Let $a, b, c, d, e, f$ be positive real numbers such that $a + b + c + d + e + f = 10.$ Find the minimum value of
\[
\frac{1}{a} + \frac{9}{b} + \frac{16}{c} + \frac{25}{d} + \frac{36}{e} + \frac{49}{f}.
\]


Given that $5^{-1} \equiv 39 \pmod{79}$, find $25^{-1} \pmod{79}$, as a residue modulo 79. (Give an answer between 0 and 78, inclusive.)


For \( -2 < s < 2 \), let \( T(s) \) denote the sum of the geometric series \[15 + 15s + 15s^2 + 15s^3 + \cdots .\] A number \( b \) between \( -2 \) and \( 2 \) satisfies \( T(b)T(-b) = 3600 \). Find \( T(b) + T(-b) \).


How many different positive six-digit integers can be formed using the digits 1, 1, 1, 5, 9, and 9?


Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 30$ and \[(x-y)^2 + (x-z)^2 + (y-z)^2 + xyz = 2xyz.\] Find $\frac{x^3 + y^3 + z^3}{xyz}.$


The graph of the line $3x - y = k$ is the perpendicular bisector of the line segment from $(2, 4)$ to $(8, 10)$. Determine the value of $k$.


Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.

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


Suppose there are three complex numbers $p+qi$, $r+si$, and $t+ui$. Given that $q=5$, $s=2q$, and $t=-p-r$, determine $u+i$ if the sum of the numbers equals $7i$.

Solution: \boxed{-8 + i}


Each successive term in the sequence $4096, 1024, 256, 64, x, y, 4, 1, \frac{1}{4},...$ is obtained by multiplying the previous term by a constant until $y$, and then the pattern changes after $y$. What is the value of $x + y$ if the multiplication constant changes after $y$?


Calculate $\sqrt{15q} \cdot \sqrt{8q^3} \cdot \sqrt{12q^5}$ and express your answer in simplest radical form in terms of $q$.

Solution: \boxed{12q^4 \sqrt{10q}}


Calculate $$8 + 77 + 666 + 5555 + 44444 + 333333 + 2222222 + 11111111$$ and find the remainder when this sum is divided by 9.


Lisa has 8 blue marbles and 12 red marbles. She selects a marble at random, notes its color, and then returns the marble to the bag. She repeats this process 8 times. What is the probability that she picks exactly four blue marbles?


For an upcoming French exam, I need to cover 750 words. The exam grading system is different: I get full points for each word I recall correctly, and for every incorrectly guessed word, I receive half the points if they are semantically or contextually related, assuming 30% of my wrong guesses are relevant. What is the minimum number of words I must learn to guarantee at least an 80% on the exam?


In a school, 40 students are enrolled in both the literature and science classes. Ten students received an A in literature and 18 received an A in science, including six who received an A in both subjects. Determine how many students did not receive an A in either subject.


If $g(x) = \frac{2ax + b}{2cx - d}$ where $abcd \neq 0$ and $g(g(x)) = x$ for all $x$ in the domain of $g$, what is the value of $2a - d$?


When a polynomial is divided by $x^3 - 5x^2 + 3x - 9,$ what are the possible degrees of the remainder?  Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least two books are in the library and at least two are checked out?


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


A paperboy delivers newspapers to 12 houses along a street. To keep his job, he ensures that he never misses four consecutive houses. Compute the number of ways the paperboy could deliver papers under this new condition.


Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$


Three faces of a right rectangular prism have areas of 72, 75, and 80 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number.


Determine the value of $a$ such that $9x^2 + 27x + a$ is the square of a binomial.


Determine how many times the functions $y = \sin x$ and $y = \left(\frac{1}{3}\right)^x$ intersect on the interval $(0,150\pi)$.


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


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{500}$?


What is the value of $n$ such that $10^n = 10^{-4} \times \sqrt{\frac{10^{60}}{1000}}$?


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{15}\right)\left(7^{18}\right)$?


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


One of the roots of $z^2 = -20 + 15i$ is $4 + 3i.$ What is the other root?

Solution: \boxed{-4 - 3i}


Determine the minimum value of \[\sqrt{x^2 + (b - x)^2} + \sqrt{(x - a)^2 + (x + b)^2}\] for $a = 2$, $b = 1$ over all real numbers $x$.


Define $h(x) = 6x - 5$. If $h(x) = f^{-1}(x) - 2$ and $f^{-1}(x)$ is the inverse of the function $f(x) = cx + d$, find $4c + 6d$.


When $0.\overline{48}$ is expressed as a common fraction in lowest terms, what is the sum of the numerator and denominator?


How many three-digit numbers are there in which the hundreds digit is greater than the sum of the tens and ones digits?


Suppose that the roots of $x^3 + 2x^2 + 5x - 8 = 0$ are $p$, $q$, and $r$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find $w$.


A rectangular prism has vertices at the corners and edges joining them similarly to a cube. The prism dimensions differ along each axis; therefore, no two adjoining sides are of the same length. If one side has a length ratio of 2:3 with another, and there are three dimensions under consideration, compute how many total diagonals (both face diagonals that lie within the surfaces and space diagonals that span the entire prism) exist.


In an equilateral triangle $ABC$ with side length $6$, point $D$ is the midpoint of $BC$. Calculate $\tan{\angle BAD}$.


Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 5,$ $\|\mathbf{w}\| = 8,$ and the angle $\theta$ between $\mathbf{v}$ and $\mathbf{w}$ is $60^\circ,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$.


Flatville has decided to modify their bicycle license plates system, which originally contained three letters with the first letter chosen from the set $\{C, H, L, P, R, S\}$, the second from $\{A, E, I, O\}$, and the third from $\{D, M, N, T, V\}$. To accommodate more bicycles, they plan to add three new letters instead of two. These new letters can be added to one set or distributed among the sets. Determine the largest possible number of ADDITIONAL license plates that can be made by adding three letters.


What is the value of the smallest positive base ten number which requires seven digits for its binary representation?


I have 7 marbles numbered consecutively from 3 to 9 in a bag. Suppose I take out three different marbles at random. What is the expected value of the sum of the numbers on these marbles?


If $x\%$ of five-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth.


Let \(y\) be an angle such that \(\tan y = \frac{2c}{3d}\) and \(\tan 2y = \frac{3d}{2c + 3d}\). Find the least positive value of \(y\) in the form \(y = \tan^{-1} m\). Compute \(m\).


Calculate the area of the parallelogram with vertices at (0, 0), (6, 3), (0, 8), and (6, 11).


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


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

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{In}\mbox{put}$ = 12",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 3",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 35",(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 7",(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("$\le 35$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 35?$",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]


Imagine you have 12 distinguishable socks in your drawer: 5 white, 4 brown, and 3 blue. In how many ways can you choose a pair of socks, provided that you get two socks of different colors and at least one of them must be blue?


What is the product of the numerator and the denominator when $0.\overline{072}$ is expressed as a fraction in lowest terms?


Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2020$, and $a^2 - b^2 + c^2 - d^2 = 2020$. Find the number of possible values of $a$.


Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]
for all positive integers $x$ and $y$. Given $g(15) = 18$ and $g(45) = 24$, find $g(675)$.


Find the seventh term of the geometric sequence with the first term $5$ and the second term $\frac{1}{5}$.


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


Suppose I have 5 shirts, 4 pairs of pants, and 5 ties. An outfit requires a shirt and pants, and can either have a tie or not have a tie. Additionally, there are 2 different pairs of shoes to choose from. How many different outfits can I create?


Solution: \boxed{40a + 20b}


The line $y = \frac{5x - 7}{6}$ is parameterized in the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{v} + t \mathbf{d},\]
so that for $x \ge 2$, the distance between $\begin{pmatrix} x \\ y \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$ is $t$. Find $\mathbf{d}$.

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


In a different setting, the 2007 AMC 10 scores by awarding $7$ points for each correct response, deducting $1$ point for each incorrect response, and awarding $2$ points for each problem left unanswered. After considering the $25$ problems, John decides to attempt the first $20$ and leave the last $5$ unanswered.

How many of the first $20$ problems must John solve correctly in order to score at least $120$ points?


Given that $$(x+y+z)(xy+xz+yz)=36$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=10$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + ab}{c^2} \geq N\]
for any $a, b, c$ that are sides of a triangle.


Suppose $q(x)$ is a monic cubic polynomial with real coefficients such that $q(2+3i) = 0$ and $q(0) = -90$.

Solution: \boxed{x^3 - \frac{142}{13}x^2 + \frac{529}{13}x - 90}


Find the integer \( m \) such that \( -180 < m < 180 \) and \( \tan m^\circ = \tan 1500^\circ \).


Calculate the sum of all integers between -10.5 and 5.7.


Evaluate the polynomial
\[ x^4 - 3x^3 - 9x^2 + 27x - 8, \]
where \( x \) is the positive number such that \( x^2 - 3x - 9 = 0 \).

Solution: \boxed{\frac{65 + 81\sqrt{5}}{2}}


$\overline{AB}\parallel \overline{CD}$ and $\angle AXE$ is $120^\circ$ less than 4 times $\angle CYX$. Find $\angle BXY$.


What is the value of $a$ for which $\frac{1}{\text{log}_5a} + \frac{1}{\text{log}_6a} + \frac{1}{\text{log}_7a} + \frac{1}{\text{log}_8a} = 2$?


