

1. A is the point (1, 0), L is the line y = kx (where k > 0). For which points P (t, 0) can we find a point Q on L such that AQ and QP are perpendicular?


2. Is there a positive integer n such that the fractional part of (3 + √5)^{n} > 0.99?


3. Show that a^{n} + b^{n} + c^{n} ≥ ab^{n1} + bc^{n1} + ca^{n1} for real a, b, c ≥ 0 and n a positive integer.


4. P_{1}, P_{2}, P_{3}, Q_{1}, Q_{2}, Q_{3} are distinct points in the plane. The distances P_{1}Q_{1}, P_{2}Q_{2}, P_{3}Q_{3} are equal. P_{1}P_{2} and Q_{2}Q_{1} are parallel (not antiparallel), similarly P_{1}P_{3} and Q_{3}Q_{1}, and P_{2}P_{3} and Q_{3}Q_{2}. Show that P_{1}Q_{1}, P_{2}Q_{2} and P_{3}Q_{3} intersect in a point.


5. Show that n divides 2^{n} + 1 for infinitely many positive integers n.


6. f(x) is defined for 0 ≤ x ≤ 1 and has a continuous derivative satisfying f '(x) ≤ C f(x) for some positive constant C. Show that if f(0) = 0, then f(x) = 0 for the entire interval.

