CGMO 2013

Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$ , where $0<t<1$ . Prove that the area of any triangle inside the region $A$ , with two vertices $P(t,t^2)$ and $Q(1,0)$ , does not exceed $\frac{1}{4}.$

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$ .

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$ ; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$ .

For any given positive numbers $a_1,a_2,\ldots,a_n$ , prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying $\sum_{i=1}^n x_i=1$ , such that for any positive numbers $y_1,y_2,\ldots,y_n$ with $\sum_{i=1}^n y_i=1$ , the inequality $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.

Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$ . Determine all possible values of $m$ .