The 7th Balkan Mathematical Olympiad
1990年第7届巴尔干地区数学奥林匹克 |
- The sequence un is defined by u1 = 1, u2 = 3, un = (n+1) un-1 - n un-2. Which members of the sequence which are divisible by 11 ?
- Expand (x + 2x2 + 3x3 + ... + nxn)2 and add the coefficients of xn+1 through x2n. Show that the result is n(n+1)(5n2 + 5n + 2)/24.
- The feet of the altitudes of the triangle ABC are D, E, F. The incircle of DEF meets its sides at G, H, I. Prove that ABC and GHI have the same Euler line (the line through the circumcenter and the centroid).
- The function f is defined on the positive integers and f(m) ≠ f(n) if m - n is prime. What is the smallest possible size of the image of f.
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