

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 s_{1} + s_{2} + ... + 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. b_{0}, b_{1}, b_{2}, ... is a sequence of positive reals such that the sequence b_{0}, c b_{1}, c^{2}b_{2}, c^{3}b_{3}, ... 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 b_{0}, ln b_{1}, ln b_{2}, ... 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} = (x^{2}  1)q(x)^{2} + 1. Show that p'(x) = nq(x).

