# Find $5$ numbers where the sum of all pairs gives the results $110, 112, 113, 114, 115, 116, 117, 118, 120, 121$

Find $$5$$ numbers where the sum of the pairs gives the results $$110, 112, 113, 114, 115, 116, 117, 118, 120, 121$$

I have been trying on this question for some time and it seems easy at first glance, but later on, I was unable to start on the question. I tried turning these into equations with $$a_1, a_2, a_3, a_4, \text{ and }a_5$$ but couldn't as it is not determined which $$2$$ numbers give which sum. I was, however, able to achieve that $$a_1+a_2+a_3+a_4+a_5=289$$ (which is obvious). Now is there anyway I could continue without guessing. Thank you.

As there are no duplicates among the pair sums, the five numbers are distinct. So wlog. $$a_1. Then $$110=a_1+a_2$$, $$112=a_1+a_3$$, which makes $$a_3=a_2+2$$. Similarly, we find $$a_4=a_3+1$$ from the last two entries $$120,121$$. So far we have found that the numbers are $$a_1, a_2, a_2+2, a_2+3, a_5.$$ This also tells us that $$a_1+a_4=113$$ and that $$a_2+a_5=118$$. Now that the three pair sums $$a_2+a_3=2a_2+2$$, $$a_2+a_4=2a_2+3$$, $$a_3+a_4=2a_2+5$$ must occur among $$114,115,116,117$$. The only possible match is that $$a_2+a_3=114$$, i.e., $$a_2=56$$. The rest follows easily from here.