5

Part of 1977 Poland - Second Round

Problems(1)

w_1(x) = x^2 - 1, w_{n+1}(x) = w_n(x)^2 - 1,

Source: Polish MO Recond Round 1977 p5

Let the polynomials

w_n

be given by the formulas: w_1(x) = x^2 - 1, w_{n+1}(x) = w_n(x)^2 - 1, (n = 1, 2, \ldots) and let

a

be a real number. How many different real solutions does the equation

w_n(x) = a

algebrapolynomial