34th Swedish 1994


1. x√8 + 1/(x√8) = √8 has two real solutions x₁, x₂. The decimal expansion of x₁ has the digit 6 in place 1994. What digit does x₂ have in place 1994?
2. In the triangle ABC the medians from B and C are perpendicular. Show that cot B + cot C ≥ 2/3.
3. The vertex B of the triangle ABC lies in the plane P. The plane of the triangle meets the plane in a line L. The angle between L and AB is a, and the angle between L and BC is b. The angle between the two planes is c. Angle ABC is 90^o. Show that sin²c = sin²a + sin²b.
4. Find all integers m, n such that 2n³ - m³ = m n² + 11.
5. The polynomial x^k + a₁x^k-1 + a₂x^k-2 + ... + a_k has k distinct real roots. Show that a₁² > 2ka₂/(k-1).
6. Let N be the set of non-negative integers. The function f:N→N satisfies f(a+b) = f(f(a)+b) for all a, b and f(a+b) = f(a)+f(b) for a+b < 10. Also f(10) = 1. How many three digit numbers n satisfy f(n) = f(N), where N is the "tower" 2, 3, 4, 5, in other words, it is 2^a, where a = 3^b, where b = 4⁵?

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