Putnam 2014 B4

B4

Let P_n(x)=\sum_{k=0}^{n}2^{k(n-k)}x^k
Evaluating P_n(-2^{-n}), P_n(-2^{2-n}), P_n(-2^{4-n}),..., P_n(-2^n)
You can see that we have a change of sign between each P_n(-2^{2q-n}) \therefore there must be at least one root over the intervals (-2^{-n}, -2^{2-n}), (-2^{2-n}, -2^{4-n})... (-2^{n-2}, -2^{n})
\therefore we have least n real roots

2 Likes

:cowboy_hat_face: