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The 3st Vietnam Mathematical Olympiad
1964年第三届越南数学奥林匹克
  1. Find cos x + cos(x + 2π/3) + cos(x + 4π/3) and sin x + sin(x + 2π/3) + sin(x + 4π/3).
  2. Draw the graph of the functions y = | x2 - 1 | and y = x + | x2 - 1 |. Find the number of roots of the equation x + | x2 - 1 | = k, where k is a real constant.
  3. Let O be a point not in the plane p and A a point in p. For each line in p through A, let H be the foot of the perpendicular from O to the line. Find the locus of H.
  4. Define the sequence of positive integers fn by f0 = 1, f1 = 1, fn+2 = fn+1 + fn. Show that fn = (an+1 - bn+1)/√5, where a, b are real numbers such that a + b = 1, ab = -1 and a > b.
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