $n=\underbrace{99999999}_{\text{8 digits}}$.

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


Consider the graphs of \( y = 2 - x^3 + x^4 \) and \( y = 1 + 2x^3 + x^4 \). Find the maximum difference between the $y$-coordinates of their intersection points.


Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$.


Calculate the value of the expression \[(2^{1501} + 5^{1502})^2 - (2^{1501} - 5^{1502})^2\] and express it as $k \cdot 10^{1501}$ for some integer $k$.


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

(i) $b_1 = 3$, and

(ii) $b_{2n} = 2n \cdot b_n + 1$ for any positive integer $n$.

Determine the value of $b_{2^{4}}$.


Let $\omega$ be a complex number such that $\omega^8 = 1$ and $\omega \ne 1.$ Find the value of
\[
\omega^{17} + \omega^{21} + \omega^{25} + \dots + \omega^{61}.
\]


Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{5}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one to six candies, leaving Tracy with four candies. How many candies did Tracy have at the start?


All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = 100^\circ$ and $\angle B = 100^\circ$. What is the degree measure of $\angle E$?


Solve
\[\frac{x - 3}{x^2 + 4x + 13} \leq 0.\]
Enter your answer using interval notation.

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


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


Joann and her brother Tom both decided to challenge each other by seeing how many lollipops they could eat over seven days. Joann starts with 10 lollipops on the first day and eats two more than the previous day thereafter. Tom eats 5 lollipops the first day but doubles the number he ate the previous day on each subsequent day. How many total lollipops did they eat combined by the seventh day?


The product of three consecutive integers is 384. What is their sum?


A basketball team has 15 players including 2 captains. In a free throw contest, each captain competes to see who can score the most free throws. For each captain, every other player on the team, including the other captain, takes turns making free throws.

How many free throws must be taken to ensure that each player has attempted free throws against each of the captains?


Let $a, b, c, d$ be positive real numbers. Find the minimum value of
\[
\frac{a + b + c}{d} + \frac{a + b + d}{c} + \frac{a + c + d}{b} + \frac{b + c + d}{a}.
\]


Compute the sum of the geometric series $-2 -6 -18 -54 -162 -486 -1458$.


If the expression $3(x^3 - x^2 + 4) - 5(x^4 - 2x^3 + x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?


A bag contains two red, two blue, and two green marbles. Four marbles are drawn without replacement. What is the probability that exactly two marbles share the same color and the other two each have different colors? Express your answer as a common fraction.


In 2010, the sum of the year's digits is 3. What is the first year after 2010 in which the sum of the digits is 5?


A dilation centered at $2 - 3i$ with a scale factor of $3$ maps which complex number to the point $1 + 2i$?

Solution: \boxed{\frac{5 - 4i}{3}}


Solution: \boxed{25 \text{ cm}^2}


If $\lceil{\sqrt{x}}\rceil=17$, how many possible integer values of $x$ are there?


The base of a triangular piece of paper $ABC$ is $15\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $25\%$ that of the area of the triangle $ABC$. What is the length of $DE$, in cm?

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


Twelve stones are arranged in a straight line. They are counted from left to right as $1,2,3, \ldots, 12$, and then from right to left, so that the stone previously counted as 11 is counted as 13. The pattern is continued to the left until the stone previously counted as 1 is counted as 23. The pattern then reverses so that the stone originally counted as 2 is counted as 24, 3 as 25, and so on. The counting continues in this manner. Which of the original stones is counted as 150? Express your answer as a single digit which corresponds to the first digit assigned to that stone.


Xanthia can read 120 pages per hour and Molly can read 60 pages per hour. If Xanthia reads a book of 360 pages and then another book of 120 pages, while Molly reads only the first book, how many more minutes than Xanthia would it take for Molly to finish reading the first book?


What is the coefficient of \( x^4 \) when $$x^5 - 2x^4 + 4x^3 - 5x^2 + 2x - 1$$ is multiplied by $$3x^4 - x^3 + 2x^2 + 6x - 5$$ and the like terms are combined?


The first and seventeenth terms of an arithmetic sequence are $\frac{3}{8}$ and $\frac{2}{3}$, respectively. What is the ninth term?


Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3),$ $(5,5),$ $(7,7),$ and $(9,9),$ determine whether there is a point where the graphs of $y=h(3x)$ and $y=3j(x)$ intersect, and find the sum of the coordinates of that point if it exists.


A designer has 4 fabric colors he may use for a dress: red, green, blue, and yellow. Five different patterns are available for the dress: stripes, polka dots, floral, geometric, and plain. Each dress design requires exactly one color and one pattern. However, the geometric pattern can only be used with red or blue fabric. How many different dress designs are possible?


Let $x$ and $y$ be real numbers such that
\[
  4 < \frac{2x - 3y}{2x + 3y} < 8.
\]
If $\frac{x}{y}$ is an integer, what is this value?


Consider an isosceles trapezoid where one base is four times the height, $x$, and the other base is three times the height, $x$. Find the expression for the area of this trapezoid in terms of the height $x$.


Calculate the following sum:
\[
\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.
\]


A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$


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}


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


Compute the sum of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] assuming all roots are real and nonnegative.


What is the sum of the integers from $-50$ to $60$, inclusive?


Determine the value of $b$ if $\log_b 216 = -\frac{3}{2}$.

Solution: \boxed{b = \frac{1}{36}}


If $x^2 + 3x + 4$ is a factor of $x^4 + Px^2 + Q,$ find $P + Q.$


A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot.


256 in base \( b \) has exactly 4 digits. What is the value of \( b \)?


The median of a set of consecutive odd integers is 155. If the greatest integer in the set is 167, what is the smallest integer in the set?


In a trapezoid, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $5:2$. If $AB + CD = 280$ cm, how long is segment $\overline{AB}$?

Solution: \boxed{200 \text{ cm}}


In triangle $ABC$, where $A$ is at (0, 4), $B$ is at (0, 0), and $C$ is at (3, 0), find $\tan A$.


Given the sequence defined by \( b_n = 2^n \cdot n! + n \) for \( n \geq 0 \), determine the maximum possible value of the greatest common divisor (gcd) of two consecutive terms of this sequence.


A certain integer has 5 digits when written in base 4. The same integer has $d$ digits when written in base 16. What is the sum of all possible values of $d$?


Solve for $z$ in the equation $3 - 2iz = 1 + 4iz$, where $i^2 = -1$. Simplify your answer as much as possible.


Let \( z \) be a complex number such that
\[ z^3 + |z|^2 = 4 + 2i. \]
Find \( |z|^2 \).


What is the least positive integer value of $x$ such that $(3x)^2 + 2 \cdot 41 \cdot 3x + 41^2$ is a multiple of 53?


The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$?


A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,3)$, and $(0,3)$. What is the probability that $x < y$?


Let $b_n = \frac{8^n - 1}{7}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


A truncated right circular cone has a large base radius of 10 cm and a small base radius of 3 cm. Attached to its smaller base is a cylindrical solid of height 10 cm and radius 3 cm. The height of the truncated cone is 8 cm. Calculate the total volume of the combined solid.

Solution: \boxed{\frac{1382\pi}{3} \text{ cm}^3}


Find the distance between the points \((3,2,-5)\) and \((7,10,-6)\).


In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac{EQ}{EF} = \frac{3}{700}$ and point $R$ is on $\overline{EH}$ such that $\frac{ER}{EH} = \frac{3}{500}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Determine $\frac{EG}{ES}$.


The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $5^{20}$?


How many integers $n$ are there such that $4 \leq n \leq 18$ and $203_n$ (the number written as $203$ in base $n$) is a perfect square?


Determine the maximum difference between the \(y\)-coordinates of the intersection points of the graphs \(y=5-x^2+2x^3\) and \(y=3+2x^2+2x^3\).


Let \(x,\) \(y,\) and \(z\) be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of \(x^2 + y^2 + z^2.\)


A trapezoid has its height denoted by $h$. One of its bases is four times the height plus an additional 2 units, and the other base is five times the height. Determine the area of the trapezoid in terms of $h$.

Solution: \boxed{\frac{9h^2 + 2h}{2}}


Find the sum of the arguments of all eighth roots of the complex number $z^8 = -81i$. Express the roots as $z_k = r_k (\cos \theta_k + i \sin \theta_k)$, where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$


Below is the graph of $y = a \csc(bx)$ for some positive constants $a$ and $b.$ The minimum positive value of the graph is 4 and it occurs at $x = \frac{\pi}{4}$. Find the values of $a$ and $b$.


A sports conference has 18 teams divided into two divisions of 9 each. How many games are in a complete season if each team must play every other team in its own division three times and every team in the other division twice?


Points \(E, F, G, H\) lie on the sides of a larger square such that each divides the sides into segments that are in the ratio 1:2 and 2:1 from one vertex. If the larger square has an area of 144 square units, what is the area of the quadrilateral formed by connecting the points \(E, F, G,\) and \(H\)?


Find the integer $n$, $0 \le n \le 180$, such that $\cos n^\circ = \cos 315^\circ$.

Solution: \boxed{n = 45}


The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability that it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1)? Express your answer as a common fraction.


Let $\mathbf{Q}$ be the matrix for reflecting over the vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}.$ Find $\mathbf{Q}^2.$

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


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


Determine the values of the center coordinates $(c,d)$ and the radius $s$ of the circle described by the equation $x^2 + 2x - 4y - 7 = -y^2 + 8x$. Find the sum $c+d+s$.

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


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{4} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$  Find $\sin \theta.$


Convert the binary number $10111001_2$ to base 4, after flipping the third bit from the right.


Triangle \( ABC \) has vertices \( A(0, 6) \), \( B(2, 0) \), \( C(8, 0) \). A line through \( B \) cuts the area of \( \triangle ABC \) in half; find the product of the slope and \( y \)-intercept of this line.


