HMMT 二月 2000 · GEN 赛 · 第 21 题
HMMT February 2000 — GEN Round — Problem 21
题目详情
- Find the area of the six-p oin ted star if all edges are of length s , all a ute angles are 60
and all obtuse angles are 240 .Æ
英文原题
Find the area of the six-p oin ted star if all edges are of length s , all a ute angles are 60 Æ and all obtuse angles are 240 Æ . 22. An equilateral triangle with sides of length 4 has an isos eles triangle with the same base and half the heigh t ut out of it. Find the remaining area. 23. What are the last t w o digits of 7 7 77 ? 24. P eter is randomly lling b o xes with andy . If he has 10 pie es of andy and 5 b o xes in a ro w lab eled A, B, C, D, and E, ho w man y w a ys an he distribute the andy so that no t w o adja en t b o xes are empt y? 25. Ho w man y p oin ts do es one ha v e to pla e on a unit square to guaran tee that t w o of them are stri tly less than 1/2 unit apart? 26. Janet is trying to nd Tim in a Cartesian forest. Janet is 5 p 2 miles from (0,0), p 41 miles from (1,0), and p 61 miles from (0,1). Tim is p 65 miles from (0,0), 2 p 13 miles from (1,0), and p 58 miles from (0,1). Ho w man y miles apart are Janet and Tim?
解析
英文解析
- It is lear that we an split the gure in to 12 equilateral triangles, all of which have sidep
2 ps 3
length s . So, sin e the area of one of these triangles is , the total area is 3 s 3 .2
p p 4
A2
eq 1 4 p 3