5th Austrian-Polish 1982


1. Find all positive integers m, n for which (n+1)^m - n and (n+1)^m+3 - n have a common factor (greater than 1).
2. C is a circle center O radius 1, and D is the interior of C (so D is the open disk center O radius 1). F is a closed convex subset of D. From any point of C there are two tangents to F, which are at an angle 60^o. Show that F must be the closed disk center O radius 1/2.
3. Let n > 1 be an integer. Let f(k) =1 + 3^k/(3ⁿ - 1), g(k) = 1 - 3^k/(3ⁿ - 1). Show that tan(f(1)π/3) tan(f(2)π/3) ... tan(f(n)π/3) tan(g(1)π/3) tan(g(2)π/3) ... tan(g(n)π/3) = 1.
4. The sequence a₁, a₂, a₃, ... satisfies a_n+1 = a_n + f(a_n), where f(m) is the product of the (decimal) digits of n. Is the sequence bounded for all a₁?
5. The closed interval [0, 1] is the union of two disjoint sets A, B. Show that we cannot find a real number k such that B = {x + k \| x belongs to A}.
6. k is a fixed integer. Let N_k = {k, k+1, k+2, ... }. Find all real-valued functions f on N_k such that f(m + n) = f(m) f(n) for all m, n such that m, n and m+n are all in N_k.
7. Find positive integers r, s, t, u, v, w such that (1) r > t > v, (2) r^s = t^u = v^w, (3) tu = vw, (4) r + s = t + u, and (5) tu is as small as possible.
8. P is a point inside a regular tetrahedron with side 1. Show that the sum of the perpendicular distances from P to each edge is at least 3/√2, with equality iff P is the center.
9. n > 2 is an integer. Let S_n be the sum of the n² values 1/√(i² + j²) for i = 1, 2, ... , n, and j = 1, 2, ... , n. Show that S_n ≥ n. Find a constant k as small as possible such that S_n ≤ kn.

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