$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$.

Solution: \boxed{n = 4}


Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.


Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$.  From each such subset choose the least element.  The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers.  Find $p + q$.


Julie is preparing a speech for her class.  Her speech must last between one-half hour and three-quarters of an hour.  The ideal rate of speech is 150 words per minute.  If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?

$\text{(A)}\ 2250 \qquad \text{(B)}\ 3000 \qquad \text{(C)}\ 4200 \qquad \text{(D)}\ 4350 \qquad \text{(E)}\ 5650$


Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.


A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$


Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).


The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$


A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

Solution: \boxed{\min(BW, 2WR, 3RB)}


The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]


Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.


The population of the United States in $1980$ was $226,504,825$. The area of the country is $3,615,122$ square miles. There are $(5280)^{2}$
square feet in one square mile. Which number below best approximates the average number of square feet per person?
$\textbf{(A)}\ 5,000\qquad \textbf{(B)}\ 10,000\qquad \textbf{(C)}\ 50,000\qquad \textbf{(D)}\ 100,000\qquad \textbf{(E)}\ 500,000$


A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?
$\mathrm{(A)}\ 24\pi - 36 \sqrt {2} \qquad \mathrm{(B)}\ 24\pi - 24 \sqrt {3} \qquad \mathrm{(C)}\ 36\pi - 36 \sqrt {3} \qquad \mathrm{(D)}\ 36\pi - 24 \sqrt {2} \qquad \mathrm{(E)}\ 48\pi - 36 \sqrt {3}$

Solution: \boxed{48\pi - 36\sqrt{3}}


The manager of a company planned to distribute a $$50$ bonus to each employee from the company fund, but the fund contained $$5$ less than what was needed.  Instead the manager gave each employee a $$45$ bonus and kept the remaining $$95$ in the company fund.  The amount of money in the company fund before any bonuses were paid was
$\text{(A)}\ 945\text{ dollars} \qquad \text{(B)}\ 950\text{ dollars} \qquad \text{(C)}\ 955\text{ dollars} \qquad \text{(D)}\ 990\text{ dollars} \qquad \text{(E)}\ 995\text{ dollars}$

Solution: \boxed{995\text{ dollars}}


Circle $C_1$ has its center $O$ lying on circle $C_2$.  The two circles meet at $X$ and $Y$.  Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$.  What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqrt{7}\qquad\textbf{(E)}\ \sqrt{30}$


Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.


The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.


Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}


Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation.  For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$.  Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.


A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.


Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$.


Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$

Solution: \boxed{(C)}


Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\text{3:00}$ AM, and $\text{7:50}$ at the former $\text{6:00}$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $\text{6:36}$ AM would set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A}+10\text{B}+\text{C}$.


Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?
$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64$


Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.


The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$.  The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s.  Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\,$, where $L \in \{A, B, C\}$, and $P_n\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\overline{P_n L}$.  Given that $P_7 = (14,92)\,$, what is $k + m\,$?


The sum of two numbers is $S$. Suppose $3$ is added to each number and then
each of the resulting numbers is doubled. What is the sum of the final two
numbers?
$\textbf{(A)}\ 2S + 3\qquad \textbf{(B)}\ 3S + 2\qquad \textbf{(C)}\ 3S + 6 \qquad\textbf{(D)}\ 2S + 6 \qquad \textbf{(E)}\ 2S + 12$

Solution: \boxed{2S + 12}


The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


For nonnegative integers $a$ and $b$ with  $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where  $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.


Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly $1$ minute to go past Jon. The westbound train takes $10$ times as long as the eastbound train to go past Steve. The length of each train is $\tfrac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.


A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.


Solution: \boxed{2S + 12}


A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.


What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]


A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$.
The perimeter of the remaining quadrilateral is

$\text{(A)} \ 6 \qquad  \text{(B)} \ 6\frac{1}{2} \qquad  \text{(C)} \ 7 \qquad  \text{(D)} \ 7\frac{1}{2} \qquad  \text{(E)} \ 8$


A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten  $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.


Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

Solution: \boxed{n = 1}


In base $R_1$ the expanded fraction $F_1$ becomes $.373737\cdots$, and the expanded fraction $F_2$ becomes $.737373\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525\cdots$, while the fraction $F_2$ becomes $.525252\cdots$. The sum of $R_1$ and $R_2$, each written in the base ten, is:
$\text{(A) } 24 \quad \text{(B) } 22 \quad \text{(C) } 21 \quad \text{(D) } 20 \quad \text{(E) } 19$


A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.


Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.


Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
$\textbf{(A)}\ 80t+100\left(\frac{8}{3}-t\right)=250 \qquad \textbf{(B)}\ 80t=250 \qquad \textbf{(C)}\ 100t=250 \qquad \textbf{(D)}\ 90t=250 \qquad \textbf{(E)}\ 80\left(\frac{8}{3}-t\right)+100t=250$

Solution: \boxed{(A)}


Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.


Let $f(x)=(x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which
\[\left |\sum_{k=1}^n\log_{10}f(k)\right|=1.\]


Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.


Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$


For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$.  For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$.  Find $f_{1988}(11)$.


In a drawer Sandy has $5$ pairs of socks, each pair a different color.  On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer.  On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random.  The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers,  Find $m+n$.


Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.


Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
$\textbf{(A)}\ 30\qquad\textbf{(B)}\ \frac{400}{11}\qquad\textbf{(C)}\ \frac{75}{2}\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ \frac{300}{7}$


Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.


