Evaluate the expression $\log_{y^4}{x}\cdot\log_{x^7}{y^3}\cdot\log_{y^5}{x^2}\cdot\log_{x^2}{y^5}\cdot\log_{y^3}{x^7}$ and express it as $b\log_y{x}$ for some constant $b$.


Lillian places 720 marbles into $m$ total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of $m$ can this be done?


Let $a$ and $b$ be relatively prime positive integers such that $\frac ab = \sum_{n=1}^{\infty} \left(\dfrac{2n-1}{2^n} + \dfrac{2n}{3^n}\right)$. Compute $a+b$.


A circle is tangent to the parabola $y = x^2$ at point $(3,9)$ and also passes through the point $(0,2)$. Find the coordinates of the center of this circle.

Solution: \boxed{\left(-\frac{141}{11}, \frac{128}{11}\right)}


The curve defined by the equation $xy = 2$ is another hyperbola. Determine the distance between the foci of this hyperbola.


$x$ is a real number such that $x+\frac{1}{x} = 3$. Define $S_m = x^m + \frac{1}{x^m}$. Determine the value of $S_5$.


Calculate the value of \( n \) for which the following complex number is a root of unity:
\[
\frac{\tan \frac{\pi}{4} + i}{\tan \frac{\pi}{4} - i}
\]
Verify if it equals to \(\cos \frac{2n \pi}{8} + i \sin \frac{2n \pi}{8}\) for some integer \( n \) between 0 and 7 inclusive.


Determine how many prime dates occurred in a leap year, such as 2012, where both the month and the day are prime numbers.


Define a function $g$ from the integers to the integers as follows:
\[g(n) = \left\{
\begin{array}{cl}
n - 4 & \text{if $n$ is odd}, \\
\frac{n}{3} & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 5.$ Find $m.$


The third and sixth terms of a geometric sequence are \(16\) and \(144\), respectively. What is the ninth term of this progression?


Cindy rolls a fair octahedral die with numbers $1,2,3,...,8$. What is the expected number of digits in the number she obtains? Express your answer as a decimal.


What is the value of $$\frac{1}{2}\times 8\times \frac{1}{8} \times 32 \times \frac{1}{32} \times 128 \times \frac{1}{128} \times 512 \times \frac{1}{512} \times 2048?$$


Find the slope \( m \) of the asymptotes for the hyperbola given by the equation \[ \frac{(y-1)^2}{16} - \frac{(x+2)^2}{9} = 1. \]


How many positive integers less than 100 that have an even number of positive divisors are not prime numbers?


Find all real numbers \(x\) which satisfy \[\frac{x-2}{x-5} \ge 3.\] (Give your answer in interval notation.)

Solution: \boxed{(5, \frac{13}{2}]}


Find the sum of the rational roots of the polynomial \( f(x) = x^3 - 9x^2 + 27x - 8 \).


Human explorers discover a solar system with 15 planets, 7 of which are Earth-like, and 8 are Mars-like. Each Earth-like planet requires 2 units of colonization effort, while each Mars-like planet requires only 1 unit. If the expedition has a total of 16 units of colonization effort available, how many different combinations of planets can be colonized given that all planets are unique?


A student, Leo, needs to complete homework assignments to earn grades. For the first four grades, he needs to complete one assignment per grade. For the next seven grades, he requires three assignments per grade. For all subsequent grades, the number of homework assignments per grade increases by two for each set of four grades (i.e., he needs 5 assignments per grade for his 12th to 15th grades, 7 assignments for 16th to 19th grades, and so on). Determine the smallest number of homework assignments Leo needs to complete a total of 27 grades.


Determine the count of possible integer values for $n$ such that the sides $\log_{5}30$, $\log_{5}50$, and $\log_{5}n$ of a triangle grant a positive area.


In a city represented by the Cartesian plane, Rachel is located at $(4, -15)$ and Sarah is located at $(-2, 12)$. They agree to meet at the midpoint between their locations and then walk directly downwards to reach their friend Max, who is positioned at $(1, -5)$. How many units downward do Rachel and Sarah walk together to get to Max?


What is the number of degrees in the acute angle formed by the hands of a clock at 3:37?


We have a triangle $\triangle DEF$ such that $DE = DF = 10$ and $EF = 12$. What is the length of the median $DM$?


Simplify
\[
\frac{1 + \sin x + \cos x}{1 + \sin x - \cos x}.
\]

Solution: \boxed{\cot \frac{x}{2}}


How many positive 3-digit numbers are multiples of 25, but not of 45?


Let $a,$ $b,$ $c,$ and $d$ be nonnegative numbers whose sum is 200. Find the largest possible value of
\[ ab + bc + cd + da. \]


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


The planet Yavin moves in an elliptical orbit around its star. At its closest approach, called perigee, it is 3 astronomical units (AU) from the star. At its farthest point, known as apogee, it is 15 AU away. Determine how far Yavin is from its star when it is exactly halfway through its orbit. Additionally, calculate the length of the semi-minor axis of Yavin's orbit.

Solution: \boxed{3\sqrt{5} \text{ AU}}


What is the sum of all the positive even factors of $630$ that are divisible by $15$?


Let $p,$ $q,$ $r$ be distinct, nonzero real numbers such that
\[p^2 + 2 \frac{1}{q} = q^2 + 2 \frac{1}{r} = r^2 + 2 \frac{1}{p}.\]
Find $|pqr|$.


Find the value of $k$ such that for any angle $\alpha$:
\[
2(\sin \alpha + \csc \alpha)^2 + 2(\cos \alpha + \sec \alpha)^2 = k + 2\tan^2 \alpha + 2\cot^2 \alpha.
\]

Solution: \boxed{k = 14}


The equations of the asymptotes of a hyperbola are $y = 3x + 4$ and $y = -3x + 6$. Given that the hyperbola passes through the point $(1, 10)$, the standard form for the equation of the hyperbola is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\] where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find $a + h.$

Solution: \boxed{\sqrt{21} + \frac{1}{3}}


Let $\mathbf{u},$ $\mathbf{v},$ $\mathbf{w}$ be vectors, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}.$  Then the determinant of the matrix whose column vectors are $\mathbf{u} \times \mathbf{v},$ $\mathbf{v} \times \mathbf{w},$ and $\mathbf{w} \times \mathbf{u}$ is equal to
\[m \cdot E^p.\]Determine the ordered pair $(m,p).$


What is the smallest four-digit positive integer that is divisible by 53 and ends with the digit 3?


Find the complex number $z$ such that
\[|z + 1| = |z - 3| = |z - 2i|.\]

Solution: \boxed{1 + \frac{1}{4}i}


A pizza shop calculates the cost of pizzas based on their area. If the price per square inch is $0.05, how much more does a 16-inch pizza cost than an 8-inch pizza?


Find the sum of the infinite series \( 1 + 3\left(\frac{1}{2000}\right) + 5\left(\frac{1}{2000}\right)^2 + 7\left(\frac{1}{2000}\right)^3 + \cdots \).


A sandwich shop offers five types of bread, seven types of meat, and six types of cheese. A customer can choose one type of bread, one type of meat, and one type of cheese for their sandwich. The shop includes options like turkey, salami, mozzarella cheese, and rye bread. The customer does not order a sandwich with a turkey/mozzarella combination, a sandwich with a rye bread/salami combination, or with white bread and chicken. How many different sandwiches could the customer potentially order?


In the nine-term sequence $A, B, C, D, E, F, G, H, I$, the value of $D$ is $7$ and the sum of any three consecutive terms is $36$. Additionally, the last term $I$ equals $10$. What is $A+I$?


How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits?


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


What is the least positive whole number divisible by four different prime numbers, where one of the primes is squared?


What is the sum of all the solutions of $x = \left|3x - |45 - 3x|\right|$?


A pet store has 10 puppies, 7 kittens, 9 hamsters, and 5 birds. Anne, Ben, Clara, and David each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Anne, Ben, Clara, and David buy pets and leave the store satisfied?


Find the number of positive integers $n,$ $1 \le n \le 2000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients.


A cone has a uniform circular base of radius $6$ feet and a slant height of $13$ feet. The cone’s side breaks, causing the height from the base to the tip along the cone's side to reduce by $2$ feet while maintaining a right angle at the vertex. Determine the new height, $H$, of the cone from the base to the tip.

Solution: \boxed{H = \sqrt{85}}


A point $(x,y)$ lies at a distance of 15 units from the $x$-axis. It is also 13 units away from the point $(5,8)$. Moreover, it is located at a distance $n$ from the origin. Assuming that $x > 5$, determine the value of $n$.

Solution: \boxed{\sqrt{370 + 20\sqrt{30}}}


Two vertices of an obtuse triangle are given as $(4, -5)$ and $(0, 0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 40 square units?

Solution: \boxed{(-16, 0)}


If \(a\) and \(b\) are non-zero real numbers such that \(a^2 + b^2 = 18ab\), find the value of \(\left|\frac{a+b}{a-b}\right|\).


Three tour guides are leading nine tourists. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?


Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 9.\)  Find the minimum value of
\[
\frac{a^2 + b^2 + c^2}{a + b + c} + \frac{b^2 + c^2}{b + c} + \frac{c^2 + a^2}{c + a} + \frac{a^2 + b^2}{a + b}.
\]


Let \( a, \) \( b, \) \( c \) be positive real numbers. Find the set of all possible values of
\[
g(a, b, c) = \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}.
\]

Solution: \boxed{(1, 2)}


What are the values of $x$ such that $3x^2 - 9x \leq 15$? Express your answer in interval notation.

Solution: \boxed{\left[ \frac{3 - \sqrt{29}}{2}, \frac{3 + \sqrt{29}}{2} \right]}


Find the interval of $x$ for which the expression \[\frac{3x-12x^2+48x^3}{27-x^3}\] is nonnegative.

Solution: \boxed{[0, 3)}


In the year 2023, the International Mathematical Olympiad will be hosted by a country. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A + B + C$?


Consider polynomials $Q(x)$ of degree at most $2$, where each coefficient is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $Q(1) = 17$?


A circle with center $O$ has a radius of $8$ units, and circle $P$ has a radius of $5$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OT$?


