| 1. ABCD is a convex quadrilateral. K, L, M, N are the midpoints of the sides AB, BC, CD, DA. BD bisects KM at Q. QA = QB = QC = QD, and LK/LM = CD/CB. Prove that ABCD is a square. | |
| 2. p > 3 is a prime. Find all integers a, b, such that a2 + 3ab + 2p(a+b) + p2 = 0. | |
| 3. If α is a real root of x5 - x3 + x - 2 = 0, show that [α6] = 3. | |
| 4. ABC is a triangle, with sides a, b, c (as usual), circumradius R, and exradii ra, rb, rc. If 2R ≤ ra, show that a > b, a > c, 2R > rb, and 2R > rc. | |
| 5. S is the set of all (a, b, c, d, e, f) where a, b, c, d, e, f are integers such that a2 + b2 + c2 + d2 + e2 = f2. Find the largest k which divides abcdef for all members of S. | |
| 6. Show that the number of 5-tuples (a, b, c, d, e) such that abcde = 5(bcde + acde + abde + abce + abcd) is odd. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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John Scholes
jscholes@kalva.demon.co.uk
30 March 2004
Last corrected/updated 31 Mar 04