APMO 2011

Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$ .

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$ . The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$ , respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$ , respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$ . Prove that $\angle BPC=90^{\circ}$ .

Let $n$ be a fixed positive odd integer. Take $m+2$ distinct points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied:

1. $P_0=(0,1),P_{m+1}=(n+1,n)$ , and for each integer $i,1\le i\le m$ , both $x$ - and $y$ - coordinates of $P_i$ are integers lying in between $1$ and $n$ ( $1$ and $n$ inclusive).
2. For each integer $i,0\le i\le m$ , $P_iP_{i+1}$ is parallel to the $x$ -axis if $i$ is even, and is parallel to the $y$ -axis if $i$ is odd.
3. For each pair $i,j$ with $0\le i<j\le m$ , line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ , where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions:

1. There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied.
2. For every pair of real numbers $x$ and $y$ , $$f(xf(y))+yf(x)=xf(y)+f(xy)$$ is satisfied.