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The length of each side of a convex quadrilateral ABCD is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.

Attempt:

I could not understand what to do, so I looked at the solution. The proposed solution is:

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I do not understand 'Here $l, m, n$ are necessarily distinct. Suppose $l = m$'

I did not understand why they say that they are distinct and assume that they are equal. Yes, it might be a proof by contradiction but we already have assumed one thing above: considering all the sides of quadrilateral are distinct.

Rather, I would like to know the approach to this problem than the solution itself.

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To spell out the part that you don't understand:

Here, we will show that $l, m, n$ are distinct. Suppose not, WLOG say $l=m$, then we get a contradiction since $la = mb \Rightarrow a = b$. Hence, $l \neq m$.

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  • $\begingroup$ I clearly misunderstood the statement. Thankyou! $\endgroup$ – Ice Inkberry Oct 3 '18 at 15:57

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