$ABCDEFG$ is a convex polygon with area 1. Points $X,Y,Z,U,V$ are arbitrary points on $AB, BC, CD, EF, FG$. Let $M, I, N, K, S$ be the midpoints of $EZ, BU, AV, FX, TE$. Find the largest and smallest possible values of the area of $AKBSCMDEIFNG$.

Simplify the fraction: $\frac{(1^4+4)\cdot (5^4+4)\cdot (9^4+4)\cdot ... (69^4+4)\cdot(73^4+4)}{(3^4+4)\cdot (7^4+4)\cdot (11^4+4)\cdot ... (71^4+4)\cdot(75^4+4)}$.

2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?

A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if: a. The field is a regular triangle? b. The field is a square?

What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?

The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?