2013 Mock AIME I Problems/Problem 3

Problem

Let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ , and let $\{x\}=x-\lfloor x\rfloor$ . If $x=(7+4\sqrt{3})^{2^{2013}}$ , compute $x\left(1-\{x\}\right)$ .

Solution

Notice that the radical conjugate of x is a positive integer from 0-1. Since the sum of powers of two radical conjugates is an integer, $1-{x}$ is just the conjugate of x to the $2^{2013}$ power. Therefore, the desired expression is just 001.

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2013 Mock AIME I Problems/Problem 3