Question

A rectangular block $L\times 100\times H$, with $L\le 100\le H$, where L and H are integers, is cut into two non-empty parts by a plane parallel to one of the faces, so that one of the parts is similar to the original. How many possibilities are there for (L, H)?

• A. 10
• B. 12
• C. 20
• D. 24

Answer 1

The correct option is B

12

We must cut the longest edges, so the similar piece has dimensions $L\times 100\times k$ for some $1\le k<H$.
The shortest edge of this piece cannot be L, so it must be k. Thus $L\times 100\times H$ and $k\times L\times 100$ are similar.
Thus, $\frac{L}{k}=\frac{100}{L}=\frac{H}{100}$.
So, HL = ${100}^2$. ${100}^2$ = $2^4\times 5^4$
So ${100}^2$ has 25 factors, of which $\frac{(25-1)}{2}=12$ pairs are $<100$.