Given the Cesaro sum of the 99-term sequence \((a_1, \dots, a_{99})\) is 1000, what is the Cesaro sum of the 100-term sequence \((2, a_1, \dots, a_{99})\)?


A jar contains five red marbles, four green marbles, twelve white marbles, and two blue marbles. One marble is drawn and then another marble is drawn without replacing the first. What is the probability that the first marble drawn is red and the second marble drawn is blue?


John's phone number and the five digits in his house number have the same sum. John's phone number is 431-782. What is the largest possible value of John's house number if all five digits are distinct?


If the three points $(2,a,b),$ $(a,3,b),$ $(a,b,4)$ are collinear, what is the value of $2a + b$?


Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} x & y \\ z & w \end{pmatrix}$ be two matrices such that $\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}.$ Assuming $4y \neq z$, find $\frac{x - w}{z - 4y}$.


Solution: \boxed{10 + 2i}


A circle with center $A$ has a radius of $10$ units, and circle $B$ has a radius of $4$ units. The circles are externally tangent to each other at point $C$. Segment $DE$ is the common external tangent to circle $A$ and circle $B$ at points $D$ and $E$, respectively. What is the length of segment $AE$? Express your answer in simplest radical form.


One digit of the decimal representation of $\frac{4}{7}$ is randomly selected. What is the probability that the digit is a 3? Express your answer as a common fraction.


Calculate $\sqrt{15q} \cdot \sqrt{10q^3} \cdot \sqrt{14q^5}$ and express your answer in simplest radical form in terms of $q$.


Oliver's phone number and his four-digit house number have the same cumulative sum for their digits. The four digits in his house number are distinct. His phone number is 563-4921. What is the largest possible value of Oliver's house number?


How many prime numbers have squares between 8000 and 12000?


How many positive integers smaller than $1{,}000{,}000$ are powers of $2$, but are not powers of $4$?


What is the value of $(10! \cdot 4! \cdot 3!) / (9! \cdot 5!)$?


Consider a triangle $\triangle ABC$, where point $X$ lies on line segment $AB$. $X$ is such that $CX$ bisects $\angle ACB$. Given that $AB = 45$, $BC = 32$ and segment $BX = 20$, find the length of segment $AX$.


A sequence of three real numbers forms an arithmetic progression with a first term of 5. If 5 is added to the second term and 30 is added to the third term, the three resulting numbers form a geometric progression. What is the highest possible value for the third term of the geometric progression?


How many values of $x$, $-10 < x < 10$, satisfy $\cos^2 x + 3\sin^2 x = 2?$ (Note: $x$ is measured in radians.)


Determine the positive integer values of $k$ for which the quadratic equation $kx^2 + 24x + 9k = 0$ has rational solutions.

Solution: \boxed{k = 4}


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

Solution: \boxed{\frac{9}{9} = 1}


One of the asymptotes of a hyperbola has equation $y = -2x$. The foci of the hyperbola have the same $x-$coordinate, which is $-4$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = 2x + 16}


Bertha has 8 daughters and no sons. Some of her daughters have 8 daughters each, and the rest have none. Bertha has a total of 48 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


Find the number of square units in the area of the shaded region. Consider a rectangular plot with vertices at points (0,0), (40,0), (40,20), and (0,20). Within this rectangle, a shaded polygon is formed with vertices at (0,0), (20,0), (40,10), (40,20), and (10,20). Determine the area of this shaded region.


Each of the numbers $b_1,$ $b_2,$ $\dots,$ $b_{75}$ is $\pm 1.$  Find the smallest positive value of
\[\sum_{1 \le i < j \le 75} b_i b_j.\]


Given three vertices of a cube: $A = (5, 9, 5)$, $B = (6, 5, -4)$, and $C = (9, 0, 4)$. Determine the surface area of the cube.


In the following diagram, $AB=72$. Find $AX$.

[asy]
import markers;

real t=.67;
pair A=(0,0);
pair B=(3,-2);
pair C=(1.5,1.5);
pair X=t*A+(1-t)*B;

draw(C--A--B--C--X);

label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SW);

markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));
markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));

label("$80$",.5*(B+C),E);
label("$40$",.5*(A+C),NW);

[/asy]


If a 4'' by 4'' square is added at each successive stage, what will be the area of the rectangle at Stage 6, in square inches?


What is the domain of the function
$$ k(x) = \frac{1}{x+9} + \frac{1}{x^2+9} + \frac{1}{x^3+9} ~?$$
Express your answer as a union of intervals.

Solution: \boxed{(-\infty, -9) \cup (-9, -\sqrt[3]{9}) \cup (-\sqrt[3]{9}, \infty)}


Five standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the probability their sum is even? Express your answer as a common fraction.


In the diagram, the grid is made up of squares and has two unshaded triangles. The grid dimensions are as follows: a 2 by 3 rectangle, a 3 by 4 rectangle, and a 4 by 5 rectangle. The first unshaded triangle is right-angled with a base of 12 and a height of 4. The second unshaded triangle is right-angled with base 3 and height 2. What is the area of the shaded region?


In triangle $ABC$, where $AB = 5$ and $AC = \sqrt{34}$, find $\tan A$.

[asy]
pair A,B,C;
B = (0,0);
A = (0,5);
C = (3,0);
draw(A--B--C--A);
draw(rightanglemark(A,B,C,7));
label("$B$",B,SW);
label("$A$",A,NW);
label("$C$",C,SE);
label("$5$",(A+B)/2,NW);
label("$\sqrt{34}$",(A+C)/2,NE);
[/asy]


Factor the following expression: $49a^3 + 245a^2 + 588a$.

Solution: \boxed{49a(a^2 + 5a + 12)}


Let $r(x)$ have a domain $\{-1, 0, 1, 2, 3\}$ and range $\{0, 1, 2, 3, 4\}$. Define $s(x)$ over the domain $\{1, 2, 3, 4, 5\}$ by the rule $s(x) = x+2$. What is the sum of all possible values of $s(r(x))$?


The median of the set of numbers $\{21, x, 56, 63, 9\}$ is 20 units more than the mean. If $x$ is a negative integer, what is the value of $x$?


Evaluate the expression $x^2 + y^2 - 3z^2 + 2xy + 2yz - 2xz$ when $x = 2$, $y = -1$, and $z = 3$.


In a particular class of students, when they attempt to take a class picture and form rows of five, there are two students left over. When they form rows of six students, there are four students left, and when they line up in rows of eight, there are six students left over. What is the fewest number of students possible in this class?


A square and four circles, each with a radius of 7 inches, are arranged similarly as in the original problem. What is the area, in square inches, of the square?


Points $A$ and $B$ are selected on the graph of $y = -x^2$ so that triangle $ABO$ is isosceles with $AB = AO = BO$. Determine the length of one side of triangle $ABO$.


Two standard decks of 52 cards are combined, making a total of 104 cards with duplicates for each rank and suit. The decks are shuffled together, and the cards are randomly arranged. What is the probability that the top card is either the King of $\heartsuit$ or the King of $\diamondsuit$?


The graphs \( y = 4(x-h)^2 + j \) and \( y = 5(x-h)^2 + k \) have \( y \)-intercepts of \( 4032 \) and \( 5040 \), respectively, and each graph has two positive integer \( x \)-intercepts. Find \( h \).


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


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


Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. The visible numbers are 51, 72, and 43. What is the average of the hidden prime numbers?


Find the number of $x$-intercepts on the graph of $y = \sin \left( \frac{3}{x} \right)$ in the interval $(0.01, 0.1).$


What is the probability that Joseph gets fewer heads than tails if he flips 12 coins?


A custom deck of 104 cards (60 black and 44 red) is dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black? Express your answer as a common fraction.


Let $u,$ $v,$ and $w$ be the roots of the equation $x^3 - 18x^2 + 20x - 8 = 0.$ Find the value of $(2+u)(2+v)(2+w).$


Sue owns 15 pairs of shoes: eight identical black pairs, four identical red pairs, and three identical white pairs. If she picks two shoes at random, what is the probability that they are the same color and one is a left shoe and the other is a right shoe? Express your answer as a common fraction.


Given the matrix for reflecting over a line $\ell$ that passes through the origin defined by \[\begin{pmatrix} \frac{3}{5} & -\frac{4}{5} \\ -\frac{4}{5} & -\frac{3}{5} \end{pmatrix},\] determine the direction vector of line $\ell$. Present your answer as $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$, and $b$ are integers, $a > 0$, and $\gcd(|a|, |b|) = 1$.

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


If \(x^2 = 2y - 6\) and \(x = 7\), then what is the value of \(y\)?


Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?


Kenton observed 2500 adults boarding a cruise ship. Half of these adults were women. On this day, 15% of the women and 12% of the men were wearing sunglasses. How many adults in total were wearing sunglasses?


Ramon runs a food stand where he sells four burritos and two quesadillas for $\$12.50$, and three burritos and four quesadillas for $\$15.00$. He also has a fixed starting fee of $\$2.00$ for every transaction. Assuming a fixed price per item and this transaction fee, determine the cost in dollars for five burritos and five quesadillas.


Jasmine rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number she obtains? Express your answer as a decimal.


How many non-similar triangles have angles whose degree measures are distinct positive integers in an arithmetic progression, with the smallest angle being at least 10°?


Find the number of real solutions to the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -1.\]


How many positive integers less than 100 that are either prime, or have an even number of positive divisors?


Let polynomial $q(x)$ be such that when divided by $x - 1$, the remainder is 10, and when divided by $x + 3$, the remainder is -8. Find the remainder when $q(x)$ is divided by $(x - 1)(x + 3)$.

Solution: \boxed{4.5x + 5.5}


Compute and simplify the expression:
\[2(i^0 + i^1 + \cdots + i^{2012}).\]


How many integers fall between $\sqrt{10}$ and $\sqrt{200}$ on a number line?


Find the remainder when $3 \times 13 \times 23 \times ... \times 93$ and $7 \times 17 \times 27 \times ... \times 97$ multiplied together is divided by $8$.


The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2017} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$.