Let
\[g(x,y) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{3xy - x + 3}{3x} & \text{if $x + y \le 5$}, \\
\dfrac{xy - y - 3}{-3y} & \text{if $x + y > 5$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]
Find $g(3,2) + g(3,4).$


Let $p,$ $q,$ $r$ be three distinct positive real numbers such that $p,$ $q,$ $r$ form a geometric sequence, and
\[\log_r p, \ \log_q r, \ \log_p q\] form an arithmetic sequence. Find the common difference of this arithmetic sequence.


A standard ten-sided fair die (with faces numbered 1 through 10) is rolled six times. The probability that the product of all six numbers rolled is a prime number is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Find all solutions to the inequality \[\sqrt[3]{x} + \frac{3}{\sqrt[3]{x} + 2} \leq 0\], giving your answer in interval notation.

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


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


Determine the distance between the planes $3x - 4y + 2z = 12$ and $6x - 8y + 4z = 6.$


Given the ten-sided regular polygon $A_1A_2A_3A_4A_5A_6A_7A_8A_9A_{10}$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1, A_2, \ldots, A_{10}\}$?


The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 27,$ $a_9 = 135,$ and for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$


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?


The planning committee has expanded to 12 members, of which 5 are teachers. A five-person subcommittee that must include at least two teachers needs to be formed from this committee. How many distinct subcommittees are possible?


What is the smallest five-digit whole number that is both a perfect square and a perfect cube?


At a bookstore, I purchased 5 different books. I have 4 identical gift bags to package these books as presents. How many ways are there for the assistant to arrange the books into 4 identical bags, assuming that she might leave some of these bags empty?


Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{\sqrt{3}}{k}\rceil - \lfloor\log_{\sqrt{3}}{k} \rfloor).$$


Suppose that all six of the numbers \[3 - \sqrt{8}, \;5+\sqrt{13}, \;16 - 3\sqrt{2}, \; -2\sqrt{3}, \; 7 - 2\sqrt{5}, \; 9 + \sqrt{7}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


What is the sum of the digits of the base $7$ representation of $999_{10}$?


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


When two fair 8-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both $a$ and $b$ are divisible by 4, and their sum $a+b$ is also divisible by 4?


Let $(x, y)$ be a solution to the system of equations
\[
\begin{aligned}
\lfloor x \rfloor - \{y\} &= 1.3, \\
\{x\} + \lfloor y \rfloor &= 3.7.
\end{aligned}
\]
Compute $|x - y|$.


A right square pyramid with base edges of length $12\sqrt{2}$ units each and slant edges of length 15 units each is cut by a plane that is parallel to its base and 4.5 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane?

Solution: \boxed{108 \text{ cubic units}}


Express $\cos (x - y) - \cos (x + y)$ as the product of trigonometric functions.

Solution: \boxed{2 \sin x \sin y}


Consider the function $f(x) = x^3 + x^2 + 2\sqrt{x}$. Evaluate $2f(3) - f(9)$.

Solution: \boxed{-744 + 4\sqrt{3}}


Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What are the values of \(d + e + f\)?


Find the largest prime divisor of the number $2102012_3$.


Find the phase shift of the graph of $y = \cos(2x + \frac{\pi}{2})$.


Determine the value of $a$ such that the following lines are perpendicular:
\[
y = -3x + 5
\]
\[
4y - ax = 16
\]


Find the angle, in degrees, between the vectors $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$.


A maintenance team needs to reach a point $O$ at the center of a circular pond. To do this, they place a 12-foot plank from $X$ to $Y$ and then an 8-foot plank from $Z$ to $O$, where $Z$ is the midpoint of $\overline{XY}$. Determine the area of the circular pond. Express your answer in terms of $\pi$.


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


If ten apples cost the same as five bananas, and three bananas cost the same as four cucumbers, how many cucumbers can Tyler buy for the price of 20 apples? Additionally, four cucumbers cost the same as six grapes. How many grapes can Tyler buy for the price of 20 apples?


Let $b_n = \frac{8^n - 1}{7}$. Define $e_n$ to be the greatest common divisor (gcd) of $b_n$ and $b_{n+1}$. Determine the maximum possible value that $e_n$ can achieve.


A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Science, and Latin. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen?


Solution: \boxed{-15x^3 - 5x^2 + 29x - 5}


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


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


What is the least natural number that can be added to 71,382 to create a palindrome?


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

Solution: \boxed{4 + 2\sqrt{3}}


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both?


Let $a, b, c, d$ be a four-term arithmetic progression where all terms are positive, and such that $abcd = 256$. Find the smallest possible value of $c$.


Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 500\}\,$. Another three numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining numbers where the sum of $b_1 + b_2 + b_3\,$ must be greater than the sum of $a_1 + a_2 + a_3$. Determine the probability $p\,$ that a brick of dimensions $a_1 \times a_2 \times a_3\,$ will fit into a box of dimensions $b_1 \times b_2 \times b_3\,$ after including the constraint on sums. If $p$ is in reduced form, what is the sum of the numerator and denominator?


In polar coordinates, find an equivalent standard polar coordinate representation for the point $\left( -1, \frac{5 \pi}{6} \right)$.

Solution: \boxed{\left(1, \frac{11 \pi}{6}\right)}


Line $l_1$ has the equation $2x + 3y = 6$ and goes through point $A = (3, -2)$. Line $l_2$ has the equation $y = 2$ and intersects line $l_1$ at point $B$. Line $l_3$ has a positive slope, passes through point $A$, and intersects $l_2$ at point $C$. The area of $\triangle ABC$ is $10$. Determine the slope of $l_3$.


How many distinct sequences of five letters can be made from the letters in TRIANGLE if each sequence must begin with G and end with E, and no letter can appear in a sequence more than once?


A classroom has enough chairs to seat $312_5$ students. If $3$ students are supposed to sit on one bench, how many benches does the classroom have?


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

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


Find the quotient when \(x^6 + 2x^4 - 5x^3 + 9\) is divided by \(x - 2\).

Solution: \boxed{x^5 + 2x^4 + 6x^3 + 7x^2 + 14x + 28}


The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse?


What is the 15th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{8}$ and $\frac{1}{9}$?


Let $g(x)=20x-4$. Find the sum of all $x$ that satisfy the equation $g^{-1}(x)=g((2x+1)^{-1})$.


Solution: \boxed{(-5, 9]}


In how many ways is it possible to arrange the digits of 1155 to get a four-digit multiple of 5?


What is the first year after 2000 for which the sum of the digits is 15?


Compute
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples \((a,b,c)\) of positive integers such that \(1 \le a < b < c\).)


Keisha's basketball team is revising their uniform policy. The sixth-graders will choose the color of the socks (red, blue, or white), and the seventh-graders will choose the color of the headband (red, white, or green). Both items' colors will be chosen independently, and each color is equally likely to be chosen. What is the probability that the socks will be a different color than the headband? Express your answer as a common fraction.


In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 5$, and $XZ = \sqrt{34}$. What is $\tan X$?


The line $y = 2x - 8$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \end{pmatrix}.\]
Determine the ordered pair $(r,m)$.


How many positive divisors do 9240 and 10010 have in common?


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


A number in the set $\{100, 101, 102, 103, ... , 999\}$ is randomly selected. What is the probability that it is an even number? Express your answer as a common fraction.


A spiral staircase turns $315^\circ$ as it rises 12 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.


In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$?


A child has a set of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, pentagon). How many blocks in the set differ from the 'metal medium blue hexagon' in exactly 2 ways?


How many even integers between 5000 and 8000 have four different digits?


The graph of \( y = \frac{r(x)}{s(x)} \) is given, where \( r(x) \) and \( s(x) \) are quadratic polynomials. It's known that there is a hole at \( x = 4 \), a zero at \( x = 0 \), a horizontal asymptote at \( y = -2 \), and a vertical asymptote at \( x = 3 \). Calculate \( \frac{r(1)}{s(1)} \).


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


In a new diagram, triangle $A'B'C'$ has an area of 36 square units. The points $A', B', C', D'$ are aligned such that $A'C' = 12$ units and $C'D' = 30$ units. What is the area of triangle $B'C'D'$?


How many 3-digit numbers have the property that the units digit is at least three times the tens digit?


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


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


How many three-digit whole numbers have at least one 8 or at least one 2 as digits?


There is a 10x10 grid, where each cell in the grid is labeled with numbers according to the formula:
\[ a_{ij} = \frac{1}{i + j - 1} \]
where \(i\) is the row index and \(j\) is the column index, starting from 1. You need to select ten cells such that no two cells are in the same row or column. Calculate the minimum possible sum of the labels of the selected cells.


Find the value of the first term in the geometric sequence $x, y, z, 16, 32, 64$.


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 6x^2 + 11x - 6 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 15x^2 + 74x - 120}


Let $G$ be the centroid of triangle $DEF.$  If $GD^2 + GE^2 + GF^2 = 72,$ then find $DE^2 + DF^2 + EF^2.$


Determine the value(s) of $x$ such that $10xy - 15y + 5x - 7.5 = 0$ is true for all values of $y$.

Solution: \boxed{x = \frac{3}{2}}


When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is $3.$ When $p(x)$ is divided by $x + 5,$ the remainder is $-9.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$

Solution: \boxed{3x + 6}


How many integers from 1 to 9 are divisors of the five-digit number 38,412?


Calculate the greatest integer less than or equal to \[\frac{3^{110}+2^{110}}{3^{106}+2^{106}}.\]


Find the remainder when $1^3 + 2^3 + 3^3 + \dots + 150^3$ is divided by 7.


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

What is the remainder when $10!$ is divided by $13$?


How many positive four-digit integers are divisible by both 7 and 6?


