Some Harder Problems

  1. Use the first part of formula 7 and its related forms along with Euler's Inequality to show $0 < \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2} = \frac{(s-a)(s-b)(s-c)}{abc}=\frac {r}{4R} \le \frac{1}{8}$ with equality if and only if the triangle is equilateral.
  2. Show that $\cos\alpha + \cos\beta + \cos\gamma = 1 + 4\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}$.
  3. Use the two previous problems to construct a proof of W3.
  4. ( 1997 Asian Pacific Mathematical Olympiad) Let triangle $ABC$ be inscribed in a circle and let

    \begin{displaymath}l_a = \frac{m_a}{M_a},\ \ l_b = \frac{m_b}{M_b},\ \ l_c = \frac{m_c}{M_c},\end{displaymath}

    where $m_a,\; m_b,\;m_c $ are the lengths of the angle bisectors (internal to the triangle) and $M_a,\;M_b,\;M_c $ are the lengths of the angle bisectors extended until they meet the circle. Prove that

    \begin{displaymath}\frac{l_a}{\sin^2\alpha} + \frac{l_b}{\sin^2\beta}+\frac{l_c}{\sin^2\gamma} \ge 3,\end{displaymath}

    and that equality holds if and only if $ABC$ is an equilateral triangle.
  5. ( 1995 Canadian Mathematical Olympiad) Let $a$, $b$, and $c$ be positive real numbers. Prove that

    \begin{displaymath}a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}.\end{displaymath}

  6. See the 33 problems from Bjorn Poonen's paper on inequalities.