HMMT 二月 2004 · 代数 · 第 1 题
HMMT February 2004 — Algebra — Problem 1
题目详情
- How many ordered pairs of integers (a,b) satisfy all of the following inequalities? 2 2 a + b < 16 2 2 a + b < 8 a 2 2 a + b < 8 b
解析
- How many ordered pairs of integers (a,b) satisfy all of the following inequalities? 2 2 a + b < 16 2 2 a + b < 8 a 2 2 a + b < 8 b Solution: 6 This is easiest to see by simply graphing the inequalities. They correspond to the (strict) interiors of circles of radius 4 and centers at (0 , 0) , (4 , 0) , (0 , 4), respectively. So we can see that there are 6 lattice points in their intersection (circled in the figure).