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0532 a_p + a_q = a_r + a_s 5th edition Round 3 p2

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May 6, 2021
number theoryalgebra5th edition

Problem Statement

Let 0<a1<a2<...<a16<1220 < a_1 < a_2 <... < a_{16} < 122 be 1616 integers. Prove that there exist integers (p,q,r,s)(p, q, r, s), with 1≤p<r≤s<q≤161 \le p < r \le s < q \le 16, such that ap+aq=ar+asa_p + a_q = a_r + a_s.
An additional 22 points will be awarded for this problem, if you can find a larger bound than 122122 (with proof).
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