Utility maximimization for unusual Leontief utility function
Above is a plot of the indifference curves for $U=1,2,...,8$ and also for $U=8.5$ that I made in MATLAB. For $P_x=P_y=1$ and $m=4$, the budget constraint is simply the the line $y=4-x$ (not pictured here, but easy to visualize). 

Notice this budget constraint overlaps the $U=4$ indifference curve (fourth contour from the origin) on the line segment connecting $(1,3)$ and $(3,1)$. Any other point on the budget line would yield $U&lt;4$, hence consuming on the line segment connecting $(1,3)$ and $(3,1)$ is optimal. But since all of these bundles correspond to $U=4$, the optimal bundle is not unique, but could be any bundle on this line segment. 

Here is the MATLAB code that generated the plot: 

u=zeros(60,60);
for i=1:60;
    for j=1:60
        u(i,j)=min([(i+j)/10 4*sqrt(i/10) 4*sqrt(j/10)]);
    end
end
figure
contour(u,[1 2 3 4 5 6 7 8 8.5]);
ylabel('Good Y')
xlabel('Good X')
grid on
grid minor
xt = get(gca, 'XTick');                                 % 'XTick' Values
set(gca, 'XTick', xt, 'XTickLabel', xt/10) 
yt = get(gca, 'YTick');                                 % 'XTick' Values
set(gca, 'YTick', yt, 'YTickLabel', yt/10)