16th Irish 2003


A1. Find all integral solutions to (m² + n)(m + n²) = (m + n)³.
A2. QB is a chord of the circle parallel to the diameter PA. The lines PB and QA meet at R. S is taken so that PORS is a parallelogram (where O is the center of the circle). Show that SP = SQ.
A3. Find ( [1^1/2] - [1_1/3] ) + ( [2_1/2] - [2_1/3] ) + ... + ( [2003^1/2 - [2003^1/3] ).
A4. A, B, C, D, E, F, G, H compete in a chess tournament. Each pair plays at most once and no five players all play each other. Write a possible arrangment of 24 games which satisfies the conditions and show that no arrangement of 25 games works.
A5. Let R be the reals and R⁺ the positive reals. Show that there is no function f : R⁺ → R such that f(y) > (y - x) f(x)² for all x, y such that y > x.
B1. A triangle has side lengths a, b, c with sum 2. Show that 1 ≤ ab + bc + ca - abc ≤ 1 + 1/27.
B2. ABCD is a quadrilateral. The feet of the perpendiculars from D to AB, BC are P, Q respectively, and the feet of the perpendiculars from B to AD, CD are R, S respectively. Show that if ∠PSR = ∠SPQ, then PR = QS.
B3. Find all integer solutions to m² + 2m = n⁴ + 20n³ + 104n² + 40n + 2003.
B4. Given real positive a, b, find the largest real c such that c ≤ max(ax + 1/(ax), bx + 1/bx) for all positive real x.
B5. N distinct integers are to be chosen from {1, 2, ... , 2003} so that no two of the chosen integers differ by 10. How many ways can this be done for N = 1003? Show that it can be done in (3·5151 + 7·1700) 101⁷ ways for N = 1002.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.