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same size same sum

Source: KJMO 2023 P7

November 4, 2023

number theory

Problem Statement

Find the smallest positive integer

N

such that there are no different sets

A, B

that satisfy the following conditions. (Here,

N

is not a power of

2

. That is,

N \neq 1, 2^1, 2^2, \dots

.)
[*]

A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}

[*]

|A| = |B| \geq 1

[*] Sum of elements in

A

and sum of elements in

B

are equal.

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