Let's color circles.

Geometry Level 4

150 concentric circles with radii 1, 2, 3, . . . , 150 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 150 can be expressed as m/n, where m and n are relatively prime positive integers. Find m + n.

The answer is 451.

1 solution

Alex Wang
Nov 2, 2014

The fraction can be expressed as $\frac{\pi(150^2-149^2+148^2-147^2+...+2^2-1^2)}{150^2 \times \pi}.$ Since $a^2 - (a - 1)^2 = 2a - 1$ , we can rewrite it as $\frac{2 \times 150-1+2 \times 148-1+...+2 \times 2-1}{150^2}$ . $=\frac{4(75 + 74 + 73 + ... + 2 + 1) - 75}{150^2}$ $=\frac{4 \times 76 \times 75/2 - 75}{150^2}$ $= \frac{151}{300}.$

Our answer is $151 + 300 = \boxed{451 }$ .

$a^2-(a-1)^2=a+(a-1)$ , or that the difference of two consecutive squares are their sum. So the numerator would simply be equal to $1+2+3+4+...+150 = \frac{(150)(151)}{2}$

Linus Setiabrata - 6 years, 7 months ago

Inductively, you can represent the n-th green area as arithmetic sequence with a=3, b = 4, and 1<=n<=75 . The first green area will be 3+(1-1)4=3, and so on...

Nico Kurniawan - 6 years, 7 months ago

I remember this problem from Math Olympiad. Great question Alex!

Ashwat Chidambaram - 6 years, 7 months ago

Let's color circles.

The answer is 451.

1 solution

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