No. Take
$$
(a_1,a_2,a_3,a_4)=(1,1,-1,-1).
$$
The full set of coefficients sums to zero, so the equation is partition regular by the <Rado theorem for one equation>. Every solution satisfies
$$
x_1+x_2-x_3-x_4=0,
$$
which can never be positive.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 130 / 3 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-130/3/i/solution