IMO Shortlist 1999 problem C7
Dodao/la:
arhiva2. travnja 2012. Let
![p >3](/media/m/6/6/8/6680785bbf6771d59ccd7677383b8927.png)
be a prime number. For each nonempty subset
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
of
![\{0,1,2,3, \ldots , p-1\}](/media/m/d/6/9/d69451a61e6f442b9876ecb1e9c99cb9.png)
, let
![E(T)](/media/m/6/3/1/6315314212d2b78baf650ab81bcbd47c.png)
be the set of all
![(p-1)](/media/m/2/1/a/21a8cb6052254bcebe1fa6227ef1fd33.png)
-tuples
![(x_1, \ldots ,x_{p-1} )](/media/m/7/5/4/754156b2142d61bb007b566824b60523.png)
, where each
![x_i \in T](/media/m/b/4/8/b487fd1bf610fbcd4ef51ff5f4d84352.png)
and
![x_1+2x_2+ \ldots + (p-1)x_{p-1}](/media/m/e/b/0/eb07cf2fd875c199a251607fef1b9be4.png)
is divisible by
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and let
![|E(T)|](/media/m/f/9/8/f980a5d67717e4716f40f4e9a41ae7c5.png)
denote the number of elements in
![E(T)](/media/m/6/3/1/6315314212d2b78baf650ab81bcbd47c.png)
. Prove that
with equality if and only if
![p = 5](/media/m/b/b/4/bb4ce19bf1cd093db859fca299b23eb9.png)
.
%V0
Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that
$$|E(\{0,1,3\})| \geq |E(\{0,1,2\})|$$
with equality if and only if $p = 5$.
Izvor: Međunarodna matematička olimpijada, shortlist 1999