Participation in the local basketball league is $15\%$ higher this year than it was last year. Also, from last year to this year, the number of males increased by $10\%$ and the number of females increased by $25\%$. There were $30$ males participating in the league last year. What fraction of the league's participants is female this year? Express your answer as a common fraction.


While walking in a park, Alex moved 100 feet closer to the fountain over a period of 40 minutes. At this rate, how many minutes will it take him to walk the remaining 90 yards to the fountain?


In $\triangle ABC, AB = 10, BC = 9, CA = 8$ and side $BC$ is extended to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.


Let $p$ and $q$ be real numbers so that the roots of
\[ z^2 - (14 + pi)z + (48 + qi) = 0 \]
are complex conjugates. Determine the ordered pair $(p, q)$.

Solution: \boxed{(0, 0)}


Determine the value of $y$ that satisfies $\sqrt[7]{y\sqrt[3]{y^5}} = 4$.


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


Find the smallest positive integer $a$ such that $x^4 + a^2 + 16$ is not prime for any integer $x$.


The value \( c^m \) has both \( c \) and \( m \) as positive integers less than or equal to 20. What is the greatest number of positive factors \( c^m \) can have?


Find the smallest positive integer that is both an integer power of 12 and is not a palindrome.


Shown below are rows 1, 2, and 3 of Pascal's triangle:

\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]

Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 3005th, 3006th, and 3007th rows, respectively, with the leftmost element occurring at $i = 0.$ Compute
\[\sum_{i = 0}^{3006} \frac{b_i}{c_i} - \sum_{i = 0}^{3005} \frac{a_i}{b_i}.\]


Let \( g \) be a function from the nonnegative integers to the nonnegative integers, such that
\[ 3g(a^2 + b^2) = g(a)g(b) + 2[g(a) + g(b)] \]
for all nonnegative integers \( a \) and \( b \).

Let \( m \) be the number of possible values of \( g(50) \), and let \( t \) be the sum of the possible values of \( g(50) \). Find \( m \times t \).


What is the units digit of the product of all the odd positive integers between 20 and 120?


What is the base five product of the numbers $203_{5}$ and $14_{5}$?


The equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\] for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$


How many different positive three-digit integers can be formed using only the digits in the set $\{3, 3, 4, 4, 4, 7, 8\}$ if no digit may be used more times than it appears in the given set of available digits?


A library has ten identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least two books must be in the library and at least two must be checked out?


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


A box contains 5 red marbles, 6 white marbles, and 7 blue marbles. Four marbles are drawn from the box (without replacement). What is the probability that they are all the same color?


In a right prism with rectangular bases, given the sum of the areas of three mutually adjacent faces (two lateral faces and one base) is 48, find the maximum volume of the prism. Assume the two lateral faces adjacent to the base are congruent.


Determine the tens digit of $13^{2023}$.


Two planes parallel to the hexagonal base of a right hexagonal pyramid cut through it, producing two cross-sections with areas of $96\sqrt{3}$ square feet and $216\sqrt{3}$ square feet. The distance between these two planes is $12$ feet. The lateral edges of the pyramid are perpendicular to its base. Determine how far the larger cross section is from the apex of the pyramid, in feet.


In the diagram below, triangle $ABC$ is isosceles with $AB = AC$ and point $D$ is the midpoint of $BC$. A point $E$ lies on segment $AD$ such that $DE = 3$ and $AE = 9$. If $BC = 16$, find the length of $AB$.


What is the greatest divisor of 540 that is smaller than 60 and also a factor of 126?


How many positive four-digit integers of the form $\_\_90$ are divisible by 90?


Let the reciprocals of the roots of the polynomial $2x^3 + 3x^2 + 5x + 7$ be $\alpha, \beta,$ and $\gamma$. Evaluate $\alpha + \beta + \gamma$.


How many different positive integers can be represented as a difference of two distinct members of the set $\{2, 3, 5, 7, 11, 13\}$?


Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^3 + b^3.$


In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\sin F$.


If $A, B,$ and $C$ are non-zero distinct digits in base $8$ such that $\overline{ABC}_8 + \overline{BCA}_8 + \overline{CAB}_8 = \overline{AAA0}_8$, find $B+C$ in base $8$.


Yan is currently between his home and the shopping mall. He can either walk directly to the shopping mall or walk back home and then ride his bicycle to the mall. His bicycle speed is 5 times his walking speed, and both choices require the same amount of time. Determine the ratio of Yan's distance from his home to his distance from the shopping mall.


If the six digits 2, 2, 3, 4, 7, and 5 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 15? Express your answer as a common fraction.


What is the digit in the tens place when \(9^{2010}\) is expressed in decimal notation?


Let \(p,\) \(q,\) and \(r\) be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \ge 0\]is true if and only if either \(x > 5\) or \(3 \le x \le 7.\) Given that \(p < q,\) find the value of \(p + 2q + 3r.\)


Let $a$ and $b$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. What is the value of $(3a-4)(6b-8)$?


Calculate the number of degrees in the acute angle formed by the hands of a clock at 3:37.


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


The graphs of three functions, labelled (1) through (3), are shown below. Note that the domain of function (1) is $$\{-3, -2, -1, 0, 1, 2, 3\}$$. Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));

real g(real x) {return x^3-2x;}
real h(real x) {return x-2;}
real j(real x) {return 2-x;}

real x;

draw(pic1,graph(g,-3,3),Arrows(4));
draw(pic2,graph(h,-8,8),Arrows(4));
draw(pic3,graph(j,-8,8),Arrows(4));

label(pic1,"(1)",(0,-9));
label(pic2,"(2)",(0,-9));
label(pic3,"(3)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(20,-20)*pic3);
[/asy]


An arithmetic sequence begins with a first term of $2$ and has a common difference of $5$. Another sequence begins with $3$ and has a common difference of $8$. What is the largest number common to both sequences within the range of $1$ to $100$?


A fenced area of a yard is a 20-foot by 18-foot rectangular region, but it has two cutouts: a 4-foot by 4-foot square and a right triangle with legs of 3 feet each. What is the area of the region within the fence?

Solution: \boxed{339.5 \text{ square feet}}


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


How many integer palindromes are there between 300 and 1000?


What is the smallest positive integer $a$ such that $a^{-1}$ is undefined modulo 99 and $a^{-1}$ is also undefined modulo 45?


Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC = 12$. Find the radius of the incircle of $\triangle ABC$.

Solution: \boxed{r = 6(2 - \sqrt{2})}


A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are 210, 345, 275, 255, \( x \), and \( y \). Find the greatest possible value of \( x+y \).


A curve is described parametrically by
\[(x,y) = (3 \cos t - 2 \sin t, 5 \sin t).\]
The graph of the curve can be expressed in the form
\[ax^2 + bxy + cy^2 = 1.\]
Find the ordered triple $(a,b,c)$.

Solution: \boxed{\left( \frac{1}{9}, \frac{4}{45}, \frac{13}{225} \right)}


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


In how many ways can the digits of $60,\!063$ be arranged to form a 5-digit number, ensuring that the number neither begins nor ends with 0?


What is the radius of the circle inscribed in triangle $ABC$ if $AB = 6, AC = 8, and BC = 10$? Express your answer in simplest radical form.


Express $7.\overline{316}$ as a common fraction in lowest terms.


How many positive divisors of 360 are not divisible by 2?


Find the minimum value of
\[(15 - x)(13 - x)(15 + x)(13 + x).\]


What is the remainder when \(11^{603}\) is divided by \(500\)?


Determine the number of angles between 0 and $2 \pi,$ other than integer multiples of $\frac{\pi}{2},$ such that $\sin \theta,$ $\cos \theta$, and $\cot \theta$ form a geometric sequence in some order.


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


The orange parabola shown is the graph of the equation \( x = ay^2 + by + c \). Find the value of \( c \). Assume that the graph has the standard unit scale and the vertex of the parabola is \((5, 3)\). The parabola also passes through the point \((0, 6)\).

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

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
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);//,above=true);

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

p=invisible);//,Arrows);

}

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));

}
};
real lowerx, upperx, lowery, uppery;
real f(real x) {return -(x-3)*(x-3)/2+5;}
lowery = -1;
uppery = 9;
rr_cartesian_axes(-10,10,lowery,uppery);
draw(reflect((0,0),(1,1))*(graph(f,lowery,uppery,operator ..)), orange);
[/asy]


Find the area in the plane contained by the graph of
\[|2x + 3y| + |2x - 3y| \le 12.\]


In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 200$, and $AD \neq BC$. The perimeter of $ABCD$ is $660$. Find $\lfloor 1000 \cos A \rfloor.$


What is the intersection point of the line \( y = 3x - 4 \) and the line perpendicular to it that passes through the point \( (3, 2) \)? Additionally, find where this perpendicular line intersects with another line parallel to \( y = 3x - 4 \) that passes through \( (0, -3) \).

Solution: \boxed{\left(\frac{9}{5}, \frac{12}{5}\right)}


Let $p(x) = x^5 - 2x^2 + 1$. Find the coefficient of the $x^4$ term in the polynomial $(p(x))^2$.


In triangle $DEF$, $\cos(2D-E)+\sin(D+E)=2$ and $DE=6$. What is $EF$?


A rectangle has vertices at \((2, z), (10, z), (2, 5)\), and \((10, 5)\). The area of the rectangle is 64 square units, and \( z \) is positive. Determine the value of \( z \).


Compute the sum of the squares of the roots of the equation \[x^{10} + 4x^5 + 5 = 0.\]


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?


How many perfect squares less than 5000 have a ones digit of 4?


