Japanese Mathematical Olympiad Preliminary 2014

Let $O$ be the circumcenter of triangle $ABC$, and let $l$ be the line passing through the midpoint of segment $BC$ which is also perpendic...

  1. Let $O$ be the circumcenter of triangle $ABC$, and let $l$ be the line passing through the midpoint of segment $BC$ which is also perpendicular to the bisector of angle $ \angle BAC $. Suppose that the midpoint of segment $AO$ lies on $l$. Find $ \angle BAC $.
  2. Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $$2^a+3^b+1=6^c.$$
  3. In a school, there are $n$ students and some of them are friends each other. (Friendship is mutual.) Define $ a, b $ the minimum value which satisfies the following conditions
    i) We can divide students into $ a $ teams such that two students in the same team are always friends.
    ii) We can divide students into $ b $ teams such that two students in the same team are never friends.
    Find the maximum value of $ N = a+b $ in terms of $n$.
  4. Let $ \Gamma $ be the circumcircle of triangle $ABC$, and let $l$ be the tangent line of $\Gamma $ passing $A$. Let $ D, E $ be the points each on side $AB, AC$ such that $ BD : DA= AE : EC $. Line $ DE $ meets $\Gamma $ at points $ F, G $. The line parallel to $AC$ passing $ D $ meets $l$ at $H$, the line parallel to $AB$ passing $E$ meets $l$ at $I$. Prove that there exists a circle passing four points $ F, G, H, I $ and tangent to line $ BC$.
  5. Find the maximum value of real number $k$ such that \[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\] holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.

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MATHEMATICAL OLYMPIAD PROBLEMS: Japanese Mathematical Olympiad Preliminary 2014
Japanese Mathematical Olympiad Preliminary 2014
MATHEMATICAL OLYMPIAD PROBLEMS
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