Real numbers non linear equations system with exponentials

Question

let $A_1 , A_2, \cdots A_p$ be any $p$ real numbers such that :

$$A_1 + A_2 +\cdots +A_p =0$$

$$e^{A_1}+ e^{A_2}+ \cdots +e^{A_p}=p$$

Do we necessarily have $A_1 = A_2 = \cdots = A_p = 0$ ?