5th Swedish 1965


1. The feet of the altitudes in the triangle ABC are A', B', C'. Find the angles of A'B'C' in terms of the angles A, B, C. Show that the largest angle in A'B'C' is at least as big as the largest angle in ABC. When is it equal?
2. Find all positive integers m, n such that m³ - n³ = 999.
3. Show that for every real x ≥ ½ there is an integer n such that \|x - n²\| ≤ √(x - ¼).
4. Find constants A > B such that f( 1/(1+2x) )/f(x) is independent of x, where f(x) = (1 + Ax)/(1 + Bx) for all real x ≠ -1/B. Put a₀ = 1, a_n+1 = 1/(1 + 2a_n). Find an expression for a_n by considering f(a₀), f(a₁), ... .
5. Let S be the set of all real polynomials f(x) = ax³ + bx² + cx + d such that \|f(x)\| ≤ 1 for all -1 ≤ x ≤ 1. Show that the set of possible \|a\| for f in S is bounded above and find the smallest upper bound.

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