IMO Shortlist 1983 problem 16
Dodao/la:
arhiva2. travnja 2012. Let
![F(n)](/media/m/6/6/6/66645d0371a25e0bbe7028cf288cb9d0.png)
be the set of polynomials
![P(x) = a_0+a_1x+\cdots+a_nx^n](/media/m/4/8/9/48971b77b5c22bfb636a05d91e081296.png)
, with
![a_0, a_1, . . . , a_n \in \mathbb R](/media/m/a/2/6/a26fa2993a2c2afc9b53297a5e2ff105.png)
and
![0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.](/media/m/a/0/4/a04ecd3f0dbfce383ac007612fd15548.png)
Prove that if
![f \in F(m)](/media/m/8/b/5/8b58a4acd87c9c52aab6e555857c0d00.png)
and
![g \in F(n)](/media/m/2/c/0/2c0292d18de45c07ca4d8428e9bde9a7.png)
, then
%V0
Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$
Izvor: Međunarodna matematička olimpijada, shortlist 1983