At the end of a game, each of the six members of a basketball team shakes hands not only with each of the six members of the opposing team but also with each other. After that, all of the players shake hands with each of the three referees. How many handshakes occur?


What is the value of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~38$ is an arithmetic sequence?


Compute without using a calculator: $10! - 9!$.


In triangle $ABC$, $\cos(3A-B)+\sin(A+B) = 1$ and $AB=5$. What is $BC$?


The average of the seven numbers in a list is 60. The average of the first four numbers is 55. What is the average of the last three numbers?


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

Solution: \boxed{96 \text{ square units}}


Suppose $p$, $q$, $r$, and $s$ are integers satisfying: $p-q+r=7$, $q-r+s=8$, $r-s+p=4$, and $s-p+q=1$. What is the value of $p+q+r+s$?


Calculate the coefficient of $x^3$ when $2(x^2 - 2x^3 + 2x) + 4(x + 3x^3 - 2x^2 + 2x^5 - x^3) - 7(2 + 2x - 5x^3 - x^2)$ is expanded.


For what base-6 digit $d$ is $2dd3_6$ divisible by the base 10 number 13?


In triangle $ABC$, $AB = 4$, $AC = 5$, and $BC = 7$. Point $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD.$


The sum of the first 6033 terms of a geometric sequence is 600. The sum of the first 12066 terms is 1140. Find the sum of the first 18099 terms.


How many numbers are in the list $$ 2, 5, 8, \ldots, 2011, 2014 ?$$


Consider a larger square with area 100. Points $P$, $Q$, $R$, and $S$ are midpoints of its sides. A rectangle is formed by joining these midpoints such that two of its adjacent sides are parallel to the diagonals of the square. What is the area of this rectangle?


What is the smallest odd number with five different prime factors?


Find the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~33$ is an arithmetic sequence.


Let $g$ be a non-constant polynomial such that
\[g(x - 1) + g(x) + g(x + 1) = \frac{[g(x)]^2}{2025x}\] for all nonzero real numbers $x$. Find the sum of all possible values of $g(1)$.


Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\]


When the greatest common divisor of two positive integers is multiplied by 20, the result equals their least common multiple. If one of the integers is 60, what is the smallest possible value of the other integer?


Given positive integers $x$ and $y$, such that $\frac{1}{x}+\frac{1}{3y}=\frac{1}{9}$, what is the least possible value of $xy$?


Calculate $\theta$ in degrees for the sum expressed in the form $r \, \text{cis} \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$, given that
\[
\text{cis } 70^\circ + \text{cis } 78^\circ + \text{cis } 86^\circ + \dots + \text{cis } 158^\circ.
\]


Given $\tan \theta = 3,$ find
\[\frac{1 - \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta}.\]

Solution: \boxed{\frac{2(\sqrt{10} - 1)}{3}}


Determine the smallest abundant number that is not a multiple of 5.


In the diagram, triangle $ABC$ has sides $AB = 30 \text{ cm}$ and $AC = 24 \text{ cm}$ with $\angle A = 90^\circ$. Adjacent to triangle $ABC$ is triangle $ABD$ where $D$ is on line extended from $B$ and $AD$ also forms a right angle with $AB$. Given $AD = 18 \text{ cm}$, what is the sum of the areas of triangles $ABC$ and $ABD$?

Solution: \boxed{630 \text{ cm}^2}


The repeating decimal for $\frac{7}{19}$ is $0.cdc\ldots$ What is the value of the sum $c+d$?


The same amount of steel used to create nine solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball?


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}$.


Let \( T(r) \) denote the sum of the geometric series \( 15 + 15r + 15r^2 + 15r^3 + \cdots \) for \( -1 < r < 1 \). Let \( b \) between \( -1 \) and \( 1 \) satisfy \( T(b)T(-b) = 2700 \). Find \( T(b) + T(-b) \).


Find the remainder when the polynomial $3x^{1010} + x^{1000}$ is divided by the polynomial $(x^2 + 1)(x - 1).$

Solution: \boxed{3x^2 + 1}


The image of the point with coordinates \((2,3)\) under the reflection across the line \(y = mx + b\) is the point with coordinates \((10,7)\). Find \(m+b\).


Two rectangles have integer dimensions, and both have a perimeter of 180 cm. What is the greatest possible difference between the areas of two such rectangles?


Vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 3 \\ -6 \\ 9 \end{pmatrix}$, where $\mathbf{a}$ is parallel to $\begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$, and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$. Find $\mathbf{b}$.

Solution: \boxed{\begin{pmatrix} 1 \\ -8 \\ 7 \end{pmatrix}}


What is the largest $4$ digit integer congruent to $12 \pmod{19}$?


A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(5,0)$, $(5,2)$, and $(0,2)$. What is the probability that $x < 2y$?


What is the largest four-digit negative integer congruent to $3 \pmod{29}?$


The point $(2,3,2)$ is first rotated $90^\circ$ about the $y$-axis, then reflected through the $xz$-plane, followed by a $90^\circ$ rotation about the $z$-axis, and finally reflected through the $yz$-plane. Find the coordinates of the point now.

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


Let $a = \underbrace{333333333}_{\text{9 digits}}$ and $b = \underbrace{666666666}_{\text{9 digits}}$.

What is $\gcd(a, b)$?


A wooden block is 6 inches long, 6 inches wide, and 1 inch high. The block is painted blue on all six sides and then cut into thirty-six 1 inch cubes. How many of the cubes have a total number of blue faces that is an odd number?


Find the cross product of $\begin{pmatrix} 3 \\ 4 \\ -2 \end{pmatrix}$ scaled by 2 and $\begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 52 \\ -26 \\ 26 \end{pmatrix}}


If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 2,$ $\|\mathbf{a} - \mathbf{b}\| = 2\sqrt{2},$ and
\[\mathbf{c} - 2\mathbf{a} - \mathbf{b} = 2 (\mathbf{a} \times \mathbf{b}),\]find $\mathbf{a} \cdot \mathbf{c}.$


For a positive integer $n$ and nonzero digits $a$, $b$, and $d$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit integer each of whose digits is equal to $d$. What is the greatest possible value of $a + b + d$ for which there are at least two values of $n$ such that $C_n - B_n = 2A_n^2$?


Yan is positioned between his home and the stadium. He can either walk directly to the stadium or walk back home and then ride his bicycle to the stadium. Riding the bicycle is 10 times as fast as walking. Both options take the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?


What is the least positive integer $n$ such that $7350$ is a factor of $n!$?


In trapezoid $ABCD$, $CD$ is three times the length of $AB$. If the area of trapezoid $ABCD$ is $18$ square units, what is the area of $\triangle ABC?$

[asy]
draw((0,0)--(1,6)--(10,6)--(15,0)--cycle);
draw((10,6)--(0,0));
label("$C$",(0,0),W);
label("$A$",(1,6),NW);
label("$B$",(10,6),NE);
label("$D$",(15,0),E);
[/asy]


The positive four-digit integers that use each of the digits $1,$ $2,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $10^{\text{th}}$ integer in the list?


What is the sum of all even positive divisors of 140?


A geometric sequence begins with 32, -48, and 72. What is the common ratio, and what is the fourth term of this sequence?


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


Given that $BDEF$ is a rectangle and $AB = BC = 1$, find the number of square units in the area of the regular hexagon inside it. The rectangle $BDEF$ is such that $BD = 2 + \sqrt{3}$ and $DE = 2$. Note that each side of the hexagon is equal to $1$.


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$ and $D$ are not necessarily distinct. If each of the two-digit primes is less than 60, how many such house numbers are possible?


In triangle $ABC,$ points $D$ and $E$ are on $\overline{AB}$ and $\overline{AC},$ respectively, and angle bisector $\overline{AT}$ intersects $\overline{DE}$ at $F.$  If $AD = 2,$ $DB = 2,$ $AE = 3,$ and $EC = 3,$ compute $\frac{AF}{AT}.$


In triangle $ABC$, $\angle B = 45^\circ$ and $\angle C = 60^\circ$. The point $D$ divides $\overline{BC}$ in the ratio $2:1$. Find
\[
\frac{\sin \angle BAD}{\sin \angle CAD}.
\]

Solution: \boxed{\frac{2 \sqrt{6}}{3}}


There are four points that are $7$ units from the line $y = 15$ and $15$ units from the point $(9, 15)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


Given that $\sec x + \tan x = \frac{5}{4},$ find all possible values of $\sin x.$


Mahmoud tosses five coins. What is the probability that he does not get two consecutive tails?


The projection of \(\begin{pmatrix} 0 \\ 4 \\ z \end{pmatrix}\) onto \(\begin{pmatrix} -4 \\ 6 \\ -2 \end{pmatrix}\) is
\[\frac{16}{56} \begin{pmatrix} -4 \\ 6 \\ -2 \end{pmatrix}.\]
Find \(z.\)


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


For what real value of $u$ is $\frac{-15-\sqrt{205}}{6}$ a root of $3x^2 + 15x + u$?


Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them. If $a^2 + b^2 = 49c^2$, find the value of
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\]


An equilateral triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = 2 \angle D$ and $\angle BAC = 60^\circ$, find the value of $k$ where $\angle BAC = k \pi$ in radians.


If $A\ \clubsuit\ B$ is defined as $A\ \clubsuit\ B = 4A + 3B + 7$, what is the value of $A$ for which $A\ \clubsuit\ 5 = 80$?


Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all the stickers among the sheets such that each sheet gets at least one sticker?


Suppose that $y^4$ varies inversely with $\sqrt[4]{z}$. If $y = 3$ when $z = 16$, find the value of $z$ when $y = 6$.


If $128^3 = 16^y$, what is the value of $2^{-3y}$? Express your answer as a common fraction.