There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.


Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
$\mathrm{(A) \ } 8\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12$


In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


For each positive integer $n$, let

$a_n = \frac{(n+9)!}{(n-1)!}$
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is
$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 9$


Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy

     (i)$f(x)\ge0$ for all $x$ in $[0, 1]$,

     (ii)$f(1) = 1$,

     (iii)     $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$
for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.


Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.


Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

Solution: \boxed{f(x) = x^2}


Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)


The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider.  Its area, in square inches, is:
$\textbf{(A)}\ 30\qquad \textbf{(B)}\ \frac{80}{3}\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ \frac{45}{2}\qquad \textbf{(E)}\ 20$


A permutation of the set of positive integers $[n] = \{1, 2, \ldots, n\}$ is a sequence $(a_1, a_2, \ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\leq k\leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$.


Positive integers $a$ and $b$ satisfy the condition
\[\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.\]
Find the sum of all possible values of $a+b$.


For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$.


Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)


Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.


Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions
(a) $0\le A<B<C\le99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\le b<a<c<p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.


Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$.


Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.


Let N = $69^{5} + 5\cdot69^{4} + 10\cdot69^{3} + 10\cdot69^{2} + 5\cdot69 + 1$. How many positive integers are factors of $N$?
$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 69\qquad \textbf{(D)}\ 125\qquad \textbf{(E)}\ 216$


In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b\ge2$. A Martian student writes down
\[3\log(\sqrt{x}\log x)=56\]
\[\log_{\log x}(x)=54\]
and finds that this system of equations has a single real number solution $x>1$. Find $b$.


Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$
$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\  44\sqrt5 \qquad\textbf{(D)}\ 70\sqrt2 \qquad\textbf{(E)}\ 100$


For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy \[|A| \cdot |B| = |A \cap B| \cdot |A \cup B|\]


Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that
\[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]


Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.


A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P$. Then $OP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Find the number of integers $c$ such that the equation \[\left||20|x|-x^2|-c\right|=21\]has $12$ distinct real solutions.


Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.


Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.


What is the value of $[\log_{10}(5\log_{10}100)]^2$?
$\textbf{(A)}\ \log_{10}50 \qquad \textbf{(B)}\ 25\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$


Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the $53^{\text{rd}}$ number said?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$


A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $a/b$ that it will take more than four draws until the third shiny penny appears and $a/b$ is in lowest terms, then $a+b=$
$\text{(A) } 11\quad \text{(B) } 20\quad \text{(C) } 35\quad \text{(D) } 58\quad \text{(E) } 66$


Suppose that $y = \frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.


The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$, which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A$. Find the remainder when $n$ is divided by $1000.$


Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$

Solution: \boxed{(C)}


If $\log_{k}{x}\cdot \log_{5}{k} = 3$, then $x$ equals:
$\textbf{(A)}\ k^6\qquad  \textbf{(B)}\ 5k^3\qquad  \textbf{(C)}\ k^3\qquad  \textbf{(D)}\ 243\qquad  \textbf{(E)}\ 125$


Jar A contains four liters of a solution that is 45% acid.  Jar B contains five liters of a solution that is 48% acid.  Jar C contains one liter of a solution that is $k\%$ acid.  From jar C, $\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B.  At the end both jar A and jar B contain solutions that are 50% acid.  Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$.


When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Find $m+n+r+s$.


Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.


Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that
(i) $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$
(ii) $x_1 + \cdots + x_n = 19$; and
(iii) $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$.
Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$, respectively. Then $\frac Mm =$
$\mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ }7$


Suppose that $(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \le i < j \le 100$ and $|a_ib_j - a_j b_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.


Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?


In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?


The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?


Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.


A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$


The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$


For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.


A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown.  Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy]
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$


Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.


Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.


Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.


Square $AIME$ has sides of length $10$ units.  Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units.  Find the length of the altitude to $EM$ in $\triangle GEM$.


The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.


Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3, 4, 12, and 13, respectively, and $\angle CBA$ is a right angle. The area of the quadrilateral is

$\text{(A)} \ 32 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 39 \qquad \text{(D)} \ 42 \qquad \text{(E)} \ 48$

Solution: \boxed{(B)}


Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?
$\textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7}$


There is a unique positive real number $x$ such that the three numbers $\log_8{2x}$, $\log_4{x}$, and $\log_2{x}$, in that order, form a geometric progression with positive common ratio.  The number $x$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$


Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?


Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$


$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at
random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\frac14$.


Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$ Find $\frac{AE}{EB} + \frac{EB}{AE}.$


There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.


Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?

$\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$


In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$, and the altitude to these bases has length $\log 16$. The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.


An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.


Lilypads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is

$\text{(A)} \ \frac{\sqrt{3}}{2} \qquad \text{(B)} \ \frac{\sqrt{3}}{3} \qquad \text{(C)} \ \frac{\sqrt{2}}{2} \qquad \text{(D)} \ \frac12 \qquad \text{(E)} \ \frac23$

Solution: \boxed{(B)}


Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?


Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. What are the last three digits of $m+n\,$?


Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$.


There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is
$\text{(A)}\ 5\frac{3}{7}\qquad\text{(B)}\ 6\qquad\text{(C)}\ 6\frac{4}{7}\qquad\text{(D)}\ 7\qquad\text{(E)}\ 7\frac{3}{7}$

Solution: \boxed{(B)}


Let $x,$ $y,$ and $z$ be positive real numbers that satisfy
\[2\log_{x}(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \ne 0.\]
The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$


The largest whole number such that seven times the number is less than 100 is
$\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ 16$

Solution: \boxed{(C)}


Find the number of five-digit positive integers, $n$, that satisfy the following conditions:

(a) the number $n$ is divisible by $5,$

(b) the first and last digits of $n$ are equal, and

(c) the sum of the digits of $n$ is divisible by $5.$


Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.


