Consider $n$ cards with marked numbers $1$ through $n$. No number have repeted, namely, each number has marked exactly at one card. They are distributed on $n$ boxes so that each box contains exactly one card initially. We want to move all the cards into one box all together according to the following instructions The instruction: Choose an integer $k(1\le k\le n)$, and move a card with number $k$ to the other box such that sum of the number of the card in that box is multiple of $k$. Find all positive integer $n$ so that there exists a way to gather all the cards in one box. Thanks to @scnwust for correcting wrong translation.
Problem
Source: kjmo 2022 P7
Tags: combinatorics
scnwust
29.10.2022 14:39
It'll be better to translate instruction part like this; Choose an integer $k(1\le k\le n)$, and move a card with number $k$ to the other box such that sum of the number of the card in that box is multiple of $k$. By the way, can you post other 2022 KJMO problems?
genius09
29.10.2022 21:38
Case 1. $n$ is even. Starting with the box with card $\frac{n}{2}+1$, add the card with number $1, \frac{n}{2}+2, 2, \cdots, n, \frac{n}{2}$ to the box containing $\frac{n}{2}+1$. We can easily calculate to show that this strategy satisfies the condition. Case 2. $n$ is odd. Gather all the cards from $1$ to $n-1$ by using the strategy in Case 1, then add the card with number $n$ the the gathered box. This satisfies the condition since $n$ divides $\frac{n(n-1)}{2}$.
mathhyhyhy
23.08.2023 12:40
What gets you to this idea.,?