Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 21"

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==Solution==
 
==Solution==
The lengths of the straight parts of the strap are <math>3\,\mathrm{m}</math>, <math>4\,\mathrm{m}</math>, and <math>5\,\mathrm{m}</math>. Their sum is <math>12\,\mathrm{m}</math>. The curved parts of the band add up to a full circumference of one of the circles, so their sum is <math>20\pi\,\mathrm{m}</math>. The total length of the strap is <math>(12+20\pi)\,\mathrm{m}\Longrightarrow\mathrm{A}</math>.
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The lengths of the straight parts of the strap are <math>3\,\mathrm{m}</math>, <math>4\,\mathrm{m}</math>, and <math>5\,\mathrm{m}</math>. Their sum is <math>12\,\mathrm{m}</math>. The curved parts of the band add up to a full circumference of one of the circles, so their sum is <math>20\pi\,\mathrm{ cm}</math>. The total length of the strap is <math>(12+\frac\pi5)\,\mathrm{m}\Longrightarrow\mathrm{D}</math>.
  
 
==See also==
 
==See also==
 
{{CYMO box|year=2007|l=Lyceum|num-b=20|num-a=22}}
 
{{CYMO box|year=2007|l=Lyceum|num-b=20|num-a=22}}

Latest revision as of 21:21, 23 July 2020

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
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