Let $n > 1$ be a fixed integer. On an infinite row of squares, there are $n$ stones on square $1$ and no stones on squares $2$, $3$, $4$, $\ldots$. Curious George plays a game in which a [i]move[/i] consists of taking two adjacent piles of sizes $a$ and $b$, where $a-b$ is a nonzero even integer, and transferring stones to equalize the piles (so that both piles have $\frac{a+b}{2}$ stones). The game ends when no more moves can be made. George wants to analyze the number of moves it takes to end the game. (a) Suppose George wants to end the game as quickly as possible. How many moves will it take him? (b) Suppose George wants to end the game as slowly as possible. Show that for all $n > 2$, the game will end after at most $\frac{3}{16}n^2$ moves. [i]Scoring note:[/i] For part (b), partial credit will be awarded for correct proofs of weaker bounds, eg. $\frac{1}{4}n^2$, $n^k$, or $k^n$ (for some $k \geq 2$).

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Let $ABCD$ be a square of side length $1$, and let $P_1, P_2$ and $P_3$ be points on the perimeter such that $\angle P_1P_2P_3 = 90^\circ$ and $P_1, P_2, P_3$ lie on different sides of the square. As these points vary, the locus of the circumcenter of $\triangle P_1P_2P_3$ is a region $\mathcal{R}$; what is the area of $\mathcal{R}$?

A convex $n{}$-gon with $n > 4$ is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the $n{}$-gon. [i]Maxim Didin[/i]

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.

Let $ABC$ be a right triangle with hypotenuse $AC$. Let $G$ be the centroid of this triangle and suppose that we have $AG^2 + BG^2 + CG^2 = 156$. Find $AC^2$.

Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

For a given integer $k \geq 1$, find all $k$-tuples of positive integers $(n_1,n_2,...,n_k)$ with $\text{GCD}(n_1,n_2,...,n_k) = 1$ and $n_2|(n_1+1)^{n_1}-1$, $n_3|(n_2+1)^{n_2}-1$, ... , $n_1|(n_k+1)^{n_k}-1$. [i]Authored by Pavel Dimovski[/i]

Given a sequence $(a_n)$ of non-negative real numbers such that $a_{n+m}\leq a_{n} a_{m} $ for all pairs of positive integers $m$ and $n,$ prove that the sequence $(\sqrt[n]{a_n })$ converges.

Suppose that $m$ and $k$ are positive integers. Determine the number of sequences $x_1, x_2, x_3, \dots , x_{m-1}, x_m$ with [list] [*]$x_i$ an integer for $i = 1, 2, 3, \dots , m$, [*]$1\le x_i \le k$ for $i = 1, 2, 3, \dots , m$, [*]$x_1\neq x_m$, and [*]no two consecutive terms equal.[/list]

Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

Let $ABC$ be an acute triangle with $AB>AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and $$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$ holds, the line $XY$ passes through $D$.

Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$

Prove the identity \[ 1 \plus{} \frac{1}{2} \minus{} \frac{2}{3} \plus{} \frac{1}{4} \plus{} \frac{1}{5} \minus{} \frac{2}{6} \plus{} \ldots \plus{} \frac{1}{478} \plus{} \frac{1}{479} \minus{} \frac{2}{480} \equal{} 2 \cdot \sum^{159}_{k\equal{}0} \frac{641}{(161\plus{}k) \cdot (480\minus{}k)}.\]

A number is called a palindromic number if its decimal representation read from the left to the right is the same as read from the right to the left. Let $(x_n)$ be the increasing sequence of all palindromic numbers. Determine all primes, which are divisors of at least one of the differences $x_{n+1} - x_n$.

There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.

As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$. [img]https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png[/img]

$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter. prove that $\widehat{OIH}$ is obtuse.

Prove the identity: a) $(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2$ b) $(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =$ $(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2$.

Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$

Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.