Solve the inequality $±x^n ± x^{n−1} ±···±x < 1/2 $

Question

Let n be an even positive integer. Prove that for any real number x there are at least $2^{n/2}$ choices of the signs + and − such that $±x^n ± x^{n−1} ±···±x < 1/2 $

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Answer 1

You can do much better than that. If a real number $a$ can be written as such a sum, then so can $-a$. Thus there are precisely $2^{n-1}$ combinations [i.e., half of the total $2^n$ combinations] so that the resulting sum is negative, which implies at least $2^{n-1} > 2^{n/2}$ combinations so that the resulting sum is smaller than $\frac{1}{2}$.