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Polynomial (x+1)^n - 1 divisible by...

Source: All-Russian Olympiad 2006 finals, problem 11.7

May 6, 2006
algebrapolynomialalgebra proposed

Problem Statement

Assume that the polynomial (x+1)n−1\left(x+1\right)^n-1 is divisible by some polynomial P(x)=xk+ck−1xk−1+ck−2xk−2+...+c1x+c0P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0, whose degree kk is even and whose coefficients ck−1c_{k-1}, ck−2c_{k-2}, ..., c1c_1, c0c_0 all are odd integers. Show that k+1∣nk+1\mid n.
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