17th Austrian-Polish 1994


1. f is a real-valued function on the reals such that f(x + 19) ≤ f(x) + 19 and f(x + 94) ≥ f(x) + 94 for all x. Show that f(x + 1) = f(x) + 1 for all x.
2. The sequences a₀, a₁, a₂, ... and b₀, b₁, b₂, ... are defined by a₀ = 1/2, a_n+1 = 2a_n/(1 + a_n²), b₀ = 4, b_n+1 = b_n² - 2b_n + 2. Show that a_n+1b_n+1 = 2b₀b₁ ... b_n.
3. Each cell of a 2 x 15 board is made into a room. The 43 interior walls are labeled from 1 to 43. Wall n has d_n doors, where d_n = 0, 1, 2 or 3. The doors are arranged so that each room has a total of three doors and it is possible to get from any room to any other room. How many possible arrangements are there for (d₁, d₂, ... , d₄₃).
4. P is a regular (n+1)-gon. One vertex is labeled 0. There are n! ways of labeling the other vertices 1, 2, ... , n. For each such labeling take the sum of the (positive) difference between the labels at the end of each side. For example, if n = 5 and the vertices are labeled 0, 3, 1, 4, 2, 5, then the sum is 3 + 2 + 3 + 2 + 3 + 5 = 18. Find the smallest possible such sum f(n) and the number of possible labelings which give it.
5. Find all integer solutions to (a + b)(b + c)(c + a)/2 + (a + b + c)³ = 1 - abc.
6. n is an odd integer and the non-negative integers a₁, a₂, ... , a_n satisfy: (a₂ - a₁)² + 2(a₂ + a₁) + 1 = n² (a₃ - a₂)² + 2(a₃ + a₂) + 1 = n² ... (a₁ - a_n)² + 2(a₁ + a_n) + 1 = n². Show that there are two consecutive a_i which are equal (we treat a_n and a₁ as consecutive).
7. Find all two digit numbers n = ab such that (x^a - x^b) is divisible by n for all integers x.
8. Let R be the reals. For each real a, b, find all functions f: R² → R which satisfy f(x, y) = a f(x, z) + b f(y, z) for all x, y, z.
9. The points A, B, C, D lie on a line in that order, with AB = a, BC = b, CD = c. Show that a point P exists such that ∠APB = ∠BPC = ∠CPD iff (a + b)(b + c) < 4ac. If the point exists, show how to construct it.

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