Home Competitions ILL 1983 Problem 33
Let F ( n ) F(n) F ( n ) P ( x ) = a 0 + a 1 x + ⋯ + a n x n P(x) = a_0+a_1x+\cdots+a_nx^n P ( x ) = a 0 + a 1 x + ⋯ + a n x n a 0 , a 1 , . . . , a n ∈ R a_0, a_1, . . . , a_n \in \mathbb R a 0 , a 1 , ... , a n ∈ R 0 ≤ a 0 = a n ≤ a 1 = a n − 1 ≤ ⋯ ≤ a [ n / 2 ] = a [ ( n + 1 ) / 2 ] . 0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}. 0 ≤ a 0 = a n ≤ a 1 = a n − 1 ≤ ⋯ ≤ a [ n /2 ] = a [( n + 1 ) /2 ] . f ∈ F ( m ) f \in F(m) f ∈ F ( m ) g ∈ F ( n ) g \in F(n) g ∈ F ( n ) f g ∈ F ( m + n ) . fg \in F(m + n). f g ∈ F ( m + n ) .
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