

1. A triangle has sides a > b > c and corresponding altitudes h_{a}, h_{b}, h_{c}. Show that a + h_{a} > b + h_{b} > c + h_{c}.


2. 6 ducks are swimming on a pond radius 5. Show that any moment there are two ducks a distance at most 5 apart.


3. x_{i} are reals. Show that for n = 3, x_{1} + x_{2} + ... + x_{n} = 0 implies x_{1}x_{2} + x_{2}x_{3} + ... + x_{n1}x_{n} + x_{n}x_{1} ≤ 0. For which n > 3 is this true?


4. p(x) is a polynomial of degree 3 with 3 distinct real zeros. How many real zeros does p'(x)^{2}  2p(x)p''(x) have?


5. Show that there is a constant c > 1 such that if the positive integers m, n satisfy m/n < √7, then 7  m^{2}/n^{2} ≥ c/n^{2}. What is the largest such c?


6. The sequence a_{1}, a_{2}, a_{3}, ... is defined by a_{1} = 1, a_{n+1} = √(a_{n}^{2} + 1/a_{n}). Show that 1/2 ≤ a_{n}/n^{c} ≤ 2 for some c (and all n).

