Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]

If $P$ is a function such that $P(2x) = 2^{-3}P(x) + 1$, find $P(0)$.

Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation $$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$ for all $x,y \in \mathbb{R}. $

For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$.

Inside the triangle $T$ there are three other triangles that do not have common points. Is it true that one can choose such a point inside $T$ and draw three rays from it so that the triangle breaks into three parts, in each of which there will be one triangle?

We have an infinite sequence of integers $\{x_n\}$, such that $x_1 = 1$, and, for all $n \ge 1$, it holds that $x_n < x_{n+1} \le 2n$. Prove that there are two terms of the sequence,$ x_r$ and $x_s$, such that $x_r - x_s = 2018$.

A positive integer $n$ is lucky if $2n+1$, $3n+1$, and $4n+1$ are all composite numbers. Compute the smallest lucky number. [i]Proposed by Michael Tang[/i]

For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?

Let points $A$ and $B$ be on circle $\omega$ centered at $O$. Suppose that $\omega_A$ and $\omega_B$ are circles not containing $O$ which are internally tangent to $\omega$ at $A$ and $B$, respectively. Let $\omega_A$ and $\omega_B$ intersect at $C$ and $D$ such that $D$ is inside triangle $ABC$. Suppose that line $BC$ meets $\omega$ again at $E$ and let line $EA$ intersect $\omega_A$ at $F$. If $ FC \perp CD $, prove that $O$, $C$, and $D$ are collinear.

What is the smallest integer $a_0$ such that, for every positive integer $n$, there exists a sequence of positive integers $a_0, a_1, ..., a_{n-1}, a_n$ such that the first $n-1$ are all distinct, $a_0 = a_n$, and for $0 \le i \le n -1$, $a_i^{a_{i+1}}$ ends in the digits $\overline{0a_i}$ when expressed without leading zeros in base $10$.

$(-1)^{5^2} + 1^{2^5} =$ $\textbf{(A)}\ -7 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 57$

For every nonempty set $M{}$ of integers denote $S(M)$ the sum of all its elements. Let $A=\{a_1,a_2,\ldots,a_{11}\}$ be a set of positive integers with the properties: 1) $a_1<a_2<\ldots<a_{11};$ 2) for every positive integer $n\leq 1500$ there is a subset $M{}$ of $A{}$ for which $S(M)=n.$ Find the smallest possible value of $a_{10}.$

Determine all triples $(x, y, z)$ of integers that satisfy the equation $x^2+ y^2+ z^2 - xy - yz - zx = 3$

Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?

A deck of $8056$ cards has $2014$ ranks numbered $1$–$2014$. Each rank has four suits - hearts, diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rank and the same suit. How many subsets of the set of cards in this deck have cards from an odd number of distinct ranks?

Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: \[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \] The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.

Let $\triangle ABC$ be an acute triangle, $H$ is its orthocenter. $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ intersect $\triangle ABC$'s circumcircle at $A',B',C'$ respectively. Find the range (minimum value and the maximum upper bound) of $$\dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'}$$

On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at least a marked square will be deleted. Kostya wants to be left as few as possible columns in the table, and Lesha wants there to be as many as possible columns in the table. What is the largest number of columns Lesha can guarantee to leave?

Rank the [i]magnitudes[/i] of the average acceleration during the ten second interval. (A) $\text{I} > \text{II} > \text{III}$ (B) $\text{II} > \text{I} > \text{III}$ (C) $\text{III} > \text{II} > \text{I}$ (D) $\text{I} > \text{II = III}$ (E) $\text{I = II = III}$

In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]"Arc $BC$ of the circumcircle" means "the arc with endpoints $B$ and $C$ not containing $A$".[/list][/hide]

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

In acute triangle $ABC$, let $D$, $E$, and $F$ be the feet of the altitudes from $A$, $B$, and $C$ respectively, and let $L$, $M$, and $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Lines $DE$ and $NL$ intersect at $X$, lines $DF$ and $LM$ intersect at $Y$, and lines $XY$ and $BC$ intersect at $Z$. Find $\frac{ZB}{ZC}$ in terms of $AB$, $AC$, and $BC$.

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum? $\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Two wheels with fixed hubs, each having a mass of $1 \text{ kg}$, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is $I = mR^2$. In order to impart identical angular accelerations about their respective hubs, how large must $F_2$ be? [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw(circle((0,0),0.5)); draw((1, 0.5)--(0,0.5)--(0,-0.5),BeginArrow); draw((-0.5,0)--(0.5,0)); draw((-0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(0.5*sqrt(2)/2,-0.5*sqrt(2)/2)); draw((0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(-0.5*sqrt(2)/2,-0.5*sqrt(2)/2)); label("$R$ = 0.5 m", (0, -0.5),S); label("$F_1$ = 1 N",(1,0.5),N); draw(circle((3,0.5),1)); draw((4.5,1.5)--(3,1.5)--(3,-0.5),BeginArrow); draw((2,0.5)--(4,0.5)); draw((3-sqrt(2)/2, 0.5+sqrt(2)/2)--(3+sqrt(2)/2, 0.5-sqrt(2)/2)); draw((3+sqrt(2)/2, 0.5+sqrt(2)/2)--(3-sqrt(2)/2,0.5-sqrt(2)/2)); label("$F_2$", (4.5, 1.5), N); label("$R$ = 1 m",(3, -0.5),S); [/asy] $ \textbf{(A)}\ 0.25\text{ N}\qquad\textbf{(B)}\ 0.5\text{ N}\qquad\textbf{(C)}\ 1\text{ N}\qquad\textbf{(D)}\ 2\text{ N}\qquad\textbf{(E)}\ 4\text{ N} $

Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt3 PS$ and the area of $PQRS$ is $28\sqrt3$, what is the length of $PC$?