Let $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ for some real numbers $a$, $b$, $c$, and $d.$ If \[\mathbf{A} + \mathbf{A}^{-1} = \mathbf{0},\] then find $\det \mathbf{A}$.


Assuming all 10 digits are equally likely to appear in numeric positions and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Texas issue than Maine, given that Texas has a license plate format of two letters followed by four numbers and one letter (LLNNNNL) and Maine has a license plate format of three letters followed by three numbers (LLLNNN)?


What is the smallest positive integer $n$ for which $8n-3$ and $6n + 4$ share a common factor greater than $1$?


Consider a new arrangement of numbers with the following rules: Each even row (2nd, 4th, 6th, etc.) contains $n \times 2$ numbers where all elements in the row are $2n$. Each odd row (1st, 3rd, 5th, etc.) contains $n + 3$ numbers where all elements are $2n$. What would be the value of the $90^{\mathrm{th}}$ number if the arrangement were continued?


Let $g(x) = \left\lceil \dfrac{1}{x+1} \right\rceil$ for $x > -1$, and $g(x) = \left\lfloor \dfrac{1}{x+1} \right\rfloor$ for $x < -1$. ($g(x)$ is not defined at $x = -1$.) Which integer is not in the range of $g(x)$?


In triangle $ABC$, $AB = 8$, $AC = 17$, and $BC = 15$. Let $I$ be the incenter. The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Find the length of $BI$.


For how many integer values of $n$ between 1 and 210 inclusive does the decimal representation of $\frac{n}{210}$ terminate?


A wire is cut into two pieces, one of length \( a \) and the other of length \( b \). The piece of length \( a \) is bent to form a square, and the piece of length \( b \) is bent to form a circle. The square and the circle have equal area. What is \( \frac{a}{b} \)?


For how many integer values of $n$ between 1 and 499 inclusive does the decimal representation of $\frac{n}{500}$ terminate?


How many rows of Pascal's Triangle contain the number $47$?


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


What is the remainder when $3^{250}$ is divided by 13?


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


How many integers $n$ satisfy the condition $150 < n < 250$ and the condition $n$ has the same remainder whether it is divided by $7$ or by $9$?


A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.


Solve for $x$ if $16^x \cdot 16^x \cdot 16^x \cdot 16^x = 64^{6}$.


What is the period of \(y = \sin(3x) + \cos(6x)\)?


Find the product of all real values of $r$ for which $\frac{1}{3x} = \frac{r - 2x}{10}$ has exactly one real solution.


For how many positive integral values of $b$ is it true that $x = 3$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
3x > 4x - 4 \\
4x - b > -8
\end{cases}
$$


Let $a$ and $b$ be real numbers. Suppose one of the roots of $x^3 + ax + b = 0$ is $2 + i \sqrt{2}.$ Find $a + b.$


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


How many three-digit positive integers are congruent to 2 (mod 4)?


Determine the value of $c$ such that $x\cdot(3x+1) < c$ if and only when $x \in \left(-\frac{5}{2}, 3\right)$.


Find the number of real solutions of the equation
\[\frac{x}{50} = \sin(2x).\]


Let \( d, e, f, g \) be the roots of \( x^4 + rx^2 + sx + t = 0 \). Compute the determinant
\[
\begin{vmatrix}
d & e & f & g \\
e & f & g & d \\
f & g & d & e \\
g & d & e & f
\end{vmatrix}.
\]


An organization starts with three leaders and a certain number of regular members. Each year, the current leaders are replaced. Each regular member recruits three new members. Additionally, three new leaders are selected from outside the organization. Initially, there are twenty-one people in the organization. Determine the total number of people in the organization after four years.


During a discussion in his math club, Alex is pondering about the properties of a quadratic equation, \[ x^2 - tx + q, \] where the roots \( a_1 \) and \( a_2 \) satisfy the equalities \[ a_1 + a_2 = a_1^2 + a_2^2 = a_1^3 + a_2^3 = \cdots = a_1^{2010} + a_2^{2010}. \] Intrigued, Alex decides to find the maximum possible value of \[ \frac{1}{a_1^{2011}} + \frac{1}{a_2^{2011}}. \]


Sandy's daughter has a playhouse in the back yard. She plans to cover the two shaded side walls and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 10-foot by 15-foot sections that cost $\$35$ each. If Sandy can cut the siding when she gets home, how many dollars will be the cost of the siding Sandy must purchase?


How many five-letter words can be formed such that the first letter is the same as the fifth, and the second letter is the same as the fourth?


Solution: \boxed{3^{123} + 81}


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


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-50^2)} = 0\] have for \(x\)?


A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment at the school.


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 4x - 8 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a - 3,$ $b - 3,$ and $c - 3.$

Solution: \boxed{x^3 + 9x^2 + 23x + 7}


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


Starting with the number 200, Shaffiq repeatedly divides his number by two and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1?


Determine how many of the first 30 rows of Pascal's triangle, excluding row 0 and row 1, consist only of even numbers (with the exception of the 1's at either end). Recall that each number in Pascal's triangle is the sum of the two numbers above it.


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


Mariana has ten socks, two of each color: red, green, blue, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color?


Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 11}.\]


If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 10000$ and $f(i) = 1 + 2\sqrt{i} + i$?


Find the integer that is closest to $500\sum_{n=2}^{5000}\frac1{n^2-1}$.


If a 4'' by 4'' square is added at each successive stage, what will be the area of the rectangle at Stage 8, in square inches?


Fifteen points are marked on a circle. How many distinct convex polygons of four or more sides can be drawn using some (or all) of the fifteen points as vertices?


A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors?


The director of a choir wants to organize the members into a formation where everyone is included and there are no gaps. If arranged in a square formation, there are 11 members left over. The director also considers a formation with 5 more rows than columns, which perfectly accommodates all members without any left over. Determine the maximum number of members that can be in this choir.


Determine the residue of $-1234 \pmod{31}$. Your answer should be an integer in the range $0,1,2,\ldots,29,30$.


Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 0.$ Find the set of all possible values of $ab + ac + ad + bc + bd + cd.$

Solution: \boxed{(-\infty, 0]}


What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper?


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

Solution: \boxed{\frac{-21 - 5d}{9}}


You recently bought more than 150 eggs. The eggs are sorted in containers that can store exactly 15 eggs. However, upon inspecting the containers, you realize that three containers each hold only 14 eggs, while all other containers hold 15 eggs. What is the smallest number of eggs you could have right now?


A polynomial product of the form
\[
(1 - z)^{a_1} (1 - z^2)^{a_2} (1 - z^3)^{a_3} \dotsm (1 - z^{16})^{a_{16}},
\]
where \( a_k \) are positive integers, has the property that if we multiply it out and discard all terms involving \( z \) to a power larger than 16, what remains is just \( 1 - z. \) Determine \( a_{16} \).


Each circle in an arrangement of four circles, which perfectly fit into a square, has a radius of 8 inches. Calculate the area, in square inches, of the square.


In how many ways can 6 people be seated around a round table if one of the seats is reserved for a specific person?


What is the sum of all the integers from -20 to 10 that are multiples of 3?


On a balance scale, $4$ red balls balance $8$ blue balls, $3$ orange balls balance $7.5$ blue balls, and $8$ blue balls balance $6$ purple balls. How many blue balls are needed to balance $5$ red, $3$ orange and $4$ purple balls?


What is the volume of a cone in liters, with a diameter of 12cm and a height of 10cm? Express your answer in terms of $\pi$.

Solution: \boxed{0.12 \pi \text{ liters}}


A right circular cone with base radius \( r \) and height \( h \) lies on its side on a flat surface. As it rolls without slipping along the table, the point where the base meets the surface traces a circular arc centered at the cone’s vertex. This arc constitutes a full circle once the cone completes \( 20 \) rotations. Calculate the ratio \( h/r \), which can be expressed in the form \( m\sqrt{n} \) where \( m \) and \( n \) are positive integers, and \( n \) is not divisible by the square of any prime. Find \( m + n \).


A conference hall is setting up chairs in rows for a meeting. Each row perfectly fits 15 chairs, and currently, there are 180 chairs in total. The hall expects 125 attendees and wants to minimize the number of unoccupied chairs while ensuring all rows are fully occupied. How many chairs should be removed?


What is the median number of moons per celestial body, considering the following data? (Include Pluto, Eris, and Ceres, in addition to the traditional planets.)
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 67 \\
Saturn & 82 \\
Uranus & 27 \\
Neptune & 14 \\
Pluto & 5 \\
Eris & 1 \\
Ceres & 0 \\
\end{tabular}


Find the product of all positive integral values of $n$ such that $n^2 - 21n + 110 = p$ for some prime number $p$. Assume there is at least one such $n$.


Define a new operation \( \diamond \) by
\[ a \diamond b = \frac{a + b}{1 + ab} \].
Compute
\[ 1 \diamond (2 \diamond (3 \diamond (\dotsb (499 \diamond 500) \dotsb))). \]


If $p,$ $q,$ $r,$ $u,$ $v,$ and $w$ are positive numbers and $p^2 + q^2 + r^2 = 49,$ $u^2 + v^2 + w^2 = 64,$ and $pu + qv + rw = 56,$ compute
\[\frac{p + q + r}{u + v + w}.\]
Also given that $p = 2q$.


In triangle $ABC$, angle $ACB$ is 45 degrees, and angle $CBA$ is 60 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.


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


Let \(\mathbf{a}\) and \(\mathbf{b}\) be nonzero vectors such that
\[\|\mathbf{a}\| = 2, \|\mathbf{b}\| = 1, \text{ and } \|\mathbf{a} + 2\mathbf{b}\| = 2.\]
Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\), in degrees.


In a press conference before the All-Star Game, ten All-Stars will be taking questions. Four are Cubs, four are Red Sox, and two are Yankees. If teammates insist on sitting together, how many ways can the ten All-Stars be seated in a row?


Simplify the expression $4 \cdot \frac{15}{7} \cdot \frac{21}{-45}$.


