Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Prove it North Macedonia

Let $1*2*3*4*5*6*7*8*9=0$ be a given equality. Is it possible to substitute some of the $*$ with $+$ and the others with $-$ to obtain a correct equality?

Solution

Sum or difference of two even or two odd numbers is an even number. Sum or difference of an even and an odd number, or an odd and an even number, is an odd number. On the left side of the equality there are $4$ even and $5$ odd numbers. Any combination of $+$ and $-$ will give an odd number, but $0$ is an even number. So equality can not be achieved no matter how we arrange the $+$ and $-$ .

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