MathDB

Given a natural number

n>4

and

2n+4

cards numbered with

1, 2, \dots, 2n+4

. On the card with number

m

a real number

a_m

is written such that

\lfloor a_{m}\rfloor=m

. Prove that it's possible to choose

4

cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than

\frac{1}{n-\sqrt{\frac{n}{2}}}

Problem Statement