Mrs. Thompson awards extra credit in her class to students with quiz grades that exceed the class average. She currently has 150 students who took the quiz. Knowing that not all students can exceed the mean, what is the maximum number of students who can receive extra credit if their scores are either of two values?


Suppose \(a\) and \(b\) are positive integers such that \((a+bi)^3 = 2 + 11i\). Determine \(a+bi\).

Solution: \boxed{2 + i}


A standard deck of cards, typically having 52 cards divided into 4 suits, now has 5 suits with a new added red suit called 'stars' ($\star$), making 3 red suits and 2 black suits. Each suit still contains 13 cards. The deck is shuffled randomly. What is the probability that the first two cards drawn from this deck are both red?


If
\[\tan x = \frac{3cd}{c^2 - d^2},\]
where $c > d > 0$ and $0^\circ < x < 90^\circ,$ then find $\cos x$ in terms of $c$ and $d.$

Solution: \boxed{\frac{c^2 - d^2}{\sqrt{c^4 + 7c^2d^2 + d^4}}}


Let $P_1$ be a regular $r$-gon and $P_2$ be a regular $s$-gon $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{61}{60}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?


The image of the point with coordinates $(2,2)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,6)$. Find $m+b$.


The first term of the sequence is 3, and each successive term is triple the sum of all the previous terms in the sequence. What is the value of the first term which exceeds 10000?


What is the smallest five-digit palindrome that is divisible by 5? (A palindrome is a number that reads the same forwards and backwards, like 12321.)


At a women's quadruples tennis tournament, there are four teams of two women each. After the tournament, women from only the first three teams shook hands, once with each of the women from the other teams except her partner. How many handshakes occurred?


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


Determine the minimum value of $z$ if $z = x^2 + 3y^2 + 8x - 6y + 30$.


A recipe for 45 cookies calls for 3 cups of all-purpose flour. Lucía decides to prepare 90 cookies for a big gathering. However, for half of the cookies, she wants to use whole wheat flour instead. How many cups of each type of flour will Lucía need?

Solution: \boxed{3 \text{ cups}}


The Euler family has two boys aged 10 and 12, and a girl aged 8. The Gauss family has three boys aged 8, 8, and 8, a girl aged 16, and a boy aged 18. What is the mean (average) age of all the children from both families combined?


In triangle $ABC$, where a circle with center $O$ is tangent to all sides, given that $\angle BAC = 70^\circ$ and $\angle ABC = 45^\circ$, find $\angle BOC$, in degrees.


Each vertex of this parallelogram has integer coordinates. The perimeter of this parallelogram is $p$ units, while the area is $a$ square units. The vertices of the parallelogram are given as $(1,3)$, $(5,6)$, $(11,6)$, and $(7,3)$. What is the value of the sum $p + a$?


For the data whose frequency histogram is shown, by how many days is the mean number of days missed per student greater than the median number of days missed per student for a classroom of 20 students? Use the following histogram:

- Days missed: 0, Number of students: 5
- Days missed: 1, Number of students: 3
- Days missed: 2, Number of students: 8
- Days missed: 3, Number of students: 2
- Days missed: 4, Number of students: 1
- Days missed: 5, Number of students: 1

Solution: \boxed{0.3 \text{ days}}


If $\mathbf{c}$ and $\mathbf{d}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $2 \mathbf{c},$ $\mathbf{d} + 3(\mathbf{d} \times \mathbf{c}),$ and $\mathbf{d}.$


Compute
\[
\left( 1 - \frac{1}{\cos 30^\circ} \right) \left( 1 + \frac{2}{\sin 60^\circ} \right) \left( 1 - \frac{1}{\sin 30^\circ} \right) \left( 1 + \frac{2}{\cos 60^\circ} \right).
\]

Solution: \boxed{\frac{25 - 10\sqrt{3}}{3}}


Find the number of real solutions to the equation:
\[
\frac{1^2}{x - 1} + \frac{2^2}{x - 2} + \frac{3^2}{x - 3} + \dots + \frac{120^2}{x - 120} = x.
\]


How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$? (Note that $B$, $C$, and $D$ are consonants, any word is valid, not just English language words, and letters may be used more than once.)


The average of Amy's, Ben's, and Chris's ages is 10. Five years ago, Chris was twice the age Amy is now. In 4 years, Ben's age will be $\frac{3}{4}$ of Amy's age at that time. Determine Chris’s current age.


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $640$?


How many integers are there between $(11.1)^3$ and $(11.2)^3$?


Let $a$ and $b$ be real numbers such that $4a + 3b \le 10$ and $3a + 6b \le 12$. Find the largest possible value of $a + b$.


A fair 8-sided die is rolled. If I roll $n$, then I win $2n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


The solutions to \( z^6 = 64i \) can be expressed in the form
\[
z_1 = r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 = r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 = r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 = r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 = r_5 (\cos \theta_5 + i \sin \theta_5), \\
z_6 = r_6 (\cos \theta_6 + i \sin \theta_6),
\]
where \( r_k > 0 \) and \( 0^\circ \le \theta_k < 360^\circ \). Find \( \theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6 \), in degrees.


Mr. and Mrs. Lopez now have three children. When they travel in their family van, two people sit in the front and three sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?


The real number $x$ satisfies $x^2 - 7x + 12 < 0.$ Find all possible values of $x^2 + 7x + 12.$

Solution: \boxed{(42, 56)}


Compute the value of
\[\frac{(1 + 13) \left( 1 + \dfrac{13}{2} \right) \left( 1 + \dfrac{13}{3} \right) \dotsm \left( 1 + \dfrac{13}{16} \right)}{(1 + 16) \left( 1 + \dfrac{16}{2} \right) \left( 1 + \dfrac{16}{3} \right) \dotsm \left( 1 + \dfrac{16}{13} \right)}.\]


Two different numbers are randomly selected from the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \). The probability that their sum is 14 would be greater if the number \( n \) had first been removed from set \( S \). What is the value of \( n \)?


Let $g(n)$ be the product of the proper positive integer divisors of $n$. For how many values of $n$ does $n$ not divide $g(n)$, given that $3 \le n \le 60$?


If 1540 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Whilst backpacking across Europe, Emily and Berengere stop in Switzerland. A cake in a local cafe costs 15 Swiss francs. Emily has 10 USD and Berengere has some euros. If the exchange rates are 1 USD = 0.90 Swiss francs and 1 euro = 1.10 Swiss francs, how many euros does Berengere need to contribute so they can buy the cake?

Solution: \boxed{5.45 \text{ euros}}


Emily has a deck of cards that she arranges in a specific pattern repeating after every 17 cards: $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, 2, 3, 4, A, 2, \cdots.$$ What is the 68th card in this sequence?


Find the absolute value of the difference of the roots of the equation \(x^2 - 7x + 12 = 0\). Also, determine the maximum value of the roots.


There are 60 chips in a box. Each chip is either small or large. If the number of small chips exceeds the number of large chips by a prime number, and this prime number also divides the number of large chips, what is the greatest possible number of large chips?


On a "modulo $m$ graph paper" for $m=17$, graph the points satisfying the congruence $$2x \equiv 3y + 2 \pmod{17}.$$ Determine the single $x$-intercept $(x_0,0)$ and the single $y$-intercept $(0,y_0)$, where $0 \leq x_0, y_0 < 17$. What is the value of $x_0 + y_0$?


The value of $\log_2{3456}$ is between which two consecutive integers $a$ and $b$? Find $a+b$.


The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number?


Find the number of solutions to the equation
\[\tan (3 \pi \cos \theta) = \cot (3 \pi \sin \theta)\]where $\theta \in (0, 2 \pi).$


Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 20 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$.


It is required to construct a right triangle in the coordinate plane so that its legs are parallel to the \( x \) and \( y \) axes, and the medians to the midpoints of the legs lie on the lines \( y = 4x + 2 \) and \( y = nx + 3 \). Determine the number of different constants \( n \) for which such a triangle exists.


Consider the lines:
\[
y = 3x + 4, \quad 4y = 12x + 20, \quad 2y = 6x - 1, \quad 3y = 2x - 6, \quad 5y = -15x + 5.
\]
Determine how many pairs of these lines are either parallel or perpendicular to each other.


It is currently $8\!:\!00\!:\!00 \text{ a.m.}$ What time will it be in $9999$ seconds?  (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)

Solution: \boxed{10\!:\!46\!:\!39 \text{ a.m.}}


What is the sum of the tens digit and the ones digit of the integer form of $(3+4)^{15}$?


A line is parameterized by a parameter $t,$ such that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -7 \\ 1 \end{pmatrix}.$ Find the vector on the line at $t = 0.$

Solution: \boxed{\begin{pmatrix} 0 \\ 9 \\ 9 \end{pmatrix}}


Half of the people in a room are seated in two-thirds of the chairs. The rest of the people are standing. If there are 8 empty chairs, how many people are in the room?


Determine the domain of the function $g(x) = \tan(\arcsin(x^2)).$


Suppose $a$, $b$, $c$, and $d$ are integers satisfying the following equations: $a - b + c = 7$, $b - c + d = 4$, $c - d + a = 2$, and $d - a + b = 1$. What is the value of $a + b + c + d$?


Michael revisits his quadratic roots question but now considers a quadratic \[x^2 - sx + p\] with roots $r_1$ and $-r_1$. He is curious about the sum of the squares of the roots for any power up to 2005, i.e., \[r_1^2 + (-r_1)^2 = r_1^4 + (-r_1)^4 = \cdots = r_1^{2006} + (-r_1)^2006.\] Determine the maximum value of \[\dfrac{1}{r_1^{2006}} + \dfrac{1}{(-r_1)^{2006}}.\]


What is the greatest common factor of 36 and 54?


The values of $a$, $b$, $c$, and $d$ are 6, 7, 8, and 9, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$, and $ad$?


The set of vectors \(\mathbf{u}\) such that
\[
\operatorname{proj}_{\begin{pmatrix} 3 \\ -4 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{9}{5} \\ -\frac{12}{5} \end{pmatrix}
\]
lie on a line. Enter the equation of this line in the form "y = mx + b".

