Goutham wrote: Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$ If $f(a)<a^2$ for some $a$, it's easy to build a straight line from $(a,f(a))$ not encountering $y=x^2$ (choose for example the parallel to tangent at $a$ to parabola), and so contradiction. So $f(a)\ge a^2$ and graph of $f(x)$ is "above" parabola $y=x^2$ If $f(a)>a^2$ for some $a$, the straight line which is tangent at the parabola at $(a,a^2)$ encounters the parabola in one point and is "below" the parabola anywhere else and so never encounters $y=f(x)$, and so contradiction. So $f(a)\le a^2$ So $f(x)=x^2$ $\forall x$