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2021 AMC 12B Question 21

AMC12 · 2021 AMC 12B

Question 21

Let $S$ be the sum of all positive real numbers $x$ for which $$
x^{2^{\sqrt2}}=\sqrt2^{\,2^x}.
$$

Which of the following statements is true?
  1. $S\lt\sqrt2$
  2. $S=\sqrt2$
  3. $\sqrt2\lt S\lt 2$
  4. $2\le S\lt 6$
  5. $S\ge 6$