Consider the matrix:
\[\mathbf{N} = \begin{pmatrix} x & 2y & 0 \\ -x & y & z \\ x & -y & z \end{pmatrix}\]
It is known that \(\mathbf{N}^T \mathbf{N} = \mathbf{I}\). Find \(x^2 + y^2 + z^2\).


Compute \(\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}^{1000}\).

Solution: \boxed{\begin{pmatrix} 1 & 0 \\ 2000 & 1 \end{pmatrix}}


A cylindrical tank with a radius of $6$ feet and a height of $15$ feet is filled with water. A solid cube with a side length of $12$ feet is placed into the tank such that one of its body diagonals is vertical. Calculate the volume of water displaced by the cube, $w$, and find $w^2$.


What is the largest number of positive, consecutive integers whose sum is 30?


Find all functions \(f : \mathbb{R} \to \mathbb{R}\) such that
\[
f(x + 2y)f(x - 2y) = (f(x) + f(y))^2 - 16y^2 f(x)
\]
for all real numbers \(x\) and \(y\).

Solution: \boxed{f(x) = 0 \text{ for all } x}


In triangle ABC below, find the length of side AB.

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18\sqrt{2}$",C/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy]


Kevin Kangaroo begins hopping on a number line at 0. He wants to get to a point 1 unit away. However, he can hop only $\frac{1}{4}$ of the distance left. Each hop tires him out, so he continues hopping $\frac{1}{4}$ of the remaining distance. How far has he hopped after six hops? Express your answer as a common fraction.


A point in space $(x,y,z)$ is randomly selected so that $-2 \le x \le 2$, $-2 \le y \le 2$, $-2 \le z \le 2$. What is the probability that $x^2 + y^2 + z^2 \le 4$ and $x+y+z \ge 0$?


Triangle $PQR$ has side lengths $PQ = 15, QR = 30,$ and $PR = 24.$ A line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ Find the perimeter of triangle $PXY$.


The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-2,5)$, $(4,5)$, and $(2,2)$. What is the $x$-coordinate of the vertex of the parabola?


How many four-digit numbers \( N \) have the property that the three-digit number obtained by removing the leftmost digit is one seventh of \( N \)?


Triangle $ABC$ has vertices $A(0,10)$, $B(4,0)$, $C(10,0)$. A vertical line intersects $AC$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 20, determine the positive difference of the $x$ and $y$ coordinates of point $R$.

Solution: \boxed{4\sqrt{10} - 10}


A school is arranging chairs in rows for a special event. Each row can accommodate $13$ chairs, and initially, there are $156$ chairs in total. The school aims to minimize empty seats, while maintaining that all rows are completely filled. If $95$ students are expected to attend the event, how many chairs should be removed?


How many values of $x$, $-25<x<105$, satisfy $\cos^2 x + 2\sin^2 x = 1?$ (Note: $x$ is measured in radians.)


Find the sum of all possible positive integer values of $b$ such that the quadratic equation $3x^2 + 6x + b = 0$ has rational roots.


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


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. What is the probability that the bug is at vertex $A$ after crawling exactly 10 meters?


Line $l_1$ has equation $4x - 3y = 2$ and goes through $A = (1, -3)$. Line $l_2$ has equation $y = 2$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $6$. What is the slope of $l_3$?


For how many integer values of $n$ between 1 and 360 inclusive does the decimal representation of $\frac{n}{350}$ terminate?


What is the smallest positive value of \( x \) such that \( x + 3456 \) results in a palindrome?


Circular arcs of radius 3 inches form a continuous pattern as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Each arc completes half of a circle.


Let $z_1$, $z_2$, $z_3$, $\dots$, $z_6$ be the 6 zeroes of the polynomial $z^{6} - 8^6$. For each $k$, let $w_k$ be one of $z_k$ or $iz_k$. Find the maximum possible value of the real part of
\[\sum_{k = 1}^{6} w_k.\]

Solution: \boxed{8 + 8\sqrt{3}}


At an airport, there are 15 gates arranged in a circular manner with exactly 80 feet between each adjacent pair of gates. Sarah's original departure gate is assigned randomly. After a short wait, she is notified that her gate has been changed to a different randomly selected gate. Calculate the probability that Sarah has to walk 320 feet or less to get to her new gate. Express the result as a fraction $\frac{p}{q}$ where $p$ and $q$ are coprime integers, and find $p+q$.


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


In the diagram, a rectangle has a perimeter of 60 and the triangle has a height of 36. The rectangle and triangle have the same area. What is the value of \( x \)? Assume the length of the rectangle is 3 times the width.
\[ \text{Diagram:} \]
\[ \text{Rectangle (Not shown): A rectangle with one unknown side labeled as } x \]
\[ \text{Triangle (Not shown): A triangle with height 36 and base labeled as } x \]


Find the coordinates of the point that divides the line segment between the points \((2, 10)\) and \((8, 4)\) in the ratio 1:3.

Solution: \boxed{(3.5, 8.5)}


Let set $\mathcal{C}$ be a 80-element subset of $\{10,11,12,\ldots,120\}$, and let $U$ be the sum of the elements of $\mathcal{C}.$ Find the number of possible values of $U$.


A shop offers an initial discount of $25\%$ on all items. A week later, an additional discount of $15\%$ is applied to the already discounted prices. Finally, an extra $10\%$ discount is applied to the new sale price on a special sale day. The shop advertises that the cumulative discount equals $50\%$ off the original prices. Compute the actual total discount and find the percentage difference between the shop's claim and the true discount.


Two circles are drawn in a 16-inch by 20-inch rectangle. Each circle has a diameter of 8 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{4\sqrt{13} \text{ inches}}


A biased die lands on 1, 2, or 3 with probability $\frac{1}{6}$ each, and on 4, 5, or 6 with probability $\frac{1}{2}$. Landing on numbers 1 through 3 gains $\$2$, but landing on 4, 5, or 6 loses $\$5$. Calculate the expected value of a single roll of this die.


Let $\mathbf{a} = \begin{pmatrix} 3 \\ 7 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}$. Find the volume of the parallelepiped with vertices at $\mathbf{0}, \mathbf{a}, \mathbf{b}$, and $\mathbf{a} + \mathbf{b}$.


Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{45} + \frac{105}{\sqrt{45} + \frac{105}{\sqrt{45} + \frac{105}{x}}}. \]


In a right triangle $XYZ$, medians are drawn from points $X$ and $Y$ to segments $\overline{YZ}$ and $\overline{XZ}$, respectively, dividing these segments in half. The lengths of the medians from $X$ and $Y$ are 5 units and $3\sqrt{5}$ units, respectively. The area of triangle $XYZ$ is 30 square units. Determine the length of segment $\overline{XY}$.


Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$ and express the result in the form $A + B\sqrt{C}$, where $A$, $B$, and $C$ are integers. Then, what is the value of $ABC$?


Calculate the units digit of the product of all the odd positive integers between 10 and 50.


Find the shortest distance between the point $(8,8)$ and the parabola given by the equation $x = \frac{y^2}{4}.$


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


Solution: \boxed{\log_5 15}


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


The line $y = \frac{5x - 9}{6}$ is parameterized in the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{v} + t \mathbf{d},
\]
so that for $x \ge 4,$ the distance between $\begin{pmatrix} x \\ y \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$ is $t.$  Find $\mathbf{d}.$

Solution: \boxed{\begin{pmatrix} \frac{6}{\sqrt{61}} \\ \frac{5}{\sqrt{61}} \end{pmatrix}}


Let
\[g(a,b) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{a^2b - a^2 + 3}{2a} & \text{if $a + b \le 4$}, \\
\dfrac{a^2b - b^2 - 3}{-2b} & \text{if $a + b > 4$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]Find \(g(1,3) + g(3,2)\).


Tiffany is constructing a fence around an irregular shaped tennis court. She now has 400 feet of fencing available. The local regulations state that the length of the fence enclosure must be at least 100 feet and the width must be at least 50 feet. Tiffany wishes to maximize the area enclosed by the fence to include seating areas and storage. How much area can she maximize under these conditions?


Solution: \boxed{1 - \frac{1}{\log_3 51}}


Consider the sequence $9720, 3240, 1080, \ldots$, formed by repeatedly dividing by 3. How many integers are in this sequence?


Triangle $PQR$ has a perimeter of 2010 units. The sides have lengths that are all integer values with $PQ < QR < PR$. What is the smallest possible value of $QR - PQ$?


Let $ABCDE$ be a convex pentagon such that $AB \parallel DE, BC \parallel AE, \angle ABC=110^\circ, AB=4, BC=6,$ and $DE = 18.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $AED$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.


In the given diagram, what is the measure of $\angle ACB$? The diagram includes:
- Point $C$ placed on a linear extension past $B$, with an external point $D$ such that $B$ is between $A$ and $D$.
- $\angle ABD = 150^\circ$
- $\angle BAC$ measures $88^\circ$.
How many degrees are there in $\angle ACB$?

Solution: \boxed{\angle ACB = 62^\circ}


At breakfast, a mid-morning snack, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, a banana, or a pear to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit?


Solution: \boxed{12z^7 + 8z^6 - 25z^5 - 2z^4 + 18z^3 + z^2 - 8z + 2}


A novel begins on page 101 and ends on page 599. Each page number is printed once in the book. How many more times is the digit 5 printed compared to the digit 2 in the page numbers?


Let $a, b, c, d, e$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 + e^2 = 5.\]
Find the maximum value of $a^3 + b^3 + c^3 + d^3 + e^3.$


As \( n \) ranges over the positive integers, what is the maximum possible value for the greatest common divisor of \( 13n+3 \) and \( 7n+1 \)?


Let $\mathbf{N}$ be a matrix such that
\[
\mathbf{N} \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}.
\]
Compute $\mathbf{N} \begin{pmatrix} 9 \\ 5 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{38}{7} \\ \frac{128}{7} \end{pmatrix}}


What is the remainder when the $80^{\mathrm{th}}$ term of the Fibonacci sequence is divided by 7?


Mary can mow a lawn in three hours, whereas Tom can mow the same lawn in four hours. Tom starts mowing the lawn and works alone for 1 hour before Mary joins him. Together, they mow for another 2 hours. How much of the lawn remains to be mowed?


