IMO Shortlist 1978 problem 1
Dodao/la:
arhiva2. travnja 2012. The set
![M = \{1, 2, . . . , 2n\}](/media/m/c/8/7/c8712c198d439c061b5ad2873131aa6e.png)
is partitioned into
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
nonintersecting subsets
![M_1,M_2, \dots, M_k,](/media/m/c/8/2/c82f97be46e2f5c52c742133ec673739.png)
where
![n \ge k^3 + k.](/media/m/e/c/8/ec829593ecddb0ce183e33a48bda567e.png)
Prove that there exist even numbers
![2j_1, 2j_2, \dots, 2j_{k+1}](/media/m/7/f/7/7f73957795ef6ae88ff7f13d667763a8.png)
in
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
that are in one and the same subset
![(1 \le i \le k)](/media/m/9/3/f/93f8bc2353978d343f2b71831fc4d36f.png)
such that the numbers
![2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1](/media/m/9/0/e/90eb827e23fe8dd9161318b90bc5edaf.png)
are also in one and the same subset
%V0
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$
Izvor: Međunarodna matematička olimpijada, shortlist 1978