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The 10th Balkan Mathematical Olympiad
1993年第10届巴尔干地区数学奥林匹克
  1. Given reals a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 ≤ a6 satisfying a1 + a2 + a3 + a4 + a5 + a6 = 10 and (a1 - 1)2 + (a2 - 1)2 + (a3 - 1)2 + (a4 - 1)2 + (a5 - 1)2 + (a6 - 1)2 = 6, what is the largest possible a6 ?
  2. How many non-negative integers with not more than 1993 decimal digits have non-decreasing digits? [For example, 55677 is acceptable, 54 is not.]
  3. Two circles centers A and B lie outside each other and touch at X. A third circle center C encloses both and touches them at Y and Z respectively. The tangent to the first two circles at X forms a chord of the third circle with midpoint M. Prove that ∠YMZ = ∠ACB.
  4. p is prime and k is a positive integer > 1. Show that we can find positive integers (m, n) ≠ (1, 1) such that (mp + np)/2 = ( (m + n)/2 )k iff k = p.
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