Olympiad problems

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Found problems: 27

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1996 Singapore Senior Math Olympiad, 2

Tags: geometry , trigonometry , angle , max , semicircle , triangle area , sum

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

2002 Estonia National Olympiad, 2

Tags: geometry , equilateral , distance , triangle area

Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.

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