Solution: \boxed{y = \frac{3}{4}x - \frac{15}{4}}


Find the matrix $\mathbf{M}$ that doubles the first row and triples the second row of a matrix. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.
\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}}


Amanda starts at the base of a lighthouse and walks 12 meters due north. She then turns due west and walks 45 feet. Finally, she turns due south and walks 12 meters plus 15 feet. Determine how many feet away from the base of the lighthouse Amanda is.

Solution: \boxed{15 \sqrt{10} \text{ feet}}


A deck of 60 cards includes 15 ranks (Ace, 2, 3, ..., 10, Jack, Queen, King, Ace2, and 2Ace) spread across 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, $\clubsuit$). Each suit has one card of each rank. What is the probability that the top card is either a King or a Queen?


Let $p(x)$ be a monic quartic polynomial such that $p(-2) = -4$, $p(1) = -1$, $p(3) = -9$, and $p(5) = -25$. Find $p(0)$.


How many ways are there to put 5 distinguishable balls into 3 distinguishable boxes if balls A and B must always be together in the same box?


The graph of $y^2 + 2xy + 25|x| = 625$ partitions the plane into several regions. Calculate the volume of the solid formed by extending the bounded region vertically to a height of $h = 10$ units.


Compute the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 1007$ such that $a_i - i$ is odd for $1 \le i \le 10$. Express your answer as ${m \choose n}$ for some $m > n$ and find the remainder when $m$ is divided by 1000.


How many different counting numbers will each leave a remainder of 3 when divided into 53 and are multiples of 4?


If the consecutive integers from $40$ to $1$ were written as $$404938373635...,$$ what would be the $45^{\text{th}}$ digit to be written?


One of the asymptotes of a hyperbola has equation \( y = -2x \). The foci of the hyperbola have the same \( x \)-coordinate, which is \( -3 \). Find the equation of the other asymptote of the hyperbola, giving your answer in the form \( y = mx + b \).

Solution: \boxed{y = 2x + 12}


Find the product of the divisors of $72$.


Maria ran 3 miles in two-thirds the time it took Liam to run 5 miles. If it took Liam 40 minutes to run 5 miles, using the same rates, how many minutes would it take Maria to run 8 miles?


The median of the set of numbers $\{20, 55, 68, x, 25\}$ is eight less than the mean. If $x$ is a negative integer, what is the value of $x$?


In the diagram, circle $O$ has a radius of 10 units. Chord $CD$ is 12 units long and parallel to segment $KB$. If $KA = 20$ units and points $K$, $A$, $O$, and $B$ are collinear, what is the area of triangle $KDC$?

Solution: \boxed{48 \text{ square units}}


Let $p$ and $q$ denote the solutions of the quadratic equation $2x^2 + 8x - 42 = 0$. Find the value of $(p-q+2)^2$.


Define a sequence of complex numbers by \(z_1 = 0\) and
\[ z_{n + 1} = z_n^2 - i \]
for all \(n \geq 1\). In the complex plane, how far from the origin is \(z_{111}\)?


A club consists of three chairpersons and some number of active members. Each year, the chairpersons step down, and every active member recruits three new members. Subsequently, three new chairpersons are selected from outside the club. Initially, the club has twenty-one members in total. How many members will the club have after four years?


A valid license plate in the land of Eldorado consists of three letters followed by four digits. How many valid license plates are possible?


Evaluate
\[
\begin{vmatrix}
\cos (\alpha + \phi) \cos (\beta + \phi) & \cos (\alpha + \phi) \sin (\beta + \phi) & -\sin (\alpha + \phi) \\
-\sin (\beta + \phi) & \cos (\beta + \phi) & 0 \\
\sin (\alpha + \phi) \cos (\beta + \phi) & \sin (\alpha + \phi) \sin (\beta + \phi) & \cos (\alpha + \phi)
\end{vmatrix}.
\]


In how many distinct ways can 8 beads be placed on a bracelet, considering the rotational symmetry but not reflection?


Compute
$\displaystyle \frac{1\cdot 2 + 2\cdot 4 - 3\cdot 8 + 4\cdot 16 + 5\cdot 32 - 6\cdot 64}{2\cdot 4 + 4\cdot 8 - 6\cdot 16 + 8\cdot 32 + 10\cdot 64 - 12\cdot 128}$.


Quadrilateral $ABCD$ is inscribed in a circle such that segment $AC$ is a diameter. If $m\angle DAC = 45^\circ$ and $m\angle BAC = 60^\circ$, calculate the ratio of the area of $ABCD$ to the area of the circle, expressed as $\frac{a+ \sqrt{b}}{c \pi}$, where $a$, $b$, and $c$ are positive integers. Determine $a+b+c$.


In a classroom containing $48$ chairs, how many different rectangular arrays of chairs are possible, given that there must be at least two chairs in every row and column, and all chairs must be included in the setup?


As $t$ takes on all real values, the set of points $(x,y)$ defined by
\[
x = t^2 - 4, \\
y = t^3 - 6t + 7
\]
forms a curve. Determine if this curve crosses itself, and if so, compute the ordered pair $(x, y)$ where this crossing occurs.

Solution: \boxed{(2, 7)}


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


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

What is the remainder when \( 13! \) is divided by \( 11 \)?


Consider a booklet with 65 pages numbered 1 to 65. Each page number is transformed as follows: if the original page number is $x$, then the new page number is found by reversing the order and increasing each resulting number by 3, i.e., $68 - x$. Determine how many pages have the same units digit in both the old page number and the transformed page number.


Given ten distinguishable rings, let $n$ be the number of possible six-ring arrangements on the five fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.


Solve for $x$: $5 \cdot 5^x + \sqrt{25 \cdot 25^x} = 50$.


Find $x$ such that $\lfloor x \rfloor + x = \dfrac{17}{4}$. Express $x$ as a common fraction.


For a certain hyperbola,
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]
where \(a > b\), the angle between the asymptotes is \(45^\circ\). Find \(\frac{a}{b}\).

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


Determine the value of $u^8 + v^8$ where $u$ and $v$ are the roots of the quadratic $x^2 - 3x\sqrt{2} + 2 = 0$.


Calculate the value of \(3x^y + 4y^x - 2xy\) when \(x = 3\) and \(y = 2\).


Points \( A, B, C, D, \) and \( E \) lie on a line in that order. If \( AB = 3 \) units, \( BC = 7 \) units, \( DE = 4 \) units, and \( AD = 17 \) units, what is the ratio of \( AC \) to \( BD \)?


What is the largest three-digit integer whose digits are distinct and form a geometric sequence, and where the tens digit is an even number?


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3\cdot5^7}$?


What is the number of square meters in the area of a circle with a diameter of 10 meters?


Let $p$ and $q$ be positive integers such that \[\frac{7}{12} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


An ellipse has foci at $(0, -3)$ and $(4, 0)$. It has two $x$-intercepts, one of which is $(0,0)$. What is the other $x$-intercept? Enter your answer as an ordered pair.

Solution: \boxed{\left(\frac{56}{11}, 0\right)}


Lucy was tasked with adding 68 to 57 and then rounding the result to the nearest ten. Opting for a shortcut, she rounded 68 to 70 and 57 to 60 before adding. What should be the correct rounded sum?


How many distinct terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i+j)?\]


Suppose the function $g(x)$ is defined on the domain $\{a, b, c\}$, so that the graph of $y=g(x)$ consists of just three points. Suppose those three points form a triangle of area $50$.

The graph of $y = 3g(3x)$ also consists of just three points. What is the area of the triangle formed by those three points?


A bookstore is deciding what price it should charge for a new novel. After market analysis, the store realizes that if the book's price is $p$ dollars (where $p \le 30$), the number of books sold per month is $150 - 4p$. Additionally, the store has a fixed cost of $200 per month for stocking this novel. What price should the store charge to maximize its profit?


Suppose $123 \cdot 156 \cdot 198 \equiv m \pmod{25}$, where $0 \le m < 25$.


A crew needs to reach a point $C$ at the center of a round park. They position a 20-foot plank from point $A$ to point $B$, and then a 15-foot plank from point $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Determine the area of the circle that forms the park. Express your answer in terms of $\pi$.

Solution: \boxed{325\pi \text{ square feet}}


How many positive $3$-digit numbers are multiples of $30$, but not of $45$?


Determine the ordered pair of relatively prime positive integers \( (p, q) \) of the successive ratios of segments cut by the curve \( y = \sin x \) when it intersects the horizontal line \( y = \sin 60^\circ \) in the interval from \( 0^\circ \) to \( 360^\circ \).

Solution: \boxed{(1, 4)}


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


What is the smallest positive integer \( n \) such that \( 17n \equiv 5678 \pmod{11} \)?


A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 9 black squares, can be drawn on the checkerboard?


A chess team has $30$ members. At the last meeting, two-thirds of the girls and all of the boys attended, totalling $20$ attendees. How many boys are on the chess team?


Calculate the expression $\left\lceil \frac{15}{8} \cdot \frac{-45}{4} \right\rceil - \left\lfloor \frac{15}{8} \cdot \left\lfloor \frac{-45}{4} \right\rfloor \right\rfloor$.


If the three lines \(4y - 3x = 2\), \(x + 3y = 3\) and \(6x - 8y = 6\) are drawn in the plane, how many points will lie at the intersection of at least two of the three lines?


I had planned to work 25 hours a week for 15 weeks during the summer to earn $\$3750$ to pay for tuition. Unfortunately, I was unable to work for the first three weeks of the summer. How many hours per week will I need to work for the remaining weeks of the summer to still meet my financial goal?


Lucy has $25$ dollars and $45$ cents. She wishes to buy as many notebooks as possible, where each notebook costs $2.15$. Additionally, for every 5 notebooks purchased, the price of the next notebook is reduced by 15%. How many notebooks can Lucy buy?


