AT-OEMO-FED1 2016

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$ .

For this $C$ , determine all $x_1, x_2, \cdots x_6$ such that equality holds.

(Walther Janous)

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$ . The point $E$ lies symmetric to $C$ with respect to the altitude $AH$ . Let $F$ be the intersection of the lines $EH$ and $AC$ . Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$ .

(Karl Czakler)

Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction.

What is the minimum number of jumps required to visit all points and return to the starting point?

(Gerd Baron)

Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$ , then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)