18th Swedish 1978


1. a > b > c > d ≥ 0 are reals such that a + d = b + c. Show that x^a + x^d ≥ x^b + x^c for x > 0.
2. Let s_m be the number 66 ... 6 with m 6s. Find s₁ + s₂ + ... + s_n.
3. Two satellites are orbiting the earth in the equatorial plane at an altitude h above the surface. The distance between the satellites is always d, the diameter of the earth. For which h is there always a point on the equator at which the two satellites subtend an angle of 90^o?
4. b₀, b₁, b₂, ... is a sequence of positive reals such that the sequence b₀, c b₁, c²b₂, c³b₃, ... is convex for all c > 0. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that ln b₀, ln b₁, ln b₂, ... is convex.
5. k > 1 is fixed. Show that for n sufficiently large for every partition of {1, 2, ... , n} into k disjoint subsets we can find a ≠ b such that a and b are in the same subset and a+1 and b+1 are in the same subset. What is the smallest n for which this is true?
6. p(x) is a polynomial of degree n with leading coefficient c, and q(x) is a polynomial of degree m with leading coefficient c, such that p(x)² = (x² - 1)q(x)² + 1. Show that p'(x) = nq(x).

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