1957 AMC 12 Problem 36

Attempt Problem 36 of the 1957 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1957 AMC 12 solutions, or check the answer key.

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36.

If

x+y=1,

then the largest value of

xy

is:

1

0.5

an irrational number about

0.4

0.25

0

Answer: D

Concepts:optimization completing the square quadratic

Difficulty rating: 1180

Small hint:

Substitute

y=1-x

into the product

Big hint:

Complete the square in

x(1-x)

Solution:

We have

\begin{aligned} xy=x(1-x) &=\frac14\\ &\quad-\left(x-\frac12\right)^2. \end{aligned}

The squared term is nonnegative, so the largest possible product is

\frac{1}{4}=0.25.

Thus, the correct answer is D.

Problem 36 in other years