 Solution

Apply Ramsey's theorem to a sufficiently large complete graph on integer vertices, colouring edge $\{i,j\}$ by the colour of $|i-j|$. Choose a monochromatic clique $y_0<y_1<\cdots <y_{n-1}$. Set $x_i=y_i-y_{i-1}$ for $1\le i\le n-1$ and $x_n=y_{n-1}-y_0$. Every $x_i$ is the difference along an edge of the clique, so all have one colour, and the differences telescope to $x_1+\cdots +x_{n-1}=x_n$.