A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$.


Find the smallest integer $k$ for which the conditions
(1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
(3) $a_9=k$
are satisfied by more than one sequence.


Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\cdot N$ contains no square of an integer.


Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.


A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$


Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations
\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}
There exist relatively prime positive integers $m$ and $n$ such that
\[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]Find $m + n$.


What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such that the links between $AB, AC$ and $BC$ are all air, all bus or all train.


The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\left(\tfrac{1+\sqrt{3}i}{2}\right)=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.


A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$


If $y=a+\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals:
$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$


In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.

[asy]  pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5);  draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]


($*$) Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

Solution: \boxed{EQ = FR}


Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is


The solutions to the system of equations

$\log_{225}x+\log_{64}y=4$
$\log_{x}225-\log_{y}64=1$
are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$.


Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left.  (Your answer should be an explicit function of $n$ in simplest form.)

Solution: \boxed{2^{n+1} - 2(n+1)}


A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$


A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.


Solution: \boxed{(D)}


Find the positive integer $n\,$ for which
\[\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994\]
(For real $x\,$, $\lfloor x\rfloor\,$ is the greatest integer $\le x.\,$)


There is a prime number $p$ such that $16p+1$ is the cube of a positive integer.  Find $p$.


Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.


Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle CDP$.


Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.


Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.


For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.


An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each
such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will
not have distinct positive real roots?
$\textbf{(A) }\frac{106}{121}\qquad \textbf{(B) }\frac{108}{121}\qquad \textbf{(C) }\frac{110}{121}\qquad \textbf{(D) }\frac{112}{121}\qquad \textbf{(E) }\text{none of these}$


Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.


Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$?


Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$.


Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$.  The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis.  In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis.  The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\alpha,\beta)$ in the new coordinate system.  Find $\alpha+\beta$.


A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$
so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is:


A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$. The minimum number of red chips is
$\textbf{(A) }24\qquad \textbf{(B) }33\qquad \textbf{(C) }45\qquad \textbf{(D) }54\qquad  \textbf{(E) }57$


Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$


A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result,  the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.


At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.

[asy] import cse5; unitsize(6mm); defaultpen(linewidth(.8pt)); dotfactor = 8; pathpen=black;  pair A = (0,0); pair B = 2*dir(54), C = 2*dir(126), D = 2*dir(198), E = 2*dir(270), F = 2*dir(342); pair G = 3.6*dir(18), H = 3.6*dir(90), I = 3.6*dir(162), J = 3.6*dir(234), K = 3.6*dir(306); pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P};  D(A--B--H--M); D(A--C--H--N); D(A--F--G--L); D(A--E--K--P); D(A--D--J--O); D(B--G--M); D(F--K--L); D(E--J--P); D(O--I--D); D(C--I--N); D(L--M--N--O--P--L);  dot(dotted);  [/asy]


Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.


What is the largest even integer that cannot be written as the sum of two odd composite numbers?


Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.


Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.


Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$


Let $S\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\,$ so that the union of the two subsets is $S\,$?  The order of selection does not matter; for example, the pair of subsets $\{a, c\},\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\},\{a, c\}.$


Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$.


Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.


Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$


As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
[asy] import graph; unitsize(5cm); pair A = (0.082, 0.378); pair B = (0.091, 0.649); pair C = (0.249, 0.899); pair D = (0.479, 0.939); pair E = (0.758, 0.893); pair F = (0.862, 0.658); pair G = (0.924, 0.403); pair H = (0.747, 0.194); pair I = (0.526, 0.075); pair J = (0.251, 0.170); pair K = (0.568, 0.234); pair L = (0.262, 0.449); pair M = (0.373, 0.813); pair N = (0.731, 0.813); pair O = (0.851, 0.461); path[] f; f[0] = A--B--C--M--L--cycle; f[1] = C--D--E--N--M--cycle; f[2] = E--F--G--O--N--cycle; f[3] = G--H--I--K--O--cycle; f[4] = I--J--A--L--K--cycle; f[5] = K--L--M--N--O--cycle; draw(f[0]); axialshade(f[1], white, M, gray(0.5), (C+2*D)/3); draw(f[1]); filldraw(f[2], gray); filldraw(f[3], gray); axialshade(f[4], white, L, gray(0.7), J); draw(f[4]); draw(f[5]); [/asy]
$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$


While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.


What is the least common multiple of ${\frac{1}{x}}$, $\frac{1}{2x}$, and $\frac{1}{3x}$ is $\frac{1}{6x}$?


Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\frac{\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p$.


For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.


The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3.  Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane.  Find the number of distinguishable acceptable arrangements.


In rectangle $ABCD$, $AB = 12$ and $BC = 10$.  Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$.  The length $EF$ can be expressed in the form $m \sqrt{n} - p$, where $m$, $n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime.  Find $m + n + p$.


Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.


Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is $10$. The area of $A 'B 'C'D'$ is


A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?


A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.


Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:

Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal.
Find the number of distinct T-grids.


An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$  What is the largest possible value of $p$?


A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$


Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$


Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined.  How many of the resulting figures can be folded into a topless cubical box?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$


Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\times$ $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.


Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.


Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is:
$\textbf{(A)}\ 4 \qquad  \textbf{(B) }\ 3 \qquad  \textbf{(C) }\ 2 \qquad  \textbf{(D) }\ 1 \qquad  \textbf{(E) }\ 0$

Solution: \boxed{(D) \ 1}


The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
$\textbf{(A)} ~18\qquad\textbf{(B)} ~27\qquad\textbf{(C)} ~36\qquad\textbf{(D)} ~45\qquad\textbf{(E)} ~54$


Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?


A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
$\textbf{(A) }48\qquad \textbf{(B) }44\qquad \textbf{(C) }40\qquad \textbf{(D) }36\qquad  \textbf{(E) }32$


At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?


A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?


A cube with side length  $100cm$  is filled with water and has a hole through which the water drains into a cylinder of radius  $100cm.$  If the water level in the cube is falling at a rate of  $1 \frac{cm}{s} ,$  how fast is the water level in the cylinder rising?


An equilateral triangle is inscribed inside of a circle of radius  $R$ . Find the side length of the triangle

Solution: \boxed{ R\sqrt{3} }


If  $0 < x < \frac{\pi}{2}$  and  $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$ , what is  $\frac{\sin 2x}{1 + \cos 2x}$ ?


Find the sum of all four-digit integers whose digits are a rearrangement of the digits  $1$ ,  $2$ ,  $3$ ,  $4$ , such as  $1234$ ,  $1432$ , or  $3124$ .


Juan wrote a natural number and Maria added a digit  $ 1$  to the left and a digit  $ 1$  to the right. Maria's number exceeds to the number of Juan in  $14789$ . Find the number of Juan.


The bisectors of the angles  $A$  and  $B$  of the triangle  $ABC$  intersect the sides  $BC$  and  $AC$  at points  $D$  and  $E$ . It is known that  $AE + BD = AB$ . Find the angle  $\angle C$ .


A point  $(x,y)$  in the first quadrant lies on a line with intercepts  $(a,0)$  and  $(0,b)$ , with  $a,b > 0$ . Rectangle  $M$  has vertices  $(0,0)$ ,  $(x,0)$ ,  $(x,y)$ , and  $(0,y)$ , while rectangle  $N$  has vertices  $(x,y)$ ,  $(x,b)$ ,  $(a,b)$ , and  $(a,y)$ . What is the ratio of the area of  $M$  to that of  $N$ ?

*Proposed by Eugene Chen*


Floyd looked at a standard  $12$  hour analogue clock at  $2\!:\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to  $247$  degrees. How many minutes after  $3\!:\!00$  was that?


Real numbers are chosen at random from the interval  $[0,1].$  If after choosing the  $n$ -th number the sum of the numbers so chosen first exceeds  $1$ , show that the expected value for  $n$  is  $e$ .

Solution: \boxed{ e }


Let  $z_1$  and  $z_2$  be the zeros of the polynomial  $f(x) = x^2 + 6x + 11$ . Compute  $(1 + z^2_1z_2)(1 + z_1z_2^2)$ .


Jerry's favorite number is  $97$ .  He knows all kinds of interesting facts about  $97$ :

- $97$  is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which  $97$  can be written as a difference of two perfect squares.
- There is only one way in which  $97$  can be written as a sum of two perfect squares.
- $\tfrac1{97}$  has exactly  $96$  digits in the [smallest] repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly  $97$  of his socks.

A repunit is a natural number whose digits are all  $1$ .  For instance, \begin{align*}&1,&11,&111,&1111,&\vdots\end{align*} are the four smallest repunits.  How many digits are there in the smallest repunit that is divisible by  $97?$


Two circles  $\mathcal{C}_1$  and  $\mathcal{C}_2$  with centers  $(1, 1)$  and  $(4, 5)$  and radii  $r_1 < r_2$ , respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of  $\mathcal{C}_1$  and  $\mathcal{C}_2$  is  $3$ . If the value of  $(r_2 - r_1)^2$  can be expressed as a common fraction in the form  $\tfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers, find  $m + n$ .


Show that there exists a composite number  $n$  such that  $a^n \equiv a \; \pmod{n}$  for all  $a \in \mathbb{Z}$ .

Solution: \boxed{ n = 561 }


Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?


For every positive integer  $k$  let  $a(k)$  be the largest integer such that  $2^{a(k)}$  divides  $k$ . For every positive integer  $n$  determine  $a(1)+a(2)+\cdots+a(2^n)$ .

Solution: \boxed{ 2^n - 1 }


In the accompanying figure  $ \overline{CE}$  and  $ \overline{DE}$  are equal chords of a circle with center  $ O$ . Arc  $ AB$  is a quarter-circle. Then the ratio of the area of triangle  $ CED$  to the area of triangle  $ AOB$  is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);

pair O = origin;
pair C = (-1,0);
pair D = (1,0);
pair E = (0,1);
pair A = dir(-135);
pair B = dir(-60);

draw(Circle(O,1));
draw(C--E--D--cycle);
draw(A--O--B--cycle);

label(" $A$ ",A,SW);
label(" $C$ ",C,W);
label(" $E$ ",E,N);
label(" $D$ ",D,NE);
label(" $B$ ",B,SE);
label(" $O$ ",O,N);[/asy]
 $ \textbf{(A)}\ \sqrt {2} : 1\qquad \textbf{(B)}\ \sqrt {3} : 1\qquad \textbf{(C)}\ 4 : 1\qquad \textbf{(D)}\ 3 : 1\qquad \textbf{(E)}\ 2 : 1$


What is the smallest possible natural number  $n$  for which the equation  $x^2 -nx + 2014 = 0$  has integer roots?


