Maths Olympiad Prep

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, 2013

Algebra Difficulty 2.4 Junior Find the answer Slovenia

Let x=22013x = 2^{2013}. Then the value of the expression
x−x2+1+1x2+1+x x - \sqrt{x^2 + 1} + \frac{1}{\sqrt{x^2 + 1} + x}
is equal to

Pick one

Solution

After finding the common denominator and rearranging the expression we get
(x+x2+1)(x−x2+1)+1x2+1+x=(x2−(x2+1))+1x2+1+x=0. \frac{(x + \sqrt{x^2 + 1})(x - \sqrt{x^2 + 1}) + 1}{\sqrt{x^2 + 1} + x} = \frac{(x^2 - (x^2 + 1)) + 1}{\sqrt{x^2 + 1} + x} = 0.
The correct answer is B.

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