## 17 Dec Few Problems – 2

1. For which $n\in \mathbb{N}$, does
$n+1 | \binom{2n}{n}$

hold?
2. For a polynomial $p(x) = a_nx^n+\ldots + a_0$ with integral coefficients, i.e. $a_i\in \mathbb{Z}$ for all $1\leq i\leq n$ with $a_n\neq 0$, if $p(\frac{r}{s})=0$ where $r$, $s$ are coprime integers with $s\neq 0$ then show that:
1. $r|a_0$
2. $s|a_n$
3. Let $ABC$ be a triangle with side-lengths $a$, $b$, $c$ corresponding to sides $BC$, $CA$ and $AB$ respectively and let $m_a$, $m_b$ and $m_c$ be the lengths of the medians from vertices $A$, $B$ and $C$ respectively. Then show that
$m_a+m_b+m_c
4. Construct an angle of $60^{\circ}$. Give reasoning as to why your construction works.
5. We call a number good if it is divisible by 5 but not by 25. How many five digit good numbers are there?

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