2014
Problem - 4588
There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes
through the origin. Compute $a$.
The tangent line of the given curve at point $(x_0, y_0)$ is
$$\frac{y+y_0}{2}=xx_0 + 1 \implies y=2x_0x + (2+y_0)$$
Plugging $(x_0, y_0)=(0, 0)$ gives
$$y_0=-2 \implies y=2x_0x$
Setting $x_0=a$ gives $y=2ax$. Meanwhile, we know the slope of the tangent line is $2$, Hence $a=\boxed{1}$.