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Sums of cubes of numbers with certain digits are equal

Source: Germany 2021, Problem 3

June 16, 2021

algebrasum of cubesalgebra proposedcombinatorics proposedDigitsDigit

Problem Statement

For a fixed

k

with

4 \le k \le 9

consider the set of all positive integers with

k

decimal digits such that each of the digits from

1

k

occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.

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