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Suppose we take a 501 element subset $S$ of the first 999 positive integers. Prove that there exists $a_x \not= a_y, a_z \in S$ such that $a_x+a_y=a_z$.

The 501 elements seems pretty close to 1000/2, so it seems to be an application of the pigeon hole principle with 500 "groups". Any hints on how to split $S$ into groups?

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  • $\begingroup$ To clarify: If $S=\{500,501, \cdots, 1000\}$ the only triple that works is, I think, $500+500=1000$, so I assume that $a_x$ might equal $a_y$? $\endgroup$ – lulu Sep 24 '19 at 20:07
  • $\begingroup$ You noted that 1000/2=500. It seems that you have to half the set. What more natural way to do it than to separate odd and even numbers? $\endgroup$ – Lucio Tanzini Sep 24 '19 at 20:08
  • $\begingroup$ @LucioTanzini ? You can't choose $S$. You have to prove you can do it whatever $S$ may be. $\endgroup$ – almagest Sep 24 '19 at 20:11
  • $\begingroup$ I didn't mean S, I meant the numbers up to 1000 $\endgroup$ – Lucio Tanzini Sep 24 '19 at 20:15
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    $\begingroup$ OK, I see you changed the question, but this is problematic... If we restrict to $a_x \neq a_y$, then the statement cannot be proved, because it is false, as shown by the counterexample of @lulu where $S=\{500, 501, \dots, 1000\}$. I.e. in that $S$, you cannot find $a_x \neq a_y, a_z$ s.t. $a_x + a_y = a_z$. OTOH, if we revert to the original question, which allows $a_x = a_y$, then the statement can be proved. Which do you want? $\endgroup$ – antkam Sep 25 '19 at 20:50
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This answer assumes $a_x = a_y$ is allowed (i.e. the OP's original wording).

HINT

Let $m$ be the maximum element in $S$. Then there are $500$ selected numbers among $A = \{1, 2, \dots, m-1\}$.

Now partition $A$ properly and apply pigeonhole principle.

If $m<1000$ you can be done right away with the right partition. For $m=1000$ you need a tiny bit more work, but not much.

HINT #2 (update 9/25/2019)

Consider $m=999$, so there are $500$ numbers selected from $A = \{1, \dots, 998\}$. You need to partition $A$ into $499$ subsets, each of size $2$. Then pigeonhole would say there is a subset where both numbers are selected. If you partition correctly, you can immediately find $a_x, a_y, a_z$ s.t. $a_x + a_y = a_z$.

Can you finish from here, or do you need more hint?

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  • $\begingroup$ I think I need another hint. Perhaps enlightening me with the partition of $m<1000$ will help? $\endgroup$ – Baker013273213 Sep 25 '19 at 20:36
  • $\begingroup$ What about the question gives away the "main idea" of the partition. I'm struggling on thinking on what kind of construction that would help to prove it. $\endgroup$ – Baker013273213 Sep 26 '19 at 1:07

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