Olym 11G. 조합 기하학
- 연습문제
Example n. Title Here
Let $S$ be a set of $a+b+3$ points on a sphere, where $a$, $b$ are nonnegative integers and no four points of $S$ are coplanar. Determine how many planes pass through three points of $S$ and separate the remaining points into $a$ points on one side of the plane and $b$ points on the other side.
BAMO 2020
Solution
Solution Here.
Example n. Title Here
Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon.
As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):
Prove that it is always possible to choose the number $h$ such that the rectangels completely cover the interior of the $n$-gon and the total area of the rectangles is no more than twice the area of the $n$-gon
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Example n. Title Here
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Example n. Title Here
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Example n. Title Here
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Example n. Title Here
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