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The 6th United States of America Mathematics Olympiad
1977年第六届美国数学奥林匹克
  1. For which positive integers a, b does (xa + ... + x + 1) divide (xab + xab-b + ... + x2b + xb + 1) ?
  2. The triangles ABC and DEF have AD, BE and CF parallel. Show that [AEF] + [DBF] + [DEC] + [DBC] + [AEC] + [ABF] = 3 [ABC] + 3 [DEF], where [XYZ] denotes the signed area of the triangle XYZ. Thus [XYZ] is + area XYZ if the order X, Y, Z is anti-clockwise and - area XYZ if the order X, Y, Z is clockwise. So, in particular, [XYZ] = [YZX] = -[YXZ].
  3. Prove that the product of the two real roots of x4 + x3 - 1 = 0 is a root of x6 + x4 + x3 - x2 - 1 = 0.
  4. ABCD is a tetrahedron. The midpoint of AB is M and the midpoint of CD is N. Show that MN is perpendicular to AB and CD iff AC = BD and AD = BC.
  5. The positive reals v, w, x, y, z satisfy 0 < h ≤ v, w, x, y, z ≤ k. Show that (v + w + x + y + z)(1/v + 1/w + 1/x + 1/y + 1/z) ≤ 25 + 6( √(h/k) - √(k/h) )2. When do we have equality ?
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