SAT 1000 problems - P4

Algebra Level 3

Let f ( x ) = x 2 1 + x 2 f(x)=\dfrac{x^2}{1+x^2} , what is the value of f ( 1 ) + f ( 2 ) + f ( 1 2 ) + f ( 3 ) + f ( 1 3 ) + f ( 4 ) + f ( 1 4 ) f(1)+f(2)+f(\dfrac{1}{2})+f(3)+f(\dfrac{1}{3})+f(4)+f(\dfrac{1}{4}) ?

The result can be expressed as a b \dfrac{a}{b} , where a , b a,b are coprime integers.

Please submit a − b a-b .


The answer is 5.

Alice Smith by Alice Smith , 20, China

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1 solution

Chris Lewis
Jul 19, 2019

f ( 1 ) = 1 2 f(1)=\frac12 , and in general, f ( x ) + f ( 1 x ) = 1 1 + x 2 + 1 1 + x − 2 = 1 1 + x 2 + x 2 1 + x 2 = 1 f(x)+f \left( \frac1x \right)=\frac{1}{1+x^2}+\frac{1}{1+x^{-2}}=\frac{1}{1+x^2}+\frac{x^2}{1+x^2}=1 , so the sum is 7 2 \frac72 and the answer 7 − 2 = 5 7-2=\boxed5 .

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