MathDB

k not integers sum among 45 sums of pairs of 2 real numbers among 10

Source: Argentina 2007 OMA L3 p4

May 12, 2024

combinatoricsnumber theory

Problem Statement

10

real numbers are given

a_1,a_2,\ldots ,a_{10}

, and the

45

sums of two of these numbers are formed

a_i+a_j

, 1\leq i<j\leq 10 . It is known that not all these sums are integers. Determine the minimum value of

k

such that it is possible that among the

45

sums there are

k

that are not integers and

45-k

that are integers.