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The 20th Vietnam Mathematical Olympiad
1982年第20届越南奥林匹克数学竞赛

A1. Find a quadratic with integer coefficients whose roots are cos 72o and cos 144o.

A2. Find all real solutions to x(x + 1)(x + 2)(x + 3) = m - 1.

A3. ABC is a triangle. A' is on the same side of BC as A, and A" is on the opposite side of BC. A'BC and A"BC are equilateral. B', B", C', C" are defined similarly. Show that area ABC + area A'B'C' = area A"B"C".

B1. Find all positive integer solutions to 2a + 2b + 2c = 2336.

B2. n is a positive integer. x and y are reals such that 0 ≤ x ≤ 1 and x n+1 ≤ y ≤ 1. Show that the absolute value of (y - x)(y - x2)(y - x3) ... (y - xn )(1 + x)(1 + x2) ... (1 + xn ) is at most (y + x)(y + x2) ... (y + xn )(1 - x)(1 - x2) ... (1 - xn ).

B3. ABCDA'B'C'D' is a cube (ABCD and A'B'C'D' are faces and AA', BB', CC', DD' are edges). L is the line joining the midpoints of BB' and DD'. Show that there is no line which meets L and the lines AA', BC and C'D'.

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