Few Problems

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<a+b+c<\frac{4}{3}(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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READ:   Math Olympiad Problem Sheet
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