Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strategy.

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.

Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$ Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules: [b](1) [/b]First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. [b](2)[/b] Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn. Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.

Let $S$ be the set of bijections \[ T\colon\{1,\,2,\,3\}\times\{1,\,2,\,\ldots,\,2024\}\to\{1,\,2,\,\ldots,\,6072\} \] such that $T(1,\,j)<T(2,\,j)<T(3,\,j)$ for all $j\in\{1,\,2,\,\ldots,\,2024\}$ and $T(i,\,j)<T(i,\,j+1)$ for all $i\in\{1,\,2,\,3\}$ and $j\in\{1,\,2,\,\ldots,\,2023\}$. Do there exist $a$ and $c$ in $\{1,\,2,\,3\}$ and $b$ and $d$ in $\{1,\,2,\,\ldots,\,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,\,b)<T(c,\,d)$ is at least $1/3$ and at most $2/3$.

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.

State consists of $2021$ cities, between some of them there are direct flights. Each pair of cities has not more than one flight, every flight belongs to one of $2021$ companies. Call a group of cities [i]incomplete[/i], if at least one company doesn't have any flights between cities of the group. Find the maximum positive integer $m$, so that one can always find an incomplete group of $m$ cities.

The circles $\omega$ and $\Omega$ touch each other internally at $A{}$. In a larger circle $\Omega$ consider the chord $CD$ which touches $\omega$ at $B{}$. It is known that the chord $AB$ is not a diameter of $\omega$. The point $M{}$ is the middle of the segment $AB{}$. Prove that the circumcircle of the triangle $CMD$ passes through the center of $\omega$. [i]Proposed by P. Bibikov[/i]

Given a circle $G$ with center $O$ and radius $r$. Let $AB$ be a fixed diameter of $G$. Let $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$. Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$. [i]Proposed by Michael Ren[/i]