IP rich ultrafilters and new partition regular equations

In 1979, using the theory of ultrafilters, N. Hindman proved that for every finite coloring of N, there exists a color that contains both additive and multiplicative IP sets. Later, in 1993, V. Bergelson and N. Hindman found an elementary proof. In this article, we prove that the partial product of these two IP sets also lies in the same color. As an immediate consequence, we have the equation a+b=c d partition regular. This was a conjecture of P. Csikv\'ari, K. Gyarmati, and A. S\'ark\"ozy, which was solved by V. Bergelson and N. Hindman independently. In this article, we separately give a very short proof of this conjecture. Then we extend these results to the flavor of Rado systems.