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HMMT 二月 1998 · ADV 赛 · 第 2 题

HMMT February 1998 — ADV Round — Problem 2

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 values of x , -19 < x < 98 satisfy cos x + 2 sin x = 1 ?
    2 3
    1 1 1
解析

英文解析

  1. How many values of x , − 19 < x < 98, satisfy
    2 2
    cos x + 2 sin x = 1?
    2 2
    Answer: 38 . For any x , sin x +cos x = 1. Subtracting this from the given equation gives sin x = 0,2
    or sin x = 0. Thus x must be a multiple of π , so − 19 < kπ < 98 for some integer k , or approximately − 6 . 1 < k < 31 . 2. There are 32 values of k that satisfy this, so there are 38 values of x that satisfycos x + 2 sin x = 1.22