2019 AMC 8 第 14 题

先试着解答 2019 AMC 8 第 14 题,然后核对你的答案与视频与精心整理的解答,解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2019 AMC 8 解答,或核对答案。

所有题目均经美国数学协会(MAA)官方合法授权使用。

14.

Isabella 有 66 张优惠券,可以在 Pete’s Sweet Treats 兑换免费冰淇淋筒。为了让优惠券用得久一些,她决定每隔 1010 天兑换一张,直到全部用完。她知道 Pete’s 星期日关门,但当她在日历上圈出这 66 个日期时,发现没有一个圈出的日期是星期日。Isabella 第一次兑换优惠券是在星期几?
Isabella has 66 coupons that can be redeemed for free ice cream cones at Pete’s Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 1010 days until she has used them all. She knows that Pete’s is closed on Sundays, but as she circles the 66 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
星期一
Monday
星期二
Tuesday
星期三
Wednesday
星期四
Thursday
星期五
Friday
答案:C
知识点:模运算日期与时间
小提示:
每隔 1010 天会让星期数前进 33 天
Every 1010 days advances the weekday by 33
大提示:
六次兑换会覆盖除一个星期几外的所有星期几
The six days hit every weekday except one
视频讲解:
解答视频缩略图
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文字解答:
逐个检查选项即可。
每隔 1010 天兑换一次,星期几会前进 33 天。
若第一次是星期 xx,六次兑换的星期依次为 x,x+3,x+6,x+2,x+5x, x+3, x+6, x+2, x+5 和 x+1x+1(模 77)。
唯一没出现的是 x+4x+4,所以星期日必须在第一次兑换日后四天。星期日前四天是星期三。
正确答案是 C。
Let us go through the answer choices.
Each redemption is 1010 days later, which advances the weekday by 33 days.
Starting on day x,x, the six redemption days are x,x+3,x+6,x+2,x+5,x, x+3, x+6, x+2, x+5, and x+1x+1 modulo 7.7.
The only missed weekday is x+4,x+4, so Sunday must be four days after the first redemption day. Four days before Sunday is Wednesday.
Thus, the correct answer is C.
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