2021美国大联盟国际夏季数学挑战活动三年级个人赛

Is it possible to place some or all of operation signs (+, –, ×, ÷) between two
consecutive digits to make the resulting expression become 100? (Note: There are 9 numbers, and there are 8 places to place operation signs.)
$1\quad 2\quad 3\quad 4\quad 5\quad 6\quad 7\quad 8\quad 9 = 100$

是否可以将运算符(+，-，×， ÷)的一部分或全部放在两个连续的数字中间使结果变成100?(注:9个数字，8个位置来填运算符)

【答案】Yse

• $1+2+3+4+5+6+7+8\times 9 = 100$
  $1+2\times 2 + 4\times 5-6+7+8\times 9 =100$

During practice, Dana’s six arrows land on the target shown. Each arrow is inside one of the regions of the target. Which of the following total scores is possible: 44, 31, 26, 16? (If an arrow lands inside the smallest circle in the center, the score Dana gets is 7. If an arrow lands inside the second circle, but not inside the smallest circle, the score Dana gets is 5. If an arrow lands inside the largest circle, but not inside the second circle or the smallest circle, the score Dana gets is 3. Assume no arrow lands on the boundary of any of the three circles.)

在练习中，戴娜的六支箭落在了图中所示的目标上。每一支箭都在目标的一个区域内。下列哪一个总分是可能的: 44, 31, 26, 16?(如果箭落在中间最小的圆圈内，Dana得7分。如果箭落在第二个圆内，但不在最小的圆内，Dana得到的分数是5。如果箭落在最大的圆内，但没有落在第二个或最小的圆内，Dana得到的分数是3。假设没有箭头落在这三个圆的边界上。)

【答案】26

• 设得$7$分的有$x$次，得$5$分的有$y$次，则得$3$分得有$6-x-y$次，那么总分是$7x+5y+3(6-x-y)=4x+2y+18$，这是一个偶数，所以不可能是$31$，当$x=6$，$y=0$时，得分最大为$4\times 6 + 18 =42$分，当$x=0$，$y=6$时得分最小为$2\times 6 + 18 = 30$分，所以只可能是$26$分

In a class of 5th grade students, 15 have pet cats, 12 have pet dogs, 5 have both pet cats and pet dogs, and 8 have neither pet cats nor pet dogs. How many total students are in the class?

在一个五年级的班级中，15人养宠物猫，12人养宠物狗，5人既养宠物猫又养宠物狗，8人既不养宠物猫也不养宠物狗。这个班总共有多少学生?

【答案】30

• $15+12-5+8=30$

Given the following clues, find out the number of gold, silver, and bronze medals that France, Italy, and Japan got in this international sports competition.
Japan has 1 more gold medal, but 3 fewer silver medals, than Italy.
Japan has 6 silver medals.
France has the most bronze medals (18), but fewest gold medals (7).
Each country has at least 6 medals of each type.
Italy has 27 medals in total.
Italy has 2 more bronze medals than gold medals.
The three countries have 38 bronze medals in total.
The three countries have 93 gold, silver, and bronze medals in total.
(a) What is the number of gold medals Italy got?
(b) What is the number of silver medals France got?
(c) What is the number of bronze medals Japan got?

根据以下线索，找出法国、意大利、日本在这次国际体育比赛中获得的金、银、铜的数量。
日本比意大利多一枚金牌，但少三枚银牌。
日本有6枚银牌。
法国获得的铜牌最多(18枚)，但金牌最少(7枚)。
每个国家至少有6枚每种类型的奖牌。
意大利总共获得27枚奖牌。
意大利的铜牌比金牌多2块。
这三个国家共获得38枚铜牌。
这三个国家共获得93枚金牌、银牌和铜牌。
(a)意大利获得了多少枚金牌?
(b)法国获得银牌的数量是多少?
(c)日本获得了多少枚铜牌?

【答案】8、10、6

• 见下表

  	意大利	法国	日本	共
  金	8	7	9
  银	9	16	6
  铜	10	18	10	38
  共	27			93

汉娜卖了价值$65$美元的烤肉票。成人票每张$4$美元，儿童票每张$3$美元。请按从小到大的顺序列出汉娜5种可能售出的成人票的数量。

【答案】2、5、8、11、14

• $4\times 2 + 3\times 19 =65$
  $4\times 5 + 3\times 15=65$
  $4\times 8 + 3\times 11 =65$
  $4\times 11 + 3\times 7 =65$
  $4\times 14 + 3\times 3 =65$

Three people went strawberry picking and picked 65 strawberries in total. At the first plant, they each picked the same amount of strawberries. At the second plant, they each collected three times the amount that they each had collected at the first plant. After picking from the third plant, they had five times the amount they had after picking strawberries from the first two plants. At the fourth plant they collected only five strawberries altogether. How many strawberries did each person collect at the first plant?

