PAMO 2015

Prove that $$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$ for all real $x$ for which the left-hand side is well defined.

A convex hexagon $ABCDEF$ is such that $$AB=BC CD=DE EF=FA$$ and $$\angle ABC=2\angle AEC \angle CDE=2\angle CAE \angle EFA=2\angle ACE$$ Show that $AD$ , $CF$ and $EB$ are concurrent.

Let $a_1,a_2,...,a_{11}$ be integers. Prove that there are numbers $b_1,b_2,...,b_{11}$ , each $b_i$ equal $-1,0$ or $1$ , but not all being $0$ , such that the number $$N=a_1b_1+a_2b_2+...+a_{11}b_{11}$$ is divisible by $2015$ .

For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum $$d(1008)+d(1009)+...+d(2015)$$

There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get?

Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).

The perpendicular from $A$ to $BC$ meets $CD$ at $K$ . The perpendicular from $A$ to $CD$ meets $BC$ at $L$ . The perpendicular from $C$ to $AB$ meets $AD$ at $M$ . The perpendicular from $C$ to $AD$ meets $AB$ at $N$ .

1. Prove that $KL$ is parallel to $MN$ .
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.