Question bank APMO 2011

Let $a,b,c$ be positive integers. Prove that it is impossible for all 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 in a plane with no three collinear. Determine the maximum possible value of the minimum of the angles $\angle A_iA_jA_k$, where $i,j,k$ are distinct integers between 1 and 5.

Illustrative configuration (new):

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

New configuration:

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 nonnegative integer, in the coordinate plane satisfying:

1. $P_0=(0,1)$, $P_{m+1}=(n+1,n)$, and for $1\le i\le m$ both coordinates of $P_i$ are integers between 1 and $n$, inclusive.
2. For $0\le i\le m$, segment $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and to the $y$-axis if $i$ is odd.
3. For every $0\le i\lt j\le m$, the segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most one point.

Determine the maximum possible value of $m$.

Determine all functions $f:\mathbb R\to\mathbb R$ satisfying:

1. There is a real number $M$ such that $f(x)\lt M$ for every real $x$.
2. For all real $x,y$,
$$f(xf(y))+yf(x)=xf(y)+f(xy).$$