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The 32nd British Mathematical Olympiad
1996年第32届英国奥林匹克数学竞赛
  1. Find all non-negative integer solutions to 2m + 3n = k2 .
  2. The triangle ABC has sides a, b, c, and the triangle UVW has sides u, v, w such that a2 = u(v + w - u), b2 = v(w + u - v), c2 = w(u + v - w). Show that ABC must be acute angled and express the angles U, V, W in terms of the angles A, B, C .
  3. The circles C and C' lie inside the circle S. C and C' touch each other externally at K and touch S at A and A' respectively. The common tangent to C and C' at K meets S at P. The line PA meets C again at B, and the line PA' meets C' again at B'. Show that BB' is a common tangent to C and C'.
  4. Find all positive real solutions to w + x + y + z = 12, wxyz = wx + wy + wz + xy + xz + yz + 27.
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