  In this short note, we present a construction of new symplectic 4-manifolds
with non-negative signature using the complex surfaces on Bogomolov-Miyaoka-Yau
line $c_1^2 = 9\chi_h$, the fake projective planes and Cartwright-Steger
surfaces. Our construction yields an infinite family of fake rational homology
$(2n-1)\CP#(2n-1)\CPb$ for any integer $3 \leq n \leq 22$.
