Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$. Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$. Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$. Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$. Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$. Prove: $MF$ and $AO$ are orthogonal to each other.

Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that \[ \sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]

Consider a $25\times25$ chessboard with cells $C(i,j)$ for $1\le i,j\le25$. Find the smallest possible number $n$ of colors with which these cells can be colored subject to the following condition: For $1\le i<j\le25$ and for $1\le s<t\le25$, the three cells $C(i,s)$, $C(j,s)$, $C(j,t)$ carry at least two different colors. (Proposed by Gerhard Woeginger, Austria)

Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime. (Proposed by Gerhard Woeginger, Austria)