HMMT 十一月 2021 · 冲刺赛 · 第 35 题

HMMT November 2021 — Guts Round — Problem 35

[20] The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. Let a be the proportion of pixels that are black. Estimate A = b 10000 a c . ⌊ ⌋ ( ) 12 A E An estimate of E will earn 20 min , points. E A https://i.imgur.com/W4IoGU3.png

解析

[20] The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let a be the proportion of pixels that are black. Estimate A = b 10000 a c . ⌊ ⌋ ( ) 12 A E An estimate of E will earn 20 min , points. E A Proposed by: Sean Li Answer: 3633 Solution: This is an area estimation problem. A good place to start is to focus on the jacket. The 2 hair adds about as much area as the hand takes away; the jacket seems to occupy about of the width 3 1 1 of the square and of the height. A crude estimate of → 3333 is already worth 7 points. One can 2 3 2 refine it some by accommodating for the fact that the jacket is a little wider than of the image. 3 381040 Exactly 381040 of the pixels are black, so a = = 0 . 36338 . . . and the answer is 3633. 2 1024