Given that $7^{-1} \equiv 55 \pmod{101}$, find $49^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


Let $x,$ $y,$ $z$ be nonnegative real numbers. Let
\begin{align*}
C &= \sqrt{x + 3} + \sqrt{y + 6} + \sqrt{z + 11}, \\
D &= \sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}.
\end{align*}
Find the minimum value of $C^2 - D^2.$


Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$

Solution: \boxed{\frac{5 + \sqrt{105}}{2}}


The polynomial $x^3 - ax^2 + bx - 1806$ has three positive integer roots. What is the smallest possible value of $a$?


A number $s$ can be expressed as a four-place decimal $0.wxyz$, where $w, x, y, z$ are digits. We want to approximate $s$ by a fraction whose numerator is 1, 2, or 3, and whose denominator is an integer. Find how many possible values for $s$ exist if the closest such fraction to $s$ is $\frac{3}{11}$.


Find the minimum value of
\[3x^2 + 2xy + y^2 - 6x + 2y + 8\]
over all real numbers $x$ and $y$.


The curve $y = \sin x$ intersects the horizontal line $y = \sin 80^\circ$. This line cuts the curve into segments having the successive ratios
\[\dots p : q : p : q \dots\] with $p < q$. Compute the ordered pair of relatively prime positive integers $(p, q)$.

Solution: \boxed{(1, 17)}


In $\triangle XYZ$, the medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\triangle XYZ$.


Jessica has a 3 by 3 grid and two colors to use for coloring it. In accordance with the rule that non-overlapping regions sharing a side cannot have the same color, how many possible ways are there for Jessica to color this grid?


In a certain base \( b \), the square of \( 34_b \) is \( 1296_b \). What is \( b \)?

Solution: \boxed{b = 10}


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 18 and an altitude of 20. The axes of the cylinder and the cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


In isosceles triangle $\triangle ABC$, where $AB=AC=5$. The altitude from $B$ meets $\overline{AC}$ at $H$. If $AH=2(HC)$, then determine $BC$.


Maria has an unfair eight-sided die. The probability that it rolls an 8 is $\frac{1}{3}$, and the probability that it rolls any other number from 1 to 7 is $\frac{2}{15}$. What is the expected value of the number shown when this die is rolled?


A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$


A square is inscribed in a circle so that each vertex of the square touches the circle. If the side of the square intersects the circle such that each intersection segment equals twice the radius of the circle, find the ratio of the area of the square to the area of the circle. Express your answer as a common fraction in terms of $\pi$.


What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 2:50? Express your answer as a decimal to the nearest tenth.


Let $h(x)$ be a function defined as \[h(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
3x-30 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ such that $h(h(h(6)))=h(h(h(a)))$.


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $90$, and $BC=10$. Find the volume of the tetrahedron.

Solution: \boxed{V = 360\sqrt{2}}


Convex hexagon $ABCDEF$ has exactly two distinct side lengths. Side $AB$ measures 8 units, and side $BC$ measures 10 units. The perimeter of hexagon $ABCDEF$ is 56 units. How many sides of hexagon $ABCDEF$ have measure 10 units?


In 2000, there were 60,000 cases of a disease reported in a country. By 2020, there were only 300 cases reported. Assume the number of cases decreased exponentially rather than linearly. Determine how many cases would have been reported in 2010.


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


Compute the sum
\[
\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 140^\circ,
\]
and express it in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


A twelve-digit integer is formed by repeating a positive three-digit integer four times. For example, 123,123,123,123 or 987,987,987,987 are integers of this form. What is the greatest common divisor of all twelve-digit integers of this form?


A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 307,307,307 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?


Solution: \boxed{\log_{216} 5 = \frac{\log_2 5}{3(1 + \log_2 3)}}


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


Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\]


Solution: \boxed{\frac{2 \log_{10}3}{\log_{10}5}}


Suppose that the graph of a certain function, $y=f(x)$, has the property that if it is shifted $30$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.

What is the smallest positive $a$ such that if the graph of $y=f\left(\frac{x}{6}\right)$ is shifted $a$ units to the right, then the resulting graph is identical to the original graph of $y=f\left(\frac{x}{6}\right)$?


A ball is dropped from a height of 800 feet and bounces back up two-thirds of the distance it just fell. After how many bounces will the ball first reach a maximum height less than 10 feet?


Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ has real roots. Find the smallest possible value of $3a + 2b$.


Milo writes down all the five-digit numbers that contain each of the digits 1, 2, 6, 7, and 8 exactly once. What is the smallest number in Milo's list that is divisible by 6?


Given that $\angle A = 30^\circ$ and $\angle AFG = \angle AGF$, calculate the sum of $\angle B + \angle D$ in the pentagon shown. Assume that $\angle BFD + \angle AFG = 180^\circ$, and that $\angle AFG + \angle AGF + \angle GAF = 180^\circ$ as in a triangle.


Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[\{-8, -6, -4, -1, 1, 3, 5, 14\}.\] What is the minimum possible value of \[
(p+q+r+s)^2 + (t+u+v+w)^2
\] given that the sum $p+q+r+s$ is at least 5?


Assume that $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$.


The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either A or E, end with P, cannot contain V, and have no letters that repeat?


In a modified version of Dean’s calculator game, there are now 45 participants sitting in a circle, including Dean. Dean has three calculators; the first shows 1, the second shows 0, and the third shows -2. Dean now performs the following: he cubes the value on the first calculator, squares the value on the second calculator, and negates the value on the third calculator. All the participants, including Dean, perform the same operations in their turn and then pass the calculators to the next person. After each calculator has been passed around the entire circle once and returns to Dean, he sums the numbers displayed on all three calculators. What sum does Dean get?


Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three mutually orthogonal unit vectors, such that
\[\mathbf{u} = s (\mathbf{u} \times \mathbf{v}) + t (\mathbf{v} \times \mathbf{w}) + k (\mathbf{w} \times \mathbf{u})\]
for some scalars $s$, $t$, and $k$, and $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 2.$ Find $s + t + k$.


Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$


Calculate the product \(\left(\frac{2}{5}\right)\left(\frac{5}{8}\right)\left(\frac{8}{11}\right)\cdots\left(\frac{2002}{2005}\right)\). Express your answer as a common fraction.


The sequence \((b_n)\) satisfies
\[kb_1 + kb_2 + kb_3 + \dots + kb_n = n^2 kb_n\] for all \(n \ge 2.\)  If \(b_{70} = 2\) and \(k=3\), find \(b_1.\)


The expression \(25x^2 - 160x - 144\) can be written as \((5x + a)(5x + b)\), where \(a\) and \(b\) are integers. What is \(a + 2b\)?


Simplify: $$\dfrac{\sqrt{448}}{\sqrt{128}}+\dfrac{\sqrt{98}}{\sqrt{49}}.$$
Express your answer as a common fraction.

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


The number of students in Michael's graduating class is more than 100 and fewer than 200. The number is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Michael's graduating class?


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 - yf(z)) = xf(x) - zf(y)\] for all real numbers $x,$ $y,$ and $z.$

Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$  Find $n \times s.$


Determine the smallest real number $\mu$ such that
\[a^2 + 4b^2 + 4c^2 + d^2 \ge 2ab + \mu bc + 2cd\]
for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$

Solution: \boxed{\mu = 6}


Find the vector $\mathbf{u}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 2 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{45}{13} \\ \frac{30}{13} \end{pmatrix}\]
and
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 4 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{32}{17} \\ \frac{128}{17} \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} -\frac{2}{5} \\ \frac{81}{10} \end{pmatrix}}


Find the product of the divisors of \( 72 \).


Factor the expression $4x(x-5) + 5(x-5) + 6x(x-2)$.

Solution: \boxed{(4x+5)(x-5) + 6x(x-2)}


Find the maximum $y$-coordinate of a point on the graph of $r = 2\sin 2\theta.$


A club has 30 members - 15 boys and 15 girls. In how many ways can we choose a president, a vice-president, and a treasurer if the president and the vice-president must be of the same gender, but the treasurer can be of any gender not holding a previous office?


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=12$. Determine the volume of tetrahedron $ABCD$.


In the expansion of $(x+2)^{30}$, what is the coefficient of the $x^3$ term?

Solution: \boxed{4060 \cdot 2^{27}}


A ball is dropped from 20 feet high and bounces back up to two-thirds of the distance it just descended. Additionally, each bounce reduces the subsequent height by an additional 5% due to air resistance. After how many bounces will the ball first reach a maximum height less than 2 feet?


Let $X$, $Y$, and $Z$ be nonnegative integers such that $X+Y+Z=15$. What is the maximum value of \[X\cdot Y\cdot Z+X\cdot Y+Y\cdot Z+Z\cdot X?\]


Find the largest constant \(C\) such that
\[x^2 + y^2 + z^2 + 2 \ge C(x + y + z - 1)\]
for all real numbers \(x\), \(y\), and \(z\).

Solution: \boxed{\frac{2 + 2\sqrt{7}}{3}}


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

Solution: \boxed{4 + 8 \sqrt{3}}


How many three-digit numbers satisfy the property that the middle digit is equal to the sum of the first and last digits minus one?


Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 72.$


The Fibonacci numbers $F_n$ are defined by the recurrence relation $F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$. Determine the units digit of $F_{F_{10}}$.


Let \( A, B, C, \) and \( D \) be positive real numbers such that
\[
\log_{10} (AB) + \log_{10} (AC) = 3, \\
\log_{10} (CD) + \log_{10} (CB) = 4, \\
\log_{10} (DA) + \log_{10} (DB) = 5.
\]
Compute the value of the product \( ABCD \).


Suppose $b$ is an integer such that $0 \le b \le 20$, and $74639281_{85} - b$ is a multiple of $17$. What is $b$?


The graphs of $y = x^3 - 4x + 3$ and $x + 3y = 3$ intersect in the points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$  If $x_1 + x_2 + x_3 = A$ and $y_1 + y_2 + y_3 = B,$ compute the ordered pair $(A,B).$


Mr. Maple teaches two different mathematics classes, one with 12 students and another with 13 students. Each class meets thrice a week. Every class period, Mr. Maple picks a random student from each class to solve a problem on the board. How many different sequences of students solving problems are possible for both classes combined in a week?