What is the measure of the smaller angle formed by the hour-hand and minute-hand of a clock when it is 3:15?


An arithmetic sequence with first term $1$ has a common difference of $8$. A second sequence begins with $4$ and has a common difference of $9$. In the range of $1$ to $250$, what is the largest number common to both sequences?


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


In a gathering, there are 8 married couples. Each person shakes hands with everyone else except their own spouse and anyone wearing the same colored shirt as themselves. No person shares a shirt color with their spouse, and four unique shirt colors are represented equally by the attendees. How many handshakes occur?


Find the sum $C + D$ (in base 10), given the following addition problem in base 6:
\[
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & 5 & C & D_{6}\\
&+& & & 3 & 2_{6}\\
\cline{2-6}
& & & 6 & 1 & C_{6}\\
\end{array}
\]


How many integers are between $(10.4)^3$ and $(10.5)^3$?


Find an ordered pair \((x, y)\) that satisfies both of the following equations:
\begin{align*}
3x - 4y &= -7, \\
4x + 5y &= 23.
\end{align*}

Solution: \boxed{\left(\frac{57}{31}, \frac{97}{31}\right)}


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


Consider the ellipse \[16(x+2)^2 + 4y^2 = 64.\] Let $C$ be one of the endpoints of its major axis, and let $D$ be one of the endpoints of its minor axis. Find the distance $CD.$


The graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.


A garden needs to be constructed adjacent to a house, using the house as one side of the boundary. The gardener has 480 feet of fencing available, which will be used to enclose the other three sides of the rectangular garden. Moreover, the length of the garden should be twice its width. What is the maximum possible area of this garden?


Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$?


Determine the remainder when \(1^4 + 2^4 + 3^4 + \dots + 50^4\) is divided by 5.


If the consecutive integers from $100$ to $50$ were written consecutively as $$100999897...50504948,$$ what would be the $150^{\text{th}}$ digit to be written?


Determine the product of the roots of the equation \[(3x^4 - 2x^3 + 5x - 15)(4x^3 + 6x^2 - 8) = 0.\]


Let \(x,\) \(y,\) and \(z\) be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of \(x^2 + y^2 + z^2.\)


Eight 8-sided dice are rolled. What is the probability that exactly four dice show even numbers, and four dice show odd numbers?


The values of $a$, $b$, $c$, and $d$ are 1, 3, 5, and 7, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$, and $da$?


The average of five different positive whole numbers is $7.$ If the difference between the largest and the smallest of these numbers is as large as possible, what is the average of the other three numbers?


Each term of a sequence, after the first term, is inversely proportional to the term preceding it, and the constant of proportionality stays the same. If the first term is 3 and the second term is 4, what is the 15th term?


Humanity discovers 13 habitable planets, of which only 5 are Earth-like and the rest are Mars-like. Earth-like planets require 2 units each for colonization, while Mars-like planets require 1 unit each. If humanity has 15 units available for colonization, how many different combinations of planets can be occupied if each planet is unique?


The decimal $0.1824$ can be expressed as a fraction $\frac{m}{n}$, where $m$ and $n$ are positive integers. What is the value of $m+n$ when the fraction is in its simplest form?


A regular dodecagon \( Q_1 Q_2 \dotsb Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (4,0) \) and \( Q_7 \) at \( (2,0) \). If \( Q_n \) is the point \( (x_n,y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).
\]


The Benton Youth Soccer Team has expanded to 25 players on the team, including reserves. This includes four goalies. Today, they are having a contest to see which goalie can block the most number of penalty kicks. For each penalty kick, a goalie stands in the net while the rest of the team (including the other goalies but excluding the one in goal) takes a shot on goal, one at a time. How many penalty kicks must be taken to ensure that everyone, including the other goalies, has a chance to shoot against each of the 4 goalies?


In how many ways can a president and a 2-person committee be chosen from a group of 10 people, if only people over 30 years old can be elected to the committee and there are 6 people over 30 in the group? (The president cannot be on the committee.)


Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac{1}{z} = 7,$ and $y + \frac{1}{x} = 20.$ Find $z + \frac{1}{y}.$


Fifteen distinct points are placed on the circumference of a circle. How many different convex hexagons can be formed if each vertex must be one of these 15 points?


Find the quadratic polynomial, with real coefficients, which has $3 - 2i$ as a root, and where the coefficient of $x$ is $2$.

Solution: \boxed{-\frac{1}{3}x^2 + 2x - \frac{13}{3}}


What is the smallest whole number $b$ such that 150 can be expressed in base $b$ using only three digits?


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $12$. Find the sum of the $x$-coordinates of the midpoints of the sides of this triangle.


Simplify $\sqrt{5\cdot 2x}\cdot \sqrt{x^3\cdot 5^3}$.


Last month, Elena hiked the Mountain Loop Trail. It took her five days to complete the trip. The first two days combined, she hiked a total of 36 miles. The second and third days she averaged 20 miles per day. The last three days combined, she hiked a total of 45 miles. The total hike for the first and fourth days was 38 miles. How many miles long was the trail?


In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?


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


Suppose \( z \) and \( w \) are complex numbers such that
\[
|z| = |w| = 2 \quad \text{and} \quad z \overline{w} + \overline{z} w = -1.
\]
Find the largest possible value of the real part of \( z + w \).


A pirate finds three chests on the wrecked ship S.S. Triumph, recorded in base 7. The chests contain $3214_7$ dollars worth of silver, $1652_7$ dollars worth of precious stones, $2431_7$ dollars worth of pearls, and $654_7$ dollars worth of ancient coins. Calculate the total value of these treasures in base 10.


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


Consider a fictional country where license plates are made using an extended Rotokas alphabet with fourteen letters: A, E, G, I, K, M, O, P, R, S, T, U, V, Z. License plates are six letters long and must start with G, end with M, and not contain the letters S, T, or Z. Additionally, no letters may repeat. How many such license plates are possible?


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

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


In the diagram, $AB$ and $CD$ are diameters of a circle with a radius of 6 that intersect at right angles at point $O$. The points $B$ and $D$ are on the horizontal diameter, and $A$ and $C$ are on the vertical diameter. Given that $\triangle OAC$ is an isosceles right triangle, find the area of the shaded regions, which include $\triangle OBD$, $\triangle OAC$, and two quarter-circle sectors: $OAB$ and $OCD$.

Solution: \boxed{36 + 18\pi}


A basketball team has 18 players, including two sets of twins: Ben & Jerry and Tom & Tim. In how many ways can we choose 5 starters if at least one member from each set of twins must be included in the starting lineup?


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


A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square?


Compute the exact value of the expression
\[|e - |e - 5||.\]
Write your answer using only integers and $e$, without any absolute value signs.

Solution: \boxed{2e - 5}


Fifty-four increased by three times a number is 75. Additionally, before adding three times the number, the fifty-four is halved. What is the number?


Given $e^{i \theta} = \frac{3 + i \sqrt{8}}{5}$, find $\cos 4 \theta$.


Eight 6-sided dice are rolled. What is the probability that four of the dice show even numbers and four show odd numbers?


Paul tosses a fair 8-sided die. If the roll is a prime number, then he wins that amount in dollars. If the roll is composite, he wins nothing unless he rolls the highest composite number on the die, in which case he loses twice that amount in dollars. Otherwise, he gains 2 dollars. What is the expected value of his winnings on one die toss? Express your answer as a dollar value to the nearest cent.


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

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

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

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

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

The value of \(F(4)\) is an integer. What is it?


What is the smallest prime divisor of $3^{15} + 11^9$?


Evaluate the expression $\dfrac{\sqrt[4]{7}}{\sqrt[7]{7}}$ and express it as 7 raised to what power?


Determine the value of the positive number $a$ such that the terms $25, a, \frac{8}{5}$ are the first, second, and third terms, respectively, of a geometric sequence.


Let $O$ be the origin. There exists a scalar $m$ so that for any points $A$, $B$, $C$, and $D$ such that
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + m \overrightarrow{OD} = \mathbf{0},\] the four points $A$, $B$, $C$, and $D$ are coplanar. Find $m$.


How many different five-letter words can be formed such that they start and end with the same letter, and the middle letter is always 'A'?


$\triangle PQR$ is inscribed in a semicircle of radius $R$ so that its base $PQ$ coincides with the diameter $PQ$. Point $R$ does not coincide with either $P$ or $Q$. Let $t = PR + RQ$. Find the maximum possible value of $t^2$.


Ben throws six identical darts. Each hits one of five identical dartboards on the wall. After throwing the six darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible?


Find $\tan Y$ in the right triangle with sides $XY = 30$, $YZ = 37$ (hypotenuse).

[asy]
pair X,Y,Z;
X = (0,0);
Y = (30,0);
Z = (0,8);
draw(X--Y--Z--X);
draw(rightanglemark(Y,X,Z,23));
label("$X$",X,SW);
label("$Y$",Y,SE);
label("$Z$",Z,N);
label("$37$",(Y+Z)/2,NE);
label("$30$",Y/2,S);
[/asy]

Solution: \boxed{\tan Y = \frac{\sqrt{469}}{30}}


The value of $\log_{10}{215}$ lies between which two consecutive integers $c$ and $d$? Find $c+d$.


Convert the point \( (2, -2) \) in rectangular coordinates to polar coordinates. Ensure the answer is in the form \( (r, \theta) \), where \( r > 0 \) and \( 0 \leq \theta < 2\pi \).

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


Two sides of a right triangle have the lengths 6 and 8. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest whole number.


Find
\[\min_{y \in \mathbb{R}} \max_{0 \le x \le 2} |x^2 - xy + 1|.\]


Two cross sections of a right triangular pyramid are obtained by cutting the pyramid with planes parallel to the triangular base. The areas of the cross sections are $144\sqrt{3}$ square cm and $324\sqrt{3}$ square cm. The two planes are $6$ cm apart. How far from the apex of the pyramid is the larger cross section, in cm?


