If $ a^x \equal{} c^q \equal{} b$ and $ c^y \equal{} a^z \equal{} d$, then $ \textbf{(A)}\ xy \equal{} qz \qquad\textbf{(B)}\ \frac {x}{y} \equal{} \frac {q}{z} \qquad\textbf{(C)}\ x \plus{} y \equal{} q \plus{} z \qquad\textbf{(D)}\ x \minus{} y \equal{} q \minus{} z$ $ \textbf{(E)}\ x^y \equal{} q^z$

Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction. Prove that $h$ is remarkable if and only if it is prime. (Recall that an common fraction has an integer numerator and a natural denominator.)

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

All vertices of a polygon $P$ lie at points with integer coordinates in the plane, and all sides of $P$ have integer lengths. Prove that the perimeter of $P$ must be an even number.

Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.

Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.

Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive. [i]Proposed by Mehtaab Sawhney[/i]

A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.

In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$

Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel. [img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]

Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and \[f(x^2-y^2)=f(x)f(y) \] for all $x,y\in\mathbb{N}_0$ with $x>y$.

Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$. Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in the red box is equal to twice the sum of the numbers of the cards in the blue box. Clarification: Some of Andrés's letters can be left out of the boxes.

Find a polynomial $P$ of lowest possible degree such that (a) $P$ has integer coefficients, (b) all roots of $P$ are integers, (c) $P(0) = -1$, (d) $P(3) = 128$.

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The natural numbers $m$ and $k$ satisfy the equality $$1001 \cdot 1002 \cdot ... \cdot 2010 \cdot 2011 = 2^m (2k + 1)$$. Find the number $m$.

Let $m$ and $n$ be positive integers. Determine the number of ways to fill each cell of an $m \times n $ table with a number from $\{-2, -1, 1, 2\}$ so that the product of the numbers written in each row and column is $-2$.

Two types of tiles, depicted on the figure below, are given. [img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img] Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\ [i]Proposed by Art Waeterschoot[/i]

Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center at $B$ intersects $\omega$ at C and $AB$ at $D$. The line $CD$ intersects $\omega$ at a point $E$ ($E\ne C$). The intersection of lines $OE$ and $BC$ is $F$. (a) Prove that triangle $OBF$ is isosceles. (b) If $D$ is the midpoint of $OB$, find the value of the ratio $\frac{FB}{BD}$.

It is given parallelogram $ABCD$. On it's sides $AB, BC, CD, DA$ are chosen points $E, F, G, H$ such that area of $EFGH$ is half of the area of $ABCD$. Show that at least one of the quadrilaterals $ABFH$ and $AEGD$ is parallelogram.

Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$.

Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.