IMO 1983 SL 16

Origin: GDR

Problem

Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn, with a0, a1, . . . , an \inR and 0 \leqa0 = an \leqa1 = an−1 \leq\cdot \cdot \cdot \leqa[n/2] = a[(n+1)/2]. Prove that if f \inF(m) and g \inF(n), then fg \inF(m + n).

Solution

Set hn,i(x) = xi + \cdot \cdot \cdot + xn−i, 2i \leqn. The set F(n) is the set of linear combinations with nonnegative coefficients of the hn,i’s. This is a convex cone. Hence, it suffices to prove that hn,ihm,j \inF(m+n). Indeed, setting p = n −2i and q = m −2j and assuming p \leqq we obtain hn,i(x)hm,j(x) = (xi + \cdot \cdot \cdot + xi+p)(xj + \cdot \cdot \cdot + xj+q) = n−i+j  k=i+j hm+n,k, which proves the claim.