29th Putnam 1968 Problem A6 Find all finite polynomials whose coefficients are all ±1 and whose roots are all real. Solution
: ±(x + 1), ±(x - 1), ±(x 2 + x - 1), ±(x 2 - x - 1), ±(x 3 + x 2 - x - 1), ±(x 3 - x 2 - x + 1). The linear and quadratic polynomials are easy to find (and it is easy to show that they are the only ones). Suppose the polynomial has degree n ≥ 3. Let the roots be k i . Then ∑ k i 2 = (∑ k i ) 2 - 2 ∑ k i k j = a 2 ± 2b, where a is the coefficient of x n-1 and b is the coefficient of x n-2 . Hence the arithmetic mean of the squares of the roots is (a 2 ± 2b)/n. But it is at least as big as the geometric mean which is 1 (because it is an even power of c, the constant term). So we cannot have n > 3. For n = 3, we must have b the opposite sign to the coefficient of x n and it is then easy to check that the only possibilities are those given above. 29th Putnam 1968 © John Scholes jscholes@kalva.demon.co.uk 14 Jan 2002