What is the sum of the last two digits of \( 9^{25} + 11^{25} \)?


What is the base ten equivalent of $54321_{7}$?


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


Simplify the fraction $\frac{156 + 72}{9000}$.


Mila writes down the smallest positive multiple of 24 that is a perfect square, the smallest positive multiple of 24 that is a perfect fourth power, and all the multiples of 24 between them. How many integers are in Mila's list?


The first and seventeenth terms of an arithmetic sequence are 6 and 40, respectively. What is the fifty-fifth term?


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 10:10?


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


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


Two distinct positive integers $u$ and $v$ are factors of 48. If $u \cdot v$ is not a factor of 48, what is the smallest possible value of $u \cdot v$?


In how many ways can 8 distinct beads, including 2 red identical beads, be placed on a bracelet? (Two arrangements are considered the same if one can be rotated or reflected to produce the other.)


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


Suppose you flip six different coins simultaneously: a penny, a nickel, a dime, a quarter, a half-dollar, and a dollar coin. What is the probability that both the penny and the half-dollar come up the same?


In triangle $ABC$, where $a$ is the side length opposite $\angle A$, $b$ opposite $\angle B$, and $c$ opposite $\angle C$, the side lengths satisfy the relations $a + b + 2c = a^2$ and $a + b - 2c = -1$. Determine the largest angle of the triangle.


The square of $a$ and the square root of $b$ vary inversely. If $a = 4$ when $b = 16$, then find $b$ when $a + b = 20$.


Compute the following expression:
\[
\left\lfloor \frac{2010! + 2007!}{2009! + 2008!}\right\rfloor.
\]
(Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)


Tim decides to change his prank strategy. On Monday, he convinces Joe to steal Nathan's spoon. On Tuesday, he can choose between Ambie, John, or Elie. On Wednesday, he doesn't use any of the individuals from Monday or Tuesday, but has four new people to choose from. On Thursday, he can't use anyone from the previous days, but there are three new people willing. Finally, on Friday, Tim decides to do the prank himself. How many different combinations of people could Tim involve in this modified prank?


In the diagram, $AB$ is a horizontal line segment and $CD$ is a vertical line segment with $C$ on $AB$. $CE$ is a line segment forming an angle with $AB$. $AB$ and $CD$ intersect at point $C$. Given $\angle ECD = 63^\circ$, and $AB$ is perpendicular to $CD$, what is the value of $x$ if it is the angle formed by $CE$ with $AB$?

[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ$",(4.75,2.25));
label("$63^\circ$",(5.5,0.75));
label("$C$",(4,0),S);
label("$D$",(4,8),N);
label("$E$",(9,7),NE);
[/asy]


Let \(\mathbf{a}\) and \(\mathbf{b}\) be vectors, and let \(\mathbf{m}\) be the midpoint of \(\mathbf{a}\) and \(\mathbf{b}\). Given \(\mathbf{m} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{a} \cdot \mathbf{b} = 10\), find \(\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2\).


If $p$, $q$, and $r$ are positive integers such that $\gcd(p,q) = 210$ and $\gcd(p,r) = 1050$, what is the smallest possible value of $\gcd(q,r)$?


A cylindrical container's actual diameter is 30 cm, and its height is 10 cm. However, the measurement of the diameter can be off by up to $10\%$. Calculate the largest possible percent error, in percent, in the computed volume of the cylinder due to this measurement error.


Find $p(x)$ if the graph of the function $\frac{x^4 - 4x^3 + 4x^2 + 8x - 8}{p(x)}$ has vertical asymptotes at $x = 1$ and $x = 3$, no horizontal asymptote, and $p(-1) = 10$.

Solution: \boxed{\frac{5}{4}x^2 - 5x + \frac{15}{4}}


Express the value of the following as a common fraction:
$$
\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right) \dotsm \left(1-\frac{1}{n+1}\right) \dotsm \left(1-\frac{1}{200}\right) \cdot \left(1 + \frac{1}{100}\right)
$$


A square has a side length of $z$ units and is divided into a smaller square of side $w$ units and four congruent rectangles. Determine the perimeter, in units, of one of these four rectangles. Express your answer in terms of $z$.


How many ways are there to put 6 balls in 2 boxes if the balls are distinguishable, the boxes are distinguishable, and each box must contain at least 2 balls?


Compute the sum of squares of cosines for half-degree increments from $0^\circ$ up to $90^\circ$, specifically:
\[
\cos^2 0^\circ + \cos^2 0.5^\circ + \cos^2 1^\circ + \dots + \cos^2 89.5^\circ + \cos^2 90^\circ.
\]


Determine all $x$ which satisfy $$\frac{x}{4} \leq 3 + x < -3(1 + x).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.

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


My three-digit code is 314. Tanya can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the positions of two digits (so 413 and 143, for example, are forbidden, but 341 is fine). Tanya can otherwise choose any three-digit code where each digit is in the set $\{0, 1, 2, ..., 9\}$. How many codes are available for Tanya?


What is the 47th digit after the decimal point in the decimal representation of $\frac{1}{17}$?


Two 12-sided dice, one red and one green, are rolled. What is the probability that the red die shows an even number and the green die shows a number that is a perfect square?


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


In right triangle $DEF$ with $\angle E = 90^\circ$, it is given that $$5\sin D = 12\cos D.$$ Find the value of $\sin D$.


What is the sum of the digits of the base $7$ representation of $999_{10}$?


Determine the diameter of the circle inscribed in triangle $DEF$ where $DE = 13,$ $DF = 14,$ and $EF = 15.$ Express your answer in simplest radical form.


The admission fee for a game night is $ \$30$ per adult and $ \$15$ per child. On a certain Friday, the total collected was $ \$2475$ from a mix of at least one adult and one child. Determine the ratio of adults to children that is closest to $1$ for that night.


Let set $\mathcal{A}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.


How many ways are there to put 7 balls into 3 boxes if the balls are indistinguishable, the boxes are indistinguishable, and each box must contain at least 1 ball?


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


Solution: \boxed{12 - 6z^2}


Let \( a \) and \( b \) be real numbers. One of the roots of
\[ x^3 + ax^2 - 2x + b = 0 \]
is \( 2 - 3i \). Find the ordered pair \( (a, b) \).

Solution: \boxed{\left(-\frac{1}{4}, \frac{195}{4}\right)}


Count the total number of four-digit odd numbers that do not contain the digit '2' anywhere (let's call this set $C$) plus all the four-digit multiples of 3 that are also less than 5000 (let's name this set $D$). Find $C+D$.


How many possible distinct arrangements are there of the letters in the word BALLOON?


Mark has $\frac{5}{6}$ of a dollar, Carolyn has $\frac{3}{10}$ of a dollar, and Jim has $\frac{1}{2}$ of a dollar. How many dollars do they have altogether? (Give your answer as a decimal.)


A circular cylindrical post with a circumference of 6 feet has a string wrapped around it, spiraling from the bottom of the post to the top. The string makes three complete loops around the post, beginning at the bottom edge and ending at the top edge. The height of the post is adjusted to 18 feet. Determine the length of the string, in feet.


Let $X$ be the greatest common factor and $Y$ be the least common multiple of 18, 27, and 36. What is the value of $X + Y$?


A rectangular prism has twelve edges and eight vertices, similar to a cube. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. How many diagonals does a rectangular prism have if it is not a cube?


What is the largest digit that is never found in the units place of an odd number?


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


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 $9$?


How many $3$-digit squares are palindromes that end with the digit 6?


The sum of all the positive factors of integer $x$ is 56. If one of the factors is 4 and $x$ is an even number, what is the value of $x$?


Factor the following expression: $81y^{19} + 162y^{38}$.

Solution: \boxed{81y^{19}(1 + 2y^{19})}


Let $a$, $b$, $c$, and $d$ be digits such that $b = a + 2$, $c = a + 4$, and $d = a + 6$. Form the numbers $abcd$ and $dcba$, where these represent four-digit numbers created by these digits. What is the greatest common divisor of all numbers of the form $abcd + dcba$?


When the greatest common divisor and least common multiple of two integers are multiplied, their product is 180. How many different values could the greatest common divisor of these two integers have?


How many even perfect square factors does $2^6 \cdot 3^4 \cdot 7^5$ have?


Find the solutions of the equation \(x(5x - 9) = -4\) and express them in the form \(\frac{m+\sqrt{n}}{p}\) and \(\frac{m-\sqrt{n}}{p}\), where \(m\), \(n\), and \(p\) are integers having a greatest common divisor of 1. Calculate \(m+n+p\).


Find the integer $n$, $-180 \le n \le 180$, such that $\sin n^\circ = \cos 522^\circ.$


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


Given that the area of the parallelogram formed by vectors $\mathbf{a}$ and $\mathbf{b}$ is 12, find the area of the parallelogram formed by the vectors $3\mathbf{a} + \mathbf{b}$ and $2\mathbf{a} - 4\mathbf{b}$.


A positive number is called $5$-primable if it is divisible by $5$ and each of its digits is a one-digit prime number. How many $5$-primable positive integers are there that are less than 10,000?


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

What is the smallest positive $b$ such that if the graph of $y=g\left(\frac{x}{10}\right)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y=g\left(\frac{x}{10}\right)$?


How many 3-digit numbers have the property that the tens digit is at least twice the units digit?


Given a block of wood that measures 6 inches long, 3 inches wide, and 2 inches high, the block is painted blue on all six sides and then cut into cubes, each measuring 1 inch on all sides. How many of these cubes have an even number of blue faces?