Consider the following system of equations for positive integers $x$ and $y$:
\begin{align*}
\log_{10}x + 2\log_{10}(\gcd(x,y)) & = 12, \\
\log_{10}y + 2\log_{10}(\text{lcm}(x,y)) & = 18.
\end{align*}
Let $p$ represent the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $q$ represent the prime factors in the prime factorization of $y$. Find the value of $p+q$.


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


A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $n$ from the origin. Given that $x>2$, what is $n$?

Solution: \boxed{\sqrt{334 + 4\sqrt{105}}}


How many values of $x$, $-25 < x < 120$, satisfy $\cos^2 x + 3\sin^2 x = 1$? (Note: $x$ is measured in radians.)


Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=7$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$.
[asy]
size(150);
defaultpen(linewidth(0.8));
pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C), H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("$P$",P,S);
label("$Q$",Q,E);
label("$H$",H,NW);
[/asy]


How many three-digit whole numbers contain at least one 6 or at least one 8?


A rectangle has dimensions of 8 by 12, and a circle centered at one of its vertices has a radius of 8. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \(\pi\).

Solution: \boxed{96 + 48\pi}


If \[f(x) =
\begin{cases}
x^2-1 &\quad \text{if } x > 0, \\
x + 2 &\quad \text{otherwise},
\end{cases}
\] then for how many values of $x$ is $f(f(x)) = 8$?


Find the greatest possible value of a three-digit number $NNM$ when the two-digit integer $MM$, where both digits are $M$, is multiplied by the integer $M$, with the condition that $M \times M$ modulo 10 must equal $M$.


Six positive integers from a list of nine positive integers are $6, 7, 2, 4, 8, 5$. What is the smallest possible value of the median of this list of nine positive integers?


Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $1$ when divided by any of $19,$ $7,$ and $3.$


Let $f(x) = \frac{x + 8}{x}$. The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[ f_n = f \circ f_{n - 1} \] for all $n \ge 2$. For example,
\[ f_2(x) = f(f(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ f_3(x) = f(f_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 80}{9x + 8}.\]
Let $S$ be the set of all real numbers $x$ such that
\[ f_n(x) = x \] for some positive integer $n.$ Find the number of elements in $S.$


What is the least natural number that can be added to 78,999 to create a palindrome?


Let $p$ and $q$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. Calculate the value of $(3p - 4)(6q - 8)$.


In triangle $XYZ$, $\angle Z=90^\circ$, $XZ=3$ and $YZ=4$. Points $W$ and $V$ are on $\overline{XY}$ and $\overline{YZ}$, respectively, and $\angle WVZ=90^\circ$. If $WV=2$, then what is the length of $WY$?


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


Find the real root of the equation \[\sqrt{x-2} + \sqrt{x+4} = 12.\]


What is the base-10 integer 678 when expressed in base 7?


What is the greatest common divisor of 1729 and 1323?


When you simplify $\sqrt[3]{60a^5b^7c^8d^2}$, what is the sum of the exponents of the variables that are outside the radical?


John is attempting a 30-question multiple choice test. Each question has six answer choices. He guesses randomly on the last five questions. What is the probability that he will answer at least two of these last five questions correctly?


Consider the function \( f(x) = \frac{x^3 + 3x^2 + 2x + 10}{x-5} \). Determine where there is a vertical asymptote.


I cycle at a constant speed, and it takes me 30 minutes to cycle to the park from my home. If the park is 5 miles away, and the library is 3 miles away from my home, how many minutes will it take me to cycle from my home to the library?

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


Evaluate the expression $\sqrt{80} - 3\sqrt{20} + \frac{\sqrt{500}}{\sqrt{5}} + 2\sqrt{45}$.

Solution: \boxed{4\sqrt{5} + 10}


A number in the set $\{60, 61, 62, ... , 1000\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction.


What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 8:20?


Determine the area enclosed by the equation $|x| + |3y| = 12$.


Calculate the value of $\frac{15!}{6!9!}$.


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 $\overline{OB}$?


Determine the domain of the function $$g(x) = \frac{1}{\lfloor x^2 - 9x + 21 \rfloor}.$$

Solution: \boxed{(-\infty, 4] \cup [5, \infty)}


What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$?


How many distinct digits can appear as the units digit of an integral perfect-square number under modulo 6?


When the base-16 number $ABCD_{16}$ is converted to base-2, how many base-2 digits (bits) does it have?


What is the area of the region defined by the equation $x^2 + y^2 + 6x - 12y - 10 = 0$?


Let $O$ be the origin. Determine the scalar $k$ such that points $A,$ $B,$ $C,$ $D,$ and $E$ are coplanar if:
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} + \overrightarrow{OE} = \mathbf{0}.\]


Yan is at a point between his house and a park. He can either walk directly to the park or walk back to his house and then take his scooter to the park. He rides his scooter 5 times as fast as he walks. Both choices take the same amount of time. What is the ratio of Yan's distance from his house to his distance from the park?


Simplify the product \[\frac{10}{5}\cdot\frac{15}{10}\cdot\frac{20}{15}\cdot \dotsm \cdot \frac{5n+5}{5n} \cdot \dotsm \frac{2510}{2505}\] and then, square the result.


Let \(x,\) \(y,\) and \(z\) be positive real numbers such that \(x + y + z = 4.\) Find the minimum value of
\[
\frac{9}{x} + \frac{1}{y} + \frac{25}{z}.
\]


Find the equation of the parabola whose graph has a vertical axis of symmetry, a vertex at \( (3, -2) \), and passes through the point \( (6, 0) \). Provide the answer in the form "$ax^2+bx+c$".

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


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


What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit $2$?


Given that \(b\) is a multiple of \(1620\), find the greatest common divisor of \(b^2 + 11b + 36\) and \(b + 6\).


Find the integer $n$ that satisfies $0 \leq n < 103$ and $$102n \equiv 74 \pmod{103}.$$


In a regular nonagon (a 9-sided polygon), 2 diagonals are chosen randomly. What is the probability that they intersect inside the nonagon?


In tetrahedron $PQRS$, the lengths of the edges are as follows: $9$, $16$, $22$, $31$, $39$, and $48$ units. If the length of edge $PQ$ is $48$, what is the length of edge $RS$?


Calculate the sum of the angles (in degrees) of all roots of the equation
\[ z^5 = \frac{1}{2} - \frac{\sqrt{3}i}{2}. \]


Let \( T \) be the set of 8-tuples \( (b_0, b_1, \dots, b_7) \), where each entry is 0 or 1, so \( T \) contains \( 2^8 \) 8-tuples. For each 8-tuple \( t = (b_0, b_1, \dots, b_7) \) in \( T \), let \( q_t(x) \) be the polynomial of degree at most 7 such that
\[ q_t(n) = b_n \] for \( 0 \le n \le 7 \).
Find
\[ \sum_{t \in T} q_t(8). \]


Convert the point $(1, -1)$ 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{2}, \frac{7\pi}{4}\right)}


Given that two of the altitudes of the scalene triangle $PQR$ have lengths $5$ and $15$, determine the largest possible integer value of the length of the third altitude.


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


Find the area of the region bounded by the graph of $r = 2\sec \theta,$ the graph of $r = 2\csc \theta,$ the $x$-axis, and the $y$-axis.


My friend Julia only likes numbers that are divisible by 8. How many different ones digits are possible in numbers that Julia likes?


Let $r = 3^s - 2s + 1$ and $s = 2^{n+1} + 2$. What is the value of $r$ when $n = 3$?


Let $p$, $q$, and $r$ be the roots of the polynomial $x^3 - 24x^2 + 50x - 8 = 0$. Compute
\[
(p+q)^2 + (q+r)^2 + (r+p)^2.
\]


A collector is hosting a competition where participants must estimate the value of a rare coin using assigned digits. The coin's value is formed from seven digits: 1, 2, 2, 4, 4, 5, and 9, and the value must start with an odd digit.

How many different possible values can the rare coin have?


The ten horizontal and ten vertical lines on a $9\times9$ chessboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$


Three interior angles $A$, $B$, and $C$ of hexagon $ABCDEF$ are $90^\circ$, $120^\circ$, and $95^\circ$. The angles $D$ and $E$ are equal, and the sixth angle $F$ is $25^\circ$ more than twice angle $D$. Find the measure of the largest angle in hexagon $ABCDEF$.


The asymptotes of a hyperbola are \(y = 2x + 3\) and \(y = -2x + 1\). The hyperbola also passes through the point \((2, 2)\). Find the distance between the foci of the hyperbola.


Suppose that all four of the numbers \[3 - \sqrt{8}, \;5+\sqrt{12}, \;16 - 3\sqrt{7}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


What is the largest three-digit multiple of 4 whose digits' sum is 12?


All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list?


Find the largest value of $c$ such that $2$ is in the range of the function $f(x) = x^2 - 7x + c$.


A safari park has five pairs of different animals, each pair consisting of one male and one female. The park ranger wants to feed these animals following a specific pattern: no two animals of the same gender can be fed consecutively. Assuming the ranger begins by feeding the male lion, in how many ways can he complete feeding all the animals?


Express $0.\overline{2}+0.\overline{05}+0.\overline{0003}$ as a common fraction.


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


Calculate $\left(4\frac{2}{3} + 5\frac{1}{4}\right) \div \left(3\frac{1}{2} - 2\frac{3}{5}\right)$. Express your answer as a mixed number.


Compute the smallest base-10 positive integer greater than 6 that is a palindrome when written in both base 2 and 4.


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


Determine the value of $x + y$ if the points $(2,x,y),$ $(x,3,y),$ and $(x,y,4)$ are collinear.


Consider the quadratic equation $5x^2 - 3x - 11$. Assuming it has two real roots, what is the sum of the squares of these roots? Present your answer as a simplified fraction.


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


A fair 8-sided die is rolled. If I roll $n$, then I win $2n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


A sequence of twelve $0$s, $1$s, and/or $2$s is randomly generated with the condition that no two consecutive characters are the same. If the probability that the generated sequence meets this condition can be expressed as $\frac{m}{n}$, where $m, n$ are relatively prime positive integers, find $m+n$.


If 15 zahs are equal to 24 tols, and 9 tols are equal in value to 15 kahs, how many zahs are equal in value to 2000 kahs?


