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A1. ABC is acute-angled. M is the midpoint of AB. A line through M meets the lines CA, CB at K, L with CK = CL. O is the circumcenter of CKL and CD is an altitude of ABC. Show that OD = OM.
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| A2. 0 ≤ k1 < k2 < ... < kn are integers. 0 < a < 1 is a real. Show that (1-a)(ak1 + ak2 + ... + akn)2 < (1+a)(a2k1 + a2k2 + ... + a2kn). | |
| A3. Find all polynomials p(x) with integer coefficients such that p(n) divides 2n - 1 for n = 1, 2, 3, ... . | |
| B1. p is a prime and a, b, c, are distinct positive integers less than p such that a3 = b3 = c3 mod p. Show that a2 + b2 + c2 is divisible by a + b + c. | |
| B2. ABCD is a tetrahedron. The insphere touches the face ABC at H. The exsphere opposite D (which also touches the face ABC and the three planes containing the other faces) touches the face ABC at O. If O is the circumcenter of ABC, show that H is the orthocenter of ABC. | |
| B3. n is even. Show that there is a permutation a1a2...an of 12...n such that ai+1 ∈ {2ai, 2ai-1, 2ai-n, 2ai-n-1} for i = 1, 2, ... , n (and we use the cyclic subscript convention, so that an+1 means a1). |
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
6 March 2004
Last corrected/updated 6 Mar 04