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All-Soviet-Union Mathematical Olympiad

Algebra Difficulty 4.6 AIME Prove it Soviet Union

Problem:
Show that $\{x\}^4 > \{x\} - 1/2$ for all real $x$ .

Solution

Solution:
$\{x\}^4 - \{x\} + 1/2 = (\{x\}^2 - 1/2)^2 + (\{x\} - 1/2)^2 \geq 0$ . We could only have equality if $\{x\}^2 = \{x\} = 1/2$ , which is impossible, so the inequality is strict.

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