The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.

Let $\omega$ be a circle with the center $O$ and $A$ and $C$ be two different points on $\omega$. For any third point $P$ of the circle let $X$ and $Y$ be the midpoints of the segments $AP$ and $CP$. Finally, let $H$ be the orthocenter (the point of intersection of the altitudes) of the triangle $OXY$ . Prove that the position of the point H does not depend on the choice of $P$. (Artemiy Sokolov)

A $5779$-dimensional polytope is call a [b]$k$-tope[/b] if it has exactly $k$ $5778$-dimensional faces. Find all sequences $b_{5780}, b_{5781}, \dots, b_{11558}$ of nonnegative integers, not all $0$, such that the following condition holds: It is possible to tesselate every $5779$-dimensional polytope with [u]convex[/u] $5779$-dimensional polytopes, such that the number of $k$-topes in the tessellation is proportional to $b_k$, while there are no $k$-topes in the tessellation if $k\notin \{5780, 5781, \dots, 11558\}$.

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

Let $t$ be the area of a regular pentagon with each side equal to $1$. Let $P(x)=0$ be the polynomial equation with least degree, having integer coefficients, satisfied by $x=t$ and the $\gcd$ of all the coefficients equal to $1$. If $M$ is the sum of the absolute values of the coefficients of $P(x)$, What is the integer closest to $\sqrt{M}$ ? ($\sin 18^{\circ}=(\sqrt{5}-1)/2$)

Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.

Some of $100$ towns of a kingdom are connected by roads.It is known that for each two towns $A$ and $B$ connected by a road there is a town $C$ which is not connected by a road with at least one of the towns $A$ and $B$. Determine the maximum possible number of roads in the kingdom.

Let $ABC$ be a scalene triangle with circumcenter $O$, and let $D$ and $E$ be points inside angle $\measuredangle BAC$ such that $A$ lies on line $DE$, and $\angle ADB=\angle CBA$ and $\angle AEC=\angle BCA$. Let $M$ be the midpoint of $BC$ and $K$ be a point such that $OK$ is perpendicular to $AO$ and $\angle BAK=\angle MAC$. Finally, let $P$ be the intersection of the perpendicular bisectors of $BD$ and $CE$. Show that $KO=KP$. [i]Proposed by Victor Domínguez[/i]

Do there exist $2022$ polynomials with real coefficients, each of degree equal to $2021$, so that the $2021 \cdot 2022 + 1$ coefficients in their product are equal?

Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.

A whole number larger than $2$ leaves a remainder of $2$ when divided by each of the numbers $3, 4, 5$ and $6$. The smallest such number lies between which two numbers? $\textbf{(A)}\ 40\text{ and }49\qquad \textbf{(B)}\ 60\text{ and }79\qquad \textbf{(C)}\ 100\text{ and }129\qquad \textbf{(D)}\ 210\text{ and }249\qquad \textbf{(E)}\ 320\text{ and }369$

Homer gives mathematicians Patty and Selma each a different integer, not known to the other or to you. Homer tells them, within each other’s hearing, that the number given to Patty is the product $ab$ of the positive integers $a$ and $b$, and that the number given to Selma is the sum $a + b$ of the same numbers $a$ and $b$, where $b > a > 1.$ He doesn’t, however, tell Patty or Selma the numbers $a$ and $b.$ The following (honest) conversation then takes place: Patty: “I can’t tell what numbers $a$ and $b$ are.” Selma: “I knew before that you couldn’t tell.” Patty: “In that case, I now know what $a$ and $b$ are.” Selma: “Now I also know what $a$ and $b$ are.” Supposing that Homer tells you (but neither Patty nor Selma) that neither $a$ nor $b$ is greater than 20, find $a$ and $b$, and prove your answer can result in the conversation above.

An $n\times n$ square ($n$ is a positive integer) consists of $n^2$ unit squares.A $\emph{monotonous path}$ in this square is a path of length $2n$ beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each $k$, $0\leq k\leq 2n-1$, let $S_k$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2n-1$.Prove that all $S_k$ contain equal number of elements.

Most enzymes are a type of $ \textbf{(A) } \text{Carbohydrate} \qquad\textbf{(B) } \text{Lipid} \qquad\textbf{(C) } \text{Nucleic Acid} \qquad\textbf{(D) } \text{Protein} \qquad $

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$. ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]

Let $m,n$ be two positive integers and $m \geq 2$. We know that for every positive integer $a$ such that $\gcd(a,n)=1$ we have $n|a^m-1$. Prove that $n \leq 4m(2^m-1)$.

Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

For a given parameter $m$, solve the equation \[x(x + 1)(x + 2)(x + 3) + 1 - m = 0.\]

Given a point $P = (p_1, p_2, \ldots, p_n)$ in $n$-dimensional space . Find point $X = (x_1, x_2, \ldots, x_n)$, such that $x_1 \leq x_2 \leq\cdots \leq x_n$ and $\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}$ is minimal.

Let $M \subset R$ be a finite set containing at least two elements. We say that the function $f$ has property $P$ if $f : M \to M$ and there are $a \in R^*$ and $b \in R$ such that $f(x) = ax + b$. (a) Show that there is at least a function having property $P$. (b) Show that there are at most two functions having property $P$. (c) If $M$ has $2003$ elements with sum $0$ and if there are two functions with property $P$, prove that $0 \in M$.

Let be a function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that satisfies the relation $$ \sqrt{x^2-x+1}\le f(x) e^{f(x)}\le \sqrt{x^2+x+1} , $$ for any positive real number $ x. $ Prove that [b]a)[/b] $ \lim_{x\to\infty } f(x)=\infty . $ [b]b)[/b] $ \lim_{x\to\infty } (1/x)^{1/f(x)} =1/e. $

If $ 1 \minus{} y$ is used as an approximation to the value of $ \frac {1}{1 \plus{} y}$, $ |y| < 1$, the ratio of the error made to the correct value is: $ \textbf{(A)}\ y \qquad \textbf{(B)}\ y^2 \qquad \textbf{(C)}\ \frac {1}{1 \plus{} y} \qquad \textbf{(D)}\ \frac {y}{1 \plus{} y} \qquad \textbf{(E)}\ \frac {y^2}{1 \plus{} y}\qquad$

(A.Zaslavsky, 9--10) Quadrilateral $ ABCD$ is circumscribed arounda circle with center $ I$. Prove that the projections of points $ B$ and $ D$ to the lines $ IA$ and $ IC$ lie on a single circle.

A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero. [i]Proposed by Morteza Saghafian[/i]

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $