China Mathematical Competition (Jiangxi)

Assume that $AB$ is on the line $y=2x-17$ and the coordinates of the other two vertices on the parabola are $C(x_1,y_1)$ and $D(x_2,y_2)$. Then $CD$ is on a line $L$ whose equation is $y=2x+b$. Combining this with the equation of the parabola, we get $x^2=2x+b\Rightarrow x_1,x_2=1\pm \sqrt{b+1}$. Assume that the length of one side of the square is $a$. Then $$\begin{array}{rl}{a}^2& =(x_1-x_2{)}^2+(y_1-y_2{)}^2\\ & =5(x_1-x_2{)}^2=20(b+1).(1)\end{array}$$

Pick a point $(6,-5)$ on the line $y=2x-17$, and the distance from the point to the line $y=2x+b$ is $a$. $$\text{So}a=\frac{|17+b|}{\sqrt{5}}.(2)$$ From (1) and (2), we get $b_1=3$, $b_2=63$, so $a^2=80$ or $a^2=1280$, and $a_{\min }^2=80$.