Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Evaluate the expression \[ \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{ 11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, . \]

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$. Show that $FG$ is perpendicular to $IO$.

Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.

Compare numbers $A=5+2\sqrt{5}$ and $B=\sqrt{45+20\sqrt{5}}$

For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle. Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game. Determine all pairs $(m,n)$ for which Renate has a winning strategy.

Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.

We have $k$ switches arranged in a row, and each switch points up, down, left, or right. Whenever three successive switches all point in different directions, all three may be simultaneously turned so as to point in the fourth direction. Prove that this operation cannot be repeated infinitely many times.

Let $ABCD$ be a cyclic quadrilateral with opposite sides not parallel. Let $X$ and $Y$ be the intersections of $AB,CD$ and $AD,BC$ respectively. Let the angle bisector of $\angle AXD$ intersect $AD,BC$ at $E,F$ respectively, and let the angle bisectors of $\angle AYB$ intersect $AB,CD$ at $G,H$ respectively. Prove that $EFGH$ is a parallelogram.

Shade in some of the regions in the grid to the right so that the shaded area is equal for each of the 11 rows and columns. Regions must be fully shaded or fully unshaded, at least one region must be shaded, and the area of shaded regions must be at most half of the whole grid. [asy] size(200); defaultpen(linewidth(0.45)); real[][] arr = { {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0},}; for (int i=0; i<11; ++i){ for (int j=0; j<11; ++j){ if(arr[10-j][i] == 1){ fill((i,j)--(i+1,j)--(i+1, j+1)--(i,j+1)--cycle, grey); } } } draw((0,0)--(4,0)--(4,3)--(3,3)--(3,1)--(1,1)--(1,2)--(2,2)--(2,1)--(2,3)--(3,3)--(1,3)--(1,2)--(1,3)--(0,3)--(0,0)--(0,5)--(1,5)--(1,4)--(4,4)--(4,3)--(6,3)--(6,2)--(5,2)--(5,1)--(4,1)--(4,0)--(6,0)--(6,3)--(9,3)--(9,0)--(8,0)--(8,2)--(7,2)--(7,1)--(8,1)--(7,1)--(7,0)--(6,0)--(11,0)--(11,2)--(10,2)--(10,1)--(9,1)--(9,3)--(11,3)--(11,2)--(11,4)--(8,4)--(8,3)--(8,4)--(6,4)--(6,3)--(6,4)--(3,4)--(3,5)--(2,5)--(2,6)--(1,6)--(1,5)--(1,7)--(0,7)--(0,5)--(0,8)--(2,8)--(2,6)--(3,6)--(3,7)--(4,7)--(4,6)--(3,6)--(4,6)--(4,5)--(3,5)--(5,5)--(5,4)--(5,5)--(7,5)--(7,4)--(7,5)--(9,5)--(9,4)--(9,5)--(11,5)--(11,4)--(11,6)--(10,6)--(10,7)--(9,7)--(9,5)--(9,7)--(8,7)--(8,5)--(8,6)--(7,6)--(7,7)--(8,7)--(6,7)--(6,6)--(7,6)--(6,6)--(6,5)--(5,5)--(5,6)--(4,6)--(4,8)--(2,8)--(2,9)--(0,9)--(0,8)--(0,10)--(1,10)--(1,9)--(1,10)--(2,10)--(2,9)--(3,9)--(3,8)--(3,11)--(0,11)--(0,10)--(0,11)--(5,11)--(5,10)--(3,10)--(4,10)--(4,8)--(5,8)--(5,6)--(5,7)--(6,7)--(6,8)--(5,8)--(5,9)--(4,9)--(6,9)--(6,10)--(5,10)--(5,11)--(8,11)--(8,10)--(6,10)--(7,10)--(7,9)--(6,9)--(7,9)--(7,8)--(6,8)--(7,8)--(7,7)--(8,7)--(8,8)--(7,8)--(8,8)--(8,10)--(9,10)--(9,9)--(8,9)--(10,9)--(10,8)--(9,8)--(9,7)--(10,7)--(10,6)--(11,6)--(11,8)--(10,8)--(11,8)--(11,10)--(9,10)--(11,10)--(11,11)--(8,11)); [/asy] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$. [i]Proposed by Marcin Kuczma, Poland [/i]

Let $(a_n)_{n=1}^{\infty}$ be a real sequence such that $a_1=1, a_3=4$ and for every $n\geq 2$, $a_{n+1}+a_{n-1}=2a_n+1$. What is $a_{2011}$? $\textbf{(A)}\ 2^{2010} \qquad\textbf{(B)}\ 2021056 \qquad\textbf{(C)}\ 1010528 \qquad\textbf{(D)}\ 3016 \qquad\textbf{(E)}\ 2011$

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that: a) $DE$ is the bisector of the $\angle FDC$ b) $FG$ is the bisector of the $\angle AFD$ c) the point $G$ is the center of the circle inscribed in the triangle $ADF$. (Ercole Suppa, Italy)

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Kolya and Dima play a game on an $8\times 8$ board, making moves in turn. During his turn, Kolya must put one cross in any empty cell (i.e., in a cell in which a cross has not yet been drawn and which has not yet been covered with a domino). Dima must cover two adjacent cells with a domino (which are not yet covered with other dominoes), in which there are an even number of crosses in total (0 or 2). The one who can't make a move loses. Which of does the player have a winning strategy, if [list=a] [*]Dima makes the first move? [*]Kolya makes the first move? [/list] [i]Proposed by M. Didin[/i]

Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Given an isosceles triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$. Let $ M$ be the midpoint of $ AB$. The line passing through $ A$ and perpendicular to $ CM$ intersects the side $ BC$ at $ P$. Prove that $ \angle AMC \equal{} \angle BMP$.

An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.

Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent: $(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$ $(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $AA',BB',CC'$ be parallel lines not lying in the same plane. Denote $U$ the intersection of the planes $A'BC,AB'C,ABC'$ and $V$ the intersection of the planes $AB'C',A'BC',A'B'C$. Show that the line $UV$ is parallel with $AA'$.

Given triangle $ABC$ in which $\angle CAB= 30^{\circ}$ and $\angle ACB=60^{\circ}$. On the ray $AB$ a point $D$ is chosen, and on the ray $CB$ a point $E$ is chosen such that $\angle BDE=60^{\circ}$. Lines $AC$ and $DE$ intersect at $F$. Prove that the circumcircle of $AEF$ passes through a fixed point, which is different from $A$ and does not depend on $D$.