Let $a,b,c,d$ be real positive numbers such that $a^2+b^2+c^2+d^2 = 1$. Prove that $(1-a)(1-b)(1-c)(1-d) \geq abcd$.

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.

A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups. Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division. For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.

Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game. For each positive integer $n$, find a winning strategy for one of the players.

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.