Olympiad

Few Problems - 2

Dec 18, 2016 • 2 min read

For which $n\in \mathbb{N}$ , does
$n+1 | \binom{2n}{n}$
hold?
For a polynomial with integral coefficients, i.e. for all with , if where , are coprime integers with then show that:
1. $r|a_0$
2. $s|a_n$
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
$m_a+m_b+m_c<a+b+c<\frac{4}{3}(m_a+m_b+m_c)$
Construct an angle of $60^{\circ}$ . Give reasoning as to why your construction works.
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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