The graphs of \(y = 2 - x^2 + x^4\) and \(y = -1 + x^2 + x^4\) intersect in multiple points. Find the maximum difference between the \(y\)-coordinates of these intersection points.


In spherical coordinates, evaluate the point $\left( 5, \frac{3 \pi}{4}, \frac{9 \pi}{4} \right)$ and find its equivalent representation in the standard spherical coordinate representation, where $\rho > 0$, $0 \le \theta < 2 \pi$, and $0 \le \phi \le \pi$.

Solution: \boxed{\left( 5, \frac{3 \pi}{4}, \frac{\pi}{4} \right)}


In triangle $\triangle ABC$ shown, $\tan A = \frac{4}{3}$. Find the length of $AC$.

[asy]
pair A,B,C;
C = (0,0);
A = (0,4);
B = (3,4);
draw(C--A--B--C);
draw(rightanglemark(C,A,B,7));
label("$C$",C,SW);
label("$A$",A,NW);
label("$B$",B,NE);
label("$3$",(A+B)/2,N);
[/asy]


Of the six points (5, 15), (10, 26), (15, 40), (22, 50), (25, 60), and (30, 75), what is the sum of the $x$-coordinates of the points that lie in the region above the line $y = 2.5x + 5$ in the coordinate plane?


A circle is circumscribed around quadrilateral $WXYZ$. The inscribed angles $\angle WXY$ and $\angle YZW$ are given as $50^\circ$ and $70^\circ$ respectively. Calculate the sum of the angles $\angle WYZ + \angle XYZ$.


Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \geq 0\]is true if and only if either $x < -6$ or $|x-30| \le 2.$ Given that $a < b,$ find the value of $a + 2b + 3c.$


A rectangular wooden block is 6 inches long, 3 inches wide, and 2 inches high. The block is painted blue on all six sides and then cut into 1 inch cubes. How many of the cubes each have a total number of blue faces that is an even number?


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


In a race, all runners start at point $A$, touch any part of a 1000-meter long wall, and stop at point $B$. The length from $A$ to the start of the wall is 400 meters, and from the end of the wall to $B$ is 600 meters. What is the minimum distance a participant must run? Assume the wall, $A$, and $B$ are aligned vertically, and the race is entirely in a vertical plane. Express your answer to the nearest meter.


How many even integers between 3000 and 6000 have four different digits?


There exist constants $b_1, b_2, b_3, b_4, b_5, b_6$ such that
\[
\cos^6 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2$.


Factor $(x^2 + 2x + 1)(x^2 + 8x + 15) + (x^2 + 6x + 5)$ as the product of two non-constant polynomials.

Solution: \boxed{(x + 1)(x + 5)(x + 2)^2}


Determine the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{HG} \perp \overline{FG}$, $EF=15$ cm, $HG=7$ cm, and $FG=18$ cm.

Solution: \boxed{40 + 2\sqrt{97} \text{ cm}}


Determine the smallest positive integer $n$ such that $n^2$ is divisible by 50 and $n^3$ is divisible by 294.


Convert the point $(8,3 \sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$

Solution: \boxed{\left(\sqrt{91}, \tan^{-1}\left(\frac{3 \sqrt{3}}{8}\right)\right)}


Find the number of real solutions of the equation
\[\frac{x}{50} = \sin x.\]


Find the number of positive solutions to
\[\csc (\arcsec( \cot (\arccsc x) ) ) = x.\]


The product of the positive integer divisors of a positive integer $n$ is 4096. Find $n$.


Triangle $PQR$ is an isosceles triangle with $PQ=QR$. Point $S$ is the midpoint of both $\overline{QR}$ and $\overline{PT}$, and $\overline{RT}$ is 15 units long. What is the length of $\overline{PS}$?  Express your answer as a decimal to the nearest tenth.


In a geometry problem, Isabella has drawn a pair of concentric circles. She starts drawing chords $\overline{CD}$, $\overline{DE}$, and so on, of the larger circle, each tangent to the smaller one. If $m\angle CDE=60^\circ$, how many chords will Isabella draw before she returns to her starting point at $C$?


The graph of the equation \[ 3x^2 + 9y^2 - 12x + 18y + 6z = m\] is a non-degenerate ellipse when projected onto the xy-plane if and only if $m > b.$ What is $b?$


Calculate
\[\cos \left( 6 \arccos \frac{2}{5} \right).\]


Define $n!!$ similarly as in the original problem. We are interested in the sum $\sum_{i=1}^{2010} \frac{(2i)!!}{(2i+1)!!}$ when it is expressed as a fraction in lowest terms. The denominator is $2^a b$ with $b$ odd. Find $\dfrac{ab}{10}$.


Let $x$ and $y$ be real numbers such that $5x + 3y \le 10$ and $3x + 6y \le 12.$  Find the largest possible value of $x + y.$


Given that \( x \) is a multiple of \( 17280 \), what is the greatest common divisor of \( g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5) \) and \( x \)?


A rectangle has a perimeter of 80 inches. Each side has an integer length, and one side length must be a multiple of 4. How many non-congruent rectangles meet these criteria?


How many of the first 512 smallest positive integers written in base 8 use the digit 5 or 6 (or both)?


Compute the multiplicative inverse of $203$ modulo $397$. Express your answer as an integer from $0$ to $396$.


Compute the multiplicative inverse of $101$ modulo $401$. Express your answer as an integer from $0$ to $400$.


A right, circular cone has a height of 10 inches and a radius of 3 inches at the opening. Filling the interior of the cone there is ice cream, capped with a cone (instead of a hemisphere) that has the same radius at the base but a height of 5 inches. Determine the total volume of ice cream.


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


For how many positive integers $n\leq 50$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit?


Three vertices of a cube are $A=(5,7,15)$, $B=(6,3,6)$, and $C=(9,-2,14)$. Determine the surface area of this cube.


Let $a_1, a_2, \ldots$ be a sequence defined by:
(i) $a_1 = 1$,
(ii) $a_{3n} = n^2 \cdot a_n$ for any positive integer $n$.
Determine the value of $a_{3^{100}}$.


Parallelepiped $PQRS TUVW$ is formed by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PT},$ as shown below. Compute
\[\frac{PU^2 + QV^2 + RT^2 + SW^2}{PQ^2 + PR^2 + PT^2}.\]


The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{3}{13} & \frac{2}{13} & \frac{6}{13} \\ \frac{2}{13} & \frac{12}{13} & -\frac{5}{13} \\ \frac{6}{13} & -\frac{5}{13} & \frac{2}{13} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

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


Find the maximum value of
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_6 + \cos \theta_6 \sin \theta_1,
\]
over all real numbers $\theta_1, \theta_2, \theta_3, \theta_4, \theta_5, \theta_6$.


Express the following as a common fraction: $\sqrt[3]{5\div 15.75}$.

Solution: \boxed{\frac{\sqrt[3]{20}}{\sqrt[3]{63}}}


Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. Suppose $M$ is a two-digit number such that $M = P(M) + S(M) + 5$. What is the units digit of $M$?


How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit?


In triangle $XYZ$, angle bisectors $\overline{XM}$ and $\overline{YN}$ intersect at $Q$. If $XY = 8$, $XZ = 13$, and $YZ = 10$, find $\frac{YQ}{QN}$.


A circle with a radius of 3 units has its center at $(0, 0)$. A second 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. Find the value of $x$. Express your answer as a common fraction.


Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$, resulting in the vector $\mathbf{v}_1$. The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, resulting in the vector $\mathbf{v}_2$. Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2$.

Solution: \boxed{\begin{pmatrix} \frac{28}{65} & \frac{14}{65} \\ \frac{42}{65} & \frac{21}{65} \end{pmatrix}}


Compute the sum of the geometric series $2 - 4 + 8 - 16 + \cdots + 1024$.


At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{\text{th}}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{\text{th}}$ prize and the winner bowls #3. The loser of this game receives $4^{\text{th}}$ prize and the winner bowls #2. The loser of this game receives $3^{\text{rd}}$ prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes?


The fraction $\frac{a}{a+50}$, where $a$ is a positive integer, equals $0.870$. What is the value of $a$?


Laura decides to train for a biathlon by biking and running. She bikes 25 miles at $3x+2$ miles per hour and then runs 4 miles at $x$ miles per hour. Laura spends 8 minutes transitioning between biking and running. Her total workout, including the transition, lasts 140 minutes. Determine Laura's running speed to the nearest hundredth of a mile per hour.

Solution: \boxed{5.17 \text{ mph}}


What is the largest five-digit integer whose digits have a product equal to the product $(9)(8)(7)(6)(5)$?


For each positive integer \( n \), let \( f(n) = n^4 + 100n^2 + 169 \). What is the sum of all values of \( f(n) \) that are prime numbers?


Calculate the remainder when the sum $1 + 9 + 9^2 + \cdots + 9^{999}$ is divided by 500.


$PQRS$ is a trapezoid with the measure of base $\overline{PQ}$ three times the measure of the base $\overline{RS}$. Point $T$ is the point of intersection of the diagonals. The measure of diagonal $\overline{PR}$ is 15. Find the length of segment $\overline{RT}$.


How many positive integer divisors of $2016^{2016}$ are divisible by exactly 2016 positive integers?


For all real numbers $x$ except $x=0$ and $x=-1$, the function $f(x)$ is defined by
\[ f\left(\frac{x}{x+1}\right) = \frac{1}{x}. \]
Suppose $0 \leq t \leq \frac{\pi}{2}$. What is the value of $f(\csc^2 t)$?

Solution: \boxed{-\cos^2 t}


Let $\overline{AB}$ be a diameter of circle $\omega$. Extend $\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\omega$ such that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\overline{AB} = 20$, and let $m$ denote the maximum possible length of segment $BP$. Find $m^{2}$ and the length of $\overline{OC}$ when $m$ is maximized.


Let $a$ and $c$ be positive real numbers. Find the maximum value of
\[2(a - x)(x + \sqrt{x^2 + c^2})\] in terms of $a$ and $c$.

