The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988).

Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.

I am trying to prove it but have gotten no idea yet.

For some small $n$'s we can see the validity of the assertion. In addition,

if $k= \frac {n}{2}$, then {$1, n$}, {$2, n-1$}, $\ldots$, {$n$,1} works, and

if $k =\frac {n+1}{2}$, then {$1, n-1$}, {$2, n-2$}, $\ldots$, {$n-1, 1$}, {$n$} works.

This observation doesn't seem to be of help for general solution.

  • $\begingroup$ It seems almost answered. $S=1+2+\cdots + n= {n(n+1)\over 2}$. Then either $n\over 2$ is integer or $n+1\over 2$ is integer. $\endgroup$ – nickchalkida Sep 17 '15 at 2:53
  • 2
    $\begingroup$ That really doesn't help, @nickchalkida $\endgroup$ – Thomas Andrews Sep 17 '15 at 2:54
  • $\begingroup$ Some thoughts: define $err = \sum_{t=1}^k |s(t) - m|$ where $s(t)$ is the sum of elements of the $t$-th set. Find a way to move elements from the biggest set to the smallest set or to interchange elements between the two to make $err$ decrease strictly by step so that after finite number of steps it goes to $0$. $\endgroup$ – corindo Sep 17 '15 at 3:04
  • $\begingroup$ Your example with $k=n/2$ generalizes somewhat. Suppose $k \mid n/2$; assign $1,2,\ldots,n$ to groups $1,2,\ldots k, k,k-1,\ldots,2,1,1,2,\ldots, k,k,\ldots,2,1,\ldots$ respectively. This solves $n=2rk$; by preceding the pattern with your $k=(n+1)/2$ solution, you can also solve the case $n=2rk-1$. $\endgroup$ – Tad Sep 17 '15 at 5:00
  • $\begingroup$ Sorry I am confused... I don't understand what a "general solution" would look like other than what you have given? There are only certain values of k and m for which the statement will be true... and you found one of them which seems good enough to answer the question. (as far as i can tell, m and k can be any combination of multiplying the prime factors of $\sum 1...n$, for example for n=11, $\sum 1...n$=66, and (m,k) can be (11,6), (22,3), (33,2), (66,1). $\endgroup$ – don bright Sep 18 '15 at 3:37

Induction on $n$. For $n=1,2$ there is nothing to prove. Assume the result is proved for $<n$ and consider the case $n$.

We establish the case $n$ by induction on $m$. If $n$ is odd the smallest possible value of $m$ is $m=n$. We have $n=(n-1)+1=(n-2)+2=\dots=\frac{1}{2}(n+1)+\frac{1}{2}(n-1)$, so the result is true for $m=n,k=\frac{1}{2}(n+1)$.

If $n$ is even, then the smallest possible value of $m$ is $n+1$. We have $n+1=(n-1)+2=\dots=(\frac{n}{2}+1)+\frac{n}{2}$, so the result is true for $m=n+1$ and $k=\frac{n}{2}$.

Now consider the case $m$ where $2n>m>n+1$ suppose it is true for $<m$. For $m$ odd we can take the sums $n+(m-n)=(n-1)+(m-n+1)=\dots=\frac{1}{2}(m+1)+\frac{1}{2}(m-1)$. These use up the numbers $m-n,m-n+1,\dots,n$. By induction the remaining numbers $1,2,\dots,m-n-1$ will give us the remaining sums of $m$.

Similarly, if $m$ is even, we can take the sums $n+(m-n)=(n-1)+(m-n+1)=\dots=(\frac{m}{2}+1)+(\frac{m}{2}-1)$. That uses up the numbers $m-n,m-n+1,\dots,n$ except for $\frac{m}{2}$. By induction the numbers $1,2,\dots,m-n-1$ can be divided into groups summing to $\frac{m}{2}$. But the numbers $1,\dots,m-n-1$ and $\frac{m}{2}$ must total a multiple of $m$, so we can divide the groups summing to $\frac{m}{2}$ into pairs which sum to $m$.

Finally consider $m\ge2n$. We can form the $k$ groups $n+(n-2k+1),(n-1)+(n-2k+2),\dots,(n-k+1)+(n-k)$ which each sum to $2n-2k+1$. But by induction we can group the remaining integers $1,2,\dots,n-2k$ into $k$ groups each with sum $m-2n+2k-1$, since $(m-2n)(m-n-1)\ge0$ and hence $m-3n-1+\frac{2n(n+1)}{m}\ge0$, so $m-3n-1+4k\ge0$ and hence $m-2n+2k=1\ge n-2k$.

That completes the induction on $m$ and hence the induction on $n$.


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