

1. Find all positive integers m, n such that 1/m + 1/n  1/(mn) = 2/5.


2. x, y are positive reals such that x  √x ≤ y  1/4 ≤ x + √x. Show that y  √y ≤ x  1/4 ≤ y + √y.


3. The sequence x_{0}, x_{1}, x_{2}, ... is defined by x_{0} = 0, x_{k+1} = [(n  ∑_{0}^{k} x_{i})/2]. Show that x_{k} = 0 for all sufficiently large k and that the sum of the nonzero terms x_{k} is n1.


4. x_{1}, x_{2}, ... , x_{8} is a permutation of 1, 2, ... , 8. A move is to take x_{3} or x_{8} and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at 1, 2, ... , 8.


5. Show that there are infinitely many odd positive integers n such that in binary n has more 1s than n^{2}.


6. Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

