Let a quardilateral $ABCD$ with $AB=AD$ and $\widehat B=\widehat D=90$. At $CD$ we take point $E$ and at $BC$ we take point $Z$ such that $AE\bot DZ$. Prove that $AZ\bot BE$

Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?

Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\pi/81$. Find the remainder when $N$ is divided by $23$.

Given are $n + 1$ points $P_1, P_2,..., P_n$ and $Q$ in the plane, no three collinear. For any two different points $P_i$ and $P_j$ , there is a point $P_k$ such that the point $Q$ lies inside the triangle $P_iP_jP_k$. Prove that $n$ is an odd number.

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

For $n\ge 2$, find the number of integers $x$ with $0\le x<n$, such that $x^2$ leaves a remainder of $1$ when divided by $n$.

Cyclic quadrilateral $ABCD$ has circumcircle $(O)$. Points $M$ and $N$ are the midpoints of $BC$ and $CD$, and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$. Let $EN$ meet $FM$ at $P$. Denote $S$ as the circumcenter of $\triangle PEF$. Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$.

No math tournament exam is complete without a self referencing question. What is the product of the smallest prime factor of the number of words in this problem times the largest prime factor of the number of words in this problem

In a convex set $S$ that contains the origin it is possible to draw $n$ disjoint unit circles such that viewing from the origin non of the unit circles blocks out a part of another (or a complete) unit circle. Prove that the area of $S$ is at least $\frac{n^2}{100}$.

Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition $f(f(x)+x+y)=2x+f(y)$ for any real $x$ and $y$.

Find $\sum_{n=2}^\infty\frac{n^2-2n-4}{n^4+4n^2+16}$.

A [i]weak binary representation[/i] of a nonnegative integer $n$ is a representation $n=a_0+2\cdot a_1+2^2\cdot a_2+\dots$ such that $a_i\in\{0,1,2,3,4,5\}$. Determine the number of such representations for $513$.

Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$

Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?

In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); real e=350,c=55; pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot); path tangent=A--B; pair P=waypoint(tangent,abs(A-D)/abs(A-B)); draw(unitcircle^^C--B--A--E); dot(A^^B^^C^^D^^E^^P,linewidth(2)); label("$O$",O,dir(290)); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,dir(120)); label("$E$",E,SE); label("$P$",P,SW);[/asy] $ \textbf{(A)} AP^2 = PB \times AB\qquad$ $\textbf{(B)}\ AP \times DO = PB \times AD\qquad$ $\textbf{(C)}\ AB^2 = AD \times DE\qquad$ $\textbf{(D)}\ AB \times AD = OB \times AO\qquad$ $\textbf{(E)}\ \text{none of these} $

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

A sphere is inscribed in a tetrahedron $ABCD$. The tangent planes to the sphere parallel to the faces of the tetrahedron cut off four smaller tetrahedra. Prove that sum of all the $24$ edges of the smaller tetrahedra equals twice the sum of edges of the tetrahedron $ABCD$.

n points are taken in the plane, no three collinear. Some of the line segments between the points are painted red and some are painted blue, so that between any two points there is a unique path along colored edges. Show that the uncolored edges can be painted (each edge either red or blue) so that all triangles have an odd number of red sides.

A starship is located in a halfspace at the distance $a$ from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than: $a)14a$ $b)13a$?

Two equal circles $\Omega_1$ and $\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays were drawn from $M$, lying in the same half-plane wrt $AB$ (see figure). The first ray intersects the circles $\Omega_1$ and $\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects them at points $Y_1$ and $Y_2$, respectively. Let $C$ be the intersection point of straight lines $AX_1$ and $BY_2$, and let $D$ be the intersection point of straight lines $AX_2$ and $BY_1$. Prove that $CD \parallel AB$. [img]https://cdn.artofproblemsolving.com/attachments/4/a/fae047c3956d8b30f15a9d88e8d12e5f4d48ec.png[/img]

The hypotenuse $AB$ of a right-angled triangle $ABC$ touches the corresponding excircle $\omega$ at point $T$. Point $S$ is symmetrical $T$ relative to the bisector of angle $C$, $CH$ is the height of the triangle. Prove that the circumcircle of triangle $CSH$ touches the circle $\omega$.

A regular tetrahedron has a square shadow of area $16$ when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron. (For example, the shadow of a sphere with radius 1 onto a flat surface is a disk of radius $1.$)

We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$ Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.

Let $ABC$ be a right triangle with right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the circumcircle of $\triangle ABC$ at points $A$ and $B$ intersect at $D$. Let $E$ be the point on side $\overline{AB}$ such that $\overline{AD} \parallel \overline{CE}$. Then $DE^2$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Awesome_guy[/i]

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.