For how many integer values of \( n \) between 1 and 380 inclusive does the decimal representation of \( \frac{n}{385} \) terminate?


What is the value of the following expression: $3 - 6 + 9 - 12 + ... + 57 - 60$ ?


Compute $\left(2\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\right)^{10}$.

Solution: \boxed{\begin{pmatrix} 1024 & 0 \\ 20480 & 1024 \end{pmatrix}}


What is the greatest positive integer that must divide the sum of the first 15 terms of any arithmetic sequence where the first term is an odd positive integer and the common difference is an even integer?


What is the result of $\frac{3}{7}$ divided by $\frac{2}{5}$, after adding $\frac{1}{3}$ to the dividend?


What is the largest integer that must divide the product of any 5 consecutive integers?


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


The current time is 7:00 a.m. What time will it be in 2687 minutes?

Solution: \boxed{3:47 \text{ a.m.}}


How many different counting numbers will each leave a remainder of 7 when divided into 59?


In a specific year, a "prime date" occurs when both the month and the day are prime numbers. Determine the total number of prime dates in a non-leap year where February has 28 days, and March, May, and July have 31 days, while November has 30 days.


For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $9$-nice nor $10$-nice.


Calculate the largest integer $n$ such that $\binom{12}{5} + \binom{12}{6} = \binom{13}{n}$.


The current time is 8:00 a.m. What time will it be in 1750 minutes from now?

Solution: \boxed{1:10 \text{ p.m.}}


Evaluate the determinant:
\[
\begin{vmatrix}
\cos \alpha \cos 2\alpha & \cos \alpha \sin 2\alpha & -\sin \alpha \\
-\sin 2\alpha & \cos 2\alpha & 0 \\
\sin \alpha \cos 2\alpha & \sin \alpha \sin 2\alpha & \cos \alpha
\end{vmatrix}.
\]


The gravitational force that a satellite exerts on an object is inversely proportional to the square of the distance between the center of the satellite and the object. If the gravitational force is 200 Newtons when an astronaut is 100 miles from the center of the satellite, find the gravitational force when the astronaut is 500 miles from the center. Assume that the satellite's mass increases, doubling the initial gravitational constant used. Express your answer as a fraction.


Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\]


Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$.


What is the greatest possible number of digits in the product of two 4-digit whole numbers?


The faces of a decagonal die are labeled with digits $1$ through $10$. What is the probability, expressed as a common fraction, of rolling a sum of $17$ with a pair of such decagonal dice?


Find the number of odd digits in the base-9 representation of $560_{10}$.


Determine the value of $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}$ if the roots of the equation $x^3 - 15x^2 + 26x + 3 = 0$ are $p$, $q$, and $r$.


Compute the multiplicative inverse of $123$ modulo $400$. Express your answer as an integer from $0$ to $399$.


Find the value(s) of $x$ such that $10xy - 15y + 3x - 4.5 = 0$ is true for all values of $y$.


In a trapezoid $WXYZ$ with $WX \parallel ZY$ and $WY \perp ZY$, given that $YZ = 15$, $\tan Z = 2$, and $\tan X = 1.5$, find the length of $XY$.


Given the function $f(x) = x^5 + x^3 + 7x$, calculate $f(3) - f(-3)$.


Jennifer has six children. What is the probability that either at least three of them are boys, or exactly half of them are girls?


Find the sum of the first five terms in the geometric sequence \( \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots \). Express your answer as a common fraction.


A solid rectangular block is constructed by fusing together $N$ identical 1-cm cubes side by side. When the block is positioned so that three faces are visible, exactly $399$ of the 1-cm cubes are hidden. Determine the smallest possible value of $N.$


Find the sum of the first seven prime numbers that have a units digit of 3.


Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be vectors such that $\mathbf{a} + 2\mathbf{b} + \mathbf{c} = \mathbf{0}$. Determine a scalar $m$ such that the equation
\[m (\mathbf{c} \times \mathbf{b}) + \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} = \mathbf{0}\]
holds.


Suppose \( n-2, n-1, n, n+1, n+2, n+3, n+4 \) are seven consecutive integers.

Determine a simplified expression for the sum of these seven consecutive integers.

Solution: \boxed{7n + 7}


Three fair dice are rolled to produce the numbers $a, b, c$. Find the probability that the product of these three numbers, $abc$, equals 72.


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


Let $\mathbf{a}, \mathbf{b}, \mathbf{c},$ and $\mathbf{d}$ be four mutually orthogonal unit vectors, such that
\[\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a}) + s (\mathbf{d} \times \mathbf{a})\] for some scalars $p,$ $q,$ $r,$ and $s,$ and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1.$ Find $p + q + r + s.$


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that its digits are all different?


Equilateral triangle $ABC$ and rectangle $BCDE$ are coplanar, with $BC = 2BE$. Point $D$ is directly below $C$, and point $E$ directly below $B$. What is the number of degrees in the measure of angle $CAD$?


A right triangle has a hypotenuse that measures 12 inches and one angle of $30^\circ$. What is the number of square inches in the area of the triangle?


Sheila was invited to a picnic, which will take place rain or shine. If it rains, there is a 30% chance that Sheila will decide to attend. If it is sunny, there is a 90% chance that she will decide to go. Additionally, there is a 10% chance that Sheila may forget about the picnic altogether. The weather forecast predicts a 50% chance of rain. What is the probability that Sheila will attend the picnic? Express your answer as a percent.


Let \( n \) be the product of all positive integers less than \( 5! \) which are invertible modulo \( 5! \). Find the remainder when \( n \) is divided by \( 5! \).


Find $x$ if $x \cdot x! + 2x! = 40320$, where $x! = x \cdot (x-1) \cdot (x-2)\cdots 2 \cdot 1$.


Compute $A(3,2)$ using the Ackermann function defined as:
\[
A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) &\text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0
\end{aligned}
\right.
\]


When the product \((5x-3y+2)(2x-5y+3)\) is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of \( y \)?


Find \(89^{-1} \pmod{90}\), as a residue modulo 90. (Give an answer between 0 and 89, inclusive.)


When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 6$.


What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity?


Let $x, y,$ and $z$ be positive real numbers such that $x + y + z = 9.$ Find the minimum value of
\[
\frac{x^2 + y^2}{3(x + y)} + \frac{x^2 + z^2}{3(x + z)} + \frac{y^2 + z^2}{3(y + z)}.
\]


Calculate the number of seven-digit palindromes.


Find the number of complex solutions to the equation
\[\frac{z^4 - 1}{z^3 + z^2 - 2z} = 0.\]


Given $ABC$ is a triangle with $A=(0,0)$, $B=(24,10)$, and both the coordinates of $C$ are integers, determine the minimum area $\triangle ABC$ can have.


Find the difference between $2222_9 + 1234_9$ and $1111_9$ in base 9.


The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, $z$ are $10$, $7$, and $4$ respectively. What is the value of $x^2+y^2+z^2$?


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

Solution: \boxed{2\sqrt{61} \text{ cm}}


What is the greatest common factor of $180$ and $294$?


Given a point has rectangular coordinates $(3, -2, 5)$ and spherical coordinates $(r, \alpha, \beta)$. Determine the rectangular coordinates of the point with spherical coordinates $(r, \alpha+\pi, \beta)$.

Solution: \boxed{(-3, 2, 5)}


Determine how many integers are in the list containing the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them.


Mira rotated point $P$ 735 degrees clockwise about point $Q$ and it landed at point $R$. Leo rotated the original point $P$ $y$ degrees counterclockwise about point $Q$ and it also landed at point $R$. If $y<360$, what is the value of $y$?


Calculate how much greater, in square centimeters, is the area of a circle with a diameter of 30 cm than the area of a circle with a radius of 10 cm. Additionally, include a smaller circle of radius 5 cm within the bigger circle and subtract its area as well in the solution.

Solution: \boxed{100\pi \text{ square centimeters}}


There exists a scalar $d$ such that
\[\mathbf{i} \times (\mathbf{v} \times 2\mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times 2\mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times 2\mathbf{k}) = d \mathbf{v}\] for all vectors $\mathbf{v}.$  Find $d$.


Given that $a$ is a multiple of $1836$, find the greatest common divisor of $2a^2 + 11a + 40$ and $a + 4$.


Line $l_1$ has equation $3x - 2y = 1$ and goes through point $A = (-1, -2)$. Line $l_2$ now has equation $y = 2$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and intersects $l_2$ at point $C$. Calculate the area of $\triangle ABC$ and determine the slope of $l_3$, given that the area is $5$ units².


Express \((3 - 4i)(-4 + 6i)\) in the form \(a+bi\), where \(a\) and \(b\) are integers and \(i^2 = -1\).

Solution: \boxed{12 + 34i}


Determine how many out of the 30 different complex numbers $z$ such that $z^{30} = 1$ have the property that $z^5$ is a real number.


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


How many pairs of positive integers $(m,n)$ satisfy $m^2 + n < 30?$


Let $m$ be the smallest positive three-digit integer congruent to 3 (mod 7). Let $n$ be the smallest positive four-digit integer congruent to 3 (mod 7). What is $n-m$?


There exists a scalar $c$ such that
\[\mathbf{i} \times (k\mathbf{w} \times \mathbf{i}) + \mathbf{j} \times (k\mathbf{w} \times \mathbf{j}) + \mathbf{k} \times (k\mathbf{w} \times \mathbf{k}) = c \mathbf{w}\]
for all vectors $\mathbf{w}$ and scalar $k$. Find $c.$


Determine the value of $b$ that satisfies the equation $295_{b} + 467_{b} = 762_{b}$.


