AIME 2006 II · 第 1 题

Solution 1

Let the side length be called $x$, so $x=AB=BC=CD=DE=EF=AF$.

The diagonal $BF=\sqrt{AB^2+AF^2}=\sqrt{x^2+x^2}=x\sqrt{2}$. Then the areas of the triangles AFB and CDE in total are $\frac{x^2}{2}\cdot 2$, and the area of the rectangle BCEF equals $x\cdot x\sqrt{2}=x^2\sqrt{2}$

Then we have to solve the equation

$$2116(\sqrt{2}+1)=x^2\sqrt{2}+x^2$$ $$2116(\sqrt{2}+1)=x^2(\sqrt{2}+1)$$ $$2116=x^2$$ $$x=46$$ Therefore, $AB$ is $\boxed{046}$.

Solution 2

Because $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent, the degree-measure of each of them is ${{720-2\cdot90}\over4}= 135$. Lines $BF$ and $CE$ divide the hexagonal region into two right triangles and a rectangle. Let $AB=x$. Then $BF=x\sqrt2$. Thus

$$\begin{aligned} 2116(\sqrt2+1)&=[ABCDEF]\\ &=2\cdot {1\over2}x^2+x\cdot x\sqrt2=x^2(1+\sqrt2), \end{aligned}$$ so $x^2=2116$, and $x=\boxed{046}$.