FKMO 2014

Suppose $x$ , $y$ , $z$ are positive numbers such that $x+y+z=1$ . Prove that $$\frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2.$$

Let $ABC$ be a isosceles triangle with $AC = BC > AB$ . Let $E, F$ be the midpoints of segments $AC, AB$ , and let $l$ be the perpendicular bisector of $AC$ . Let $l$ meets $AB$ at $K$ , the line through $B$ parallel to $KC$ meets $AC$ at point $L$ , and line $FL$ meets $l$ at $W$ . Let $P$ be a point on segment $BF$ . Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$ . Line $FJ$ meets $l$ at $M$ . Prove that $AW = PW$ if and only if $B$ lies on the circumcircle of $EFM$ .

There are $n$ students sitting on a round table. You collect all of $n$ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation: The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him. Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.

Let $ABC$ be a isosceles triangle with $AC=BC$ . Let $D$ a point on a line $BA$ such that $A$ lies between $B, D$ . Let $O_1$ be the circumcircle of triangle $DAC$ . $O_1$ meets $BC$ at point $E$ . Let $F$ be the point on $BC$ such that $FD$ is tangent to circle $O_1$ , and let $O_2$ be the circumcircle of $DBF$ . Two circles $O_1 , O_2$ meet at point $G ( \ne D)$ . Let $O$ be the circumcenter of triangle $BEG$ . Prove that the line $FG$ is tangent to circle $O$ if and only if $DG \bot FO$ .

Let $p>5$ be a prime. Suppose that there exist integer $k$ such that $k^2 + 5$ is divisible by $p$ . Prove that there exist two positive integers $m,n$ satisfying $p^2 = m^2 + 5n^2$ .

In an island there are $n$ castles, and each castle is in country $A$ or $B$ . There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them. Prove that the following two statements are equivalent: (1) If some commanders from country $B$ move to attack an adjacent castle in country $A$ , some commanders from country $A$ could appropriately move in defense to adjacent castles in country $A$ so that in every castle of country $A$ , the number of country $A$ 's commanders defending that castle is not less than the number of country $B$ 's commanders attacking that castle. (Each commander can defend or attack only one castle at a time.) (2) For any arbitrary set $X$ of castles in country $A$ , the number of country $A$ 's castles that are in $X$ or adjacent to at least one of the castle in $X$ is not less than the number of country $B$ 's castles that are adjacent to at least one of the castles in $X$ .