2012 x 2012 isn't huge, right?

Algebra Level 3

1 + 201 1 2 + 2011 2 201 2 2 + 2011 2012 \sqrt { 1+2011^{ 2 }+\frac { { 2011 }^{ 2 } }{ 2012^{ 2 } } } +\frac { 2011 }{ 2012 }

Evaluate the expression above and write your answer to 3 decimal places.


The answer is 2012.000.

Steven Jim by Steven Jim , 17, Vietnam

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1 solution

x = 1 + 201 1 2 + 201 1 2 201 2 2 + 2011 2012 Let a = 2012 = 1 + ( a − 1 ) 2 + ( a − 1 ) 2 a 2 + a − 1 a = 1 + a 2 − 2 a + 1 + 1 − 2 a + 1 a 2 + 1 − 1 a = a 2 + 2 + 1 a 2 − 2 ( a + 1 a ) + 1 + 1 − 1 a = ( a + 1 a ) 2 − 2 ( a + 1 a ) + 1 + 1 − 1 a = ( a + 1 a − 1 ) 2 + 1 − 1 a = a + 1 a − 1 + 1 − 1 a = a = 2012 \begin{aligned} x & = \sqrt{1+2011^2+\frac {2011^2}{2012^2}} + \frac {2011}{2012} & \small \color{#3D99F6} \text{Let }a = 2012 \\ & = \sqrt{1+(a-1)^2+\frac {(a-1)^2}{a^2}} + \frac {a-1}a \\ & = \sqrt{1+a^2-2a+1+1-\frac 2a + \frac 1{a^2}} + 1 - \frac 1a \\ & = \sqrt{a^2 + 2 + \frac 1{a^2} -2\left(a+\frac 1a\right) + 1} + 1 - \frac 1a \\ & = \sqrt{\left(a+\frac 1a\right)^2 -2\left(a+\frac 1a\right) + 1} + 1 - \frac 1a \\ & = \sqrt{\left(a+\frac 1a - 1\right)^2} + 1 - \frac 1a \\ & = a+\frac 1a - 1 + 1 - \frac 1a \\ & = a = \boxed{2012} \end{aligned}

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Nice solution!

Steven Jim - 4 years, 1 month ago

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