三个人去摘草莓，一共摘了65个草莓。在第一株植物中，他们每人采摘了相同数量的草莓。在第二株植物上，他们每人收集到的量都是在第一株植物上收集到的量的三倍。从第三棵树上摘草莓后，他们的收获量是前两棵树上摘草莓后的五倍。在第四株植物上，他们总共只采集了五颗草莓。第一次采摘时，每个人摘了多少草莓?

【答案】1

• 前三棵一共摘了$65-5=60$个，那么前两棵一共$60\div 5=12$个，每人$12\div 3 =4$个，而第二棵树是第一棵树的$3$倍，所以第一棵树上每个人摘了$4\div (3+1)=1$颗草莓。

What is the 5-digit mystery number that fits all the conditions below?
The sum of the first two digits (counting from left to right, same below) is one smaller than the third digit.
The third digit is double the fourth digit.
The fourth digit is double the last digit.
The third digit is the product of the fourth and fifth digits.
The second digit is five more than the first digit.
The first digit is one-eighth of the third digit and also one-fourth of the fourth
digit.

符合下面所有条件的5位数是什么?
前两位数字的和(从左到右计数，下同)比第三位数字小$1$。
第三位数是第四位数的两倍。
第四个数字是最后一个数字的两倍。
第三位是第四位数和第五位数的乘积。
第二个数字比第一个数字大5。
第一个数字是第三个数字的八分之一，也是第四个数字的四分之一

【答案】16842

• 第一个数字是第三个数字的八分之一，也是第四个数字的四分之一，所以第一个数字肯定是$1$，第三个数字是$8$，第四个数字是$4$，前两位数字的和比第三位数字小$1$，那么第二个数字是$8-1-1=6$，第三位是第四位数和第五位数的乘积，所以第五位数字是$8\div 4=2$，那么这个数是16842

If two sides of a square field were increased by five feet, as seen in the diagram below, not drawn to scale, the area of the field would increase by 245 square feet. Find the area of the original square.

如下图所示，如果一块正方形土地的两边增加5英尺，如果不按比例画，这块土地的面积将增加245平方英尺。求原来正方形的面积。

【答案】484

• $245-5\times 5=220$，$220\div 2=110$，$110\div 5=22$，所以原来正方形的面积是$22\times 22 =484$平方英尺。

A girl and a boy play Rock, Paper, Scissors ten times, where Rock beats Scissors,
Scissors beats Paper, and Paper beats Rock. The boy uses Rock three times, Scissors six times, and Paper once. The girl uses Rock twice, Scissors four times, and Paper four times. None of the ten games was a tie. How many games has the boy won?

一个女孩和一个男孩玩十次石头剪刀布游戏，石头打败了剪刀，剪刀打败布，布打败石头。男孩用石头三次，剪刀六次，布一次。女孩用石头两次，剪刀四次，布四次。十场比赛没有一场是平局。那么这个男孩赢了多少场比赛?

【答案】7

• 如下表，因为没有平局，所以女孩的4次剪刀不能对男孩的剪刀，只能对男孩的石头或者布，而男孩的石头和布合起来刚好是$4$，因此男孩的$6$次剪刀对女孩的$2$次石头和$4$次布，那么男孩一共赢了$3+4=7$次

  男孩	石头3	剪刀6	布1
  女孩	石头2	剪刀4	布4

When a clock turned from 08:32 to 08:33, the digit 3 appeared 3 times. Once when it was 08:32, and the second and third appearance is when it was 08:33. On a digital clock showing 24-hour time, over a whole day, the clock will display 1440 different times, from 0:00 to 23:59. Of all the 1440 different times displayed, how many times does the digit 5 appear in total? (For example, when the clock displays 5:55, which is 5:55 AM in the morning, the digit 5 appears three times. For another example, when the clock displays 15:55, which is 3:55 PM in the afternoon, the digit 5 appears three times.)

当时钟从08:32转到08:33时，数字3出现了3次。第一次是在08:32，第二次和第三次出现是在08:33。一个显示24小时时间的数字时钟上，在一天中，时钟会显示1440个不同的时间，从0:00到23:59。在显示的1440个不同的次数中，数字5总共出现了多少次?(例如，当时钟显示5:55时，也就是早上5:55，数字5出现了三次。例如，当时钟显示15:55时，也就是下午3点55分，数字5出现了3次。)

【答案】504

• 我们记一天的时间为$AB:CD$，$A$不可能是$5$
  若$B=5$，则$A$可以取$0$、$1$，$CD$可以取$00\sim59$，共$2\times 60=120$次
  若$C=5$，则$AB$可取$00\sim 23$，$D$可取$0\sim9$，共$24\times 10 =240$次
  若$D=5$，则$AB$可取$00\sim 23$，$C$可取$0\sim5$，共$24\times 6 =144$次
  共出现$120+240+144=504$次