Line segment $\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 4:5.$  Then
\[\overrightarrow{P} = s \overrightarrow{A} + v \overrightarrow{B}\] for some constants $s$ and $v.$ Determine the ordered pair $(s, v).$

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


What is the smallest positive integer value of $y$ such that $(3y)^2 + 3 \cdot 43 \cdot 3y + 43^2$ is a multiple of 53?


In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$.


If 35,867,413 hot dogs are packaged in sets of 6, how many will be left over?


Sandy is now planning to cover two shaded exterior walls and the two triangular faces of the roof with a special siding. Each wall measures 10 feet by 6 feet, and each triangular roof face has a base of 10 feet and a height of 7 feet. The siding is sold in 10-foot by 15-foot sections costing $35 each. Determine the total cost of the siding Sandy must purchase.


If \( a, b, c \) are non-negative integers less than \( 9 \) such that
\begin{align*}
a + 3b + 2c &\equiv 0 \pmod{9}, \\
2a + 2b + 3c &\equiv 3 \pmod{9}, \\
3a + b + 2c &\equiv 6 \pmod{9},
\end{align*}
then determine the remainder when \( abc \) is divided by \( 9 \).


What is the sum of the squares of the even, positive integers less than 100?


The operation $\star$ is defined as $a \star b = a + \frac{a}{b}$. Determine the result of $(5 \star 2) - 1$.


How many integer values of \( n \) satisfy \( -100 < n^3 < 100 \)?


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

Solution: \boxed{\begin{pmatrix} 5 \\ 3 \\ -1 \end{pmatrix}}


Compute the distance between the parallel lines given by
\[\begin{pmatrix} 3 \\ 1 \end{pmatrix} + t \begin{pmatrix} 4 \\ -6 \end{pmatrix}\]and
\[\begin{pmatrix} 2 \\ 4 \end{pmatrix} + s \begin{pmatrix} 4 \\ -6 \end{pmatrix}.\]


Maria has been tasked with writing all the possible rearrangements of the letters in her name. If she can write eight rearrangements of her name every minute, how many hours does it take to write all the possible rearrangements of her name?

Solution: \boxed{0.125 \text{ hours}}


In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=150$, and $AD \ne BC$. The perimeter of $ABCD$ is 580. Find $\cos A$.


Each letter in the alphabet is assigned a random integer value, and $G = 15$. The value of a word is the sum of its letters' values. If $FLAG$ equals 50 points, $LEAF$ equals 65 points, and $FEEL$ equals 58 points, what is the value of $A$?


Calculate $\sqrt{42q} \cdot \sqrt{7q} \cdot \sqrt{3q}$. Express your answer in simplest radical form in terms of $q$.

Solution: \boxed{21q \sqrt{2q}}


Calculate $68\cdot 97\cdot 113 \equiv n \pmod{25}$, where $0 \le n < 25$.


A wizard is creating a magic elixir. In order to do so, he must select one of four enchanted herbs and one of six mystical crystals. However, two of the crystals are incompatible with three of the herbs. How many valid combinations can the wizard use to concoct his elixir?


Factor $x^2+6x+9 - 100x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a<d$.

Solution: \boxed{(-10x^2 + x + 3)(10x^2 + x + 3)}


Find the sum of the first five prime numbers that have a units digit of 3.


Calculate the product of the sequence $\frac{1}{4}\cdot\frac{16}{1}\cdot\frac{1}{64}\cdot\frac{256}{1}\cdot\frac{1}{1024}\cdot\frac{4096}{1}\cdot\frac{1}{16384}\cdot\frac{65536}{1}$.


For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal?


Janice bought 40 items, each priced at 50 cents, $1.50, or $3.00. If her total purchase price was $\$$45.00, how many 50-cent items did she purchase?


The numbers $a, b, c, d$ are equal to 2, 3, 4, 5, in some order. Find the largest possible value of
\[
(a + b)(c + d).
\]


Determine the least positive multiple of 11 that is not a lucky integer.


Winnie now has 24 blue balloons, 38 red balloons, 44 white balloons, 74 green balloons, and 92 chartreuse balloons. She wants to distribute as many balloons as possible to her 5 best friends evenly. How many balloons will Winnie keep for herself after distributing them?


For any positive integer \( n \), the value of \( n! \) is the product of the first \( n \) positive integers. Find the greatest common divisor of \( 8! \) and \( 10! \).


In a coordinate plane, points A, B, and C have coordinates $(-2, 3)$, $(5, 3)$, and $(5, -2)$, respectively. Verify if triangle ABC is a right triangle and if yes, calculate the perimeter of triangle ABC.

Solution: \boxed{12 + \sqrt{74}}


A new model of a printer prints 25 pages per minute. How many minutes will it take to print 300 pages if the printer requires an additional minute of maintenance after every 50 pages printed?

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


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


Let points $A, B, C, D,$ and $E$ be collinear in that order, with distances $AB = 1, BC = 2, CD = 3,$ and $DE = 4$. If $P$ is any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?


Find the ratio of the area of $\triangle BDY$ to the area of $\triangle ADY$ in the diagram if $DY$ bisects $\angle ADB$. [asy]
import markers;

real t=35/(35+40);
pair A=(-17.57,0);
pair B=(11.43,0);
pair D=(0,30);
pair Y=t*A+(1-t)*B;

draw(D--A--B--D--Y);

label("$A$",A,SW);
label("$B$",B,E);
label("$D$",D,N);
label("$Y$",Y,NE);

label("$28$", .5*(B+A), S);
label("$35$", .5*(B+D), NE);
label("$40$", .5*(A+D), NW);
[/asy]


Determine the number of solutions to
\[3\sin^3 x - 7 \sin^2 x + 3 \sin x + 1 = 0\]in the range $0 \le x \le 2 \pi$.


A right triangle has an area of 180 square units, and one leg length of 30 units. Find the perimeter of the triangle.

Solution: \boxed{42 + 2\sqrt{261}}


A museum curator is documenting relics from an ancient civilization that used base 5 for their numbering system. In one exhibit, there are $1432_{5}$ clay tablets, $2041_{5}$ bronze sculptures, and $232_{5}$ stone carvings. Calculate the total number of items in the exhibit, expressed in base 10.


What is the period and amplitude of $y = 3\cos\left(\frac{x}{3} + \frac{\pi}{4}\right)$?


When the base-16 number $ABCDEF_{16}$ is written in base 2, how many base-2 digits (bits) does it have?


Let
\[f(x) =
\begin{cases}
9x + 3 & \text{if } x \text{ is an integer}, \\
\lfloor x \rfloor + 6 & \text{if } x \text{ is not an integer}.
\end{cases}
\]
Find $f(\sqrt{50})$.


Suppose an amoeba is placed in a container one day, and on that day it splits into three amoebas. Each subsequent day, every surviving amoeba splits into three new amoebas. However, at the end of every second day starting from day two, only half of the amoebas survive. How many amoebas are in the container at the end of one week (after seven days)?


In the coordinate plane, construct a right triangle with legs parallel to the $x$ and $y$ axes such that the medians to the midpoints of the legs lie on the lines $y = 2x + 3$ and $y = mx + b$. If the intersection of these medians is required to coincide with the fixed point $(4, 11)$, determine the number of different possible values for the slope $m$ and the intercept $b$.


Let $\mathcal{P}$ be the parabola given by the equation \( y = x^2 \). Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are \((-4,16)\), \((1,1)\), and \((6,36)\), find the sum of the distances from the directrix of the parabola to all four intersection points.


Calculate the number of digits to the right of the decimal point in the expression $\frac{5^7}{10^5 \cdot 125}$ when it is expressed as a decimal.


Let \( u \) and \( v \) be real numbers greater than 1 such that
\[
(\log_3 u)^3 + (\log_5 v)^3 + 6 = 6 (\log_3 u)(\log_5 v).
\]
Compute \( u^{\sqrt{3}} + v^{\sqrt{3}} \).


Let
\[\mathbf{B} = \begin{pmatrix} \cos \frac{\pi}{4} & 0 & -\sin \frac{\pi}{4} \\ 0 & 1 & 0 \\ \sin \frac{\pi}{4} & 0 & \cos \frac{\pi}{4} \end{pmatrix}.\]
Compute $\mathbf{B}^{2023}.$

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


Find $n$ such that $2^6 \cdot 3^3 \cdot n = 9!$.


Tom has a collection of marbles: one red, one green, one blue, one purple, and three identical yellow marbles. How many different groups of two marbles can Tom choose?


When the product $(5x + 3y + 2)(2x + 5y + 3)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?


How many ways are there to put 6 indistinguishable balls into 2 indistinguishable boxes?


An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. What is the smallest abundant number that is not a multiple of 4?


Two rays with a common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, and point $B$ on the other ray, with $AB=1$. What is the maximum possible length of $\overline{OB}$?


Either increasing the radius or the height of a cylinder by ten inches will result in the same volume. The original height of the cylinder is four inches. What is the original radius in inches?

Solution: \boxed{r = 4 + 2\sqrt{14}}


If \( x + 5 \) is a factor of \( dx^3 + 17x^2 - 2dx + 50 \), find the constant \( d \).


For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 3.$$


Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\]


Alex has 6 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 12 friends if each friend can receive one or more than one problem?


A list of five positive integers has all of the following properties:
- The only integer in the list that occurs more than once is 7.
- Its median is 10.
- Its average (mean) is 12.

What is the largest possible integer that could appear in the list?


In $\triangle ABC$, the medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 20$ and $BN = 15$, determine the length of side $\overline{AB}$.


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


The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 8x + 9.$ Find the leading coefficient of $g(x).$


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


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


A state modifies its boat license system such that the license must consist of either the letter A, B, M, or V followed by any seven digits. However, the first digit cannot be 0 or 1. What is the number of groups of letters and numbers available for these boat licenses?


