Let the number given be . If itself is a power of 2, nothing more to be done. Else, we need to show that there exists some such that there exists a power of 2 between and . Taking log to the base 2, we need to show that there is some such that there exists an integer between and . The gap between these two numbers is an increasing function of and so is lower bounded by the value when . If the fractional part of the first number is less than this lower bound, the interval must include an integer. But this follows from the denseness of the fractional part of since is irrational (the proof of this was part of another recent puzzle post also from Russian Olympiad).

Let the number given be . If itself is a power of 2, nothing more to be done. Else, we need to show that there exists some such that there exists a power of 2 between and . Taking log to the base 2, we need to show that there is some such that there exists an integer between and . The gap between these two numbers is an increasing function of and so is lower bounded by the value when . If the fractional part of the first number is less than this lower bound, the interval must include an integer. But this follows from the denseness of the fractional part of since is irrational (the proof of this was part of another recent puzzle post also from Russian Olympiad).