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Annual Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:
Suppose that $x$ , $y$ , and $z$ are non-negative real numbers such that $x + y + z = 1$ . What is the maximum possible value of $x + y^{2} + z^{3}$ ?

Solution

Solution:
Since $0 \leq y, z \leq 1$ , we have $y^{2} \leq y$ and $z^{3} \leq z$ . Therefore $x + y^{2} + z^{3} \leq x + y + z = 1$ . We can get $x + y^{2} + z^{3} = 1$ by setting $(x, y, z) = (1, 0, 0)$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.