Compute the number of positive integers less than or equal to  $10000$  which are relatively prime to  $2014$ .


Suppose  $xy-5x+2y=30$ , where  $x$  and  $y$  are positive integers. Find the sum of all possible values of  $x$


Let  $\triangle ABC$  have  $\angle ABC=67^{\circ}$ . Point  $X$  is chosen such that  $AB = XC$ ,  $\angle{XAC}=32^\circ$ , and  $\angle{XCA}=35^\circ$ . Compute  $\angle{BAC}$  in degrees.

*Proposed by Raina Yang*


Paul fills in a  $7\times7$  grid with the numbers  $1$  through  $49$  in a random arrangement. He then erases his work and does the same thing again, to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?


 $\textbf{Problem 5.}$  Miguel has two clocks, one clock advances  $1$  minute per day and the other one goes  $15/10$  minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously?


A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime integers, find  $m+n$ .


Let  $a_1 = 1$  and  $a_{n+1} = a_n \cdot p_n$  for  $n \geq 1$  where  $p_n$  is the  $n$ th prime number, starting with  $p_1 = 2$ . Let  $\tau(x)$  be equal to the number of divisors of  $x$ . Find the remainder when  $$ \sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d) $$  is divided by 91 for positive integers  $d$ . Recall that  $d|a_n$  denotes that  $d$  divides  $a_n$ .

*Proposed by Minseok Eli Park (wolfpack)*


A group of women working together at the same rate can build a wall in  $45$  hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?


Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after  $t$  minutes he is at position  $(t,\sin t)$ .

Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for  $m$  minutes, his position is  $(m,\cos t)$ .

What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?


In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order  $1\  2\  3\  4\  5$ , which of the following could **not** be the order in which the secretary types them? $ \textbf{(A)}\ 1\ 2\ 3\ 4\ 5  \qquad\textbf{(B)}\ 2\ 4\ 3\ 5\ 1 \qquad\textbf{(C)}\ 3\ 2\ 4\ 1\ 5 \qquad\textbf{(D)}\ 4\ 5\ 2\ 3\ 1  \qquad\textbf{(E)}\ 5\ 4\ 3\ 2\ 1 $

Solution: \boxed{(D)}


Let  $ a$ ,  $ b$ ,  $ c$ ,  $ x$ ,  $ y$ , and  $ z$  be real numbers that satisfy the three equations
\begin{align*}
  13x +  by +  cz &= 0
   ax + 23y +  cz &= 0
   ax +  by + 42z &= 0.
\end{align*}Suppose that  $ a \ne 13$  and  $ x \ne 0$ .  What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?\]


Let  $T  =  TNFTPP$ .  When  properly  sorted,   $T  -  35$   math  books  on  a  shelf  are  arranged  in  alphabetical order  from  left  to  right.  An  eager  student  checked  out  and  read  all  of  them.  Unfortunately,  the student  did  not  realize  how  the  books  were  sorted,  and  so  after  finishing  the  student  put  the  books  back  on  the  shelf  in  a  random  order.  If  all  arrangements  are  equally  likely,  the  probability  that exactly   $6$   of  the  books  were  returned  to  their  correct  (original)  position  can  be  expressed  as   $\frac{m}{n}$ ,  where   $m$   and   $n$   are  relatively  prime  positive  integers.  Compute   $m  +  n$ .

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, here's the value of T$T=44$.


Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches?


Let  $m,n$  be natural numbers such that  $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of  $m+n$ .


Find value of   $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$   if  $x$ ,  $y$  and  $z$  are real numbers usch that  $xyz=1$


Suppose that  $N$  is a three digit number divisible by  $7$  such that upon removing its middle digit, the remaining two digit number is also divisible by  $7$ . What is the minimum possible value of  $N$ ?

*2019 CCA Math Bonanza Lightning Round #3.1*


Garfield and Odie are situated at  $(0,0)$  and  $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point  $(9, 12)$  at speeds of  $7$  and  $10$  units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as  $\frac{m}{\sqrt{n}}$  for relatively prime positive integers  $m$  and  $n$ . Find  $m+n$ .

*Proposed by **Th3Numb3rThr33***

Solution: \boxed{ 159 }


ABC  is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0).
(a) Let P  be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P.
(b) Let Q be the point on line BC  for which PQ is perpendicular to AB. Determine the length of AQ.
(c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.


In a bag are all natural numbers less than or equal to  $999$  whose digits sum to  $6$ . What is the probability of drawing a number from the bag that is divisible by  $11$ ?


Find all pairs of real numbers  $(a,b)$  so that there exists a polynomial  $P(x)$  with real coefficients and  $P(P(x))=x^4-8x^3+ax^2+bx+40$ .

Solution: \boxed{(28, -48), (2, 56)}


On a  $3 \times 3$  board the numbers from  $1$  to  $9$  are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.


Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?


For positive integers  $n$  and  $k$ , let  $\mho(n,k)$  be the number of distinct prime divisors of  $n$  that are at least  $k$ . For example,  $\mho(90, 3)=2$ , since the only prime factors of  $90$  that are at least  $3$  are  $3$  and  $5$ . Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

*Proposed by Daniel Zhu.*


Find the largest value of the expression   $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$  , where  $p,R, r$  is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.


