Track / Stage 4 / 30 of 340 #290 of 2000

Problem 290

AMC 12 late, AIME early

Algebra Difficulty 4.5 Find the answer

4. Given $x=t^{\frac{1}{1}}, y=t^{\frac{t}{1}}(t>0, t \neq 1)$ . Then the relationship between $x, y$ is ( ).
(A) $y^{x}=x^{\frac{1}{y}}$
(B) $y^{\frac{1}{x}}=x^{2}$
(C) $y^{x}=x^{y}$
(D) $x^{x}=y^{x}$

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.