HMMT 十一月 2009 · 冲刺赛 · 第 3 题
HMMT November 2009 — Guts Round — Problem 3
题目详情
英文原题
- [ 5 ] Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside whichis inscribed a circle, and so on, with the outermost square having side length 1. Find the differencebetween the sum of the areas of the squares and the sum of the areas of the circles.
2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd
解析
英文解析
- [ 5 ] Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside whichis inscribed a circle, and so on, with the outermost square having side length 1. Find the differencebetween the sum of the areas of the squares and the sum of the areas of the circles.
Answer: 2 −π
The ratio of the area of each square and the circle immediately inside it is . The total sum of the 42
1 1 4πareas of the squares is 1 + + + . . . = 2. Difference in area is then 2 − 2 · .
2 4 π
2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd