Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME Prove it United States

Problem:

The numbers $1$ to $10$ are written on a blackboard in a row. Determine whether it is possible to put nine $+$ and $-$ signs between them (one between each pair of adjacent numbers) to make the resulting sum $0$ .

Solution

Solution:

It is not possible. Note that the sum of the numbers is
$1+2+3+\cdots+9+10=55$
an odd number. Thus, if we put all $+$ signs, the sum is odd. Every time we change a $+$ to a $-$ , we change an $n$ term to a $-n$ term for some $n$ , which subtracts $2n$ from the sum. This changes the sum by an even number, so the sum will still be odd. Thus, no matter what, the sum is odd and therefore cannot be $0$ .

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