Two problems with prime numbers 
Problem 1. Prove that there exists $n\in\mathbb{N}$ such that in interval $(n^2, \ (n+1)^2)$ there are at least $1000$ prime numbers.
Problem 2. Let $s_n=p_1+p_2+...+p_n$ where $p_i$ is the $i$-th prime number. Prove that for every $n$, there exists $k\in\mathbb{N}$ such that $s_n<k^2<s_{n+1}$.

I've found these two a while ago and they interested me. But don't have any ideas.

Problem 2:
For any positive real $x$, there is a square between $x$ and $x+2\sqrt{x}+2$. Therefore it will suffice to show that $p_{n+1}\geq 2\sqrt{s_n}+2$.  We have $s_{n}\leq np_n$ and $p_{n+1}\geq p_n+2$, so we just need to show $p_n\geq 2\sqrt{np_n}$, i.e., $p_n\geq 4n$.  That this holds for all sufficiently large $n$ follows either from a Chebyshev-type estimate $\pi(x)\asymp\frac{x}{\log(x)}\,$ (we could also use PNT, but we don't need the full strength of this theorem), or by noting that fewer than $\frac{1}{4}$ of the residue classes mod $210=2\cdot3\cdot5\cdot7$ are coprime to $210$.  We can check that statement by hand for small $n$.
There have already been a couple of answers, but here is my take on problem 1:
Suppose the statement is false. It follows that $\pi(x)\leq 1000\sqrt{x}$ for all $x$.  This contradicts Chebyshev's estimate $\pi(x)\asymp \frac{x}{\log(x)}$
