What is the integer part of the following fraction: $\dfrac{2012^{2013}+2013^{2014}}{2012^{2012}+2013^{2013}}$.
This is a competition problem for 7th grade students.
The answer to this question is $2012$.
Is there any way to simplify/evaluate it so we can see the integer part clearly? Or can we just estimate?
Best Answer
$2012=\frac{2012^{2012}\times2012+2013^{2013}\times2012}{2012^{2012}+2013^{2013}}<\frac{2012^{2013}+2013^{2014}}{2012^{2012}+2013^{2013}}<\frac{2012^{2012}\times2013+2013^{2013}\times2013}{2012^{2012}+2013^{2013}}=2013$