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HMMT 二月 2004 · 代数 · 第 1 题

HMMT February 2004 — Algebra — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. 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

英文解析

  1. 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. Sowe can see that there are 6 lattice points in their intersection (circled in the figure).