Question
A rectangular block L×100×H, with L≤100≤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)?
Answer: Option B
:
B
We must cut the longest edges, so the similar piece has dimensions L×100×k for some 1≤k<H.
The shortest edge of this piece cannot be L, so it must be k. Thus L×100×H and k×L×100 are similar.
Thus, Lk=100L=H100.
So, HL = 1002. 1002 = 24×54
So 1002 has 25 factors, of which (25−1)2=12 pairs are <100.
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:
B
We must cut the longest edges, so the similar piece has dimensions L×100×k for some 1≤k<H.
The shortest edge of this piece cannot be L, so it must be k. Thus L×100×H and k×L×100 are similar.
Thus, Lk=100L=H100.
So, HL = 1002. 1002 = 24×54
So 1002 has 25 factors, of which (25−1)2=12 pairs are <100.
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