MathDB

For a positive integer

n

and a subset

S

\{1, 2, \dots, n\}

, let

S

be "

n

-good" if and only if for any

x

y\in S

(allowed to be same), if

x+y\leq n

, then

x+y\in S

. Let

r_n

be the smallest real number such that for any positive integer

m\leq n

, there is always a

m

-element "

n

-good" set, so that the sum of its elements is not more than

m\cdot r_n

. Prove that there exists a real number

\alpha

such that for any positive integer

n

|r_n-\alpha n|\leq 2024.

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