Solution: \boxed{a^2 + c^2}


During a parade, you counted 25 heads and a total of 68 legs belonging to some group of acrobats, elephants, and bicycles. Assuming each acrobat has 1 head and 2 legs, each elephant has 1 head and 4 legs, and each bicycle has 0 heads and 2 legs, determine how many bicycles were in the parade.


Charlyn walks completely around the boundary of a square whose sides are each 10 km long. From any point on her path, she can see exactly 2 km horizontally in all directions in the shape of a square (rather than a circle). What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers?

Solution: \boxed{160 \text{ km}^2}


Define a positive integer $n$ as $15$-pretty if $n$ has exactly $15$ positive divisors and $n$ is divisible by $15$. Let $T$ be the sum of positive integers less than $2023$ that are $15$-pretty. Compute $\tfrac{T}{15}$.


As $x$ ranges over all real numbers, find the range of the function:
\[ g(x) = \sin^3 x + \cos^2 x. \]

Solution: \boxed{[-1, 1]}


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


Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied?


If $\csc y,$ $\sec y,$ $\cot y$ form a geometric sequence, in this order, then find $\tan^6 y - \tan^2 y.$


Determine the number of possible values for \( m \) such that the lengths of the sides of a triangle are \( \ln 20 \), \( \ln 60 \), and \( \ln m \), and the triangle has a positive area.


Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\triangle ACE$. Compute the ratio of the area of $\triangle ABJ$ to the area of $\triangle ACE$.


Two positive integers differ by 8 and their product is 272. What is the larger integer?


Regular decagon \( ABCDEFGHIJ \) has its center at \( K \). Each of the vertices and the center are to be associated with one of the digits \( 1 \) through \( 10 \), with each digit used exactly once, in such a way that the sums of the numbers on the lines \( AKF \), \( BKG \), \( CKH \), \( DKI \), and \( EKJ \) are all equal. Find the number of valid ways to associate the numbers.


Let $x, y, z$ be real numbers satisfying the equations:
1. $(x+y+z)(xy+xz+yz) = 45$
2. $x^2(y+z) + y^2(x+z) + z^2(x+y) = 14$
3. $(x+y+z)^2 = 25$

What is the value of $xyz$?


Find the number of ordered 19-tuples \((a_1, a_2, a_3, \dots, a_{19})\) of integers such that the cube of any number in the 19-tuple is equal to three times the sum of the other 18 numbers.


In an extended professional bowling tournament, the top 6 bowlers enter into a playoff. Initial game starts with #6 bowling against #5. The loser takes 6th place and the winner goes on to bowl against #4. The subsequent losers will take 5th, 4th, and 3rd places consecutively, as they lose against the sequential winners from previous matches respectively (#3, then #2). Finally, the remaining two bowlers compete, where the loser gets 2nd prize and the winner gets 1st prize. Determine how many orders can bowlers #1 through #6 receive the prizes.


A car travels due east at 1 mile per minute on a straight road. At the same time, a circular storm, whose radius is 60 miles, moves southeast at $\frac{\sqrt{2}}{2}$ mile per minute. At time $t=0$, the center of the storm is 150 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac{1}{2}(t_1+t_2)$.


Among all the roots of
\[z^{12} - z^9 + z^6 - z^3 + 1 = 0,\]
find the maximum imaginary part of a root which can be expressed as $\sin \theta$, where $-90^\circ \le \theta \le 90^\circ$. Determine $\theta.$


Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral.


Let $p, q, r, s, t, u, v, w$ be distinct elements from the set $\{-6, -4, -1, 0, 3, 5, 7, 12\}$. Determine the minimum possible value of
\[
(p+q+r+s)^2 + (t+u+v+w)^2.
\]


A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 6x + \sin x + 3\]
Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S$.


Determine $M$, the number of 8-digit positive integers such that the digits are in strictly increasing order, using only the digits 1 through 8. Calculate the remainder of $M$ when divided by 1000.


Let $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ be a constant vector, and consider the equation
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) + \mathbf{a} \times \mathbf{v} = d \mathbf{v}\]
for all vectors $\mathbf{v}$. Find the scalar $d$.


Suppose $a$ is an integer such that $0 \le a \le 20$, and $56831742_{85}-a$ is a multiple of $17$. What is $a$?


Let **v** be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of $\|\mathbf{v}\|$.

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


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


Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, suppose that 0, $z_1,$ and $z_2$ form the vertices of a right-angled triangle with the right angle at $z_1$. Find $\frac{a^2}{b}.$


Let $T$ be the sum of all positive integers $n$ such that $n^2 + 12n - 208$ is a perfect square. Find the remainder when $T$ is divided by $100$.


A positive five-digit integer is in the form $AB,DBA$; where $A$, $B$, and $D$ are each distinct digits. What is the greatest possible value of $AB,DBA$ that is divisible by eleven?


John has recorded completion times, in seconds, of 100, 108, 112, 104, and 110 for running a 100-meter dash. After another race, he realized his median time dropped to 106 seconds. What was his time, in seconds, for the latest race?


In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=36$. Find the perimeter of quadrilateral $ABCD.$

Solution: \boxed{40.5 + 31.5\sqrt{3}}


Three spherical balls, each with a diameter of 12 cm, are stacked in a triangular pyramid formation inside a box, as shown in the diagram below. Determine the height, \( h \), of this stack of three balls.

[asy]
draw(circle((10,10),12),black+linewidth(1));
draw(circle((34,10),12),black+linewidth(1));
draw(circle((22,32.7846),12),black+linewidth(1));
draw((50,0)--(50,45.568),black+linewidth(1));
draw((49,0)--(51,0),black+linewidth(1));
draw((49,45.568)--(51,45.568),black+linewidth(1));
label("$h$",(50,0)--(50,45.568),E);
[/asy]

Solution: \boxed{12 + 6\sqrt{3} \text{ cm}}


Suppose a parabola has vertex $\left(\frac{3}{2},-\frac{25}{4}\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$.


Let $A$ and $B$ be points on the coordinate plane with coordinates $(2t-4, -3)$ and $(-6, 2t+5)$, respectively. The square of the distance between the midpoint of $\overline{AB}$ and an endpoint of $\overline{AB}$ is equal to $4t^2 + 3t$. Find the value of $t$.

Solution: \boxed{\frac{7 - \sqrt{185}}{4}}


Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 3} + \frac{1}{y + 3} = \frac{1}{4}.\] Find the minimum value of $x + 2y.$

Solution: \boxed{3 + 8 \sqrt{2}}


Milton spilled ink on another part of his homework covering a different quadratic equation. Again, he knows the equation has two distinct negative, integer solutions. This time, he remembers that the product of the roots is 24. What is the sum of all distinct possible integers that could be under the ink stain now?


Each pack of hot dogs contains 9 hot dogs, and each pack of hot dog buns contains 7 buns. After hosting a barbecue, Alex finds that he has 6 hot dogs left over. What is the SECOND smallest number of packs of hot dogs Alex could have bought?


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x + y) + xg(y) - 2xy - x + 2 \] for all real numbers \( x \) and \( y. \)
Let \( n \) be the number of possible values of \( g(1) \), and let \( s \) be the sum of all possible values of \( g(1) \). Find \( n \times s \).


Let \(\triangle ABC\) be an isosceles triangle such that \(BC = 24\) and \(AB = AC\). The triangle has an incenter \(I\) with distance \(IC = 20\). Determine the length of the inradius of \(\triangle ABC\).


Find all the solutions to
\[\sqrt{(1 + \sqrt{2})^x} + \sqrt{(1 - \sqrt{2})^x} = 2.\]


The sum of three consecutive one-digit, positive, odd integers is one-eighth of the product of the same three integers. What is the middle integer when the three integers are listed in ascending order?


Let
\[ g(x) = x^3 - 3x^2 + 9x - 5. \]
The graphs of \( y = g(x) \) and \( y = g^{-1}(x) \) intersect at exactly one point \((c,d)\). Enter the ordered pair \((c,d)\).

Solution: \boxed{(1, 1)}


Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms.


What is the perimeter, in cm, of quadrilateral $ABCD$ if $\overline{AB} \perp \overline{BC}$, $\overline{DC} \perp \overline{BC}$, $AB=6$ cm, $DC=6$ cm, and $BC=7$ cm?

Solution: \boxed{26 \text{ cm}}


Triangles $\triangle ABC$ and $\triangle PQR$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,10)$, $C(20,0)$, $P(30,0)$, $Q(30,20)$, $R(50,0)$. A rotation of $n$ degrees counterclockwise around the point $(x,y)$ where $0<n<180$, will transform $\triangle ABC$ to $\triangle PQR$. Find $n+x+y$.


Let \( g(x) \) be a polynomial of degree 2010 with real coefficients, and let its roots be \( s_1, s_2, \ldots, s_{2010} \). Assume there are exactly 1008 distinct values among
\[ |s_1|, |s_2|, \dots, |s_{2010}|. \]
What is the minimum number of real roots that \( g(x) \) can have?


Let parallelepiped $PQRSXYZW$ be defined by vectors $\overrightarrow{PQ}, \overrightarrow{PR},$ and $\overrightarrow{PS}$ in a 3D coordinate system. Find the value of:
\[
\frac{QR^2 + PX^2 + SZ^2 + TW^2}{PQ^2 + PR^2 + PS^2}.
\]


In triangle $XYZ$, $XY = 5$, $XZ = 9$, $YZ = 11$, and point $W$ lies on $\overline{YZ}$ such that $\overline{XW}$ bisects $\angle YXZ$. Calculate $\cos \angle YXW$.


Given \( ab+bc+ca+db = 42 \) and \( b+c = 6 \), find \( a+d \).


The degree measures of the angles in a convex 20-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.


A dodecahedron is constructed by attaching two identical pentagonal pyramids at their pentagonal bases. An ant starts at the top vertex of the dodecahedron and randomly walks to one of the five adjacent vertices, which it calls vertex A. From vertex A, the ant randomly chooses one of the five adjacent vertices, which it calls vertex B. What is the probability that vertex B is the bottom vertex of the dodecahedron? Express your answer as a common fraction.