If $\mathbf{B} = \begin{pmatrix} 2p & 2q \\ 2r & 2s \end{pmatrix},$ then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} 2p & 2r \\ 2q & 2s \end{pmatrix}.\] Given that $\mathbf{B}^T = 4\mathbf{B}^{-1},$ find $p^2 + q^2 + r^2 + s^2.$


In quadrilateral $PQRS$, $PQ = 6$, $QR = 10$, and $RS = 25$ units. Angles $Q$ and $R$ are both right angles. Determine the length of segment $PS$.


A line segment begins at $(1, 3)$ and is 12 units long. It ends at the point $(-7, y)$ where $y > 0$. What is the value of $y$?

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


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

Solution: \boxed{6 + 8 \sqrt{3}}


A pentagon is formed by cutting a triangular corner from a rectangular piece of cardboard. The five sides of the pentagon have lengths $9,$ $16,$ $30,$ $40,$ and $41,$ in some order. Additionally, two sides of the pentagon are parallel. Determine the area of the pentagon, assuming right angles at the vertices where the triangle was removed.


How many positive, three-digit integers contain at least one $7$ as a digit but do not contain a $2$ as a digit?


Let $Q$ be the plane passing through the origin with normal vector $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}.$ Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto plane $Q.$

Solution: \boxed{\begin{pmatrix} \frac{5}{9} & -\frac{2}{9} & \frac{4}{9} \\ -\frac{2}{9} & \frac{8}{9} & \frac{2}{9} \\ \frac{4}{9} & \frac{2}{9} & \frac{5}{9} \end{pmatrix}}


Sandy plans to cover one shaded exterior wall, which she has extended to be 8 feet wide and 8 feet high, and the two rectangular sides of the roof, each 8 feet wide and 5 feet high, with a special siding. The siding can be cut and is sold in sections of 10 feet by 12 feet that cost $\$30$ each. What is the total cost of the siding Sandy must purchase?


Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{50}=50 \quad \text{and} \quad
b_{51}+b_{52}+ \cdots + b_{100}=150.
$$ What is the value of $b_2 - b_1$? Express your answer as a common fraction.


Albert now decides to extend his list to the 2000th digit. He writes down positive integers in increasing order with a first digit of 1, such as $1, 10, 11, 12, \ldots$. Determine the three-digit number formed by the 1998th, 1999th, and 2000th digits.


Calculate the result of the expression \((3^5 \div 3^2) \cdot 2^7\).


A choir has 96 members who will all be performing in a concert. They need to line up in rows with the same number of members per row. If there must be between 5 and 20 members per row, in how many possible row-lengths can the choir be arranged?


Convert the decimal number $256_{10}$ into its equivalent in base five.


A notebook contains 75 pages, numbered from 1 to 75. The pages are then renumbered from 75 to 1. Determine how many pages have the same units digit in both the original and the reversed numbering.


Mary decides to walk from her home to the nearby park before heading to school. It takes her 45 minutes to walk uphill 1.5 km to the park, and then another 15 minutes to walk from the park to school covering another 0.5 km. The total distance from home to school via the park remains the same as walking directly. When Mary returns, it takes her 20 minutes to walk the same route from school to home. What is her average speed, in km/hr, for the entire round trip?

Solution: \boxed{3\text{ km/hr}}


What is the smallest positive four-digit number divisible by 6 which has three even and one odd digit?


Project the vectors $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ onto a vector $\mathbf{v}$. Both result in the same projection vector $\mathbf{p}$. Find $\mathbf{p}$.

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


Sandy is updating her backyard playhouse. She wants to replace siding on one exterior wall measuring 10 feet wide by 8 feet high and on two inclined roof sections, each inclined at a 30-degree angle with a base of 10 feet and an average height of 7 feet. The siding is only available in 10-foot by 10-foot sections at a cost of \$30 each. How much will the siding for these renovations cost?


How many four-digit numbers are divisible by 11?


Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers needed at intersections.


Find the sum of the solutions of the equation $-16x^2 + 72x - 108=0$.


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?


Let set $\mathcal{A}$ be a 60-element subset of $\{1, 2, 3, \ldots, 120\}$. Let $S$ be the sum of the elements of set $\mathcal{A}$. Find the number of possible values of $S$.


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


Solution: \boxed{6x^{28} - x^{26} - 24x^{12} + 4x^{10} - 48x^5 + 8x^3 + 12x^2 - 2}


Determine the smallest possible positive integral value for the constant term "e" in the polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
with integer coefficients and roots $-3$, $7$, $10$, and $-1/4$.


Each of the first ten prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number less than 30?


The positive four-digit integers that use each of the digits $1,$ $2,$ $3,$ and $5$ exactly once are ordered from least to greatest. What is the $16^{\text{th}}$ integer in the list?


Let $m=\underbrace{55555555}_{\text{8 digits}}$ and $n=\underbrace{111111111}_{\text{9 digits}}$.

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


Derek's history test consisted of 80 questions: 20 on ancient history, 25 on medieval history, and 35 on modern history. Derek scored 60% on ancient history, 56% on medieval history, and 70% on modern history questions. He failed because he got less than 65% of the questions right overall. How many more questions would he have needed to answer correctly to reach a 65% passing grade?


Calculate the value of $\frac{3^3 \cdot 3^{-5} \cdot 2^2}{3^{-2} \cdot 2^3 \cdot 2^{-5}}$.


Find the product of the divisors of $72$.


I have modified my walking game. On move 1, I still do nothing, but for each move $n$ where $2 \le n \le 30$, I walk one step backward if $n$ is prime and three steps forward if $n$ is composite. After all 30 moves, I assess how far I am from my starting point. How many total steps would I need to walk to return?


Determine the distance between the vertices of the parabolas formed by the equation:
\[
\sqrt{x^2+y^2} + |y-3| = 5
\]


Segment $s_1$ has endpoints at $(6, -2)$ and $(-4, 6)$. Segment $s_2$ is obtained by translating $s_1$ by $3$ units to the left and $4$ units down, and also scaling it by a factor of $2$ from its midpoint. Calculate the midpoint of segment $s_2$. Express your answer as $(a,b)$ where $a$ and $b$ are integers.

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


A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum.

Solution: \boxed{10\pi \sqrt{61} + 68\pi}


A projection takes $\begin{pmatrix} 2 \\ -4 \end{pmatrix}$ to $\begin{pmatrix} 3 \\ -3 \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} -8 \\ 2 \end{pmatrix}$ to?

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


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$, the second card is a $\diamondsuit$, and the third card is a $\spadesuit$?


Each of the nine letters in "CALCULATE" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "VALUE"? Express your answer as a common fraction.


Bob has now changed his password format slightly. It still consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number. What is the probability that Bob's password starts with an odd single-digit number, is followed by a letter, and ends with a prime digit?


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


Find the sum $1_6 + 2_6 + 3_6 + \cdots + 55_6$ and express your answer both in base 6 and base 10.


What is the greatest common factor of 36, 54, and 81?


Rectangle $ABCD$ is inscribed in triangle $PQR$ such that side $AD$ of the rectangle is along side $PR$ of the triangle, as displayed. The triangle's altitude from $Q$ to side $PR$ measures 8 inches, and $PR = 12$ inches. The length of segment $AB$ is twice the length of segment $AD$. Determine the area of rectangle $ABCD$.

Solution: \boxed{18 \text{ square inches}}


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


In the diagram, rectangle $ABCD$ is divided into four identical squares. If the area of rectangle $ABCD$ is $400$ square centimeters, what is its perimeter?
[asy]
size(4cm);
pair a = (0, 2); pair b = (2, 2); pair c = (2, 0); pair d = (0, 0);
draw(a--b--c--d--cycle);
draw(shift(1) * (a--d)); draw(shift(1,0) * (a--b));
label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW);
[/asy]


A rectangular prism has dimensions of 8 inches by 15 inches by 12 inches. Determine the length of the diagonal connecting point A and point B of the prism, where point A is at one vertex of the top face and point B is the opposite vertex on the bottom face. Express your answer in simplest radical form.

Solution: \boxed{\sqrt{433} \text{ inches}}


For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?


Calculate the angle $\theta$ in degrees for the expression:
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 140^\circ\]
expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$.


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


Let \( x, y, z \) be positive real numbers such that \( xyz = 12 \). Find the minimum value of \( 2x + 3y + 6z \).


Define a function $g(x)$ such that $g(13)=40$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and:

$g(b)=3b+5$ if $b$ is odd

$g(b)=\frac{b}{2}$ if $b$ is even.

What is the smallest possible number of integers in the domain of $g$?


Determine how many even and odd natural-number factors are present in $n = 2^2 \cdot 3^2 \cdot 7^2$.


In an isosceles trapezoid, the longer base \(AB\) is 24 units, the shorter base \(CD\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \(AC\)?


John has 15 marbles, including two red, two green, and two blue marbles. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are from the set of red, green, or blue marbles?


Find the distance between the planes \(x - 4y + 4z = 10\) and \(2x - 8y + 8z = 4.\)


What is the 25th number in the row of Pascal's triangle that has 31 numbers?


What is the area, in square units, of a triangle with vertices at $A(2, 2), B(7, 2), C(4, 9)$?


For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(4,6)--(13,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(13,2),dashed);
label("7",(2,3),NW);
label("11",(8.5,4),NE);
label("17",(10.5, -2.5),SE);
label("13",(4,-2.5),SW);
[/asy]


Given an equilateral triangle $ABC$ and a square $BCFG$ such that they share a common side $BC$, find the measure of $\angle FAG$ where point $F$ is on the extension of side $AB$ and point $G$ is on the extension of side $AC$. Assume $F$ and $G$ are such that $BF = AB$ and $CG = AC$.


