Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha+1)$$ in the first quadrant is __________.

If $$a_{\alpha}$$ is the greatest term in the sequence $$\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots$$, then $$\alpha$$ is equal to _____________.

Let a curve $$y=f(x), x \in(0, \infty)$$ pass through the points $$P\left(1, \frac{3}{2}\right)$$ and $$Q\left(a, \frac{1}{2}\right)$$. If the tangent at any point $$R(b, f(b))$$ to the given curve cuts the $$\mathrm{y}$$-axis at the point $$S(0, c)$$ such that $$b c=3$$, then $$(P Q)^{2}$$ is equal to __________.

The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.