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The 31st Canadian Mathematical Olympiads
1999年第三十一届加拿大数学奥林匹克
  1. Find all real solutions to the equation 4x2 - 40[x] + 51 = 0 .
  2. ABC is equilateral. A circle with center on the line through A parallel to BC touches the segment BC. Show that the length of arc of the circle inside ABC is independent of the position of the circle .
  3. Find all positive integers which equal the square of their number of positive divisors .
  4. X is a subset of eight elements of { 1 , 2 , 3 , ... , 17 } . Show that there are three pairs of (distinct) elements with the same difference .
  5. x , y , z are non-negative reals with sum 1 , show that x2y + y2z + z2x ≤ 4/27 . When do we have equality ?
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