For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd. Proposed by Navid Safaie

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

There are n stations $1,2,...,n$ in a broken road (like in Cars) in that order such that the distance between station $i$ and $i+1$ is one unit. The distance betwen two positions of cars is the minimum units needed to be fixed so that every car can go from its place in the first position to its place in the second (two cars can be in the same station in a position). Prove that for every $\alpha<1$ thre exist $n$ and $100^n$ positions such that the distance of every two position is at least $n\alpha$.

In a trapezoid $ABCD$ with $AD$ parallel to $BC$ points $E, F$ are on sides $AB, CD$ respectively. $A_1, C_1$ are on $AD,BC$ such that $A_1, E, F, A$ lie on a circle and so do $C_1, E, F, C$. Prove that lines $A_1C_1, BD, EF$ are concurrent.

A $9\times 9$ table is filled with zeroes.In every step we can either take a row add $1$ to every cell and shift it one unit to right or take a column reduce every cell by $1$ and shift it one cell down. Can the table with the top right $-1$ and bottom left $+1$ and all other cells zero be reached?

For all $n>1$. Find all polynomials with complex coefficient and degree more than one such that $(p(x)-x)^2$ divides $p^n(x)-x$. ($p^0(x)=x , p^i(x)=p(p^{i-1}(x))$) Proposed by Navid Safaie