What is the integer part of the following fraction: $\frac{2012^{2013}+2013^{2014}}{2012^{2012}+2013^{2013}}$ 
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?

Observe that for any $a,b > 0,$ we have $\dfrac{a^{a+1}+b^{b+1}}{a^a + b^b} = \dfrac{a(a^a + b^b)+(b-a)b^{b}}{a^a + b^b} = a + (b-a)\dfrac{b^b}{a^a+b^b}.$ Clearly, $b^b < a^a + b^b.$ So in order for $\dfrac{b^b}{a^a+b^b}$ to be a fraction, we need $\dfrac{b^b}{a^a+b^b} < \frac{1}{b-a}.$ Here we have $a = 2012, b = 2013.$ Hence $(b-a)\dfrac{b^b}{a^a+b^b} = \dfrac{b^b}{a^a+b^b} < 1,$ so the integer part is $a = 2012.$
