Consider an exchange economy with $2$ goods and $2m$ identical Households, but in this case each household has utility function $u(x_1,x_2)=x_{1}^2+x_{2}^2$, and endowments $w_1=w_2$. Show that there exist prices at which agents can engage in mutually beneficial trades.
My thinking: 
Here $MRS_{x_1,x_2}$ is not equal to $\frac{p_1}{p_2}$ because the utility is not convex.
$MRS_{x_1,x_2} = \frac{2x_1}{2x_2} = \frac{w_1}{w_2} = 1$.
Any suggestions on where I can go from here?
As you can see the $MRS$ increases as you move along an indifference curve, $x_1\nearrow$ and $x_2\searrow$. That means the consumer likes specialization, the more he or she gets of $x_1$ the more she is willing to give-up of good $x_2$ to get more of good $x_1$ and also the other way around.
Consumers have the same amount of $x_1$ and $x_2$ but they like specialization (without any particular preference of good $x_1$ over good $x_2$).
This suggests than an optimal trade will involve one consumer giving all of her $x_1$ in exchange for all the $x_2$ of another consumer. As the number of consumers is even, this can work. The (implicit) prices in this trade are $p_1=p_2$.
Now, you have to prove formally that the suggested trade does in fact make everyone better off.