KMO 2006

On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$

Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.

Prove that for any positive real numbers $x,y$ and $z,$ $xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$

Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$

$27$ students are given a number from $1$ to $27.$ How many ways are there to divide $27$ students into $9$ groups of $3$ with the following condition?

(i) The sum of students number in each group is $1\pmod{3}$ (ii) There are no such two students where their numbering differs by $3.$