2007 Cyprus MO/Lyceum/Problem 21

Problem

In the figure, three equal cycles of diameter $20\,\mathrm{ cm}$ represent pulleys, that are connected with a strap. If the distances between any two pulley center points are $AB=3\,\mathrm{m}$ , $AC=4\,\mathrm{m}$ and $BC=5\,\mathrm{m}$ , then the length of the strap is

$\mathrm{(A) \ } (12+20\pi)\,\mathrm{m}\qquad \mathrm{(B) \ } (12 + \pi)\,\mathrm{m}\qquad \mathrm{(C) \ } (12+ 4\pi)\,\mathrm{m}\qquad \mathrm{(D) \ } \left(12+\frac\pi5\right)\,\mathrm{m}\qquad \mathrm{(E) \ } \mathrm{None\,of\,these}$

Solution

The lengths of the straight parts of the strap are $3\,\mathrm{m}$ , $4\,\mathrm{m}$ , and $5\,\mathrm{m}$ . Their sum is $12\,\mathrm{m}$ . The curved parts of the band add up to a full circumference of one of the circles, so their sum is $20\pi\,\mathrm{m}$ . The total length of the strap is $(12+20\pi)\,\mathrm{m}\Longrightarrow\mathrm{A}$ .

2007 Cyprus MO, Lyceum (Problems)
Preceded by Problem 20	Followed by Problem 22
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2007 Cyprus MO/Lyceum/Problem 21

Problem

Solution

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