EMC 2015

Dag 1

Vraag 1 Opgelost!

$A=\{ a,b,c\}$ is een verzameling bestaande uit drie natuurlijke getallen.
Bewijs dat we twee elementen $x,y$ uit $A$ kunnen vinden zodat voor alle oneven natuurlijke getallen $m, n$ geldt dat
$$10 | x^my^n - x^ny^m.$$

Vraag 2

Zij $a,b,c$ postieve reele getallen zodat geldt dat $abc=1$. Bewijs dat
$$ \frac{a+b+c+3}{4} \geq \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$

Vraag 3 Opgelost!

Circles $k_1$ and $k_2$ intersect in points $A$ and $B$, such that $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ in points $K$ and $O$ and $k_2$ in points $L$ and $M$, such that the point $L$ is between $K$ and $O$. The point $P$ is orthogonal projection of the point $L$ to the line $AB$. Prove that the line $KP$ is parallel to the $M$-median of the triangle $ABM$.

Vraag 4

A group of mathematicians is attending a conference. We say that a mathematician is $k$-content if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in a same room then they are all at least $3k+1$-content. Prove that you can assign everyone into one of $2$ rooms in a way that everyone is at least $k$-content in his room and neither room is empty.

Admiration is not necessarily mutual and no one admires himself.