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{17}$?


Alex and Jamie are planning to meet at a point that is exactly halfway between them. Alex is located at $(0,8)$ and Jamie is positioned at $(-5,-2)$. What are the coordinates at which they will meet?

Solution: \boxed{(-2.5, 3)}


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ f(f(x) + y) = f(x^2 - y) + kf(x)y \]
for all real numbers \( x \) and \( y \), and for some constant \( k \). Determine the number of possible values of \( f(2) \), and let \( s \) be the sum of all possible values of \( f(2) \). Find the product of the number of possible values and the sum, i.e., \( n \times s \), where \( n \) is the number of possibilities for \( f(2) \).


In $\triangle XYZ$, angle XZY is a right angle. There are three squares constructed such that each side adjacent to angle XZY has a square on it. The sum of the areas of these three squares is 512 square centimeters. Also, XZ is 20% longer than ZY. What's the area of the largest square?


Give an example of a quadratic function that has zeroes at $x=1$ and $x=5$, and that takes the value $8$ when $x=2$.

Solution: \boxed{-\frac{8}{3}x^2 + 16x - \frac{40}{3}}


Sasha has $\$3.80$ in U.S. coins. She has the same number of quarters as nickels, and twice as many dimes as quarters. What is the greatest number of quarters she could have?


Let $a$, $b$, and $c$ be the roots of the cubic equation $$
5x^3 - 2019x + 4029 = 0.
$$ Compute $(a + b)^3 + (b + c)^3 + (c + a)^3.$


Find the smallest possible value of the sum $\lvert x + 3\rvert + \lvert x + 4\rvert + \lvert x + 6\rvert$.


Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q| = 3$, $|q-r| = 4$, and $|r-s| = 5$. What is the sum of all possible values of $|p-s|$?


If a school bus leaves school with 60 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?


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


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

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


In a group of 10 people who attend a theater show, 4 of them are only 1/3 confident that they will stay for the entire performance, while the rest are certain they will stay. What is the probability that, by the end of the show, at least 9 people stayed for the entire show?


Let \(g : \mathbb{R} \to \mathbb{R}\) be a function such that
\[g(x) g(y) - g(xy) = x^2 + y^2\] for all real numbers \(x\) and \(y\).

Let \(m\) be the number of possible values of \(g(3)\), and let \(t\) be the sum of all possible values of \(g(3)\). Find \(m \times t\).


An argon gas is contained in a 4.56 liter container at a pressure of 10 kPa. If we transfer all of the argon to a 2.28 liter container while keeping temperature constant, what will the new pressure of argon be in kPa?

Solution: \boxed{20 \text{ kPa}}


Thirteen girls are standing around a circle. A ball is passed clockwise around the circle. The first girl, Bella, starts with the ball, skips the next four girls, and throws to the sixth girl. This girl then skips the next four girls and throws the ball accordingly. If this throwing pattern continues, including Bella's initial throw, how many total throws are necessary for the ball to return to Bella?


In spherical coordinates, determine the equivalent standard representation of the point $ \left( 5, \frac{19\pi}{6}, \frac{29\pi}{12} \right)$. 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(5, \frac{7\pi}{6}, \frac{5\pi}{12}\right)}


How many three-character license plates can be formed if each consists of two consonants followed by a vowel? (Assume Y is considered a vowel.)


In Brenda's garden, there are three trees: a pine tree that is $14\frac{1}{4}$ feet tall, a birch tree that is $18\frac{1}{2}$ feet tall, and a cedar tree that is $20\frac{5}{8}$ feet tall. How much taller is the cedar tree compared to the pine tree and the birch tree, expressed as simplified mixed numbers?

Solution: \boxed{2\frac{1}{8}\text{ feet}}


There are 3 math teams in a district, with 6, 9, and 10 students respectively. Each team has three co-captains. If a team is randomly selected, and then three members of that team are randomly selected to receive a copy of $\emph{Introduction to Geometry}$, what is the probability that all three people who receive books are co-captains?


Let $c$ be a complex number. Suppose there exist distinct complex numbers $p$, $q$, and $u$ such that for every complex number $z$, we have
\[
(z - p)(z - q)(z^2 - u) = (z - cp)(z - cq)(z^2 - cu).
\]
Compute the number of distinct possible values of $c$.


What is the domain of the function $f(x)=\log_5(\log_7(\log_8x))$?

Solution: \boxed{(8, \infty)}


Given a polynomial with integer coefficients,
\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\]
find the number of different possible rational roots of this polynomial.


Bobby takes 30 free throws and makes 60% of them. After an additional 10 throws, his overall shooting percentage increases to 64%. How many of the last 10 throws did he make?


Let $P$ be a point on the parabola $y = x^2 - 4x + 7,$ and let $Q$ be a point on the line $y = 2x - 5.$ Find the shortest possible distance $PQ.$


Consider the curve $y = \cos x$ and the line $y = \cos 30^\circ$. Determine the ordered pair of relatively prime positive integers $(p, q)$ that represents the ratio of lengths of segments when the curve cuts the line, maintaining successive ratios: $\dots p : q : p : q \dots$ with $p < q$.

Solution: \boxed{(1, 5)}


What is the area, in square units, of a regular octagon inscribed in a circle whose area is $400\pi$ square units? Express your answer in simplest radical form.

Solution: \boxed{800 \sqrt{2}}


When \( n \) is divided by 8, a remainder of 3 is given. What is the remainder when \( n+2040 \) is divided by 8?


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


You have $6$ blue shirts, $6$ yellow shirts, $7$ pairs of pants, $9$ blue hats, and $9$ yellow hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?


A specific weighted die, when rolled, shows a 6 with probability $\frac{1}{4}$, any number from 1 to 5 (each with equal probability) with the remaining probability. If it shows 6, you lose 2 dollars. If it shows an odd number (1, 3, 5), you win 2 dollars. If it shows an even number (2, 4), you win 4 dollars. What is the expected winnings from rolling this die? Express your answer as a dollar value.


Suppose \( x^2 \) varies inversely with \( y^5 \). If \( x = 5 \) when \( y = 2 \), find the value of \( x^2 \) when \( y = 4 \).

Solution: \boxed{x^2 = \frac{25}{32}}


A sphere intersects the $xy$-plane in a circle centered at $(3, -2, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, -2, -10),$ with radius $r.$  Find $r.$


The sum of two positive integers $x$ and $y$ is 780. What is the largest possible value of $\gcd(x, y)$?


Find the matrix $\mathbf{M}$ that adds 3 to every element in the second column of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b+3 \\ c & d+3 \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


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

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-6.25,ymax=7.25;

pen cqcqcq=rgb(0.75,0.75,0.75);

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

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

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f1(real x){return (x-2)*(x)*(x+1)/6+x+2;}
draw(graph(f1,-3.25,3.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=f(x)$",(3.5,0.6),E);
[/asy]

If \( f(a)=c \) and \( f(c)=3 \), then what is the value of \( a-c \)?


In my closet, there are 5 shirts, 7 pairs of shorts, and 8 pairs of socks. If I randomly pick four articles of clothing, what is the probability that I get two shirts, one pair of shorts, and one pair of socks?


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[3]{\sqrt{\frac{1}{15625}}}}$$


The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$.


Find the area of the triangle with vertices \((0,0)\), \((4,3)\), and \((-2,5)\).


Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 8.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\]


Two standard dice are rolled. What is the expected number of 6's obtained? Express your answer as a common fraction.


Jane visits a local coffee shop that sometimes offers a special cinnamon blend. She plans to visit the coffee shop for 7 days. Each day, the shop has a 3/4 chance of offering the cinnamon blend. What is the probability that the shop offers the cinnamon blend exactly 5 out of the 7 days she visits?


Compute the sum of the squares of the roots of the equation \[x^{10} + 5x^7 + 2x^2 + 20 = 0.\]


A line contains the points $(5,10)$, $(-3, k)$ and $(-11, 5)$. What is the value of $k$?

Solution: \boxed{k = 7.5}


**
How many subsets of the set $\{1, 2, 3, 4, 5, 6\}$ contain either the number 5 or 6, but not both?

**


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


A student receives an 80 points score on a teacher's 100-point test after 4 hours of preparation. Given that the score varies directly with the time spent preparing, determine the score the student would receive if they spent 6 hours preparing for another test.

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


In how many ways can 10 people sit around a round table if two people, Alice and Bob, must sit next to each other? (Two seatings are considered the same if one is a rotation of the other.)


How many different 8-digit positive integers exist? (Note that we don't allow "8-digit" integers that start with 0, such as 01234567; this is actually a 7-digit integer.)


The graphs of $y=5-2x^2+x^4$ and $y=3+2x^2+x^4$ intersect at several points. Find the maximum difference between the $y$-coordinates of these intersection points.


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z) + 1) = xf(x) + zf(y) + 1\]
for all real numbers $x,$ $y,$ and $z.$ Determine all possible functions and find $f(5)$.

Solution: \boxed{f(5) = 5}


Determine the smallest positive integer that satisfies all these conditions:
a) When divided by 3, the remainder is 2.
b) When divided by 5, the remainder is 4.
c) When divided by 7, the remainder is 6.
d) When divided by 11, the remainder is 10.


Let $T = 1 - 2 + 3 - 4 + \cdots + 2047 - 2048$. What is the residue of $T$, modulo 1024?


What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity?


If 18 bahs are equal to 27 rahs, and 12 rahs are equal in value to 18 yahs, how many bahs are equal in value to 1200 yahs?

Solution: \boxed{\frac{1600}{3} \text{ bahs}}


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

Solution: \boxed{\sin x = \frac{ab}{\sqrt{a^4 - a^2b^2 + b^4}}}


In a certain sequence, the first term is $a_1=1001$ and the second term is $a_2=1002$. The values of the subsequent terms are chosen so that $a_n+a_{n+1}+a_{n+2}=n$ for all $n \geq 1$. Determine $a_{1000}$.


In the diagram, $\triangle ABC$ is right-angled at $A$, with $AB=90$ and $AC=120$. The point $D$ is on $BC$ such that $AD$ is perpendicular to $BC.$ Determine the length of $DB.$


A square of side length 10 units contains two smaller squares, each with sides of 4 units. The first small square overlaps completely with the second small square. What is the area of the region within the large square not covered by the small squares?


Consider a monic quartic polynomial $h(x)$ such that $h(-2) = -4$, $h(1) = -1$, $h(3) = -9$, and $h(5) = -25$. Find the value of $h(0)$.


In the diagram, $AOB$ is a sector of a circle with $\angle AOB=90^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ The radius $OY$ of the circle is given as 10 units. What is the length of $XY$?

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


A 12-foot by 15-foot room is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles with a radius of 1 foot centered at each corner of the tile. Calculate how much of the room's floor is shaded.

Solution: \boxed{180 - 45\pi}


Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000.


What is the sum of all the integers from 100 to 1999?


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


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.


What is the integer whose square is less than triple its value?

Solution: \boxed{1 \text{ and } 2}


Let $y$ be a real number. Consider the following five statements:
1. $1 < y^2 < 4$
2. $y^2 > 4$
3. $-2 < y < 0$
4. $0 < y < 2$
5. $0 < y^3 - y^2 < 4$

What is the maximum number of these statements that can be true for any value of $y$?


What is the largest three-digit integer whose digits are distinct and form a geometric sequence, given that the hundreds digit is at most 8?


In how many ways can 8 distinct beads be placed on a bracelet where the arrangement is indistinguishable under rotation and reflection?


Solve for $t$ if $\sqrt{3\sqrt{t-3}} = \sqrt[4]{10 - t}$.


In triangle $\triangle ABC$, $\tan B = \frac{4}{3}$. Given that $AB = 3$ units, find the length of side $BC$.

[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]


A pet store has 20 puppies, 10 kittens, 12 hamsters, and 5 rabbits. Alice, Bob, Charlie, and David each want to buy a different kind of pet. How many ways can the four customers buy one pet each such that they all have different types and leave the store satisfied?


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$ Consider also that $\mathbf{u}$ and $\mathbf{v}$ define a parallelogram with sides characterized by these vectors. If $\frac{\|\mathbf{p}\|}{\|\mathbf{u}\|} = \frac{3}{4},$ find the value of $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$


The Euler family has four girls aged $6$, $6$, $9$, and $11$, and two boys aged $13$ and $16$. What is the mean (average) age of the children?


A geometric series $a + ar + ar^2 + \cdots$ has a sum of $20$. The terms involving odd powers of $r$ sum up to $8$. Determine the value of $r$.


What is the remainder when $4x^8 - x^7 + 5x^6 - 7x^4 + 3x^3 - 9$ is divided by $3x - 6$?


Let $\mathbf{C}$ and $\mathbf{D}$ be matrices such that
\[\mathbf{C} + \mathbf{D} = \mathbf{C} \mathbf{D}.\] If $\mathbf{C} \mathbf{D} = \begin{pmatrix} 10 & 2 \\ -4 & 4 \end{pmatrix},$ find $\mathbf{D} \mathbf{C}.$

Solution: \boxed{\begin{pmatrix} 10 & 2 \\ -4 & 4 \end{pmatrix}}


If $A\,938\,B67$ is divisible by $9$, where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B$?


Vanessa and Liam each think of a polynomial. Each of their polynomials is monic, has degree 3, and has the same positive constant term and the same coefficient of $x$. The product of their polynomials is \[x^6 + 4x^5 + 6x^4 + 6x^3 + 5x^2 + 8x + 9.\] What is the constant term of Liam's polynomial?


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

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

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

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

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return sgn(2x-1)*abs(2x-1)^(1/3)+2*sgn(2x-1)*abs(2x-1)^(1/5)+abs(2x-1)^(1/7);}
draw(graph(f2,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=q(x)$",(xmax+0.25,f2(xmax)),E);
[/asy]

If the value of $q(2)$ is an integer, what is it?


Given that $z$ is a complex number such that $z+\frac{1}{z}=2\cos 5^\circ$, find $z^{1500} + \frac{1}{z^{1500}}$.


A stock investment appreciated by 40% in one year. Assuming it starts the following year at this increased value, what percent decrease is required in the second year to return the stock’s value to its original price at the beginning of the first year?


A math teacher asks Chloe to do one homework assignment for each of the first three homework points she wants to earn; for each of the next three homework points, she needs to do two homework assignments; and so on, such that to earn the $n^{\text{th}}$ homework point, she has to do $\lceil n\div3 \rceil$ (rounded up) homework assignments. What is the smallest number of homework assignments necessary to earn a total of 30 homework points?


Find the quotient of the division $(5z^5 - 3z^4 + 4z^3 - 7z^2 + 6z - 1) / (z + 2)$.

Solution: \boxed{5z^4 - 13z^3 + 30z^2 - 67z + 140}


Find the minimum value of $\log_{a^3} \left(\frac{a}{b}\right) + \log_{b^3} \left(\frac{b}{c}\right)$ given $a \ge b \ge c > 1$.


At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives 6th prize and the winner bowls #4 in another game. The loser of this game receives 5th prize and the winner bowls #3. The loser of this game receives 4th prize and the winner bowls #2. The loser of this game receives 3rd 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?


Find the volume of the region in space defined by
\[ |x + y + 2z| + |x + y - 2z| \le 12 \]
and \(x, y, z \ge 0\).


A circular park includes a central circular pond, a ring-shaped seating area, a flower garden ring, and an outer ring path for walkers. The pond has a diameter of 20 feet. The seating area is 4 feet wide, the flower garden is 10 feet wide, and the walking path is 8 feet wide. What is the diameter of the circle that forms the outer boundary of the walking path?


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


The product $N$ of three positive integers is $8$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.


A father purchases 4 yellow cups, 3 blue cups, and 2 red cups. He wants to arrange these cups in a circular pattern on a tray. How many ways can he arrange these cups if he does not want the 2 red cups to be next to each other?


Five plastic bottles are required to create a new, recycled bottle. How many new bottles can eventually be made from 125 initial plastic bottles? Remember to consider recycled bottles for further recycling processes but do not include the original count of 125 bottles in your total.


Let \( x \) and \( y \) be positive real numbers, and \( z \) be a constant such that \( x^2 + y^2 = z \). Find the minimum value of
\[
\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 2020 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 2020 \right).
\]


Piravena needs to travel from city $X$ to city $Y$, from $Y$ to city $Z$, and then back to $X$. The cities form a right triangle with $Z$ a distance of 3600 km from $X$ and $Y$ a distance of 3900 km from $X$. The cost structure for travel includes:
- Bus: $\$0.20$ per kilometer.
- Airplane: $\$120$ booking fee plus $\$0.12$ per kilometer.

Calculate the least expensive total cost for Piravena to complete her journey.


Marius now has eight snow leopards of different heights to arrange in a row for a photo contest. The two shortest leopards still have inferiority complexes and must be placed at the ends of the row. Additionally, the tallest leopard refuses to stand next to either of the shortest leopards. How many ways can Marius arrange these leopards?

Solution: \boxed{960 \text{ ways.}}


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


Anna has 4 red pens and 4 blue pens. She places them in a line on her desk randomly and then randomly picks 4 of them to lend to her friend. What is the probability that the leftmost pen is blue and not picked, and the rightmost pen is red and picked?


For some positive integer $k$, when 120 is divided by $k^2$, the remainder is 8. What is the remainder when 150 is divided by $k$?


In a configuration where triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 30^\circ$, there is another smaller circle $O'$ inside triangle $PAB$ tangent to all three sides. Find the $\angle AOB$.


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


$F, G, H, I,$ and $J$ are collinear in that order such that $FG = 2, GH = 1, HI = 3,$ and $IJ = 7$. If $P$ can be any point in space, what is the smallest possible value of $FP^2 + GP^2 + HP^2 + IP^2 + JP^2$?


April has five different basil plants and three different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that for all real numbers \( x \) and \( y \):
\[ f(f(x) + y) = f(x) - f(f(y) + f(-x)) + x \]
Determine \( f(2) \).

Solution: \boxed{f(2) = -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}}


Let $\alpha$, $\beta$, and $\gamma$ be roots of the equation $x^3 - 7x^2 + 11x - 13 = 0$. Compute $\frac{\alpha \beta}{\gamma} + \frac{\beta \gamma}{\alpha} + \frac{\gamma \alpha}{\beta}$.


The positive integers $A, B$, and $C$ form an arithmetic sequence while the integers $B, C,$ and $D$ form a cubic sequence (i.e., $D - C = (C - B)^2$). If $\frac{C}{B} = \frac{4}{3}$, find the smallest possible value of $A + B + C + D$.


Find a positive integer that is divisible by 18 and has a square root between 28 and 28.2.


Point $C$ has coordinates $(3, 4)$. When Point $C$ is reflected over the $y$-axis, it lands on Point $D$. What is the sum of the four coordinate values of points $C$ and $D$?


In triangle ABC, angle BAC is a right angle. Three squares are constructed externally on each side of the triangle. The sum of the areas of the three squares is 450 square centimeters. Given that the area of the square on side AB is 100 square centimeters, find the number of square centimeters in the area of the largest square.


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


A triangle $\triangle DEF$ is inscribed in a circle of radius $r$ such that its side $DE$ is the diameter of the circle. Point $F$ lies on the circle but is not coincident with points $D$ or $E$. Let $t = DF + EF$. Determine the maximum value of $t^2$ in terms of $r$.


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


For a given positive integer $n > 3^2$, what is the greatest common divisor of $n^3 + 3^3$ and $n + 3$?

Solution: \boxed{n + 3}


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


A circular park is designed with several concentric features: a central circular pond, a surrounding ring-shaped garden, a grassy area, and an outer-ring path for walkers. The pond has a diameter of 12 feet. Surrounding the pond is a garden that is 10 feet wide. Beyond the garden is a grassy area that is 4 feet wide. Finally, there is a walking path that is 6 feet wide. What is the diameter of the circle that forms the outer boundary of the walking path?


