All three runners finish together. <p/>

<!---->
Let the radius of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>R</mi></mrow></math>`s track be 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>r</mi></mrow></math> and let the radius of the 
first semicircle of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>P</mi></mrow></math>`s track be 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>p</mi></mrow></math>; then the radius of the 
second semicircle of this track is 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow></math>. The total length of 

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>P</mi></mrow></math>`s track is 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>&#x03C0;</mi><mi>p</mi><mo>+</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>&#x03C0;</mi><mi>r</mi></mrow></math>, the same as the 
length of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>R</mi></mrow></math>`s track. <p/>

<!---->
By a simlar argument, the length of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>Q</mi></mrow></math>`s 
track is also 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>&#x03C0;</mi><mi>r</mi></mrow></math>, so all three runners finish at the same time.   

<!----><!---->

    


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