VMO 1982

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

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

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".

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

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 - $x^2$)(y - $x^3$) ... (y - $x^n$ )(1 + x)(1 + $x^2$) ... (1 + $x^n$ ) is at most (y + x)(y + $x^2$) ... (y + $x^n$ )(1 - x)(1 - $x^2$) ... (1 - $x^n$ ).

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'.