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combinatorics of the last two digits

Source: Problem 3 of Russian Regional Olympiad 2011, grade 10

September 1, 2011
number theory proposednumber theory

Problem Statement

a1,a2,…,a14a_1,a_2,\dots,a_{14} are different positive integers. All 196 numbers of the form ak+ala_k+a_l with 1≤k,l≤141\leq k,l\leq 14 are written on a board. Is it possible that for any two-digit combination, there exists a number among all 196 that ends with that combination (i.e., there exist numbers ending with 00,01,…,9900, 01, \dots, 99)? (Author: P. Kozhevnikov)
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