A collection of 7 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 7.$ A tower is to be built using all 7 cubes under the following rules:
- Any cube may be the bottom cube in the tower.
- The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+1.$
Let $S$ be the number of different towers that can be constructed. What is the remainder when $S$ is divided by 100?


George has an unfair eight-sided die. The probability that it rolls an 8 is $\frac{3}{8}$, and the probabilities for the numbers 1 through 7 are each $\frac{5}{56}$. What is the expected value of the number shown when this die is rolled?


For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ \cdots + n$. What is the greatest possible value of the greatest common divisor of $8T_n$ and $n+1$?


Determine how many integer values of $b$ are possible for exactly two integers $x$ satisfying
\[x^2 + bx + 5 \le 0.\]


A rectangular tile measures 3 inches by 5 inches. What is the fewest number of these tiles needed to completely cover a rectangular region that is 3 feet by 4 feet?


Let $f(n)$ denote the sum of all divisors of a positive integer $n$. Define $n$ as superdeficient if $f(f(n)) = n + 3$. Determine how many superdeficient integers exist under this definition.


A collection of seven positive integers has a mean of 5.7, a unique mode of 4, and a median of 6. If a 10 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.


Determine the number of real solutions to the equation
\[\frac{6x}{x^2 + x + 4} + \frac{8x}{x^2 - 8x + 4} = 3.\]


In the new diagram, a point F is added to create more triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Modify the diagram as follows, with an additional filled region:

[asy]
draw((0,0)--(2,0)--(0,2)--(0,0)--cycle,linewidth(1));  // Big triangle AFG
draw((0,0)--(1,0)--(1,1)--(0,0)--cycle,linewidth(1));  // Smaller triangle AED
draw((1,0)--(2,0)--(1,1)--cycle, linewidth(1));  // Smaller triangle DFC
label("A",(0,2),NW);
label("E",(.6,.6),NE);
label("D",(1,0),S);
label("F",(2,0),SE);
label("G",(0,0),SW);
filldraw((1,0)--(2,0)--(1,1)--cycle, gray, black);  // fill triangle DFC
filldraw((0,0)--(1,0)--(.5,.5)--cycle, gray, black); // fill triangle AED part
[/asy]


The equation $x^2 - 2Ax + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist?

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


There is a unique two-digit positive integer \(s\) for which the last two digits of \(13 \cdot s\) are \(52\).


Let \( Q_1 \) be a regular \( t \)-gon and \( Q_2 \) be a regular \( u \)-gon \((t \geq u \geq 3)\) such that each interior angle of \( Q_1 \) is \( \frac{60}{59} \) as large as each interior angle of \( Q_2 \). What is the largest possible value of \( u \)?


In the diagram, if $\angle PQR = 90^\circ$, what is the measure of $\angle PMN$?
```asy
size(6cm);

pair p = (0, 0);
pair m = dir(180 - 45);
pair n = dir(180 + 45);
pair r = 1.3 * dir(45);
pair q = 1.3 * dir(-45);

label("$M$", m, SW);
label("$R$", r, NE);
label("$P$", p, 2*S);
label("$N$", n, SE);
label("$Q$", q, SE);

draw(m--q--r--cycle);
draw(n--p--m);

circle(p, 1.3);

add(pathticks(m--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(q--p, 2, spacing=0.9, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
```


Find the least positive integer \( n \) such that
$$
\frac{1}{\sin 30^\circ \sin 31^\circ} + \frac{1}{\sin 32^\circ \sin 33^\circ} + \cdots + \frac{1}{\sin 148^\circ \sin 149^\circ} = \frac{1}{\sin n^\circ}.
$$

Solution: \boxed{n = 1}


Find \( x \) such that \( \lceil x \rceil \cdot x = 210 \). Express \( x \) as a decimal.


If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2 = e^b$, find the median of the set $\{0, 0.5, a, b,
1/b\}$ under the condition $b < e^b$.


In the diagram, point $E$ lies on line segment $CD$, and triangles $AED$ and $BEC$ are isosceles. Also, $\angle AEC$ is three times $\angle DEA$. What is the measure of $\angle EBC$ in degrees? Assume $\angle DEA = 50^\circ$.


Suppose that the angles of triangle $PQR$ satisfy:
\[\cos 4P + \cos 4Q + \cos 4R = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side.


Square $WXYZ$ has an area of $144$. Point $P$ lies on side $\overline{WY}$ such that $WP:PY = 2:1$. Points $Q$ and $R$ are the midpoints of $\overline{WP}$ and $\overline{ZP}$, respectively. Quadrilateral $WQRP$ has area $25$. What is the area of triangle $RYZ$?


Let $g(x) = \frac{2x + 8}{x}$. The sequence $(g_n)$ of functions is defined by $g_1 = g$ and
\[g_n = g \circ g_{n - 1}\] for all $n \ge 2.$ For example,
\[g_2(x) = g(g(x)) = \frac{\frac{2x + 8}{x} + 8}{\frac{2x + 8}{x}} = \frac{10x + 8}{2x + 8}\]
and
\[g_3(x) = g(g_2(x)) = \frac{\frac{10x + 8}{2x + 8} + 8}{\frac{10x + 8}{2x + 8}} = \frac{26x + 72}{10x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[g_n(x) = x\]for some positive integer $n.$  Find the number of elements in $T.$


A 4-dimensional hyperparallelepiped $ABCDEFGHJKLMN$ is generated by vectors $\overrightarrow{AB},$ $\overrightarrow{AD},$ $\overrightarrow{AE},$ and $\overrightarrow{AF},$ with $A$ at the origin. Compute
\[
\frac{AG^2 + BK^2 + CM^2 + DN^2}{AB^2 + AD^2 + AE^2 + AF^2}.
\]


Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. The length of segment $PT$ can be expressed in the form $\sqrt{k}-m$ units. What is the integer value of $k+m$?


Positive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \[
2x + y = 2023\quad\text{and}\quad y = |x-a| + |x-b| + |x-c|
\]has exactly one solution. Determine the minimum value of $c$.


Let $x,$ $y,$ $z$ be distinct, nonzero real numbers such that
\[x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} + 1.\]Find $|xyz|$.


The first digit of a string of 3003 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 17, 19, or 31. What is the largest possible last digit in this string?


In triangle $PQR$, $\angle Q$ is a right angle, and the altitude from $Q$ meets $\overline{PR}$ at $S$. The lengths of the sides of triangle $PQR$ are integers, and $PR=3^{5}, PS=3^3$. If $\tan P = m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.


Four congruent isosceles triangles $DAO$, $AOB$, $BOC$, and $COD$ are arranged to consistently form a rhombus $ABCD$, with $DA = AO = OB = BC = CD = OD = 15$ and $AB = DC = 20$. Point $P$ is situated exactly at the midpoint of diagonal $AC$, with $O$ being the center of the rhombus.

Point $X$ is the midpoint of $ DA $ and point $Y$ is the midpoint of $BC$. When lines $XO$ and $OY$ are drawn, they divide the rhombus into four smaller triangular regions. Calculate the ratio of the area of triangle $ AXO $ to the area of triangle $ OYC $ in simplified form, and find $p+q$ if the ratio is expressed as $p:q$.


In the equation $w^3 + x^3 + y^3 = kz^3$, where $w$, $x$, $y$, and $z$ are distinct, consecutive positive integers and $k$ is a positive integer factor, find the smallest possible value of $z$ that satisfies the equation for $k = 2$.

Solution: \boxed{z = 6}


A regular octagon is inscribed in a square such that each side of the square is trisected by the vertices of the octagon. The perimeter of the square is now 144 centimeters. What is the area of the octagon inscribed in the square?


Consider sequences of positive real numbers of the form $x, 3000, y, \dots$ in which every term after the first is 2 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 3002 appear somewhere in the sequence?


Let \(a,\) \(b,\) \(c,\) be nonzero real numbers such that \(a + b + c = 1.\) Find all possible values of
\[\frac{a^2 b^2}{(a^2 + bc)(b^2 + ac)} + \frac{a^2 c^2}{(a^2 + bc)(c^2 + ab)} + \frac{b^2 c^2}{(b^2 + ac)(c^2 + ab)}.\]


A quadrilateral in the plane has vertices at \((0, 0)\), \((0, 2)\), \((3, 2)\), and \((5, 5)\). Calculate the area of this quadrilateral in square units.


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


In triangle $ABC$, $5 \sin A + 3 \cos B = 7$ and $3 \sin B + 5 \cos A = 3$. Find all possible values of $\angle C,$ in degrees. Enter all possible values, separated by commas.


Solution: \boxed{x = -\frac{27}{2}}


A $5$-foot by $10$-foot rectangular piece of plywood will be cut into $5$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?


If $p$ and $q$ are odd integers, how many terms in the expansion of $(p+q)^8$ are odd?


How many integers between 400 and 600 have the sum of their digits equal to 17?


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


Find the smallest positive four-digit number that is divisible by each of the five smallest prime numbers.


Phil rolls 8 fair 6-sided dice. What is the probability that at least two dice show the same number?


How many two-digit positive integers are congruent to 1 (mod 4)?


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


For what base is the representation of $500_{10}$ a four-digit number whose final digit is odd?


For which positive integer values of $k$ does $kx^2+16x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order.

Solution: \boxed{k = 8}


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


The second hand on a clock is 8 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.


Let $p$ and $q$ be real numbers. One of the roots of $x^3 + px + q = 0$ is $2 + i$. Find $p + q$.


Let $z$ be a complex number such that
\[|z^2 + 9| = |z(z + 3i)|.\]Find the smallest possible value of $|z + i|$.


Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


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


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


What is the value of $\dfrac{5 \times 7}{10}$?


Three years ago, there were 30 trailer homes on Maple Street with an average age of 15 years. At that time, a group of brand new trailer homes was added to Maple Street. Today, the average age of all the trailer homes on Maple Street is 12 years. How many new trailer homes were added three years ago?


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


Let $x = (3 + 2\sqrt{2})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


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


An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.


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


A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.


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


If the least common multiple of \( X \) and \( Y \) is \( 180 \), and the ratio of \( X \) to \( Y \) is \( 2:5 \), then what is their greatest common divisor?


