Let $ABC$ be a triangle with incenter $I.$ The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA,$ and $AB$ at points $D, E,$ and $F,$ respectively. Let $D'$ be the reflection of $D$ over $I.$ Let $P$ be a point on $\omega$ such that $\angle{ADP}=90^\circ.$ $\mathcal{H}$ is a hyperbola passing through $D', E, F, I,$ and $P.$ Given that $\angle{BAD}=45^\circ$ and $\angle{CAD}=30^\circ,$ the acute angle between the asymptotes of $\mathcal{H}$ can be expressed as $\left(\tfrac{m}{n}\right)^\circ,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number n when read from left to right. Compute the expected value of $n$.

$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information. [I]Proposed by Dominic Yeo, United Kingdom.[/i]

Prove the inequality $2\sqrt{(p-b)(p-c)}\le a$, where $a, b, c$ are the lengths of the sides, and $p$ is the semiperimeter of some triangle..

Find all values of the real parameter $a$, for which the system $(|x| + |y| - 2)^2 = 1$ $y = ax + 5$ has exactly three solutions

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$. Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$. What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?

If $x,y,z$ are sides of a triangle, prove that \[ | x^2(y-z) + y^2(z-x) + z^2(x-y) | < xyz. \]

Already the complete problem: Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers: $\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$ One idea? :arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]

Let $p$ be a nonnegative integer and let $n=2^p$. Consider all subsets $A$ of the set $\{1,2,\ldots,n\}$ with the property that, whenever $x\in A$, $2x\notin A$. Find the maximum number of elements that such a set $A$ can have.

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.