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


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


Cara is sitting at a circular table with her seven friends. If Mark is always one of Cara's neighbors, how many different pairs of people could Cara be sitting between?


The equation of a parabola is $y^2 + 8y + 2x + 11 = 0.$ Find the vertex of the parabola.

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


Determine the value of $m$ such that $10^m = 10^{-3} \times \sqrt{\frac{10^{85}}{0.0001}}$.


What is the greatest common divisor of $7!$ and $\frac{10!}{5!}$? Express your answer as an integer.


How many four-digit perfect cubes are divisible by 16?


Let $b_n = \frac{7^n - 1}{6}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


In triangle \( \triangle PQR \), medians \( \overline{PM} \) and \( \overline{QN} \) are perpendicular. If \( PM = 18 \) and \( QN = 24 \), find the area of \( \triangle PQR \).


Find the ratio of line segments for the curve $y = \sin x$ when it cuts the line $y = \sin 60^\circ$. Assume the curve intersects this line in a pattern yielding segments with ratios \[\dots p : q : p : q \dots\] where $p < q$. Compute the ordered pair $(p,q)$ of relatively prime positive integers representing these ratios.

Solution: \boxed{(1, 5)}


$x = 1 + \frac{\sqrt{3}}{1+\frac{\sqrt{3}}{1+...}}$. Compute $\frac{1}{(x+2)(x-3)}$. When your answer is in the form $\frac{A+\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, find $|A|+|B|+|C|$.


Suppose 5 different integers are randomly chosen from between 10 and 59, inclusive. What is the probability that they each have a different tens digit?


In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A+B+C$?


A point $(x, y)$ is a distance of 15 units from the $x$-axis and 13 units from the point $(2, 8)$. It is a distance $m$ from the origin. Given that $x > 2$, what is $m$?

Solution: \boxed{\sqrt{349 + 8\sqrt{30}}}


I am playing a modified walking game. On move 1, I do nothing, but on move $n$ where $2 \le n \le 30$, I take one step forward if $n$ is prime and three steps backwards if the number is composite. After all 30 moves, I stop and walk back to my original starting point. How many steps long is my walk back?


Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]


A pet store has 12 puppies, 8 kittens, 10 hamsters, and 5 birds. Alice, Bob, Charlie, and David each want to buy a pet. To keep variety, they each decide to buy a different kind of pet. How many ways can Alice, Bob, Charlie, and David buy their pets and leave the store satisfied?


I had planned to work 25 hours a week for 15 weeks during the fall to earn \$3750 to fund a project. Unfortunately, I was unable to work for the first three weeks of fall. How many hours a week will I have to work for the remaining weeks if I still want to fund the project?

Solution: \boxed{31.25 \text{ hours per week}}


Let $\mathbf{p}$ and $\mathbf{q}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{p}} \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix},$ determine $\operatorname{proj}_{\mathbf{q}} \begin{pmatrix} 4 \\ 2 \end{pmatrix}.$

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


Let $\triangle ABC$ be an isosceles triangle such that $BC = 40$ and $AB = AC$. The incenter $I$ of $\triangle ABC$ satisfies $IC = 26$. Determine the length of the inradius of the triangle.


Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\log_2(S)$.


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

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


Let \( r, s, \) and \( t \) be the roots of the equation \( x^3 - 15x^2 + 13x - 6 = 0 \). Find the value of \( \frac{r}{\frac{1}{r}+st} + \frac{s}{\frac{1}{s}+tr} + \frac{t}{\frac{1}{t}+rs} \).


What is the smallest positive integer that can be written in the form $2017m + 48576n$, where $m$ and $n$ are integers?


Suppose the function $f(x)$ is defined on the domain $\{x_1, x_2, x_3\}$, such that the graph of $y=f(x)$ consists of just three points, forming a triangle of area $27$. Calculate the area of the triangle formed by the graph of $y = 3f(3x)$, which also consists of just three points.


Cube each integer $n$ in the range $1 \leq n \leq 10$ and find the remainders when the cubes are divided by $13$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $13$?


Given that $9!=362880$, what is the value of $9!\div4!$?


There is a committee composed of six women and four men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. The meeting requires that the first and last seat in the row must be a rocking chair. How many distinct ways are there to arrange the six rocking chairs and four stools for a meeting?


Find the sum of the real roots of the equation \(x^4 - 6x^2 - x + 6 = 0.\)


Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $6$ steps at a time, there are $5$ steps left at the top. When he does it $7$ steps at a time, there are $4$ steps left at the top. What is the smallest possible value of $n$ that is greater than $20$?


Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded.

Solution: \boxed{576 - 108\pi}


Solution: \boxed{2x^{11} + 13x^{10} + 10x^9 + 3x^8 + x^7 + 4x^4 + 7x + 9}


In a pentagon $FGHIJ$, two interior angles $F$ and $G$ are $50^{\circ}$ and $75^{\circ}$. Two of the remaining angles, $H$ and $I$, are equal, and the fifth angle $J$ is $10^{\circ}$ more than twice $H$. Determine the measure of the largest angle.


Calculate the sum of the following expression: $\frac{1}{2}-\frac{1}{8}+\frac{1}{32}-\frac{1}{128}+\frac{1}{512}-\frac{1}{2048}$. Express your answer as a common fraction.


Determine the maximum number of real roots of the polynomial
\[ 2x^n + (2-2)x^{n-1} + (2-4)x^{n-2} + \dots + (2-2j)x^{n-j} + \dots + (2-2n)x^0 = 0, \]
where \( n \) is a positive integer and \( j \) ranges from \( 1 \) to \( n \).


Consider the sequence starting at $10800$, where each term is obtained by dividing the previous term by 2. How many integers are in this sequence?


A circular cylindrical post with a circumference of 6 feet has a string wrapped around it, spiraling from the bottom of the post to the top of the post. The string evenly loops around the post exactly six times, starting at the bottom edge and finishing at the top edge. The height of the post is 18 feet. What is the length, in feet, of the string?


I have a drawer with 6 shirts, 7 pairs of shorts, and 8 pairs of socks. If I randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one pair of socks?


Five concentric circles are drawn with radii of 1, 4, 6, 8, and 10. The innermost circle is painted black, the next ring is white, followed by a black ring, another white ring, and the outermost ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


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


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


What is the area, in square units, of the interior region formed by the lines \( y = 3x - 6, y = -2x + 18 \), and the \( y \)-axis?


Suppose \(x\) is an integer that satisfies the following congruences:
\[
4 + x \equiv 2^2 \pmod{3^3}, \\
6 + x \equiv 3^2 \pmod{5^3}, \\
8 + x \equiv 5^2 \pmod{7^3}.
\]
What is the remainder when \(x\) is divided by \(105\)?


There exists a scalar $k$ such that for any coplanar vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ satisfying $\mathbf{a} + \mathbf{b} + \mathbf{c} = 2\mathbf{a},$ the equation
\[k (\mathbf{b} \times \mathbf{a}) + (k+1)(\mathbf{b} \times \mathbf{c}) + (\mathbf{c} \times \mathbf{a}) = \mathbf{0}\]
holds. Find $k$.


Alex is at the candy store buying jellybeans. He wants to buy at least 150 jellybeans. He also wants to ensure that after dividing the jellybeans among 15 people, he has exactly 14 jellybeans leftover. Additionally, when he shares them among 17 people, there should be 16 jellybeans remaining. How many jellybeans should Alex buy?


Add $A85_{12}$ and $2B4_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


If 5 knicks equal 3 knacks and 4 knacks equal 7 knocks, how many knicks are equivalent to 28 knocks?


A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300?


What is the modulo 7 remainder of the sum $1+2+3+ \ldots +194+195+196+197+198?$


What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 8 units and lateral edges measuring 10 units?


Consider the three lines in a plane given by the equations $2y - 3x = 4$, $x + 3y = 3$, and $6x - 4y = 8$. Determine how many points lie at the intersection of at least two of these lines.


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least two books are in the library and at least two are checked out?


If 20 bahs are equal to 36 rahs and 12 rahs equal 20 yahs, how many bahs are equivalent to 1500 yahs?

Solution: \boxed{500 \text{ bahs}}


The formula for the total surface area of a cylinder is \( SA = 2\pi r^2 + 2\pi rh \), where \( r \) is the radius and \( h \) is the height. A particular solid right cylinder has a radius of 3 feet and a total surface area of \( 27\pi \) square feet. What is the height of this cylinder?


A fair 12-sided die has each face numbered from 1 to 12. One dot is removed at random from a face chosen such that the face with $n$ dots is as likely to be chosen as a face with $m$ dots for any $n$ and $m$. The die is then rolled. What is the probability that the top face shows an odd number of dots?


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


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


Angela has $a$ marbles, Brian has three times as many marbles as Angela, Caden has four times as many marbles as Brian, and Daryl has six times the number of marbles Caden has. If in total Angela, Brian, Caden, and Daryl have 186 marbles, what is the value of $a?$


Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_2 x & \quad \text{ if } \log_2 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$


When \( n \) is divided by 4, the remainder is 3. What is the remainder when \( 7n \) is divided by 5?


A math conference is planning a lecture series with six different lecturers named Dr. X, Dr. Y, Dr. Z, Dr. L, Dr. M, and Dr. N. If Dr. X's lecture depends on Dr. Y's lecture (Dr. X must be after Dr. Y), and Dr. L's lecture depends on Dr. N's lecture (Dr. L must be after Dr. N), in how many orders can the six lecturers be scheduled?