Find the integer $n$, $10 \le n \le 15$, such that \[n \equiv 12345 \pmod{7}.\]


Calculate the product \(\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{2004}{2008}\right)\). Express your answer as a common fraction.


Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 45^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$.

Solution: \boxed{12 - 6\sqrt{2}}


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


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


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


What is the value of $x$ if a cube's volume is $8x$ cubic units and its surface area is $2x$ square units?


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


What are the rightmost four digits of $5^{2023}$?


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


The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both red? Express your answer as a common fraction.


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


The product of two 2-digit numbers is $4536$. What is the smaller of the two numbers?


When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 216$.


In triangle $XYZ$, $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$.


How many perfect squares are factors of 450?


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


Evaluate $\left|\frac{3}{4} + 3i\right|$.


How many four-digit perfect squares are palindromes?


Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{2000} \right\rfloor, \ \left\lfloor \frac{2^2}{2000} \right\rfloor, \ \left\lfloor \frac{3^2}{2000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{2000} \right\rfloor.\]


Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{2}{n}\]for all real numbers $x.$


What is the largest three-digit multiple of 12 whose digits' sum is 24?


Find the sum of the coefficients in the polynomial $3(x^8 - x^5 + 2x^3 - 6) - 5(x^4 + 3x^2) + 2(x^6 - 5)$.


For how many different digits $n$ is the three-digit number $25n$ divisible by $n$?


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


Tom has a red marble, a green marble, a blue marble, and four identical yellow marbles. How many different groups of three marbles can Tom choose?


If $y < 0$, find the range of all possible values of $y$ such that $\lceil y \rceil \cdot \lfloor y \rfloor = 72$. Express your answer using interval notation.

Solution: \boxed{(-9, -8)}


What is the base ten equivalent of $23456_{7}$?


Compute $\tan 30^\circ + 4 \cos 30^\circ.$

Solution: \boxed{\frac{7 \sqrt{3}}{3}}


How many 6-digit numbers have at least two zeros?


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


How many positive divisors does 8! have?


What is the largest integer $n$ for which $\binom{9}{4} + \binom{9}{5} = \binom{10}{n}$?


Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \]


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


The sum of the first 3000 terms of a geometric sequence is 500. The sum of the first 6000 terms is 950. Find the sum of the first 9000 terms.


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{2}{3}\] and $q$ is as small as possible. What is $q-p$?


Find the smallest composite number that has no prime factors less than 15.


Find the number of distinct numbers in the list:
\[\left\lfloor \frac{1^2}{2000} \right\rfloor, \ \left\lfloor \frac{2^2}{2000} \right\rfloor, \ \left\lfloor \frac{3^2}{2000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{2000} \right\rfloor.\]


Determine the residue of $-1235 \pmod{29}$. Your answer should be an integer in the range $0,1,2,\ldots,27,28$.


For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?


For what value of $k$ does the equation $(x+5)(x+2) = k + 3x$ have exactly one real solution? Express your answer as a common fraction.


What is the fourth power of the cube of the third smallest prime number?


Simplify
\[\frac{\tan^3 15^\circ + \cot^3 15^\circ}{\tan 15^\circ + \cot 15^\circ}.\]


Fully factor the following expression: $3x^2 - 75$.

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


Simplify and rationalize the denominator: $$\frac{1}{1+ \frac{1}{\sqrt{5}+2}}.$$

Solution: \boxed{\frac{\sqrt{5} + 1}{4}}


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


How many three-digit whole numbers have at least one 8 or at least one 9 as digits?


Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\).

Solution: \boxed{n = 2100}


What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?


If a school bus leaves school with 64 students on board, and one-third of the students get off the bus at each of the first four stops, how many students remain on the bus after the fourth stop?


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?


If the least common multiple of two 5-digit integers has 9 digits, then their greatest common divisor has at most how many digits?


Find a positive integer that is divisible by 24 and whose cube root is a number between 8 and 8.2.


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?


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


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


Point $P$ is the center of an ellipse with major axis $\overline{XY}$ and minor axis $\overline{WZ}.$ Point $G$ is one focus of the ellipse. If $PG = 8$ and the diameter of the inscribed circle of triangle $PWG$ is 4, compute the product $(XY)(WZ).$


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


A point has rectangular coordinates \((3, -4, 12)\) and spherical coordinates \((\rho, \theta, \phi)\). Find the rectangular coordinates of the point with spherical coordinates \((\rho, -\theta, \phi)\).

Solution: \boxed{(3, 4, 12)}


If $B$ is an angle such that $\tan B + \sec B = 3,$ find all possible values of $\cos B.$


A regular octagon is inscribed in a circle of radius 3 units. In square units, what is the area of the octagon? Express your answer in simplest radical form.


How many non-congruent triangles with perimeter 12 have integer side lengths?


What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{2310}$ is defined?


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $729x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2 + q^2 + r^2 + s^2 + t^2 + u^2$?


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


The surface area of a sphere is $256\pi\text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of $\pi$.

Solution: \boxed{\frac{2048}{3}\pi \text{ cm}^3}


For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?


What is the coefficient of $a^3b^2$ in $(a+b)^5 \left(c + \frac{1}{c}\right)^8$?


Compute without using a calculator: $\dfrac{10!}{7!3!}$.


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


Below is the graph of \(y = a \cos(bx + c) + d\) for some positive constants \(a,\) \(b,\) \(c,\) and \(d.\) The graph oscillates between 5 and 1. Find \(d.\)


A regular polygon has interior angles of 160 degrees, and each side is 4 units long. How many sides does the polygon have, and what is its perimeter?


In a round-robin chess tournament, 10 players each play every other player exactly once. Additionally, 3 matches are pre-determined as draws. How many matches will be held during this tournament?


Manny, Nick, Olivia, and Paula weigh themselves in pairs. Together, Manny and Nick weigh 320 pounds, Nick and Olivia weigh 295 pounds, and Olivia and Paula weigh 310 pounds. How many pounds do Manny and Paula weigh together?


What is the distance on a Cartesian coordinate plane from $(2, 3)$ to $(8, -5)$?


Alice believes that the amount of time she studies for a test and her score on that test are directly related. On her first exam, she studied for three hours and scored 60. To the nearest tenth, how many hours does Alice believe she must study the night before her second exam so that the average of her two exam scores is a 75?


What is the volume of a round conical tent that is 20 feet in diameter at its base and 10 feet tall? Express your answer in terms of $\pi$.


The least common multiple of two numbers is 5040, and the greatest common divisor is 24. Given that one of the numbers is 240, what is the other number?


Ella, Frank, Greta, and Hank weigh themselves in pairs. Together Ella and Frank weigh 280 pounds, Frank and Greta weigh 230 pounds, and Greta and Hank weigh 260 pounds. How many pounds do Ella and Hank weigh together?


Write the expression $\frac{6+2c}{7} + 3$ as a single fraction.


What is the sum of the digits of the base $5$ representation of $588_{10}$?


Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 5,$ $\|\mathbf{b}\| = 8,$ the angle $\theta$ between them is 60 degrees, find $\|\operatorname{proj}_{\mathbf{b}} \mathbf{a}\|$.


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


A fair 8-sided die is rolled. What is the probability that the number rolled is a divisor of 10?


Calculate the product and the sum of all positive integer divisors of 18.


Arthur walks 8 blocks east, 10 blocks north, 3 blocks south, and 5 blocks west. If each block represents one-quarter of a mile, how many miles did Arthur walk?


In convex hexagon $ABCDEF$, angles $A$, $B$, and $C$ are congruent, and angles $D$ and $E$ are also congruent. If the measure of angle $A$ is \(30^\circ\) less than the measure of angle $D$, and angle $F$ is equal to angle $A$, what is the measure of angle $D$?


A digit is written to the right of the units digit of $682$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?


Xanthia buys hot dogs in packages of five, and hot dog buns in packages of seven. What is the smallest number of hot dog packages she must buy to have an equal number of hot dogs and hot dog buns?


Calculate $7 \cdot 7! + 5 \cdot 5! + 6 \cdot 6!$.


The base-10 numbers 326 and 57 are multiplied. The product is then written in base-7. What is the units digit of the base-7 representation?


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


Ten points are marked on the circumference of a circle. How many different chords can be drawn by connecting two of these ten points, excluding those that connect directly adjacent points?


What is the largest divisor of 360 that is also a factor of 315?


Three years ago, there were 30 trailer homes on Oak Road with an average age of 15 years. At that time, a group of brand new trailer homes was added to Oak Road. Today, the average age of all the trailer homes on Oak Road is 12 years. How many new trailer homes were added three years ago?


What is the largest divisor of 540 that is also a factor of 462?


When a polynomial is divided by \(3x^3 - 4x^2 + x - 8\), what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


What is the area, in square units, of a triangle with sides of 5 units, 3 units, and 4 units? Express your answer in simplest radical form.


Find the equation of the directrix of the parabola $x = -\frac{1}{4} y^2$.

Solution: \boxed{x = 1}


The statue of Liberty measures 151 feet in height. A scale model of the statue is 5 inches tall. How many feet of the statue does one inch of the model represent?


The Eiffel Tower in Paris is 300 meters tall. If a scale model of the Eiffel Tower is 25 centimeters tall, how many meters of the tower does one centimeter of the model represent?


**Emilio's International Treasures**

Emilio collects stamps and organizes his collection by country and the decade in which they were issued. He paid the following prices at a stamp collector's market: Italy and Japan, $7$ cents each; Germany and China, $5$ cents each. Emilio's stamps from Europe are Italy and Germany, and from Asia are China and Japan.

Here are the amounts of stamps per decade:
- 1950s: Italy (5 stamps), Germany (8 stamps), China (10 stamps), Japan (6 stamps)
- 1960s: Italy (9 stamps), Germany (12 stamps), China (5 stamps), Japan (10 stamps)
- 1970s: Italy (11 stamps), Germany (14 stamps), China (7 stamps), Japan (8 stamps)
- 1980s: Italy (6 stamps), Germany (9 stamps), China (10 stamps), Japan (5 stamps)

Calculate the total cost of Emilio's European stamps (from Italy and Germany) issued in the 1950s and 1960s in dollars and cents.


