Weighted Sum with restricted weights
The name of the model you are looking for is: Constrained Estimation of Ordinary Least Squares, this model allows you to introduce linear constrains to your estimation by just modifying the matrix expression of the OLS estimation.

The resulting model should satisfy the expression

$$Q^T\beta = c$$

It is solved with the equation:
$$\hat{\beta^c} = \hat{\beta}-(X^TX)^{-1}Q(Q^T(X^TX)^{-1}Q)^{-1}(Q^T\hat{\beta}-c)$$

For your problem, $Q=[1,1,1]$, $c=1$, $X$ and $Y$ are the matrixes for your data.