| A1. There are n2 students in a class. Each week they are arranged into n teams of n players. No two students can be in the same team in more than one week. Show that the arrangement can last for at most n+1 weeks. |
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| A2. Find all integers n for which x2 + nxy + y2 = 1 has infinitely many distinct integer solutions x, y. |
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| A3. X lies on the line segment AD. B is a point in the plane such that ∠ABX > 120o. C is a point on the line segment BX, show that (AB + BC + CD) ≤ 2AD/√3. |
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| A4. Xk is the point (k, 0). There are initially 2n+1 disks, all at X0. A move is to take two disks from Xk and to move one to Xk-1 and the other to Xk+1. Show that whatever moves are chosen, after n(n+1)(2n+1)/6 moves there is one disk at Xk for |k| ≤ n. | |
| A5. Find all real-valued functions f(x) such that xf(x) - yf(y) = (x-y) f(x+y) for all real x, y. |
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| B1. Show that for every positive integer n, nn ≤ (n!)2 ≤ ( (n+1)(n+2)/6 )n. |
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| B2. a, b, c are complex numbers. All roots of z3 + az2 + bz + c = 0 satisfy |z| = 1. Show that all roots of z3 + |a|z2 + |b|z + |c| = 0 also satisfy |z| = 1. |
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| B3. S is the square {(x,y) : 0 ≤ x, y ≤ 1}. For each 0 < t < 1, Ct is the set of points (x, y) in S such that x/t + y/(1-t) ≥ 1. Show that the set ∩ Ct is the points (x, y) in S such that √x + √y ≥ 1. |
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| B4. Given points P, Q, R show how to construct a triangle ABC such that P, Q, R are on BC, CA, AB respectively and P is the midpoint of BC, CQ/QA = AR/RB = 2. You may assume that P, Q, R are positioned so that such a triangle exists. |
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| B5. n < 1995 and n = abcd, where a, b, c, d are distinct primes. The positive divisors of n are 1 = d1 < d2 < ... < d16 = n. Show that d9 - d8 ≠ 22. |
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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
12 Oct 2003
Last updated/corrected 12 Oct 03