CombinatorialIdentity Putnam

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2687   
Compute the value of $$\displaystyle\sum_{k=1}^n k^2\binom{n}{k}$$

2693   
Let $n$ be a positive integer. Show that the smallest integer that is larger than $(1+\sqrt{3})^{2n}$ is divisible by $2^{n+1}$.

4172   

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$


4297   

Let $\lfloor{x}\rfloor$ be the largest integer not exceeding real number $x$. Show that $$\sum_{k=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\left(\left(1-\frac{2k}{n}\right)\binom{n}{k}\right)^2=\frac{1}{n}\binom{2n-2}{n-1}$$


4328   

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$


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