Let there be $B \subset \{10,11,...,99\}$, with $|B| =10$. I need to prove, using the pigeonhole principle, that there are at least two disjoint subsets of $B$ (and none empty) where the sums of their elements are identical. I.e. $\exists B_1 \ne \varnothing, B_2 \ne \varnothing$ such that $B_1 \bigcup B_2 = B$, $B_1 \bigcap B_2 =\varnothing$ and $$\sum_{a \in B_1} a = \sum_{b \in B_2} b$$
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1This looks related to http://math.stackexchange.com/questions/320555/a-question-related-to-pigeonhole-principle?rq=1 – Arnaud D. Jan 07 '17 at 15:41
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What does "a set of the integers {10-99}" mean? Is $B={10,11,12,\ldots, 99}$ or is $B$ an unspecified subset of ${10,11,12,\ldots, 99}$? In the first case, you can pick ${10,99}$ and ${11,98}$. In the second case, you have problems if $B={97,98,99}$, for example. - You may want to check back with the original problem statement (I suspect that $B\subset{1,2,3,\ldots,99}$ and $|B|=10$) – Hagen von Eitzen Jan 07 '17 at 15:41
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You're correct, I forgot to mention that B is a set of exactly 10 integers. Ive edited the original question. Arnaud, it seems similar but I have a requirment that subsets will be foreign where as in that question from what I understand two different groups may have two people the same age. – user403156 Jan 07 '17 at 15:49
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I think you mean 'disjoint' when you say 'foreign': two sets $S$ and $T$ are disjoint if $S \cap T = \varnothing$. Please edit your question if this is the case. – TonyK Jan 07 '17 at 15:58
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@TonyK Thanks. done. – user403156 Jan 07 '17 at 16:02
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Rather similar problem: Prove that there are two subsets of any cardinality-10 subsets of ${1,2\dots,106}$ that sum to the same number. – Martin Sleziak Jan 14 '17 at 19:58
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A $10$-element subset $B$ of $\{10,\ldots,99\}$ has a sum of at most $90+\cdots+99=945$. This means that there can be at most 946 possible values $(0,\ldots,945)$ for the sum of elements in a subset of $B$.
But $B$ has $2^{10}=1024$ subsets. So at least two subsets, say $S$ and $T$, must have the same sum.
Obviously neither of these can be the empty set, because then the other (non-empty) set must have a non-zero sum. And if $S$ and $T$ are not disjoint, just remove the common elements from each set: $S\setminus T$ and $T\setminus S$. They will still have the same sum, and they will still be non-empty.
This argument goes through unaltered if we allow $B$ to be a subset of $\{0,\ldots,106\}$.
TonyK
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@THELONEWOLF.: I am counting the empty set, which is why I say there are $1024$ subsets. (Later I explain why neither of $S$ and $T$ can in fact be empty.) – TonyK Jan 07 '17 at 17:00
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But we have to choose sets with 10 elements then why empty set?? – Vidyanshu Mishra Jan 07 '17 at 17:01
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@THELONEWOLF.: because we are looking at subsets of the $10$-element set $B$. Pay attention. – TonyK Jan 07 '17 at 17:02
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Sorry.... apologies.. Got it now... Gotta work on it. Thanks +1 – Vidyanshu Mishra Jan 07 '17 at 17:04
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This was a IMO 1972 problem 1, by the way.
The original problem is as follows:
Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.
The solution can also be found here: https://artofproblemsolving.com/wiki/index.php/1972_IMO_Problems/Problem_1?srsltid=AfmBOophW2LA6TM8HKjSjaQOYq0RjdEckKmsHtWqbO4FDJa56ldd9dBA
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