IMO Shortlist 1993 problem C2
Dodao/la:
arhiva2. travnja 2012. Let
![n,k \in \mathbb{Z}^{+}](/media/m/1/7/0/170841bc881180ddc366d53f308ff611.png)
with
![k \leq n](/media/m/7/6/d/76d61635b245248a9b2b412db0147304.png)
and let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be a set containing
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
distinct real numbers. Let
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
be a set of all real numbers of the form
![x_1 + x_2 + \ldots + x_k](/media/m/2/5/2/25243f50ec2ef494f6ee695fe833bc16.png)
where
![x_1, x_2, \ldots, x_k](/media/m/1/a/6/1a68fd6761a471d954d25d7630daae54.png)
are distinct elements of
![S.](/media/m/3/7/7/3772accbdc4fffed2efa17d53f141907.png)
Prove that
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
contains at least
![k(n-k)+1](/media/m/a/b/7/ab79e5692004f8aef6c14d7ddfa8950a.png)
distinct elements.
%V0
Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.
Izvor: Međunarodna matematička olimpijada, shortlist 1993