### A note on Sierpiński's problem related to triangular numbers

We show that the system of equations ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r}$, where ${t}_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r},{t}_{x}+{t}_{y}+{t}_{z}={t}_{s}$ has infinitely many rational two-parameter solutions.