prove that for every 8 choosen numbers from 10 to 36 you can always make equalities.

Question

number can be used once. examples. let say that the choosen numbers are 10, 11, 12, 15, 18, 25, 32, 36 you can write 11+25=36 or 10+12+18=15+25. i tried to prove for summation for every 2 numbers $$\binom{8}{2}=28 $$ but there are 51 different summation that's the closest i got

Answer 1

You ask for a proof for "every" 8 chosen numbers.

I choose 10, 11, 12, 13, 14, 15, 16, 17.

None of these numbers can be expressed as the sum of some of the other 7.

The contrary example negates the conclusion.

The first number than does satisfy is 21.