Two toads named Gamakichi and Gamatatsu are sitting at the points  $(0,0)$  and  $(2,0)$  respectively. Their goal is to reach  $(5,5)$  and  $(7,5)$  respectively by making one unit jumps in positive  $x$  or  $y$  direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?


Let  $a_1 < a_2 < \cdots < a_k$  denote the sequence of all positive integers between  $1$  and  $91$  which are relatively prime to  $91$ , and set  $\omega = e^{2\pi i/91}$ . Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that  $S$  is a positive integer, compute the number of positive divisors of  $S$ .


Let  $f(x) = x^3 - 3x + b$  and  $g(x) = x^2 + bx -3$ , where  $b$  is a real number. What is the sum of all possible values of  $b$  for which the equations  $f(x)$  = 0 and  $g(x) = 0$  have a common root?


There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$  in the interval  $(0,\frac{\pi}{2})$  all of the form  $\frac{m_k}{n_k}\pi$ where  $m_k$  and  $n_k$  are relatively prime positive integers, for  $k = 1, 2, 3, . . . , s$ . Find  $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$ .


Let  $ABC$  be an acute triangle.  $PQRS$  is a rectangle with  $P$  on  $AB$ ,  $Q$  and  $R$  on  $BC$ , and  $S$  on  $AC$  such that  $PQRS$  has the largest area among all rectangles  $TUVW$  with  $T$  on  $AB$ ,  $U$  and  $V$  on  $BC$ , and  $W$  on  $AC$ . If  $D$  is the point on  $BC$  such that  $AD\perp BC$ , then  $PQ$  is the harmonic mean of  $\frac{AD}{DB}$  and  $\frac{AD}{DC}$ . What is  $BC$ ?

Note: The harmonic mean of two numbers  $a$  and  $b$  is the reciprocal of the arithmetic mean of the reciprocals of  $a$  and  $b$ .

*2017 CCA Math Bonanza Lightning Round #4.4*


A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.


Find all positive integers  $n$ , such that  $n$  is a perfect number and  $\varphi (n)$  is power of  $2$ .

*Note:a positive integer   $n$ , is called perfect if the sum of all its positive divisors is equal to  $2n$ .*

Solution: \boxed{ n = 6 }


For a real number  $x,$  let  $\lfloor x\rfloor$  denote the greatest integer less than or equal to  $x,$  and let  $\{x\} = x -\lfloor x\rfloor$  denote the fractional part of  $x.$  The sum of all real numbers  $\alpha$  that satisfy the equation   $$ \alpha^2+\{\alpha\}=21 $$   can be expressed in the form   $$ \frac{\sqrt{a}-\sqrt{b}}{c}-d $$   where  $a, b, c,$  and  $d$  are positive integers, and  $a$  and  $b$  are not divisible by the square of any prime. Compute  $a + b + c + d.$


Forty teams play a tournament in which every team plays every other team exactly once.  No ties occur, and each team has a  $50 \%$  chance of winning any game it plays.  The probability that no two teams win the same number of games is  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $\log_2 n.$


Suppose that  $(a_n)$  is a sequence of real numbers such that the series  $$ \sum_{n=1}^\infty\frac{a_n}n $$  is convergent. Show that the sequence  $$ b_n=\frac1n\sum^n_{j=1}a_j $$  is convergent and find its limit.


What is the smallest positive integer  $n$  such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true?

*2019 CCA Math Bonanza Team Round #5*


A triangle has sides of length  $48$ ,  $55$ , and  $73$ .  Let  $a$  and  $b$  be relatively prime positive integers such that  $a/b$  is the length of the shortest altitude of the triangle.  Find the value of  $a+b$ .


Two cubes  $A$  and  $B$  have different side lengths, such that the volume of cube  $A$  is numerically equal to the surface area of cube  $B$ . If the surface area of cube  $A$  is numerically equal to six times the side length of cube  $B$ , what is the ratio of the surface area of cube  $A$  to the volume of cube  $B$ ?


Jamie, Linda, and Don bought bundles of roses at a ﬂower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a ﬁxed price which was higher than the price that the ﬂower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made  $60$  proﬁt. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 proﬁt. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars proﬁt did Don make?


  $$ \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n}, $$   where  $\frac{m}{n}$  is irreducible.
a) Find  $m+n.$ b) Find the remainder of division of  $(m+3)^{1444}$  to  $n{}$ .


Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove  $2$ ,  $3$ ,  $7$ , or  $8$ marbles. If they can’t make a move, because there's  $0$  or  $1$  marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of  $n$  from  $1$  to  $1434$  does Evin lose the game?

*Proposed by Evin Liang*

Solution*Solution.*  $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such
that the total number of marbles removed is  $10$ . Thus, when the number of marbles is a multiple of  $10$ , the first player loses the game. We analyse this game based on the number of marbles modulo  $10$ :

If the number of marbles is  $0$  modulo  $10$ , the first player loses the game

If the number of marbles is  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ , the first player wins the game by moving to  $0$  modulo 10

If the number of marbles is  $5$  modulo  $10$ , the first player loses the game because every move leads to  $2$ ,  $3$ ,  $7$ , or  $8$  modulo  $10$ In summary, the first player loses if it is  $0$  mod 5, and wins if it is  $2$  or  $3$  mod  $5$ . Now we solve the remaining cases by induction. The first player loses when it is  $1$  modulo  $5$  and wins when it is  $4$  modulo  $5$ . The base case is when there is  $1$  marble, where the first player loses because there is no move. When it is  $4$  modulo  $5$ , then the first player can always remove  $3$  marbles and win by the inductive hypothesis. When it is  $1$  modulo  $5$ , every move results in  $3$  or  $4$  modulo  $5$ , which allows the other player to win by the inductive hypothesis.
Thus, Evin loses the game if n is  $0$  or  $1$  modulo  $5$ . There are  $\boxed{573}$  such values of  $n$  from  $1$  to  $1434$ .