Find the number of solutions to the equation
\[
\tan \left(\frac{3\pi}{2} \cos \theta \right) = \cot \left(\frac{3\pi}{2} \sin \theta \right)
\]
where \(\theta \in (0, 2\pi)\).


How many non-congruent triangles with perimeter 11 have integer side lengths?


A domino set is expanded such that each rectangular tile composed of two squares now includes integers between 0 and 13 on each square. Each integer from 0 to 13 is paired with every other integer from 0 to 13 exactly once, forming a complete set of dominoes. A $\textit{double}$ is a domino that has the same integer on both its squares. What is the probability that a randomly selected domino from this set will be a $\textit{double}$? Express your answer as a common fraction.


The median of the set $\{n, n + 2, n + 7, n + 10, n + 16\}$ is 10. What is the mean?


Suppose $185\cdot 944\equiv n\pmod{60}$, where $0\le n< 60$.

What is the value of $n$?


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


The product of two positive integers plus their sum is 255. One of the integers is odd, and each is less than 30. What is the sum of the two integers?


In the revised diagram, $EF$ and $GH$ are straight lines. In the new setup:
- $\angle VEK = 70^\circ$
- $\angle KEW = 40^\circ$
- $\angle EVG = 110^\circ$

What is the value of $\theta?$ [asy]
draw((0,0)--(12,0)); // GH
draw((0,5)--(12,5)); // EF
draw((4,0)--(5,5)--(8,0)); // Lines intersecting from GH to EF at point K
label("$70^\circ$",(5,4.5),W);
label("$40^\circ$",(5.5,4.5),E);
label("$E$",(0,5),W);
label("$G$",(0,0),W);
label("$F$",(12,5),E);
label("$H$",(12,0),E);
label("$110^\circ$",(4,0),NW);
label("$\theta^\circ$",(7.5,0),N); // Variable theta angle
[/asy]


I have seven apples and twelve oranges. If a fruit basket must contain at least one piece of fruit, how many different fruit baskets can I make? (The apples are identical, and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and the arrangement of the fruit in the basket does not matter.)


If a square starting from 3'' by 3'' is added at each successive stage, and the side of each new square increases by 1 inch from the previous square, what will be the area of the shape at Stage 5?


Find all real numbers \(x\) such that \[2 \le \frac{x}{3x-7} < 5.\] (Give your answer in interval notation.)

Solution: \boxed{\left(\frac{5}{2}, \frac{14}{5}\right]}


For $\mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 8 \\ \frac{16}{7} \end{pmatrix}.\]Find $y$.


John draws a regular six-pointed star in the sand, and at each of the 6 outward-pointing points and 6 inward-pointing points, he places one of twelve different sea shells. How many ways can he place the shells, if rotations of the arrangement are considered equivalent but reflections are not?


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


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Jack and the next two cards are both Queens? Assuming the cards are dealt without replacement.


Simplify $\sqrt{72a^5b^8c^{13}}$, and find the sum of the exponents of the variables that are outside the square root.


If \( 81^4 = 27^x \), what is the value of \( 3^{-x} \)? Express your answer as a common fraction.


What is \(1\tfrac{1}{2}\) divided by \(\tfrac{5}{6}\)?


Parallelogram $EFGH$ with $E(3,6)$, $F(5,10)$, $G(7,6)$, and $H(5,2)$ is reflected across the $x$-axis to $E'F'G'H'$ and then $E'F'G'H'$ is reflected across the line $y = x + 2$ to $E''F''G''H''$. Determine the ordered pair of $H''$ in the coordinate plane.

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


A swimming pool can be filled by any of three hoses X, Y, or Z. Hoses X and Y together take 5 hours to fill the pool. Hoses X and Z together take 6 hours to fill the pool. Hoses Y and Z together take 7 hours to fill the pool. How many hours does it take hoses X, Y, and Z working together to fill the pool? Express your answer as a decimal to the nearest hundredth.


How many distinct four-digit numbers are divisible by 3 and have 47 as their last two digits?


What value of $m$ must be chosen so that the point $(3, -2)$ lies on the line represented by the equation $2m - my = 3x + 1$?


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

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


In the function machine shown, the input is 7. What is the output?

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{In}\mbox{put}$ = 7",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 3",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 20",(0,0));
draw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,"Add 10",(width1/2,height/2));
draw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,"Subtract 8",(width1/2,height/2));
draw(f,(0,0)--(widthE,0)--(widthE,0)+lengthE*dir(angleE)--lengthE*dir(angleE)--cycle);
label(f,"$\mbox{Out}\mbox{put}$ = ?",lengthE/2*dir(angleE) + (widthE/2,0));
add(shift(width1/2*left)*a); draw((0,0)--(0,-2),EndArrow(4));
add(shift(5*down + width2/2*left)*b);
add(shift((7+heightC/2)*down)*c); draw((0,-5)--(0,-7),EndArrow(4));
pair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);
draw("$\le 20$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 20?$",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));
rightpt = rightpt + 2.5E+2S;
leftpt = leftpt + 2.5W+2S;
add(shift(leftpt+height*down+.3*width3*left)*d);
add(shift(rightpt+height*down+.7*width4*left)*e);
rightpt = rightpt+.75height*down+.7*width4*left;
leftpt = leftpt+.75height*down+.7*width3*right;
draw(leftpt--rightpt);
pair midpt = (leftpt+rightpt)/2;
draw(midpt--(midpt+2down),EndArrow(4));
add(shift(midpt+.65widthE*left+(2+lengthE*Sin(angleE))*down)*f);[/asy]


How many perfect squares are factors of 450?


Calculate the product of these fractions: $\frac{4}{2} \times \frac{8}{4} \times \frac{9}{3} \times \frac{18}{6} \times \frac{16}{8} \times \frac{24}{12} \times \frac{30}{15} \times \frac{36}{18}$.


What is the greatest common factor of 36 and 54?


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 3 \\ -4 \end{pmatrix}}


For how many values of \( b \) is it true that:
(1) \( b \) is a positive integer such that \( b \leq 100 \).
(2) the quadratic equation \( x^2 + (b+3)x + \left(\frac{b+3}{2}\right)^2 = 0 \) has two integer solutions?


There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+4x+4$ and $y=mx+b$ at two points which are exactly $4$ units apart. If the line $y=mx+b$ passes through the point $(2,8)$, and $b\neq 0$, determine the equation of the line. Enter your answer in the form "$y = mx + b$".

Solution: \boxed{y = 12x - 16}


Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 8,$ and $y + \frac {1}{x} = 29.$ Find the value of $z + \frac {1}{y}.$


Solve
\[\arccos (\cos x) = 2x.\] Enter all the solutions, separated by commas.


Calculate \(\sqrt{48y} \cdot \sqrt{18y} \cdot \sqrt{50y}\). Express your answer in simplest radical form in terms of \(y\).

Solution: \boxed{120y \sqrt{3y}}


Amerigo Vespucci has updated his map of America on the complex plane, ensuring no distortions in distance. In this new map, San Francisco corresponds to the origin $0$, Miami is located at $3120i$, and a new city, Atlantis, is situated at $1300 + 3120i$. Compute the straight line distance from San Francisco to Atlantis on this complex plane.


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is $24$, determine the area of $\triangle ABC$.


Find the ordered pair $(a, c)$ such that the matrix
\[\begin{pmatrix} a & \frac{20}{49} \\ c & \frac{29}{49} \end{pmatrix}\]
is a projection matrix.

Solution: \boxed{\left(\frac{20}{49}, \frac{29}{49}\right)}


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

Solution: \boxed{0, 1, 2, 3, 4, 5}


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


Lisa is looking to purchase a custom-made wetsuit and needs to specify her chest size in centimeters on the order form. There are $12$ inches in a foot and $31$ centimeters in a foot. If her chest size is $36$ inches, adjusting for a comfortable fit requires adding an extra $2$ inches. What chest size should she specify on the form, in centimeters? (You may use a calculator for this problem; answer to the nearest tenth.)


In regular hexagon $ABCDEF$, diagonal $AD$ is drawn. Given that each interior angle of a regular hexagon measures 120 degrees, what is the measure of angle $DAB$?


What is the greatest number of points of intersection that can occur when $3$ different circles and $1$ straight line are drawn on the same piece of paper?


The function $g$ is graphed below. Each small box has width and height 1.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-1,9,-1,9);

dot((0,0),red+5bp);
dot((2,1),red+5bp);
dot((4,2),red+5bp);
dot((6,3),red+5bp);
dot((8,4),red+5bp);
dot((1,9),red+5bp);
dot((3,7),red+5bp);
dot((5,5),red+5bp);
dot((7,3),red+5bp);
dot((9,1),red+5bp);

[/asy]

Susan writes the number 4 on her thumb. She then applies $g$ to 4 and writes the output on her next finger. If Susan continues this process of applying $g$ and writing the output on a new finger, what number will Susan write on her tenth finger?


Five of the following test scores are Cyprian's and the other four are Margaret's: 84, 86, 90, 92, 93, 95, 97, 96, 99. Cyprian's mean score is 92. What is Margaret's mean score?


Triangle $XYZ$ has a perimeter of 2010 units. The sides have lengths that are all integer values with $XY < YZ \leq XZ$. What is the smallest possible value of $YZ - XY$?


Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pr &=& qs &=& 1200.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \).


Evaluate the expression \(8^{15} \div 64^5\).


What is the minimum value of $3x^2 - 18x + 2023$?


Calculate the sum of the first 5 terms of the geometric series $2 + 2\left(\frac{1}{4}\right) + 2\left(\frac{1}{4}\right)^2 + 2\left(\frac{1}{4}\right)^3 + 2\left(\frac{1}{4}\right)^4$.


In triangle $XYZ$, medians $\overline{XU}$ and $\overline{YV}$ are perpendicular. If $XU = 10$ and $YV = 24$, and the altitude from vertex $Z$ to $\overline{XY}$ is $16$, calculate the area of triangle $XYZ$.