

1. Solve x^{2}x1  2  3  4  5 = x^{2} + x  30.


2. Circle C center O touches externally circle C' center O'. A line touches C at A and C' at B. P is the midpoint of AB. Show that ∠OPO' = 90^{o}.


3. Find nonnegative integers a, b, c, d such that 5^{a} + 6^{b} + 7^{c} + 11^{d} = 1999.


4. An equilateral triangle side x has its vertices on the sides of a square side 1. What are the possible values of x?


5. x_{i} are nonnegative reals. x_{1} + x_{2} + ... + x_{n} = s. Show that x_{1}x_{2} + x_{2}x_{3} + ... + x_{n1}x_{n} ≤ s^{2}/4.


6. S is any sequence of at least 3 positive integers. A move is to take any a, b in the sequence such that neither divides the other and replace them by gcd(a,b) and lcm(a,b). Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.

