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The 5th Balkan Mathematical Olympiad
1988年第5届巴尔干地区数学奥林匹克
  1. ABC is a triangle area 1. AH is an altitude, M is the midpoint of BC and K is the point where the angle bisector at A meets the segment BC. The area of the triangle AHM is 1/4 and the area of AKM is 1 - (√3)/2. Find the angles of the triangle.
  2. Find all real polynomials p(x, y) such that p(x, y) p(u, v) = p(xu + yv, xv + yu) for all x, y, u, v.
  3. The sum of the squares of the edges of a tetrahedron is S. Prove that the tetrahedron can be fitted between two parallel planes a distance √(S/12) apart.
  4. xn is the sequence 51, 53, 57, 65, ... , 2n + 49, ... Find all n such that xn and xn+1 are each the product of just two distinct primes with the same difference.
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