25th Swedish 1985


1. Show that (a-b)²/8a < (a+b)/2 - √(ab) < (a-b)²/8b for reals a > b > 0.
2. Find the smallest positive integer n such that if the first digit is moved to become the last digit, then the new number is 7n/2.
3. BA = BC. D is inside the circle through A, B, C, such that BCD is equilateral. AD meets the circle again at E. Show that DE equals the radius of the circle.
4. p(x) is a polynomial of degree n with real coefficients and is non-negative for all x. Show that p(x) + p'(x) + p''(x) + ... (n+1 terms) ≥ 0 for all x.
5. A (a, 0), B (0, b), C (c, d) is a triangle with a, b, c, d, > 0. Show that its perimeter is at least 2 CO (where O is the origin).
6. A town has several clubs. Given any two residents there is exactly one club that both belong to. Given any two clubs, there is exactly one resident who belongs to both. Each club has at least 3 members. At least one club has 17 members. How many residents are there?

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