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The 23rd Canadian Mathematical Olympiads
1991年第二十三届加拿大数学奥林匹克
  1. Show that there are infinitely many solutions in positive integers to a2 + b5 = c3 .
  2. Find the sum of all positive integers which have n 1s and n 0s when written in base 2 .
  3. Show that the midpoints of all chords of a circle which pass through a fixed point lie on another circle .
  4. Can ten distinct numbers a1, a2, b1, b2, b3, c1, c2, d1, d2, d3 be chosen from {0, 1, 2, ... , 14}, so that the 14 differences |a1 - b1|, |a1 - b2|, |a1 - b3|, |a2 - b1|, |a2 - b2|, |a2 - b3|, |c1 - d1|, |c1 - d2|, |c1 - d3|, |c2 - d1|, |c2 - d2|, |c2 - d3|, |a1 - c1|, |a2 - c2| are all distinct ?
  5. An equilateral triangle side n is divided into n2 equilateral triangles side 1 by lines parallel to its sides. How many parallelograms can be formed from the small triangles ? (For example, if n = 3, there are 15, nine composed of two small triangles and six of four.)
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