An oreo shop now sells $6$ different flavors of oreos and $4$ different flavors of milk. Alpha and Beta decide to purchase some oreos and milk. Alpha will not order more than 1 of the same flavor and is willing to buy milk. Beta will only order oreos, with repeats of flavors allowed. How many ways could they have left the store with $4$ products collectively?


If the product $(5x^3 - 3x^2 + x - 8)(8 - 3x)$ can be written in the form $ax^4 + bx^3 + cx^2 + dx + e$, where $a, b, c, d, e$ are real numbers, then find $16a + 8b + 4c + 2d + e$.


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 17 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of $n$?


For some positive integer $k$, when 90 is divided by $k^2$, the remainder is 6. What is the remainder when 130 is divided by $k$?


Suppose that $9$ is a factor of $a$, $a$ is a divisor of $18$, and $a$ is positive. What is the number of possible values of $a$?


Square PQRS has its center at $(5, -5)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of 3. What are the coordinates of the vertex of the image of square PQRS that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(21, -21)}


Find the maximum value of
\[
\frac{\sin^4 x + \cos^4 x + 2}{\sin^6 x + \cos^6 x + 2}
\]
over all real values \(x\).


What is the value of $\sqrt{49+49} - \sqrt{36+25}$?

Solution: \boxed{7\sqrt{2} - \sqrt{61}}


Factor completely the expression: \[(3x^3 - 67x^2 - 14) - (-8x^3 + 3x^2 - 14)\]

Solution: \boxed{x^2(11x - 70)}


Regular hexagon $PQRSTUV$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area?


The school's debate team includes 20 members: 8 boys and 12 girls. A 4-person debate panel is to be selected at random. What is the probability that the panel includes at least 1 boy and at least 1 girl?


Each of Natalie's blueberry bushes yields ten containers of blueberries. If she can trade four containers of blueberries for one zucchini, and if Natalie needs to keep 20 containers for herself, how many bushes must she pick to end up with 60 zucchinis?


At the start of my jogging, I run at 10 miles per hour. As I start to feel exhausted, my speed drops to 6 miles per hour. If I run a total of 64 miles in a total time of 8 hours, for how many hours did I run energetically? Express your answer as a common fraction.


Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 6x^2 + 11x - 6 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 15x^2 + 74x - 120}


Find the intersection of the lines \(3x - 2y = 8\) and \(5x + 3y = 27\). Express your answer as an ordered pair \((x,y)\).

Solution: \boxed{\left(\frac{78}{19}, \frac{41}{19}\right)}


How many three-digit numbers are there if we exclude all three-digit numbers in which there are two adjacent digits the same, regardless of the third digit?


Let $\mathbf{S}$ be the matrix that scales by a factor of 3 and rotates about the origin counterclockwise by an angle of $45^\circ$. Find $\det \mathbf{S}$.


Given that \( x+\cos y=3005 \) and \( x+3005 \sin y=3004 \), where \( 0 \leq y \leq \frac{\pi}{2} \), find the value of \( x+y \).


Let $a$ and $b$ be positive real numbers such that $a + b = 2$ and let $c$ be a positive real constant. Find the set of all possible values of $\frac{c}{a} + \frac{c}{b}$.

Solution: \boxed{[2c, \infty)}


Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fifth row is 14. The sum of the interior numbers of the sixth row is 30. What is the sum of the interior numbers of the eighth row?


Tracy had a bag of candies. She first ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. Afterward, her mom and Tracy each ate 10 candies. Her cousin then took one to three candies, leaving Tracy with exactly 5 candies. How many candies did Tracy have at the beginning?


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


Joe and JoAnn each bought 20 ounces of coffee in a 24-ounce cup. Joe drank 3 ounces of his coffee and then added 3 ounces of cream. JoAnn added 3 ounces of cream, stirred the coffee well, and then drank 3 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? Express your answer as a fraction.


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


When the base-10 integers 250 and 800 are expressed in base 2, how many more digits does 800 have than 250?


Each of the numbers $a_1, a_2, \dots, a_{100}$ is either $+1$ or $-1$. Find the smallest possible positive value of
\[
\sum_{1 \leq i < j \leq 100} a_i a_j.
\]


Compute the infinite series:
\[
\sum_{n = 1}^\infty \frac{3n}{n(n + 1)(n + 2)}.
\]


Determine the product of all possible values for $a$ if the length of the segment between the points $(3a, a-5)$ and $(7, 0)$ is $3\sqrt{10}$ units.


Determine how many integer values of $b$ are possible such that the quadratic inequality
\[ x^2 + bx + 5 \leq 0 \]
has exactly three integer solutions for $x$.


Determine the greatest number of positive factors that $b^n$ can have if $b$ and $n$ are positive integers, with $b \leq 20$ and $n = 10$.


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


Evaluate the determinant of the following matrix:
\[
\begin{vmatrix}
0 & \cos \alpha & \sin \alpha \\
-\cos \alpha & 0 & \cos \beta \\
-\sin \alpha & -\cos \beta & 0
\end{vmatrix}.
\]


Solve for $x$: $x = \displaystyle\frac{\sqrt{8^2+15^2}}{\sqrt{49+36}}$. Express as a common fraction.


Our club has 24 members, 12 boys and 12 girls. They are also evenly divided into two age groups: young and old. In how many ways can we choose a president and a vice-president if they must be of different gender and from different age groups?


From point $A$, Jenna walked 50 yards south, 30 yards west, 15 yards north, and 10 yards east to point $C$. What is the length, in yards, of $\overline{AC}$?


If $x^2 + 4x + k$ is a factor of $x^4 + Px^2 + Q$, find the values of $P + Q$ assuming $k = 3$.


A solid right prism $PQRSTU$ has a height of 20. The bases of the prism are equilateral triangles with a side length of 10. Points $V$, $W$, and $X$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $VWX$.

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


What is the sum of all of the positive even factors of $630$?


Simplify the expression: $i^0 + i^1 + \cdots + i^{2013}$.

Solution: \boxed{1 + i}


Given that \( b \) and \( n \) are positive integers with the values of \( b \) and \( n \) both less than or equal to 20, what is the maximum number of positive factors that \( b^n \) can have?


When the greatest common divisor and least common multiple of two integers are multiplied, their product is 252. How many different values could the greatest common divisor of the two integers be?


How many of the 300 smallest positive integers are congruent to 5 (mod 7)?


The Fibonacci sequence is given as 1, 1, 2, 3, 5, $\ldots$ where each term is the sum of the two preceding ones. Calculate the remainder when the $70^{\mathrm{th}}$ term of the Fibonacci sequence is divided by 10.


Suppose \(a\) and \(b\) are positive integers such that \(\gcd(a, b)\) is divisible by exactly \(5\) distinct primes and \(\mathop{\text{lcm}}[a, b]\) is divisible by exactly \(20\) distinct primes.

If \(b\) has fewer distinct prime factors than \(a\), then \(b\) has at most how many distinct prime factors?


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1600$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.


The degree measures of the angles in a convex 15-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $648$?


I run at a constant pace, and it takes me 24 minutes to run to the store from my house. The store is 3 miles away from my house. If my friend's house is 1.5 miles away from my house, how many minutes will it take me to run from my house to my friend's house?

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


In a circle with a center $O$, $\angle MAO$ is $45^\circ$ and $OM=8$ cm. Calculate the length of arc $MA$ in terms of $\pi$.

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


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 20 and altitude 24, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


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


Determine the set of vectors \(\mathbf{u}\) such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ -1 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{9}{2} \\ -\frac{3}{2} \end{pmatrix}\]
and find the equation of the line on which these vectors lie, in the form "y = mx + b".

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


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 44. If one integer is 60, what is the smallest possible value of the other integer?


Factor the following expression: $43b^2 + 129b$.

Solution: \boxed{43b(b + 3)}


Homewood Middle School now has 1500 students, and 900 of these students attended a winter festival. If two-thirds of the girls in the school and one-half of the boys in the school attended the festival, how many girls attended the festival?


How many of the positive divisors of 5,940 are multiples of 6?


Find the greatest common divisor of $12347$ and $30841$.


Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a perfect square. His leash is exactly long enough to reach the midpoint of each side of the square yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest form as a fraction of $\pi$, what is the value of the denominator when the fraction is expressed in lowest terms?


Let $\{a_k\}$ be a sequence of integers such that $a_1=2$ and $a_{m+n}=a_m+a_n+mn,$ for all positive integers $m$ and $n.$ Find $a_{10}.$


Determine the greatest possible value of $x$ if the least common multiple of $x$, $12$, and $18$ is $180$.


Circle $D$ has radius 9 cm. How many square centimeters are in the area of the largest possible equilateral triangle inscribed in Circle $D$?

Solution: \boxed{\frac{243\sqrt{3}}{4} \text{ square centimeters}}


Compute $(2 \cos 30^\circ + 2i \sin 30^\circ)^{10}.$ Enter your answer in rectangular form.

Solution: \boxed{512 - 512i\sqrt{3}}


Solution: \boxed{x^4 + 23x^3 - 78x^2 + 39x + 34}


In the following diagram, \( AB = 80 \). 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("$84$",.5*(B+C),E);
label("$42$",.5*(A+C),NW);

[/asy]


The graphs of a function \( f(x) = 2x + b \) and its inverse \( f^{-1}(x) \) intersect at the point \( (-4, a) \). If \( b \) and \( a \) are integers, determine the value of \( a \).


