RO-NMO 2009 Problem 94

Let be two natural numbers $m,n\ge 2,$ two increasing finite sequences of real numbers $\left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} ,$ and the set $$\left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} .$$ Show that the set above has $n+m-1$ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.