Is there a number $c>0$ such that: 


For any $n$ there is a polynomial $p(x) = a\_nx^n +\cdots + a\_0$ where the coefficients are $-1, 0$ or $1$ such that the number of repetition of the root $x=1$ in $p$ is at least equal to $c\sqrt{\frac{n}{\log n}}$.



On the number of repeated roots
Yes, there is a cute combinatorial argument by Bloch and Polya



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> "On the roots of certain algebraic equations", Proc. London Math. Soc. 33 (1932), 102-114.
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Consider the points $c=(c\_1,\dots,c\_k)\in \mathbb Z^k$ defined by $c\_i=\sum\_{r=0}^{n} \binom{r}{i-1}b\_r$ for $i=1,2,\dots,k$, where the $b\_r$'s are allowed to take values in $\{0,1\}^{n+1}$. The $c$ points are bounded in the region $0\le c\_i\le \binom{n+1}{i}$. A simple estimate shows that for $k<\sqrt{\frac{n}{\log\_2 n}}$ the number of $c$ points is less than $2^n$. That means that there exist two vectors $b,b'$ which give rise to the same $c$. Now let $a\_r=b\_r-b'\_r$ for $r=0,\dots,n$. Define $P(x)=\sum\_{r=0}^n a\_r x^r$, which has coefficients in $\{-1,0,1\}$, then we have $0=P(1)=P'(1)=\cdots=P^{(k-1)}(1)$.