Solution: \boxed{5 + 30x}


It takes five painters working at the same rate 1.5 work-days to finish a job. If only four painters are available, how many work-days will it take them to finish the job, working at the same rate? Express your answer as a mixed number.

Solution: \boxed{1\frac{7}{8} \text{ work-days}}


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


What is the smallest positive integer $n$ such that $253n \equiv 989n \pmod{15}?$


The equation $x^2 - 6x + 14 = 31$ has two solutions, $c$ and $d$, with $c \geq d$. What is the value of $c+2d$?

Solution: \boxed{9 - \sqrt{26}}


Planet Zircon encircles a star in an elliptical path, where the star is at one of the focus points. The closest point (perigee) of Zircon is 3 astronomical units (AU) from the star, and the farthest point (apogee) is 10 AU away. Determine the distance from Zircon to the star when it is midway along its orbit.

Solution: \boxed{6.5 \text{ AU}}


The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?


Bob rolls a fair eight-sided die each morning. The die sides are numbered from 1 to 8. If Bob rolls a composite number (4, 6, 8), he eats sweetened cereal. If he rolls a prime number (2, 3, 5, 7), he eats unsweetened cereal. If he rolls a 1 or an 8, then he rolls again. In a leap year (366 days), what is the expected number of times Bob will roll his die?


Four more than four times a number $B$ is equal to 33. What is the value of $B$?


Consider a "particular integer" defined as a positive integer which is divisible by the square of the sum of its digits. Find the smallest positive multiple of 7 that is not a particular integer.


Compute $\sin 870^\circ$ and $\cos 870^\circ$.


Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$.

Solution: \boxed{\frac{9\pi - 18}{2}}


In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 15$, $\tan Z = 2$, and $\tan X = \frac{3}{2}$, what is the length of $XY$?

[asy]
pair WW,X,Y,Z;
Z = (0,0);
Y = (15,0);
WW = (15,30); // assuming
X= (22.5,30); // adjusting to new tangents
draw(WW--X--Y--Z--WW);
label("$W$",WW,N);
label("$X$",X,N);
label("$Y$",Y,S);
label("$Z$",Z,S);
[/asy]


Find the remainder when $3x^8 - 2x^5 + 5x^3 - 9$ is divided by $x^2 - 2x + 1$.

Solution: \boxed{29x - 32}


How many different positive, seven-digit integers can be formed using the digits 2, 2, 2, 5, 5, 9, and 9 such that the number is divisible by 5?


One line is parameterized by
\[\begin{pmatrix} 2 + s \\ 5 - ks \\ 3 + ks \end{pmatrix}.\]
Another line is parameterized by
\[\begin{pmatrix} 2t \\ 4 + 2t \\ 6 - 2t \end{pmatrix}.\]
Determine the value of \( k \) for which the lines are coplanar.


A line containing the points $(3, 20)$ and $(-9, -6)$ intersects the $y$-axis at what point?

Solution: \boxed{(0, \frac{27}{2})}


A large sphere has a volume of $675\pi$ cubic units. A smaller sphere's volume is $20\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a simplified fraction.


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


Find the number of odd digits in the base-4 representation of $317_{10}$.


The volume of a sphere is $288\pi$ cubic inches. Find the total surface area of a spherical cap of height $h=2$ inches that sits atop this sphere, in terms of $\pi$.


In a diagram, $\triangle ABC$ and $\triangle BDC$ are right-angled, with $\angle ABC = \angle BDC = 45^\circ$, and $AB = 16$. Determine the length of $BC$.


What is the modular inverse of $17$, modulo $392$?

Express your answer as an integer from $0$ to $391$, inclusive.


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 10$, what is $b_2b_7$?


Express as a common fraction: $$\cfrac{ \frac{3}{7} + \frac{5}{8} }{ \frac{5}{12} + \frac{2}{9} }.$$


For which positive integer values of \( k \) does \( kx^2 + 12x + k = 0 \) have rational solutions? Express your answers separated by commas and in increasing order.


Consider two numbers, 324 and 67. First, square the smaller number, then multiply the result by the larger number, and finally express the product in base-6. What is the units digit of this base-6 representation?


A side view of three pipes placed in a triangular stacking configuration is shown. Each pipe has a diameter of 12 cm. The configuration forms an isosceles triangle, where the base is formed by two pipes and the third pipe is placed on top of these two supporting pipes.

Determine the total height, \(H\), of this stack of three pipes.

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


George has an unfair eight-sided die. The probabilities are as follows: rolling an 8 has a probability of $\frac{3}{8}$, and each of the numbers from 1 to 7 has an equal probability of occurring. What is the expected value of the number shown when this die is rolled?


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


In a new race setup, all runners start at point \( A \), touch any part of the 800-meter wall, and stop at point \( B \). The distance from \( A \) directly down to the left end of the wall is 200 meters. The distance from the right end of the wall directly vertically up to \( B \) is 400 meters. Determine the number of meters in the minimum distance a participant must run, rounding to the nearest meter. Illustrate using a diagram similar to the original problem's setup:

[asy]
import olympiad; import geometry; size(250);
defaultpen(linewidth(0.8));
draw((0,2)--origin--(8,0)--(8,4));
label("200 m",(0,2)--origin,W); label("800 m",(0,0)--(8,0),S); label("400 m",(8,0)--(8,4),E);
draw((0,2)--(4,0)--(8,4),linetype("3 3")+linewidth(0.7));
label("$A$",(0,2),N); label("$B$",(8,4),N);
[/asy]


How many times does the digit 9 appear in the list of all integers from 1 to 300?


In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 10. What is the greatest possible perimeter of the triangle?


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

Solution: \boxed{f(3) = 0}


Let \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) be unit vectors such that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\), and the angle between \(\mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}\) is also \(\alpha\). If \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{8}\), find the smallest possible value of \(\alpha\), in degrees.


A circle having center $(0,k)$, where $k<-6$, is tangent to the lines $y=x$, $y=-x$ and $y=-6$. What is the radius of this circle?


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


A swimming pool can be filled by any of three hoses A, B, or C. Hoses A and B together take 3 hours to fill the pool. Hoses A and C together take 4 hours to fill the pool. Hoses B and C together take 5 hours to fill the pool. How many hours does it take hoses A, B, and C working together to fill the pool? Express your answer as a decimal to the nearest hundredth.


If $x$ is doubled, increased by $7$, and then divided by $6$, the result is $13$. What is the value of $x$?

Solution: \boxed{x = 35.5}


What is the value of $a$ for which $\frac{1}{\text{log}_5a} + \frac{1}{\text{log}_6a} + \frac{1}{\text{log}_7a} = 1$?


Find $\tan X$ for a right triangle where one leg measures 40 units and the hypotenuse measures 41 units. Assume the triangle is labeled with vertices $X$, $Y$, and $Z$, where $X$ and $Z$ are on the ground, $Y$ is at the right angle, and $XZ$ is the hypotenuse.


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 $m$?


In a parallelogram, the sides are measured as $6y-2$ units, $12$, $3x+4$ units, and $9$ units respectively. What is the value of $x+y$?


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


The Little Twelve Basketball Conference has restructured with three divisions this year, each containing four teams. Each team plays each of the other teams in its own division twice and teams in all other divisions once. Calculate the total number of conference games scheduled under this new arrangement.


A sphere is inscribed in both a cube and a right circular cylinder. If the edge of the cube is 8 inches, determine the volume of the inscribed sphere. Express your answer in terms of $\pi$.

Solution: \boxed{\frac{256}{3} \pi}


Evaluate the expression
\[
\left((a^2+b)^2 - (a^2-b)^2\right)^3,
\]
if \(a=5\) and \(b=3\).


Mrs. Thompson awards extra credit on a test to her students with test grades that exceed the class mean. Given that 200 students take the same test, what is the largest number of students who can be awarded extra credit?


Consider an isosceles triangle $XYZ$ where $XY = YZ$ and $\angle YXZ = \angle YZX = 50^\circ$. Line segments $\overline{JK}$, $\overline{LM}$, and $\overline{NO}$ are drawn parallel to base $\overline{XZ}$ such that $YJ = JL = LM = MO$. What is the ratio of the area of trapezoid $NOXZ$ to the area of triangle $XYZ$?


A geometric sequence starts with $10$, $-15$, $22.5$, $-33.75$. What is the common ratio of this sequence?


The function $q(x) = (x^2 - 2)^2$ is defined for all real numbers $x$. Determine the range of $q(x)$.


Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring?


Find the largest prime divisor of $17^2 + 30^2$.


At HappyTails Training Center, cats can learn to do three tricks: jump, fetch, and spin. Of the cats at the center:
- 40 cats can jump
- 25 cats can fetch
- 30 cats can spin
- 15 cats can jump and fetch
- 10 cats can fetch and spin
- 12 cats can jump and spin
- 5 cats can do all three
- 7 cats can do none

How many cats are in the center?


In the diagram below, point \(S\) is on the horizontal diameter of a circle centered at \(P\), and \(R\) is another point on the circumference such that $\sin \angle RPQ = \frac{3}{5}$. If \(Q\) is the right endpoint of the diameter, what is $\cos \angle RPS$?

[asy]
pair R, P, Q, S;
S = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(3/5))*(2,0);
dot("$S$", S, W);
dot("$Q$", Q, E);
dot("$R$", R, N);
dot("$P$", P, S);
draw(Q--S);
draw(P--R);
[/asy]


A storm in Paris, France, caused €45 million in damage. At that time, 1.2 Euros were equivalent to 1 British pound. Determine the number of British pounds of damage the storm caused.