HMMT February

Algebra Difficulty 4.8 AIME Find the answer

Find $(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots$ , where $|x|<1$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $S=(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots=1+x+x^{2}+x^{3}+\cdots$ . Since $x S=x+x^{2}+x^{3}+x^{4}+\cdots$ , we have $(1-x) S=1$ , so $S=\frac{1}{1-x}$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.