2007 Cyprus MO/Lyceum/Problem 21

Problem

2007 CyMO-21.PNG

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 $20\pi\,\mathrm{ cm}$. The total length of the strap is $(12+\frac\pi5)\,\mathrm{m}\Longrightarrow\mathrm{D}$.

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Invalid username
Login to AoPS