- Set of $16$ numbers are chosen from $[1,29]$ , show that there are always atleast $3$ numbers in that set so that sum of two of them results in the third number .
My first method :. I thought of trying PHP , but i am really confused what hole type should be considered , i know it should be having property of what its being needed , so i thought of ${1,2,3}$ as a box where property os being satisfied $1+2=3$ , but still it will give us only about $11$ boxes we want $15$ , i dont know what to do now . Although i think these types of box will not work since i could chose later 1,3 only and not $2$ .
- My second method : i realize mahbe better would be do like taking all possible pairs of $a_i ,a_j$ both belonging to set $S$ and taking sum of the pairs each and then making that as a set of $\binom{16}{2}$ elements and then trying to get a counter example but finally showing that one element out of that would indeed be in the set S itself .
By either method if possible i need a solution as such i dont get how to fully solve it from these ideas .
- Note : @GregMartin idea of largest element,i was able to prove it , but how do we prove it without observing that critical idea ?