5th Swedish Mathematical Society Problems 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. 

Solutions

Problem 2
Find all positive integers m, n such that m³ - n³ = 999.
Answer
10³ - 1³, 12³ - 9³
Solution
The fastest way to do this is simply to write down the small cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197 (those at least you should know), 2744, 3375, 4096, 4913, 5832, 6859. Note that 6859 - 5832 > 999 and (n+1)³ - n³ is obviously an increasing function of n, so there are no solutions for m ≥ 19. We find the solutions for m < 19 by inspection.


Problem 3
Show that for every real x ≥ ½ there is an integer n such that |x - n²| ≤ √(x - ¼).
Solution
|x - n²| ≤ √(x - ¼) is equivalent to x² - (2n²+1)x + n⁴ + 1/4 ≤ 0. The roots are n² ± n + 1/2, so the inequality is satisfied for n² - n + 1/2 ≤ x < n² + n + 1/2. But these intervals cover the reals ≥ 1/2, because n² + n + 1/2 = (n+1)² - (n+1) + 1/2