1990 APMO Problems/Problem 2

We have to prove \[S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n.\]

Let, Sk=i=1(nk)tiSnk=(m=1nam)(i=1(nk)1ti) Thus, SkSnk=(i=1(nk)ti)(m=1nam)(i=1(nk)1ti)=(m=1nam)[i=1(nk)1+i=1(nk)j=1(nk)ijtitj]=(m=1nam)[(nk)+i,j(titj+tjti)](m=1nam)[(nk)+2(nk)2(nk)2]=(nk)2m=1nam

In the fourth line, we used the AM-GM inequality and used the fact that there are $\dfrac{1}{2}\left({n \choose k}^2-{n \choose k}\right)$ terms in the sum of third line.