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


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:2$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 24, determine the area of $\triangle ABC$.


At a conference with $30$ businessmen, fifteen businessmen drank coffee, thirteen drank tea, and eight drank soda. Six businessmen drank both coffee and tea, two drank both coffee and soda, and three drank both tea and soda. One businessman drank all three beverages. How many businessmen drank none of the beverages?


Complex numbers \( a, b, c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 42 \), find \( |ab + ac + bc| \).


The graph of the line $3x-y=c$ is a perpendicular bisector of the line segment from $(2,4)$ to $(6,8)$. What is the value of $c$?


How many positive divisors does 8! have?


A segment with endpoints at $A(1, -1)$ and $B(11, 3)$ is extended through $B$ to point $C$. If $BC = \frac{2}{3} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair.

Solution: \boxed{\left(\frac{53}{3}, \frac{17}{3}\right)}


Rachel has two identical rose plants and an orchid. She has two identical blue lamps and two identical green lamps. Each plant must be under exactly one lamp, but more than one plant can be under a single lamp. How many ways can Rachel arrange her plants under these lamps?


Find the sum of all integral values of $c$ with $c \leq 30$ for which the equation $y = x^2 - 9x - c$ has two rational roots.


Find the matrix $\mathbf{P}$ that triples the first column of a matrix and simultaneously rotates the second column by 90 degrees counterclockwise. Specifically, for any matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the transformation should result in $\begin{pmatrix} 3a & -d \\ 3c & b \end{pmatrix}$. If no such matrix $\mathbf{P}$ exists, then enter the zero matrix.

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


A polynomial with integer coefficients is given by
\[7x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 18 = 0.\] Determine the number of different possible rational roots of this polynomial.


In triangle $DEF$, $DE = 6$, $EF = 10$, and the length of median $DM$ is 5. Find $DF$.


Evaluate the largest possible value of $x$ in the form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{4x}{5}-2 = \frac{5}{x}$, where $a, b, c,$ and $d$ are integers. What is $\frac{a \cdot c \cdot d}{b}$?


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


Lara is trying to solve the following equation by completing the square: $$100x^2 + 60x - 49 = 0.$$ She seeks to rewrite the above equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$?


Let $A$, $B$, and $C$ be distinct digits where $A < 7$, $B < 7$, and $C < 7$. Suppose we have ${AB_7} + {C_7} = {C}0_7$ and ${AB_7} + {BA_7} = {CC_7}$. Find the three-digit number ${ABC}$.


Two angles of a triangle measure 60 and 30 degrees. If the side opposite the 30-degree angle measures $8\sqrt3$ units, what is the sum of the lengths of all sides of the triangle? Express your answer as an exact value.

Solution: \boxed{24\sqrt3 + 24}


Using the digits 1, 2, 3, 4, 5, 6, how many even three-digit numbers less than 600 can be formed if each digit can be used more than once?


What is the smallest prime divisor of $6^{15} + 9^{11}$?


Joe and JoAnn each bought 15 ounces of coffee in a 20-ounce cup. Joe drank 3 ounces of his coffee and then added 3 ounces of cream. JoAnn added 3 ounces of cream, stirred the coffee well, and then drank 3 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? Express your answer as a fraction.


The Great Eighteen Hockey League consists of three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions once. How many league games are scheduled?


What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{1020}$ is defined?


Let $Q(x) = x^2 - 5x + 4$. A real number $x$ is chosen at random from the interval $10 \le x \le 20$. Determine the probability that $\lfloor\sqrt{Q(x)}\rfloor = \sqrt{Q(\lfloor x \rfloor)}$.


The areas of three of the faces of a right rectangular prism are $30 \mathrm{cm}^2$, $45 \mathrm{cm}^2$, and $54 \mathrm{cm}^2$. What is the volume of the prism, in cubic centimeters?

Solution: \boxed{270 \text{ cubic centimeters}}


Define $X \star Y$ as $X \star Y = \frac{(X+Y)}{4}$. What is the value of $(3 \star 8) \star 6$?


How many different four-letter arrangements can be formed using the seven letters $A, B, C, D, E, F$, and $G$, if the first letter must be $D$, one of the other letters must be $A$, and no letter can be used more than once in the arrangement?


Joel selected two different acute angles $x$ and $y$ (both strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, $\tan x$, and $\sec y$ on four different cards. Then he gave those cards to four students, Malvina, Paulina, Georgina, and Bella, one card to each. They shared the values on their cards with each other. Only Bella was able to surely identify which trigonometric function produced the value on her card. Compute the sum of all possible values that Joel wrote on Bella's card.


An infinite geometric series has a first term of $20$ and a second term of $5.$ A second infinite geometric series has the same first term of $20,$ a second term of $5+n,$ and a sum of double that of the first series. Find the value of $n.$


You recently bought more than 150 eggs. The eggs are sorted in containers that can store exactly 18 eggs each. However, upon inspecting the containers, you realize that three containers each hold only 16 eggs and one container holds 17 eggs, while all the other containers hold the full 18 eggs. What is the smallest number of eggs you could have right now?


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


A circular park consists of an outer-ring path for walkers (white), a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is six feet wide at all points, the garden ring is now ten feet wide at all points, and the fountain has a diameter of 12 feet. What is the radius, in feet, of the circle that forms the outer boundary of the flower garden?

Solution: \boxed{16 \text{ feet}}


What is the remainder when \(8x^4 - 20x^3 + 28x^2 - 32x + 15\) is divided by \(4x - 8\)?


Find the square root of $\dfrac{9!}{210}$. (Note that $n!$ (n factorial) is the product of all positive integers up to $n$.)


Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^3 x - \cos^3 x = \frac{1}{\cos x} - \frac{1}{\sin x}.\]


Let \( A, B, \) and \( C \) be constants such that the equation \[\frac{(x+B)(2Ax+56)}{(x+C)(x+14)} = 2\] has infinitely many solutions for \( x \). For these values of \( A, B, \) and \( C \), it turns out that there are only finitely many values of \( x \) which are not solutions to the equation. Find the sum of these values of \( x \).

Solution: \boxed{-14 + (-28) = -42}


Solution: \boxed{14 + e^2}


Add $36_7 + 274_7.$ Express your answer in base 7.


Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'?


Suppose we roll two standard fair 6-sided dice. What is the probability that the sum of the numbers rolled is a prime number?


What is the largest divisor of 408 that is also a factor of 340?


Consider the polynomial
\[ a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 \]
where all coefficients are integers. All its roots $r_1, r_2, r_3, r_4, r_5$ are also integers. Additionally, the roots of the polynomial
\[ a_0 x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 = 0 \]
are the same as the first polynomial's roots. Find the number of possible multisets $S = \{r_1, r_2, r_3, r_4, r_5\}$.


Solve \[\arccos (\cos x) = \frac{x}{2}.\]
Enter all the solutions within the domain \[-\frac{\pi}{3} \le x \le \frac{\pi}{3}.\]


Compute the largest integer $k$ such that $2025^k$ divides $(2025!)^2$.


Ed and Sue both cycle, jog, and swim at equal and constant rates. They perform these activities at distinct rates but equal times in the given scenario. Ed covers 88 kilometers after cycling for 3 hours, jogging for 4 hours, and swimming for 2 hours. Sue covers 104 kilometers after jogging for 3 hours, swimming for 4 hours, and cycling for 2 hours. Their cycling, jogging, and swimming rates are even whole numbers of kilometers per hour. Find the sum of the squares of Ed's cycling, jogging, and swimming rates.


How many numbers in the set $\{5, 17, 29, 41, \ldots, 197\}$ can be written as the difference of two prime numbers, where one of the primes is 3?


A paperboy delivers newspapers alongside a lane containing 12 houses. The paperboy does not have to deliver every newspaper, but to keep his customers somewhat satisfied, he must ensure that he never misses three consecutive houses with a delivery. Additionally, he must deliver at least to the last house to leave a good impression. Calculate the number of ways he can distribute the newspapers under these conditions.


In rectangle $ABCD$, side $AB$ measures $10$ units and side $BC$ measures $4$ units. Points $F$ and $G$ are on side $CD$ with segment $DF$ measuring $2$ units and segment $GC$ measuring $3$ units. Lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$?


What is the value of \( x \) for which \( (2015 + x)^2 = x^2 \)?


In an international sprinting event, there are 10 sprinters, including 4 Americans. The medals awarded are gold, silver, and bronze. Determine the number of ways the medals can be awarded if exactly two Americans must win medals.


Let \( g(x) = x^2 - 4x \). How many distinct real numbers \( c \) satisfy \( g(g(g(g(c)))) = 2 \)?


The cost of four pencils and one pen is $\$2.60$, and the cost of one pencil and three pens is $\$2.15$. Find the cost of three pencils and two pens.


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


In a set of seven consecutive integers, the largest integer is less than three times the average of the seven integers. What is the smallest integer that could be in the set?


There are three real numbers $x$ that are not in the domain of $$g(x) = \frac{1}{2+\frac{1}{2+\frac 1x}}.$$ What is the sum of those three numbers?


The graph of \(y = \frac{p(x)}{q(x)}\) displays a horizontal asymptote at \(y = 3\) and a vertical asymptote at \(x = -3\). A hole is detected at \(x = 6\). This graph crosses the x-axis at \(x = 2\). Determine \(\frac{p(4)}{q(4)}\).


Mary takes 45 minutes to walk uphill 1.5 km from her home to school, but it takes her only 5 minutes to return home downhill along the same route. What is her average speed, in km/hr, for the uphill journey, the downhill journey, and the entire round trip?

Solution: \boxed{3.6 \text{ km/hr}}


In triangle $XYZ$, sides $XY$, $YZ$, and $ZX$ are tangent to a circle with center $O$. Given that $\angle XYZ = 75^\circ$ and $\angle YZO = 25^\circ$, find $\angle YXZ$, in degrees.


