49-th Mathematical Olympiad in Poland Second Round, February 27–28, 1998

First Day

1. Let A_{n} = 1, 2, …, n. Prove or disprove the following statement:

for all integers n ≥ 2 there exist functions f : A_{n} → A_{n} and g : A_{n} → A_{n} which satisfy f(f(k)) = g(g(k)) = k for k = 1, 2, …, n; g(f(k)) = k + 1 for k = 1, 2, …, n − 1 .

2. In triangle ABC the angle ^{}B̸CA is obtuse and ^{}B̸AC = 2^{}A̸BC . The line through B and perpendicular to BC intersects line AC in D. Let M be the midpoint of AB. Prove that ^{}A̸MC = ^{}B̸MD.

3. a) Assume that nonnegative numbers a , b , c , d , e , f with sum equal to 1

satisfy

ace + bdf ≥ \frac{1}{108} .

Show that

abc + bcd + cde + def + efa + fab ≤ \frac{1}{36} .

b) Do there exist six different positive numbers a , b , c , d , e , f with sum equal to 1 for which the two above inequalities become equalities?

Second Day

4. Find all pairs (x, y) of integers which satisfy the equation x^{2} + 3y^{2} = 1998x.

5. Suppose that nonnegative numbers a_{1}, a_{2}, …, a_{7}, b_{1}, b_{2}, …, b_{7} satisfy

a_{i} + b_{i} ≤ 2 for i = 1, 2, …, 7 .

Prove that there exist two different indices k, m ∈ {1, 2, …, 7} for which

|a_{k} − a_{m}| + |b_{k} − b_{m}| ≤ 1 .

6. Prove that in tetrahedron ABCD the edge AB is perpendicular to the edge CD if and only if there exists a parallelogram CDP Q such that P A = P B = P D and QA = QB = QC.