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

Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]

Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of a regular $2013$-gon. Adjacent islands are joined with exactly two bridges. Christine starts on one of the islands with the intention of burning all the bridges. Each minute, if the island she is on has at least one bridge still joined to it, she randomly selects one such bridge, crosses it, and immediately burns it. Otherwise, she stops. If the probability Christine burns all the bridges before she stops can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the remainder when $m+n$ is divided by $1000$. [i]Evan Chen[/i]

Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$. [i]Based on a proposal by Matthew Babbitt[/i]

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$ Prove that (A) $\lambda _n$ is an integer (B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .

In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x \plus{} 1$ and $ \frac {x}{x \plus{} 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

If $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$, where $a,b,c$ are positive integers with no common factor, prove that $(a +b)$ is a square.

A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$? [asy]size(200); defaultpen(fontsize(10)); label("1", origin); label("3", (2,0)); label("5", (4,0)); label("$\cdots$", (6,0)); label("97", (8,0)); label("99", (10,0)); label("4", (1,-1)); label("8", (3,-1)); label("12", (5,-1)); label("196", (9,-1)); label(rotate(90)*"$\cdots$", (6,-2));[/asy]

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$. If the chosen sequence is $a_1, a_2, ..., a_k$ ($k$ is not fixed), then at the $i^{th}$ step, we choose the first a$_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$. The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Find all positive integers $n$ such that for all positive integers $m$, $1<m<n$, relatively prime to $n$, $m$ must be a prime number.

Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is $\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$

Let $k$ be a positive integer. Prove that there exists a positive integer $\ell$ with the following property: if $m$ and $n$ are positive integers relatively prime to $\ell$ such that $m^m\equiv n^n \pmod{\ell}$, then $m\equiv n \pmod k$.

Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.