Baltic Way, 2015, Stockholm

The competition took place November 5-9 2015 in Stockhom, Sweden. There were 12 teams from Denmark, Estonia, Finland, Germany, Iceland, Latvia, Lithuania, Norway, Poland, S:t Petersburg, Sweden and The Netherlands. The team from S:t Petersburg won.

I was one of coordinators in algebra, 5 problems (of 20). Here are the problems which solutions we must read and evaluate in all languages, and only then discuss with team leaders in English. As a Russian I chose to evaluate Poland, S:t Petersburg and Lithuania.

1. For n ≥ 2, an equilateral triangle is divided into n2 congruent smaller equilateral triangles. Determine all ways in which real numbers can be assigned to the (n+1)(n+2)/2 vertices so that three such numbers sum to zero whenever the three vertices form an equilateral triangle with edges parallell to the sides of the big triangle. (Denmark)

2. Let a1,..., an be real numbers, fulfilling 0 ≤ ai ≤ 1 for i = 1,..., n. Prove the inequality
(1 - a1n)(1 - a2n)...(1 - ann) ≤ (1 - a1a2...an)n.

3. Let n > 1. Find all non-constant real polynomials P(x) fulfilling, for any real x, the identity
P(x)P(x2)P(x3)...P(xn) = P(xn(n+1)/2).

4. A family wears three colours of clothing: red, blue and green, with a separate laundry bin for each colour. Each week, the family generates a total of 10 kg laundry (the proportion of each colour is subject of variation). The laundry is first sorted by colour and disposed of in the bins. Next, the heaviest bin is emtied and its content washed. What is the storing capacity required of the laundry bins if they must never overflow? (Norway)

5. Find all real functions f satisfying, for all real x and y, the equation
|x|f(y) + y f(x) = f(xy) + f(x2) + f(f(y)).

All problems(pdf)