Rationalize the denominator:
$$\frac{1}{\sqrt[3]{3} + \sqrt[3]{81}}$$


A bag contains 4 red, 7 green, 9 yellow, and 10 blue jelly beans. A jelly bean is selected at random. What is the probability that it is not yellow?


If the product $(2x^2 - 3x + 5)(5 - 3x)$ can be written in the form $Ax^3 + Bx^2 + Cx + D$, where $A, B, C, D$ are real numbers, then determine $A+ B + C + D$.


Cathy, David, and Elena start jogging around a park at 6:00 AM. Cathy completes a lap every 5 minutes, David finishes a lap every 9 minutes, and Elena completes a lap every 8 minutes. What is the earliest time when all three joggers meet back at the starting point, and how many laps has Cathy completed by then?


Two circles of radius 12 cm overlap such that one circle's center is 8 cm from the center of the other circle. How long, in cm, is the common chord (dotted segment) of the two circles?

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


A 599-page book prints each page number exactly once. From page 1 to page 599, how many more times does the digit '5' appear than the digit '9'?


A torus having an inner radius of $3$ and an outer radius of $5$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center of the torus so that the ball still touches the horizontal plane and is tangent to the side of the torus? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-plane centered at $(4,0,1)$ with radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis and rests on the table or the donut.)


How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6?


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


Lily is at a candy store buying Skittles. She decides she wants to buy at least 150 Skittles and needs to have exactly 17 leftover after she shares them equally among 19 friends. What is the minimum number of Skittles Lily should buy?


Let $S$ be the set of positive integer divisors of $15^7.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$


Let \( F_1 = \left( -2, 2 - \frac{\sqrt{3}}{2} \right) \) and \( F_2 = \left( -2, 2 + \frac{\sqrt{3}}{2} \right) \). The set of points \( P \) such that
\[
|PF_1 - PF_2| = \sqrt{2}
\]
forms a hyperbola. The equation of this hyperbola can be written as
\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,
\]
where \( a, b > 0 \). Find \( h + k + a + b \).

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


Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number, and the tens digit is 2.


The Cookie Monster now encounters another cookie whose boundary is described by the equation $x^2 + y^2 + 10 = 6x + 12y$. Determine the radius of this cookie.


What is the greatest integer less than 200 for which the greatest common divisor of that integer and 30 is 10?


Find the seventh term of the geometric sequence with first term $3$ and second term $-\frac{1}{2}$.


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


Determine the ordered pair $(x, y)$ that satisfies the equations:
\begin{align*}
6x + 3y &= -12, \\
4x &= 5y + 10.
\end{align*}

Solution: \boxed{\left(-\frac{5}{7}, -\frac{18}{7}\right)}


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


If $5x + 8 \equiv 3 \pmod{14}$, what is $3x + 10$ congruent to $\pmod{14}$?


A right cylinder and a right circular cone are both inscribed in a sphere. The base radius of the cylinder is 4 units, and the sphere has a radius of 7 units. The cone shares the base with the cylinder and its vertex points toward the top of the cylinder. Calculate the total volume, in cubic units, of the space inside the sphere and outside the combined volume of the cylinder and cone. Let this volume be $X\pi$. Find $X$, as a common fraction.

Solution: \boxed{\frac{1372 - 128\sqrt{33}}{3}}


Two standard 8-sided dice are rolled. What is the probability that the sum rolled is a perfect square?


Suppose that the following numbers are roots of the same nonzero polynomial with rational coefficients: \[3 - \sqrt{7}, \;5+\sqrt{11}, \;17 - 2\sqrt{10}, \;-\sqrt{3}\]. What is the smallest possible degree of the polynomial?


Compute the smallest positive value of $x,$ in degrees, for which the function
\[f(x) = \sin \frac{x}{5} + \sin \frac{x}{13}\] achieves its maximum value.


If $x$ is a positive integer, what is the value of $x$ for the equation $(x! - (x-4)!) = 120$?


Compute the series $\displaystyle \sum_{n=2}^\infty \sum_{k=1}^{n-1} \frac{k}{3^{n+k}}$.


Solve for $z$ in the equation $2 - 3iz = -4 + 5iz$.


What is the sum of the greatest common divisor of $60$ and $45045$ and the least common multiple of $60$ and $45045$?


Find the reflection of the vector \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\) over the vector \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\) after translating the initial vector by \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\).

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


Suppose $13^6 \equiv m \pmod{11}$, where $0 \le m < 11$.

What is the value of $m$?


What is the matrix $\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$, where $A$ goes to $A'$, $B$ goes to $B'$, $C$ goes to $C'$, and $D$ goes to $D'$? Specifically, $ABCD$ is rotated by $30^\circ$ clockwise and scaled by a factor of 2.

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


Determine the maximum possible value of the greatest common divisor of two consecutive terms of the sequence given by \( a_n = n^2! + n \), where \( n \geq 0 \).


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


How many different three-letter sets of initials are possible using the unique letters from $A$ to $I$?


How many non-congruent triangles with perimeter 8 have integer side lengths?


If the consecutive integers from $65$ to $1$ were written as $$6564636261...,$$ what would be the $79^{\text{th}}$ digit to be written?


How many integers $n$ satisfy $(n-3)(n+5)<0$?


A collection of 150 pens is sold through a website for $\$24.50$. Shipping is an additional $\$7.50$. Including the charges for both the pens and the shipping, what is the average cost, in cents, for each pen? Express your answer rounded to the nearest whole number.


The numbers 1, 2, 4, 5, 7, 8 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?

\[
\begin{array}{c||c|c|c|}
\times & a & b & c \\ \hline \hline
d & & & \\ \hline
e & & & \\ \hline
f & & & \\ \hline
\end{array}
\]


What is the greatest common factor of 150, 225, and 300?


Find the third-smallest three-digit palindromic prime.


There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed among these six people, with each person getting one letter, what is the probability that exactly three people get the right letter?


Suppose \( n \) yields a remainder of 3 when divided by 7. What will be the remainder when \( n + 2947 \) is divided by 7?


If $3(x^2+x+1) = y - 6$ and $x = -3$, what is the value of $y$?


Seventy-five percent of a ship's passengers are women, and fifteen percent of those women are in first class. What is the number of women in first class if the ship is carrying 300 passengers?


Find the minimum value of
\[ \frac{(x-1)^2}{y-2} + \frac{(y-1)^2}{x-2} \]
for real numbers \( x > 2 \) and \( y > 2 \).


A rectangle has its length increased by $30\%$ and its width by $15\%$. By what percentage does the area of the rectangle increase?


Both $c$ and $d$ are integers and $d > 1$. When $c^d$ is the greatest possible value less than 630, what is the sum of $c$ and $d$?


Given vectors $\mathbf{a}$ and $\mathbf{b}$, let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\]
Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$ Find the ordered pair $(t,u)$.

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


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 --- where $A$ 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 $\it{good\ words}$ are there?


Name the largest whole number less than $150$ that has exactly 9 positive factors.


In pentagon $FGHIJ$, the interior angles $F$ and $G$ are $70^{\circ}$ and $110^{\circ}$ respectively. Angles $H$ and $I$ are equal, and the fifth angle $J$ is $25^{\circ}$ more than twice $H$. Determine the measure of the largest angle.


Evaluate the expression $3^{13} \div 3^3 + 2^3$.


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


A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria?


Compute $\sin 225^\circ$ and $\cos 225^\circ$.


Find the $2 \times 2$ matrix $\mathbf{N}$ such that $\mathbf{N} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ -7 \end{pmatrix}$ and $\mathbf{N} \begin{pmatrix} 4 \\ 3 \end{pmatrix} = \begin{pmatrix} 20 \\ 21 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 3.5 & 2 \\ 0 & 7 \end{pmatrix}}


Determine two **positive** integers \(d\) such that the quadratic equation \(7x^2 + 13x + d = 0\) has rational solutions. What is the product of these \(d\) values?


Find the maximum value of
\[
\cos (\theta_1 + \phi) \sin (\theta_2 + \phi) + \cos (\theta_2 + \phi) \sin (\theta_3 + \phi) + \cos (\theta_3 + \phi) \sin (\theta_4 + \phi) + \cos (\theta_4 + \phi) \sin (\theta_5 + \phi) + \cos (\theta_5 + \phi) \sin (\theta_1 + \phi),
\]
where $\phi$ is a fixed real number and $\theta_1, \theta_2, \theta_3, \theta_4, \theta_5$ are variables.


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


Jordan and his three friends are on a relay team, where runners need to complete four consecutive laps with Jordan finishing the last one due to his speed. For variation in team strategy, Jordan decides to take either the third or fourth lap while the fastest of the remaining three friends must start the race. How many different orders can the four team members run under these conditions?


Two of the roots of the equation \[ax^3 + bx^2 + cx + d = 0\] are $4$ and $-3.$ Additionally, $d = 0$ and $a \neq 0.$ Compute $\frac{b+c}{a}.$


Find the number of positive integers $n \leq 500$ such that $21n$ is a perfect square.


Solve the inequality:
\[|x - 2| + |x + 3| < 8.\]

Solution: \boxed{\left(-\frac{9}{2}, \frac{7}{2}\right)}


Caroline can make nine lassis out of three mangoes and one banana. How many lassis can she make if she has fifteen mangoes and five bananas?


Let \[g(x) = \left\{ \begin{aligned}
3x + 12 & \quad \text{if } x < -1 \\
x^2 - 1 & \quad \text{if } -1 \le x < 1 \\
2x + 1 & \quad \text{if } x \ge 1
\end{aligned} \right.\]
Find all solutions to the equation $g(x) = 12.$


What is $5^{-1} + 5^{-2} \pmod{31}$? Express your answer as an integer from $0$ to $30$, inclusive.


