HMMT February

Algebra Difficulty 4.9 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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