Herzog Competition - 2001

The 29th Herzog Competition

Problem 1

Let p and q be two consecutive odd primes (like 5 and 7 or 23 and 29). Prove that p + q can be represented as a product of 3 integer factors, each of the factors being greater than 1.

Problem 2

An n by n square filled with numbers 1, 2, …, n² is called magic if the sum of the numbers in each row, each column, and each of the two diagonals is the same. Does there exists a magic 4 by 4 square that contains numbers 2, 3, and 4 in the positions shown below?

Problem 3

During the last 11 weeks John played at least one game of chess every day but not more than 12 games during any week. Prove that there was a period of several consecutive days during which John played exactly 22 games.

Problem 4

Find (with proof) all mappings F from Z₂₃₉ (the ring of residues modulo 239) into itself satisfying

$$F(x+y)=F\left(x\right)F\left(y\right)$$

for all x,y ∈ Z₂₃₉.

Problem 5

Prove that for every x > 0,

$$e^{-x}+e^{-\frac{1}{x}}\ge \frac{2}{e}$$