2024 IMO Shortlist C6

Let $n$ and $T$ be positive integers. James has $4n$ marbles with weights $1,2,\ldots,4n$. He places them on a balance scale so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most $T$.

Find, in terms of $n$, the minimum positive integer $T$ such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.

设 $n$ 和 $T$ 为正整数。James 有 $4n$ 颗弹珠，重量分别为 $1,2,\ldots,4n$。他把这些弹珠放在天平两侧，使两侧总重量相等。Andrew 每次可以把一颗弹珠从天平一侧移到另一侧，但要求移动后两侧重量差的绝对值始终不超过 $T$。

用 $n$ 表示满足下述条件的最小正整数 $T$：无论 James 最初怎样放置弹珠，Andrew 都能通过一系列移动，使每颗弹珠最终都到达天平的另一侧。