How many numbers can you get by multiplying two or more distinct members of the set $\{2,3,7,13\}$ together?


Four friends have a total of 8 identical pencils, and each friend has at least one pencil. Additionally, the first friend always has at least two pencils. In how many ways can this happen?


In the trapezoid shown, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $8:2$. If the total length of $AB + CD = 150$ cm, and the segment $\overline{AB}$ is three times the length of segment $\overline{CD}$, how long is segment $\overline{AB}$? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
pair A = (0,0), B = (8,0), C = (5,2), D = (2,2);
draw(A--B--C--D--cycle--C);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW);
[/asy]

Solution: \boxed{120 \text{ cm}}


Calculate the remainder when the sum $1 + 7 + 7^2 + \cdots + 7^{100}$ is divided by $500$.


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


In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal, and the height from $F$ to $GH$ is 5 units. If the length of the base $GH$ is 10 units and the length of the base $EF$ is 4 units, what is the perimeter of $EFGH$?

Solution: \boxed{14 + 2\sqrt{34}}


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


When the base-16 number $1A1A1_{16}$ is written in base 2, how many base-2 digits (bits) does it have?


Let $g(x) = \frac{2ax - b}{dx - 2b}$ where $a, b, d \neq 0$. If $g(g(x)) = x$ for all $x$ in the domain of $g$, determine the value of $2a - 2b$.


The sides of a triangle have lengths $7, 15,$ and $k$, where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?


For a specific weekend, the weather forecast states a $60\%$ probability of rain on Saturday and, if it rains on Saturday, a $70\%$ probability of rain on Sunday. If it does not rain on Saturday, the probability of rain on Sunday is $40\%$. What is the probability it will rain at least one day during the weekend? Express your answer as a percentage.


For integers $a$, $b$, $c$, and $d$, factorize the quartic polynomial given by $(x^2+ax+b)(x^2+cx+d)$ to match the polynomial $x^4 + x^3 - 5x^2 + x - 6$. What is the value of $a + b + c + d$?


If $g(x)=\frac{x-3}{x^2+cx+d}$, and $g(x)$ has vertical asymptotes at $x=2$ and $x=-1$, find the sum of $c$ and $d$.


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


If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are unit vectors in $\mathbb{R}^3$, find the smallest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{c} - \mathbf{a}\|^2.
\]


Triangle $PQR$ is an obtuse, isosceles triangle. Angle $P$ measures 30 degrees. What is the number of degrees in the measure of the largest interior angle of triangle $PQR$?


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 3$, what is $\left|\frac{1}{z} + \frac{1}{w}\right|$?


Chloe wants to buy a book that costs $\$47.50$. She has four $\$10$ bills, five half-dollars, and a pile of nickels. What is the minimum number of nickels that must be in her pile so she can pay for the book?


Let \( a, b, \) and \( c \) be three positive real numbers such that \( a + b + c = 3 \). Assume that no one of these numbers is more than three times any other. Find the minimum value of the product \( abc \).


Consider three coins where two are fair and a third coin lands on heads with a probability of $\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ and $q$ are coprime integers. Find $p + q$.


In triangle $PQR$, you have $PQ = QR = RS = SP$ forming a quadrilateral, and the measure of angle $PQR$ is $90^\circ$. Determine the measure of angle $QRS$.


Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)?


The sum of two positive integers is 60, and their difference is 16. What is the value of the positive difference of the squares of the integers? Additionally, find the product of the two integers.


A line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} + t \begin{pmatrix} 4 \\ 3 \end{pmatrix}.
\]
A second line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ 15 \end{pmatrix} + u \begin{pmatrix} 2 \\ 5 \end{pmatrix}.
\]
Find the cosine of the acute angle $\phi$ formed by the two lines.


Three cubes with faces numbered from 1 to 6 are tossed. What is the probability that the sum of the numbers on the top faces is exactly 10?


In a hypothetical deck of 65 cards evenly divided into 5 suits, Max picks two different cards without replacement. What is the probability that the cards are of different suits?


Let $T_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{T_{4n}}{T_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term.


Let $x$ and $y$ be two positive real numbers such that $x + y = 50$. Find the ordered pair $(x, y)$ for which $x^3 y^4$ is maximized.

Solution: \boxed{\left(\frac{150}{7}, \frac{200}{7}\right)}


Sean adds up all the odd integers from 1 to 500, inclusive. Julie adds up all the integers from 1 to 200, inclusive. What is Sean's sum divided by Julie's sum?


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


What is the area enclosed by the graph of $|5x| + |2y| = 10$?


Let's define an integer $N$ as delicious if there exist several consecutive integers, including $N$, that add up to 2023. Find the smallest delicious integer.


A rectangular box has a volume of $4320$ cubic inches and a surface area of $1704$ square inches. The sum of the lengths of its $12$ edges is $208$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{6240 \text{ cubic inches}}


In how many ways can a president, a 2-person committee, and a secretary be chosen from a group of 10 people, where the order of the committee members does not matter and the president and the secretary cannot be on the committee?


A round pizza is $\frac{1}{2}$ of an inch thick and has a diameter of 18 inches. It is cut into 16 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\pi$.


Find a positive integer that is divisible by 24 and whose cube root is a number between 7.9 and 8.


What is the remainder of $9^6 + 8^8 + 7^9$ divided by 5?


What is the greatest 4-digit base 7 positive integer that is divisible by 7?  (Express your answer in base 7.)


Find a positive integer that is divisible by 21 and whose cube root is a number between 9 and 9.1.


In triangle $XYZ$, the sides are in the ratio of $3:4:5$. Segment $XE$ is the angle bisector of $\angle X$, drawn to the longest side, dividing it into segments $YE$ and $ZE$. What is the length, in units, of the shorter subsegment of the longest side if the length of the longest side is $15$ units? Express your answer as a common fraction.


Solution: \boxed{x^2 + y^2 = 2(3002^{3002} + 3002^{-3002})}


A domino is a rectangular tile composed of two squares. Each square may contain an integer from 0 to 12, and each integer within this range is paired with every other integer exactly once to form a complete domino set. A $\textit{double}$ is a domino that has the same integer on both of its squares. What is the probability that a domino randomly selected from the set will be a $\textit{double}$? Express your answer as a common fraction.


On the first day, Vanessa Magica used her magical staff to make an object's length increase by $\frac{1}{3}$. On the second day, she increased the object's longer length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day of performing this trick, Vanessa will make the object's length 50 times its original length. What is the value of $n$?


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that its digits are all different?


Solution: \boxed{1 - \frac{1}{501} + \frac{1}{2}H_{500}}


Five pens and four pencils cost $\$2.90$. Three pens and six pencils cost $\$2.18$. No prices include tax. In cents, what is the cost of a pencil?


The equations of the asymptotes of a hyperbola are $y = 3x + 4$ and $y = -3x + 6$. Given that the hyperbola passes through the point $(1, 9)$, find the standard form for the equation of the hyperbola: \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\] where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find $a - h.$

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


Find the magnitude of \(\left(\frac{4}{7} + \frac{3}{7}i\right)^7\).


A school has between 160 and 210 students enrolled. Every afternoon, all the students gather for a music rehearsal. The students are divided into eight distinct groups. If one student is absent, the groups can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Let $x$ and $y$ be positive real numbers such that $x + 3y = 1.$ Find the minimum value of
\[\frac{1}{x} + \frac{3}{y}.\]


Suppose $a < 0$ and $a < b < c$. Determine which of the following statements must be true:
1. $ab > bc$
2. $ac > bc$
3. $ab > ac$
4. $a + b < b + c$
5. $c/a > 1$


Alice walks five blocks south and eight blocks west. If each block is one-quarter of a mile, how many miles did Alice walk?

Solution: \boxed{3.25 \text{ miles}}


The value of $\log_{10}{865}$ is between which two consecutive integers $c$ and $d$? Find $c+d$.


You have a whole pizza in the refrigerator. On your first trip, you eat one-third of the pizza. On each successive trip, you eat one-third of what was left from the previous trip. After six trips to the refrigerator, what fractional part of the pizza have you eaten?


Calculate one-fourth of the difference between the product of 9 and 5 and the sum of 7 and 3.


If $x^{3y} = 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


Three cubes, each with faces numbered 1 through 6, are tossed. What is the probability that the sum of numbers on the top faces is a multiple of 3? Express your answer as a common fraction.


What is the coefficient of $x^4$ in the expansion of $$(x+3\sqrt{2})^9?$$


Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits, where the hundreds digit is 8.


What is the remainder when $(x - 1)^{2028}$ is divided by $x^2 - x + 1$?


Given $\tan \theta = 3,$ find
\[
\frac{1 + \cos \theta}{\sin \theta} - \frac{\sin \theta}{1 - \cos \theta}.
\]


A traffic light cycles repeatedly in the following order: green for 45 seconds, then yellow for 5 seconds, and red for 50 seconds. Cody picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching?


Quadrilateral $EFGH$ is a parallelogram. The angle $G$ is given as $80^\circ$. What is the degree measure of angle $E$?


John draws a regular seven-pointed star in the sand, and at each of the 7 outward-pointing points and 7 inward-pointing points, he places one of fourteen different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent?


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


Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 15$ and $X$ is an integer, what is the smallest possible value of $X$?


What is the largest five-digit number that is divisible by 8?


A teacher distributes candy to her class of 40 students, with a mean of 4 pieces of candy per student. If every student takes at least 2 pieces of candy, what is the greatest number of pieces one student could have taken?


Let $p$, $q$, $r$, and $s$ be the roots of the polynomial equation \[x^4 + 6x^3 - 4x^2 + 7x + 3 = 0.\] Calculate the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\]


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