Find the largest possible value in the real numbers of the term   $$ \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} $$   with  $x^2 + y^2 \ne 0$ .


The diagram shows a semicircle with diameter  $20$  and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is  $N\pi$ , where  $N$  is a positive integer. Find  $N$ .

Solution: \boxed{ N = 25 }


A square paper of side  $n$  is divided into  $n^2$  unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of  $n$ , the largest possible total length of the walls.

Solution: \boxed{ (n-1)^2 }


Let  $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime  $p$  satisfying  $f(p) = 418{,}195{,}493$ .

*Proposed by Eugene Chen*


Let  $ \alpha_1$ ,  $ \alpha_2$ ,  $ \ldots$ ,  $ \alpha_{2008}$  be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]


Let  $ABCD$  be a trapezoid of bases  $AB$  and  $CD$  . Let  $O$  be the intersection point of the diagonals  $AC$  and  $BD$ . If the area of the triangle  $ABC$  is  $150$  and the area of the triangle  $ACD$  is  $120$ , calculate the area of the triangle  $BCO$ .


Find the number of integer  $n$  from the set  $\{2000,2001,...,2010\}$  such that  $2^{2n} + 2^n + 5$  is divisible by  $7$ (A):  $0$ , (B):  $1$ , (C):  $2$ , (D):  $3$ , (E) None of the above.


Determine the smallest positive number  $a$  such that the number of all integers belonging to  $(a, 2016a]$  is  $2016$ .


Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers.  Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together.  The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?

*Proposed by Jacob Weiner*


A rectangular pool table has vertices at  $(0, 0) (12, 0) (0, 10),$  and  $(12, 10)$ . There are pockets only in the four corners. A ball is hit from  $(0, 0)$  along the line  $y = x$  and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.


A  $9 \times 9$  square consists of  $81$  unit squares. Some of these unit squares are painted black, and the others are painted white, such that each  $2 \times 3$  rectangle and each  $3 \times 2$  rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.


Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

*Proposed by Andrew Wu*


Let be given a triangle  $ABC$  and points  $D,M,N$  belong to  $BC,AB,AC$ , respectively. Suppose that  $MD$  is parallel to  $AC$  and  $ND$  is parallel to  $AB$ . If  $S_{\vartriangle BMD} = 9$  cm  $^2, S_{\vartriangle DNC} = 25$  cm $^2$ , compute  $S_{\vartriangle AMN}$ ?

Solution: \boxed{15 \, \text{cm}^2}


Let  $P(x,\ y)\ (x>0,\ y>0)$  be a point on the curve  $C: x^2-y^2=1$ . If  $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$ , then find the area bounded by the line  $OP$ , the  $x$  axis and the curve  $C$  in terms of  $u$ .

Solution: \boxed{\frac{1}{2} u}


Let  $T_1$  be an isosceles triangle with sides of length 8, 11, and 11.  Let  $T_2$  be an isosceles triangle with sides of length  $b$ , 1, and 1.  Suppose that the radius of the incircle of  $T_1$  divided by the radius of the circumcircle of  $T_1$  is equal to the radius of the incircle of  $T_2$  divided by the radius of the circumcircle of  $T_2$ .  Determine the largest possible value of  $b$ .


Given that the polynomial  $P(x) = x^5 - x^2 + 1$  has  $5$  roots  $r_1, r_2, r_3, r_4, r_5$ . Find the value of the product  $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$ , where  $Q(x) = x^2 + 1$ .


Let  $P$  be the portion of the graph of  $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$  located in the first quadrant (not including the  $x$  and  $y$  axes). Let the shortest possible distance between the origin and a point on  $P$  be  $d$ . Find  $\lfloor 1000d \rfloor$ .

*Proposed by **Th3Numb3rThr33***


Let  $a_1,a_2,\ldots$  be a sequence defined by  $a_1=a_2=1$  and  $a_{n+2}=a_{n+1}+a_n$  for  $n\geq 1$ .  Find \[\sum_{n=1}^\infty \dfrac{a_n}{4^{n+1}}.\]


Let  $n$  be an integer greater than two, and let  $A_1,A_2, \cdots , A_{2n}$  be pairwise distinct subsets of  $\{1, 2, ,n\}$ . Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where  $A_{2n+1}=A_1$  and  $|X|$  denote the number of elements in  $X.$

Solution: \boxed{ n }


In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers?  Regard  $m \cdot n$  and  $n \cdot m$  as the same product.


In equilateral triangle  $ABC$ , the midpoint of  $\overline{BC}$  is  $M$ . If the circumcircle of triangle  $MAB$  has area  $36\pi$ , then find the perimeter of the triangle.

*Proposed by Isabella Grabski*


Let  $S$  be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that  $S$  consists of precisely the integers of the form  $4^a(8b+7)$  where  $a$  and  $b$  are nonnegative integers. Find the smallest  $n\in\mathbb N$  such that  $n$  and  $n+1$  are both in  $S$ .


