| A1. D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the side AC such that AE/EC = BD/(AD-BC). Show that AD > BE. | |
| A2. Given 101 distinct non-negative integers less than 5050 show that one choose four a, b, c, d such that a + b - c - d is a multiple of 5050. | |
| A3. Show that one can find 50 distinct positive integers such that the sum of each number and its digits is the same. | |
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B1. For which n do the equations have a solution in integers:
x12 + x22 + 50 = 16x1 + 12x2 x22 + x32 + 50 = 16x2 + 12x3 ... xn-12 + xn2 + 50 = 16xn-1 + 12xn xn2 + x12 + 50 = 16xn + 12x1 |
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| B2. Show that ∑1≤i<j≤n (|ai-aj| + |bi-bj|) ≤ ∑1≤i<j≤n |ai-bj| for all integers ai, bi. | |
| B3. The convex hexagon ABCDEF satisfies ∠A + ∠C + ∠E = 360o and AB·CD·EF = BC·DE·FA. Show that AB·FD·EC = BF·DE·CA. |
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