16th Austrian-Polish 1993


1. Find all positive integers m, n such that 2^m - 3ⁿ = 7.
2. Find all tetrahedra ABCD such that area ABD + area ACD + area BCD ≤ 1 and the volume of the tetrahedron is as large as possible.
3. Let N be the set of positive integers. Define f: N → N by f(n) = n+1 if n is a prime power and = w₁ + ... + w_k when n is a product of the coprime prime powers w₁, w₂, ... , w_k. For example f(12) = 7. Find the smallest term of the infinite sequence m, f(m), f(f(m)), f(f(f(m))), ... .
4. The Fibonacci numbers are defined by F₀ = 1, F₁ = 1, F_n+2 = F_n+1 + F_n. The positive integers A, B are such that A¹⁹ divides B⁹³ and B¹⁹ divides A⁹³. Show that if h < k are consecutive Fibonacci numbers then (AB)^h divides (A⁴ + B⁸)^k.
5. Find all real solutions to: x + y = 3x + 4 2y³ + z = 6y + 6 3z³ + x = 9z + 8.
6. Show that for non-negative real numbers x, y we have: ( (√x + (√y)/2)² ≤ (x + (x²y)^1/3 + (xy²)^1/3 + y)/4 ≤ (x + (√(xy) + y)/3 ≤ ( (x^2/3 + y^2/3)/2)^3/2.
7. The integer sequence a₀ = 1, a₁, a₂, ... is defined by a_n+1 = [ (a_n + n)^1/3]³. Find an explicit formula for a_n and find all n such that a_n = n.
8. Find all real polynomials p(x) such that there is a unique real polynomial q(x) with (1) q(0) = 0, (2) x + q(y + p(x) ) = y + q(x + p(y) ) for all x, y.
9. ABC is an equilateral triangle. On the line AB take a point P such that A is between P and B. Let r be the inradius of the triangle PAC and r' the exradius of triangle PBC opposite P. Find r + r' in terms of the side length a of ABC.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.