# 2017-06 Powers of 2

Does there exist infinitely many positive integers $$n$$ such that the first digit of $$2^n$$ is $$9$$?

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# 2011-2 Power

Prove that for all positive integers m and n, there is a positive integer k such that $(\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.$

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# 2009-6 Sum of integers of the fourth power

Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist $$y_1,y_2,\ldots,y_k$$ with $$k\le 53$$ such that $$x=\sum_{i=1}^k y_i^4$$.

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