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last digit of sums of alternate sums of every subset of {1, 2, ...,10}

Source: 1999 Greece Junior p4

March 17, 2020

number theoryLast digitSubsets

Problem Statement

Define alternate sum of a set of real numbers

A =\{a_1,a_2,...,a_k\}

with

a_1 < a_2 <...< a_k

, the number

S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1

(for example if

A = \{1,2,5, 7\}

then

S(A) = 7 - 5 + 2 - 1

) Consider the alternate sums, of every subsets of

A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}

and sum them. What is the last digit of the sum obtained?