**Problem 1.**Let G be a finite group of order |G|. Let H be its subgroup, such that the index (G:H) is the smallest prime factor of |G|. Prove that H is a normal subgroup.

**Problem 2.**Consider a procedure: given a polygon in a plane, the next polygon is formed by the centers of its edges. Prove that if we start with a polygon and perform the procedure infinitely many times, the resulting polygon will converge to a point. In another variation, instead of using the centers of edges to construct the next polygon, use the centers of gravity of k consecutive vertices.

**Problem 3.**Find numbers a

_{n}such that 1 + 1/2 + 1/3 + … + 1/k = ln k + γ + a

_{1}/k + … + a

_{n}/k

^{n}+ …

**Problem 4.**For x

_{1}not equal to zero, let x

_{k}= sin x

_{k-1}. Find the asymptotic behavior of x

_{k}.

**Problem 5.**Calculate the integral from 0 to 1 of x

^{−x}over x with the precision 0.001.