APMO 2012 - Asian Pacific Mathematics Olympiad

Let $P$ be a point inside triangle $ABC$. Let $D,E,F$ be the intersections of line $AP$ with side $BC$, line $BP$ with side $CA$, and line $CP$ with side $AB$, respectively. Prove that the area of triangle $ABC$ is 6 if each of triangles $PFA,PDB,PEC$ has area 1.

Newly drawn configuration:

Each cell of a $2012\times2012$ square grid contains a real number in $[0,1]$. Split the grid into two nonempty rectangles of cells by a line parallel to a horizontal or vertical side. Suppose that for every such split, at least one resulting rectangle has sum at most 1. Determine the maximum possible sum of all entries.

Determine all pairs $(p,n)$, where $p$ is prime and $n$ is a positive integer, for which $\frac{n^p+1}{p^n+1}$ is an integer.

Let $ABC$ be an acute triangle. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $M$ the midpoint of $BC$, and $H$ the orthocenter. Let $E$ be the intersection of the circumcircle $\Gamma$ of $ABC$ with ray $MH$, and $F\ne E$ the other intersection of line $ED$ with $\Gamma$. Prove
$$\frac{BF}{CF}=\frac{AB}{AC},$$
where $XY$ denotes the length of segment $XY$.

Newly drawn configuration:

Let $n\ge2$ be an integer. Prove that if real numbers $a_1,\ldots,a_n$ satisfy $a_1^2+\cdots+a_n^2=n$, then
$$\sum_{1\le i\lt j\le n}\frac1{n-a_i a_j}\le\frac n2.$$