10.3

Part of 2011 All-Russian Olympiad Regional Round

Problems(1)

combinatorics of the last two digits

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

a_1,a_2,\dots,a_{14}

are different positive integers. All 196 numbers of the form

a_k+a_l

1\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, \dots, 99

)? (Author: P. Kozhevnikov)

number theory proposednumber theory