Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
Let $a_1$, $a_2$, … be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1$, $a_2$, …, $a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence.
Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $\overline{BC} \nparallel \overline{DA}$. Let two variable points $E$ and $F$ lie on the sides $BC$ and $DA$, respectively, and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.
In a mathematical competition $6$ problems were posed to the contestants. Each pair of problems was solved by more than $\frac{2}{5}$ of the contestants. Nobody solved all 6 problems. Show that there were at least $2$ contestants who each solved exactly $5$ problems.