Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

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 $P(x)=x^3+x^2-r^2x-2020$ be a polynomial with roots $r,s,t$. What is $P(1)$? [i]Proposed by James Lin.[/i]

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

Across all polynomials $P$ such that $P(n)$ is an integer for all integers $n$, determine, with proof, all possible values of $P(i)$, where $i^2=-1$.

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

Let $ABC$ be a scalene acute triangle with orthocenter $H$. The circle with center $A$ and radius $AH$ meets the circumcircle of $BHC$ at $T_{a} \neq H$. Define $T_{b}$ and $T_{c}$ similarly. Show that $H$ lies on the circumcircle of $T_{a}T_{b}T_{c}$. [i]Authored by Nikola Velov[/i]

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for $\forall$ $x\in (-\infty , m]$.

Krut and Vert go by car from point $A$ to point $B$. The car leaves $A$ in the direction of $B$, but every $3$ km of the road Krut turns $90^o$ to the left, and every $7$ km of the road Vert turns $90^o$ to the right ( if they try to turn at the same time, the car continues to go in the same direction). Will Krut and Vert be able to get to $B$ if the distance between $A$ and $B$ is $100$ km?

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

$100$ students came to a party. The students who did not have friends among other students left the party first. Then those with one friend among remaining students left. Then those with $2,3, \ldots 99$ friends among remaining students left. What is the maximal number of students that can still remain at the party after that? (If $A$ is a friend of $B$, then $B$ is a friend of $A$).

For how many positive integers $m$ is \[\dfrac{2002}{m^2-2}\] a positive integer? $\textbf{(A) }\text{one}\qquad\textbf{(B) }\text{two}\qquad\textbf{(C) }\text{three}\qquad\textbf{(D) }\text{four}\qquad\textbf{(E) }\text{more than four}$

Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.

An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?

On the coordinate plane is given a polygon $\mathcal P$ with area greater than $1$. Prove that there exist two different points $(x_1,y_1)$ and $(x_2,y_2)$ inside the polygon $\mathcal P$ such that $x_1-x_2$ and $y_1-y_2$ are both integers.