18th Swedish 1978

1.  a > b > c > d ≥ 0 are reals such that a + d = b + c. Show that xa + xd ≥ xb + xc for x > 0.
2.  Let sm be the number 66 ... 6 with m 6s. Find s1 + s2 + ... + sn.
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 90o?
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.

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© John Scholes
26 September 2003
Last corrected/updated 26 Sep 03