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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:

If $x \geq 0$ , $y \geq 0$ are integers, randomly chosen with the constraint $x+y \leq 10$ , what is the probability that $x+y$ is even?

Solution

Solution:

For each $p \leq 10$ , if $x+y=p$ , $x$ can range from $0$ to $p$ , yielding $p+1$ ordered pairs $(x, y)$ . Thus there are a total of $1+2+3+\cdots+11$ allowable ordered pairs $(x, y)$ , but $1+3+5+\cdots+11$ of these pairs have an even sum. So the desired probability is

$\frac{1+3+5+\cdots+11}{1+2+3+\cdots+11}=\frac{6^{2}}{11 \cdot 6}=\frac{6}{11}.$

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