Express $2.\overline{02}$ as a reduced fraction, given that $0.\overline{01}$ is $\frac{1}{99}$.


Let $A$ be the greatest common factor and let $B$ be the least common multiple of 10, 15, and 25. Calculate the value of $A + B$.


Rosie can make two apple pies out of nine apples and three pear pies out of fifteen pears. If she has twenty-seven apples and thirty pears, how many total pies can she make?

Solution: \boxed{12 \text{ pies}}


If $256^8 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.


How many values of $x$, $-12 < x < 128$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$? (Note: $x$ is measured in radians).


How many four-digit whole numbers have no 7's, 8's, or 9's as digits?


The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers?


The polynomial \( Q(x) = x^3 + px^2 + qx + d \) has the property that the mean of its zeros, twice the product of its zeros, and the sum of its coefficients are all equal. If the \(y\)-intercept of the graph of \(y = Q(x)\) is 4, what is \(q\)?


A right cone has a base with a circumference of $18\pi$ inches and a height of 36 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $270\pi$ cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fraction.


How many different counting numbers will each leave a remainder of 4 when divided into 53?


Jasmine is driving her colleague home from a conference. Her car's odometer reads 85,412 miles when she picks up her colleague and 85,448 miles when she drops her colleague off. Jasmine's car has a fuel efficiency of 32 miles per gallon and the current price of gas is $\$4.20$ per gallon. How much did Jasmine spend on gas for this trip?


Jessica works at a home appliance store and claims that the popularity of a blender, measured in number of sales, is influenced by both its cost and its energy efficiency rating. The number of customers who buy the blender (p) is inversely proportional to its cost (c) and directly proportional to its energy efficiency rating (e). If a blender costing $400 with an efficiency rating of 10 units is bought by 50 customers, how many customers would buy a blender costing $800 with an efficiency rating of 20 units?


The prime factorization of the number 3430 is $2 \times 5 \times 7^3$. How many ordered pairs of positive integers $(x, y)$ satisfy the equation $xy = 3430$?


Laura and her three younger siblings each want a new cellphone for their birthdays. Laura insists on having a service from Verizon, while her siblings want different providers from each other and none of them wants Verizon. There are in total 25 service providers. In how many ways can their parents purchase the cellphones satisfying all conditions?


The measure of each exterior angle of a regular polygon is $15$ degrees. How many sides does the polygon have, and what is the sum of the measures of the interior angles, in degrees?


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 5$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right |$?


What is the sum of the digits of the base-2 expression for $511_{10}$?


The sum of all the positive factors of integer $x$ is 18. If one of the factors is 2, what is the value of $x$?


How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides \(9\), \(40\), and \(41\)?


The diameter, in centimeters, of a sphere with three times the volume of a sphere of radius 5 centimeters can be expressed in the form $c\sqrt[3]{d}$ where $c$ and $d$ are positive integers and $d$ contains no perfect cube factors. Compute $c+d$.


Topsoil costs $7$ per cubic foot. Calculate the cost, in dollars, for topsoil required to cover a rectangular garden that measures 8 feet long, 4 feet wide, and 6 inches deep.


Using the digits 1, 2, 3, 4, 5, 6, how many even three-digit numbers less than 700 can be formed if each digit can be used more than once?


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


For how many non-negative real values of $x$ is $\sqrt{196-\sqrt[3]{x}}$ an integer?


$\textbf{Laura's Historical Stamps}$

Laura sorts her stamp collection by continent and era in which they were issued. She paid these amounts at an antique market: Brazil and Spain, $7$ cents each, France $9$ cents each, and Peru $6$ cents each (where Brazil and Peru are from South America and Spain and France are from Europe). The number of stamps she has by country and decade is as follows:
- $80$s: Brazil $12$, France $10$, Spain $18$, Peru $15$
- $90$s: Brazil $14$, France $12$, Peru $20$, Spain $16$

Calculate in dollars and cents, how much her European stamps from the $80\text{'s}$ and $90\text{'s}$ cost her.


A rhombus has diagonals of lengths 18 units and 16 units. Calculate both the area and the length of one side of the rhombus.


Find the product of $111_6$ and $2$, and then add that result to $202_6$ in base 6.


A shop is deciding the price for a special edition toy. If the toy's price is $p$ dollars (where $p \le 20$), then the number of toys sold per month is given by $100 - 4p$. Determine the price that maximizes the shop's monthly revenue.


BoatsRUs made 10 canoes in February of this year and then each subsequent calendar month, they increased production by a factor of 3 from the previous month. How many total canoes were built by BoatsRUs by the end of July of this year?


Convert the base-10 integer 947 to base 8.


If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$


In how many different ways can five students stand in a straight line if two of them refuse to stand next to each other?


The expression $x^2 + 18x + 80$ can be written as $(x + d)(x + e)$, and the expression $x^2 - 20x + 96$ can be written as $(x - e)(x - f)$, where $d$, $e$, and $f$ are integers. What is the value of $d + e + f$?


Abby, Bart, Cindy, and Damon weigh themselves in pairs. Together Abby and Bart weigh 280 pounds, Bart and Cindy weigh 255 pounds, and Cindy and Damon weigh 290 pounds. How many pounds do Abby and Damon weigh together?


Two lines, one with a slope of $3$ and another with a slope of $\frac{1}{2}$, intersect at the point $P(2, 3)$. Determine the area of the triangle formed by these lines and the x-axis, represented as $\triangle PQR$.


Solve the equation \(9^{x+6} = 10^x\) and express \(x\) in the form \(x = \log_b 9^6\). What is \(b\)?


You have a certain number of gold coins that you planned to distribute among 13 friends. After dividing your gold coins into 13 equal piles, it turns out that 3 people would end up receiving one extra gold coin each. You have less than 110 gold coins. What is the largest number of gold coins you could have for this distribution to be possible?


The Cookie Monster now finds another cookie with the boundary given by the equation $x^2 + y^2 - 10 = 2x + 6y$. He is not just curious about the size of this cookie but also wants to know the area it covers. How large is this cookie in terms of its radius and what is its area?


What is the smallest five-digit palindrome that is divisible by 5? (A palindrome reads the same forwards and backwards.)


How many distinct, positive factors does $1320$ have?


Let $f(x) = -2x^3 - 4x^2 + 2x - 5$, $g(x) = -3x^2 + 4x - 7$, and $h(x) = 6x^2 + 3x + 2$. Express $f(x) + g(x) + h(x)$ as a single polynomial, with the terms in order by decreasing degree.

Solution: \boxed{-2x^3 - x^2 + 9x - 10}


Find $\frac{9}{23} - \frac{5}{69}$ and reduce your answer to the simplest form.


In triangle $DEF$, the measure of $\angle D$ is $75$ degrees. The measure of $\angle E$ is $15$ degrees more than four times the measure of $\angle F$. Determine the measure, in degrees, of $\angle F$.

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


Both $c$ and $d$ are positive integers and $d > 1$. When $c^d$ is the greatest possible value less than 500, what is the sum of $c$ and $d$?


To qualify for a geometry class, a student needs to have an average of at least $85\%$ over five different tests. If Jamie scored $80\%$, $90\%$, and $78\%$ on the first three tests, what is the minimum average score she must achieve on the next two tests to be eligible for the class?


An 8-sided die is rolled. If the even number rolled is 4 or less, you win that amount in dollars; if the number is 6, you win twice that amount ($\$12$); if the number is 8, you win half the amount ($\$4$). If the roll is odd, you still win nothing. What is the expected value of your winnings? Express your answer as a dollar value.


The polynomial \( Q(x) = x^3 + px^2 + qx + d \) has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the \( y \)-intercept of the graph of \( y = Q(x) \) is 3, find the value of \( q \).


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

[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.966,.259);
C=(-.707,-.707);
D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.5);
label("Books", W, N);
label("$\frac{1}{4}$", W, S);
X=(-.55, 0);
label("Games", X, S);
Y=(-.17,-.7);
label("Snacks", Y, N);
label("$\frac{1}{5}$", Y, S);
Z=(.5, -.15);
label("Music", Z, N);
label("$\frac{2}{5}$", Z, S);
[/asy]


An infinite geometric series has a common ratio of \( \frac{1}{4} \) and its sum is \( 40 \). What is the second term of the sequence?


Find the minimum value of the function $3x^2 - 18x + 2023$.


Eight green balls and two red balls are in a bag. A ball is taken from the bag, its color recorded, then placed back in the bag. A second ball is taken, and its color is recorded. What is the probability that the two balls are the same color?


How many distinct prime factors are in the prime factorization of $91\cdot92\cdot93\cdot94$?


In the row of Pascal's triangle that starts with 1 and then 13, what is the fifth number?


If \(x+y=12\) and \(xy=20\), what is the value of \(x^3+y^3\)? Also, determine \(x^2 + y^2\).


Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\circ}$ and $90^{\circ}$. Two of the remaining angles, $C$ and $D$, are equal, and the fifth angle $E$ is $10^{\circ}$ more than twice $C$. Find the measure of the largest angle.


Mike transforms the quadratic $4x^2 - 16x + 15$ into the form $(px + q)^2 + r$, where $p$, $q$, and $r$ are integers. What is $pq$?


Three standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction.


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

Solution: \boxed{50 - 10q}


Solution: \boxed{\left(\frac{51}{32}\right)^{1/7}}


Calculate $\sqrt{15q} \cdot \sqrt{20q^3} \cdot \sqrt{12q^5}$ and express your answer in simplest radical form in terms of $q$.


Andy and Alexa together bake 30 cookies. Andy eats \( x \) cookies, and Alexa eats a prime number multiple of the cookies eaten by Andy. Assuming all cookies are eaten, find the maximum number of cookies Andy could have eaten.


The points $(1, -2)$ and $(-3, 5)$ are adjacent vertices of a square. What is the area of the square?


Determine the value of $x$ for which $(10 - x)^2 = (x - 2)^2 + 8$.


If a positive integer $n$ is a perfect square, and the product of its positive integer divisors is $1024$, find $n$.


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

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