| A1. Find all pairs (n,r) with n a positive integer and r a real such that 2x2+2x+1 divides (x+1)n - r. | |
| A2. P is a point inside the triangle ABC such that ∠PBC = ∠PCA < ∠PAB. The line PB meets the circumcircle of ABC again at E. The line CE meets the circumcircle of APE again at F. Show that area APEF/area ABP does not depend on P. | |
| A3. ai, xi are positive reals such that a1 + a2 + ... + an = x1 + x2 + ... + xn = 1. Show that 2 ∑i<j xixj ≤ (n-2)/(n-1) + ∑ aixi2/(1-ai). When do we have equality? | |
| B1. ABCD is a tetrahedron with ∠BAC = ∠ACD and ∠ABD = ∠BDC. Show that AB = CD. | |
| B2. Let p(k) be the smallest prime not dividing k. Put q(k) = 1 if p(k) = 2, or the product of all primes < p(k) if p(k) > 2. Define the sequence x0, x1, x2, ... by x0 = 1, xn+1 = xnp(xn)/q(xn). Find all n such that xn = 111111. | |
| B3. Let S be the set of permutations a1a2...an of 123...n such that ai ≥ i. An element of S is chosen at random. Find all n such that the probability that the chosen permutation satisfies ai ≤ i+1 exceeds 1/3. |
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
1 March 2004
Last corrected/updated 1 Mar 04