

1. Show that x^{6}  x^{5} + x^{4}  x^{3} + x^{2}  x + 3/4 has no real zeros.


2. ABCD is a quadrilateral area S. Its diagonals meet at X. Area ABX = X_{1}, area CDX = X_{2}. Show that √X_{1} + √X_{2} ≤ √X, with equality iff AB is parallel to CD.


3. N is a positive integer > 2. Show that there are the same number of pairs of positive integers a < b ≤ N such that b/a > 2 and such that b/a < 2.


4. Show that the only solution to x + y^{2} + z^{3} = 3, y + z^{2} + x^{3} = 3, z + x^{2} + y^{3} = 3 in positive reals is x = y = z = 1.


5. In an m x n array of reals the difference between the smallest and largest number in each row is at most d > 0. We now rearrange each column in decreasing order. Show that after the rearrangement the difference between the smallest and largest number in each row is still at most d.


6. A finite number of intervals cover [0, 1]. Show that one can find a subset of pairwise disjoint intervals with total length at least 1/2.

