A rectangular array has $m$ rows and $n$ columns, where $m < n$. Some cells of the array contain stars, in such a way that there is at least one star in each column. Prove that there is at least one such star such that the row containing it has more stars than the column containing it. (A. Razborov, Moscow)

You have $4$ game pieces, and you play a game against an intelligent opponent who has $6$. The rules go as follows: you distribute your pieces among two points a and b, and your opponent simultaneously does as well (so neither player sees what the other is doing). You win the round if you have more pieces than them on either $a$ or$ b$, and you lose the round if you only draw or have fewer pieces on both. You play the optimal strategy, assuming your opponent will play with the strategy that beats your strategy most frequently. What proportion of the time will you win?

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

Le be a real number $ |a|<1, $ a natural number $ n\ge 2, $ and a $ 2\times 2 $ real matrix $ A $ that verifies $$ \det \left( A^{2n} -aA^{2n-1} -aA+I \right)=0. $$ Show that $ \det A=1. $

Solve the equation for positive integers $m, n$: \[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.

Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) Let $p$ be a prime. Consider the graph whose vertices are the ordered pairs $(x, y)$ with $x, y \in\{0, 1, . . . , p-1\}$ and whose edges join vertices $(x, y)$ and $(x' , y')$ if and only if $xx'+yy'\equiv 1 \pmod{p}$ . Prove that this graph does not contain $C_{4}$ . (b) Prove that for infinitely many values $n$ there is a graph $G_{n}$ with $e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n$ that does not contain $C_{4}$.

[u]Calculus Round[/u] [b]p1.[/b] Evaluate $$\int^2_{-2}(x^3 + 2x + 1)dx$$ [b]p2.[/b] Find $$\lim_{x \to 0} \frac{ln(1 + x + x^3) - x}{x^2}$$ [b]p3.[/b] Find the largest possible value for the slope of a tangent line to the curve $f(x) = \frac{1}{3+x^2}$ . [b]p4.[/b] An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point when the water reaches a height of $10$ meters. [b]p5.[/b] Compute $$\int^{\frac{\pi}{2}}_0 \sin (2016x) \cos (2015x) dx$$ [b]p6.[/b] Let $f(x)$ be a function defined for $x > 1$ such that $f''(x) = \frac{x}{\sqrt{x^2-1}}$ and $f'(2) =\sqrt3$. Compute the length of the graph of $f(x)$ on the domain $x \in (1, 2]$. [b]p7.[/b] Let the function $f : [1, \infty) \to R$ be defuned as $f(x) = x^{2 ln(x)}$. Compute $$\int^{\sqrt{e}}_1 (f(x) + f^{-1}(x))dx$$ [b]p8.[/b] Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$. [b]p9.[/b] Compute $$\int^e_1 \frac{ln (x)}{(1 + ln (x))^2} dx$$ [b]p10.[/b] For $x \ge 0$, let $R$ be the region in the plane bounded by the graphs of the line $\ell$ : $y = 4x$ and $y = x^3$. Let $V$ be the volume of the solid formed by revolving $R$ about line $\ell$. Then $V$ can be expressed in the form $\frac{\pi \cdot 2^a}{b\sqrt{c}}$ , where $a$, $b$, and $c$ are positive integers, $b$ is odd, and $c$ is not divisible by the square of a prime. Compute $a + b + c$. [u]Calculus Tiebreaker[/u] [b]Tie 1.[/b] Let $f(x) = x + x(\log x)^2$. Find $x$ such that $xf'(x) = 2f(x)$. [b]Tie 2.[/b] Compute $$\int^{\frac{\sqrt2}{2}}_{-1} \sqrt{1 - x^2} dx$$ [b]Tie 3.[/b] An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle. PS. You should use hide for answers.

A group of $n$ people play a board game with the following rules: 1) In each round of the game exactly $3$ people play 2) The game ends after exactly $n$ rounds 3) Every pair of players has played together at least at one round Find the largest possible value of $n$

Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$.

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

Let $a, b, c \in \mathbb R$. Prove that \[(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.\] When does the equality hold?

Let $\mathcal F$ be the set of all $2013 \times 2013$ arrays whose entries are $0$ and $1$. A transformation $K : \mathcal F \to \mathcal F$ is defined as follows: for each entry $a_{ij}$ in an array $A \in \mathcal F$, let $S_{ij}$ denote the sum of all the entries of $A$ sharing either a row or column (or both) with $a_{ij}$. Then $a_{ij}$ is replaced by the remainder when $S_{ij}$ is divided by two. Prove that for any $A \in \mathcal F$, $K(A) = K(K(A))$. [i]Proposed by Aaron Lin[/i]

Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type [img]http://oi53.tinypic.com/v3pqoh.jpg[/img].

Let $ABC$ be a non-isosceles acute triangle with $(I)$ be it's incircle. $D, E, F$ are the touchpoints of $(I)$ and $BC, CA, AB,$ respectively. $P$ is the perpendicular projection of $D$ on $EF.$ $DP$ intersects $(I)$ at the second point $K,L$ is the perpendicular projection of $A$ on $IK.$ $(LEC), (LFB) $ intersects $(I)$ the second time at $M, N,$ respectively. Prove that $M, N, P$ are collinear.

Find all real numbers $x,y,z$ such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$

An integer $n$ such that $\Bigl\lfloor \frac{n}{9} \Bigr\rfloor$ is a three digit number with equal digits, and $\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor$ is a $4$ digit number with the digits $2,0,2,4$ in some order. What is the remainder when $n$ is divided by $100$?

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that: $\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.