20th Vietnam 1982 problems

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'.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
23 July 2002