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MATH / One hundred concentric circles with radii 1,2,3,…,100 are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is…

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problem

One hundred concentric circles with radii 1,2,3,…,1001,2,3,\ldots,100 are drawn in a plane. The interior of the circle of radius 11 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
Plain-text mathematical notation (without MathML)
One hundred concentric circles with radii 1,2,3,…,100 are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as m/n, where m and n are relatively prime positive integers. Find m+n.
Original LaTeX notation
One hundred concentric circles with radii $1,2,3,\ldots,100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

level

Level 5

type

Geometry

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Source and history

task_id
70c6d118-8d60-54e2-889f-fedcd63ee4b2
task_key
geometry--test--70c6d118-8d60-54e2-889f-fedcd63ee4b2
task_revision_id
1
config / split
geometry / test

Official source

initial import