- Let x
_{1}, x_{2}, . . . , x_{n} be positive real numbers, and S =x_{1}+ x_{2}+ . . . + x_{n} . Prove that
(1 + x_{1}) ... (1 + x_{n}) ≤ 1 + S + S^{2}/2! + ... + S^{n}/n! .
- Prove that the equation: 6(6a^2 + 3b^2 + c^2) = 5n^2
has no solutions in integers except a = b = c = n = 0.
- Let A
_{1}, A_{2}, A_{3} be three points in the plane, and for convenience, let A_{4} = A_{1}, A_{5} = A_{2}. For n = 1, 2, and 3, suppose that B_{n} is the midpoint of A_{n}A_{n+1}, and suppose that C_{n} is the midpoint of AnBn. Suppose that A_{n}C_{n+1} and B_{n}A_{n+2} meet at D_{n}, and that A_{n}B_{n+1} and C_{n}A_{n+2} meet at E_{n}. Calculate the ratio of the area of triangle D_{1}D_{2}D_{3} to the area of triangle E_{1}E_{2}E_{3}.
- Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least m(4m-n^2)/(3n) triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.
- Determine all functions f from the reals to the reals for which
(1) f(x) is strictly increasing,
(2) f(x) + g(x) = 2x for all real x,
where g(x) is the composition inverse function to f(x).
(Note: f and g are said to be composition inverses if f(g(x)) = x and g(f(x)) = x for all real x.) |