Suppose  $a_1, a_2, a_3, \dots$  is an increasing arithmetic progression of positive integers.  Given that  $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen*


There are  $ 2010 $  people sitting around a round table. First, we give one person  $ x $  a candy. Next, we give candies to  $1$  st person,  $1+2$  th person,  $ 1+2+3$  th person,  $\cdots$  , and  $1+2+\cdots + 2009 $  th person clockwise from  $ x $ . Find the number of people who get at least one candy.


Tatjana imagined a polynomial  $P(x)$  with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer  $k$  and Tatjana tells her the value of  $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.


Find the smallest natural number, which divides  $2^{n}+15$  for some natural number  $n$  and can be expressed in the form  $3x^2-4xy+3y^2$  for some integers  $x$  and  $y$ .


For a table  $n \times 9$  ( $n$  rows and  $9$  columns), determine the maximum of  $n$  that we can write one number in the set  $\left\{ {1,2,...,9} \right\}$  in each cell such that these conditions are satisfied:

1. Each row contains enough  $9$  numbers of the set  $\left\{ {1,2,...,9} \right\}$ .

2. Any two rows are distinct.

3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

Solution: \boxed{ 8! }


Circles  $k_1$  and  $k_2$  with radii  $r_1=6$  and  $r_2=3$  are externally tangent and touch a circle  $k$  with radius  $r=9$  from inside. A common external tangent of  $k_1$  and  $k_2$  intersects  $k$  at  $P$  and  $Q$ . Determine the length of  $PQ$ .


For a positive integer  $n$ , let  $d_n$  be the units digit of  $1 + 2 + \dots + n$ . Find the remainder when
\[\sum_{n=1}^{2017} d_n\]
is divided by  $1000$ .


Inside of the square  $ABCD$  the point  $P$  is given such that  $|PA|:|PB|:|PC|=1:2:3$ . Find  $\angle APB$ .


Consider the set  $S$  of  $100$  numbers:  $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers,  $a$  and  $b$ , are eliminated in  $S$ , and the number  $a+b+ab$  is added. Now, there are  $99$  numbers on  $S$ .
After doing this operation  $99$  times, there's only  $1$  number on  $S$ . What values can this number take?


You roll three fair six-sided dice.  Given that the highest number you rolled is a  $5$ , the expected value of the sum of the three dice can be written as  $\tfrac ab$  in simplest form.  Find  $a+b$ .


Let  $a,b$  be positive reals such that  $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$  and  $(a-b)^2=4(ab)^3$ . Find  $\log_a b$ .


Let  $p$  be an odd prime number less than  $10^5$ . Granite and Pomegranate play a game. First, Granite picks a integer  $c \in \{2,3,\dots,p-1\}$ .
Pomegranate then picks two integers  $d$  and  $x$ , defines  $f(t) = ct + d$ , and writes  $x$  on a sheet of paper.
Next, Granite writes  $f(x)$  on the paper, Pomegranate writes  $f(f(x))$ , Granite writes  $f(f(f(x)))$ , and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of  $p$ , and the player who wrote the most recent number wins. Find the sum of all  $p$  for which Pomegranate has a winning strategy.

*Proposed by Yang Liu*


The number  $2^{1997}$  has  $m$  decimal digits, while the number  $5^{1997}$  has  $n$  digits. Evaluate  $m+n$ .


The Fibonacci sequence is defined as follows:  $F_0=0$ ,  $F_1=1$ , and  $F_n=F_{n-1}+F_{n-2}$  for all integers  $n\ge 2$ . Find the smallest positive integer  $m$  such that  $F_m\equiv 0 \pmod {127}$  and  $F_{m+1}\equiv 1\pmod {127}$ .


Consider the  $10$ -digit number  $M=9876543210$ . We obtain  a new  $10$ -digit number from  $M$  according to the following rule: we can choose one or more disjoint pairs of adjacent digits in  $M$  and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from  $M=9\underline{87}6 \underline{54} 3210$  by interchanging the  $2$  underlined pairs, and keeping the others in their places, we get  $M_{1}=9786453210$ . Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from  $M$ .


Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.

*Anonymous Proposal*


Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?

*2022 CCA Math Bonanza Lightning Round 4.3*


Let  $M\subset \Bbb{N}^*$  such that  $|M|=2004.$ If no element of  $M$  is equal to the sum of any two elements of  $M,$ find the least value that the greatest element of  $M$  can take.

Solution: \boxed{ 4007 }


The sequence of real numbers  $\{a_n\}$ ,  $n \in \mathbb{N}$  satisfies the following condition:  $a_{n+1}=a_n(a_n+2)$  for any  $n \in \mathbb{N}$ . Find all possible values for  $a_{2004}$ .

Solution: \boxed{[-1, \infty)}


$ABC$  is a triangle with  $AB = 33$ ,  $AC = 21$  and  $BC = m$ , an integer. There are points  $D$ ,  $E$  on the sides  $AB$ ,  $AC$  respectively such that  $AD = DE = EC = n$ , an integer. Find  $m$ .


Find the number of permutations of the letters  $ABCDE$  where the letters  $A$  and  $B$  are not adjacent and the letters  $C$  and  $D$  are not adjacent. For example, count the permutations  $ACBDE$  and  $DEBCA$  but not  $ABCED$  or  $EDCBA$ .