<s>[INST] 2 problems related to the number 2015 
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*Let $p=\underbrace{11\cdots1}_\text{2015}\underbrace{22\cdots2}_\text{2015}$. Find $n$, where $n(n+1) = p$

*Prove that $\frac{1}{2^2} + \frac{1}{3^2} + \cdots +  \frac{1}{2015^2} < \frac{2014}{2015}$
For 1, I tried dividing in various ways until I got a simpler expression, but no result. For 2, I know that $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$, but the proofs i found are over the elementary level the problem is aimed for. I also proved that it's smaller than $\frac{1}{2} + \frac{1}{2^2} + \cdots + \frac{1}{1024^2} < 1$, but that obviously is missing $\frac{1}{2015}$
 [/INST] For 1 you can use the quadratic equation:
$n^2+n-p=0 \rightarrow n=\frac{-1 \pm \sqrt{4 \cdots 4 8 \cdots 89}}{2} $.
Now show $\sqrt{4 \cdots 4 8 \cdots 89}=\underbrace{66\cdots67}_\text{2015}$ and take the positive root to get $n=\underbrace{33\cdots3}_\text{2015}$
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