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1、2021 美国大联盟(Math League)国际夏季数学挑战活动2021 Math League International Summer ChallengeGrade 4, Individual Questions & SolutionsQuestion 1:There is a table in front of you with one hundred quarters on it. You have been blindfolded and are wearing a thick pair of gloves. You are not able to see whether the
2、quarters are heads or tails because you are blindfolded. And you are not able to feel whether the quarters are heads or tails because of the thick gloves. Your friend tells you that twenty of these quarters are tails and remaining eighty are heads, but you do not know which are which. He tells you t
3、hat if you are able to split the quarters into two piles where the number of tails quarters is the same in each pile, you will win all of the quarters. You are free to move the quarters, flip them over, and arrange them into two piles of any number. For you to win all of the quarters, how many quart
4、ers are in each of the two piles? (Please enter your answers in ascending order.)Note:heads: the front side of a coin. tails: the back side of a coin.Figure below, heads and tails of a quarter.Answer:82080Solution:You can win this game with just one elegant move: Take any twenty quarters from the hu
5、ndred, put them into a separate pile, and flip them over.No matter how many of the quarters are tails in the group flip over, you will always wind up with two groups of quarters with the same number of tails. If you happened to select all twenty of the quarters that were already tails, you would sep
6、arate them and flip them, resulting in two groups of quarters with zero tails in them. If you happened to select twenty quarters with only six tails among them, you would leave fourteen tails in the original pile, and after flipping the twenty quarters you selected, you would have fourteen tails and
7、 six heads in the new pile. This will always work as long as you only take twenty quarters, put them into their own pile, and flip them all over.Question 2:The pages of a book are consecutively numbered from 1 through 384. How many times does the digit 8 appear in this numbering?Answer: 73 Solution:
8、One way: We can look at the ones and tens digits separately:The ones digit of 8 from 1 through 384: since it occurs one time in every set of 10 consecutive numbers, there are 38 complete sets of 10 consecutive numbers. So, the digit 8 appears 38 times as a ones digit.The tens digits of 8 from 1 thro
9、ugh 384: since it occurs 10 times in every set of 100 consecutive numbers, there are 3 complete sets of 100 (1-100, 101-199, 200-299). The digit 8 appears 30 times as a tens digit. In addition, the numbers 380-384 contain 5 more tens digit of 8. In all, the digit 8 appears a total of 38 + 30 + 5 = 7
10、3 times.Or if you are looking at the first hundred numbers there will be 20 digits 8 in them: 8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 98.Since we have 384 pages it means 20 3 + 13 = 73 times digit 8 will be happening in the page numeration 1-384.Question 3:Annabella is
11、 visiting her grandma who resides 295 miles away from her. She started her driving trip at a speed of 65 mph (miles per hour) for the first 3 hours. For the rest of the trip, she drove at a speed of 50 mph. How many hours did she take to drive to her grandma?Answer: 5 Solution:For the first 3 hours,
12、 she completes 65 mph 3 hours = 195 miles. She then has 295 195 = 100 miles to go. Since her speed is 50 mph, it will take her 100 50 = 2 hours to complete the rest of her journey. Total hours: 2 + 3 = 5 hours.Question 4:Sarah is using popsicle sticks to build some grids for her city planning projec
13、t. She needs 4 popsicle sticks to make a 1 by 1 grid, and 12 sticks to make a 2 by 2 grid as shown below. How many sticks does she need to make a 10 by 20 grid?Answer: 430 Solution:Analyze the 2 by 2 grid. To create it Sarah used 3 rows of 2 sticks each and 3 columns of 2 sticks each. Similar to it,
14、 there would be 11 rows and there should be 20 sticks in each row. So, there are 20 11 = 220 sticks (placed horizontally).There would also be 21 columns and there should be 10 sticks in each. Therefore, there are 10 21 = 210 sticks placed vertically.Total = 220 + 210 = 430 popsicle sticksQuestion 5:
15、The sides of the large rectangle are 20 m and 16 m, figure below, not drawn to scale.Answer:192All six shaded rectangles are identical. What is the total area of all the shaded regions, in square meters?Answer: 192 Solution:Find the dimension of the small rectangle. The length: 16 2 = 8 m. The width
16、: 20 8 8 = 4 m. The total area of all the shaded regions: 6 (8 4) = 192 m2.Question 6:Alice, Bibi, and Clary are three intelligent girls. One day, their teacher decides to play a simple game and award some prize to the first girl who solves it:A black or white hat is placed on each girls head, and e
17、ach girl can see the other girls hats, but not her own. They are instructed to raise their hands if they see at least one black hat. And the first girl to tell the teacher what color hat she is wearing and how she figured this out will get the prize. After the hats are placed on Alice, Bibi, and Cla
18、rys heads, each girl raises her hand. After a few seconds, Alice says: “I have figured it out!”What color hat is Alice wearing? Three choices:(a) Black(b) White(c) Non-deterministic Answer: (a)Solution:Alice knows that she is wearing a black hat and this is how she is able to deduce it: It is clear,
19、 because all three persons raised their hands, that there are at least two black hats. However, if there were two black hats and one white hat, either of the people with the black hats would see the white hat, and see that everyone had their hand raised which would allow her to instantly deduce the
20、color of their hat is black.After a few seconds had passed without anyone speaking about the color of their hat, it became clear to Alice, the brightest of the bunch, that they must all be wearing black hats, otherwise either Bibi or Clary, who are intelligent, would have said something by now.Quest
21、ion 7:You have a balance scale and six weights. There are two red weights, two orange weights, and two blue weights. In each pair of colored weights, one weight is slightly heavier than the other, but is otherwise identical. The three heavier weights all weigh the same, and the three lighter weights
22、 all weigh the same.What is the fewest number of times you need to use the balance scale in order to positively identify the heavier weight in each pair?Answer: 2 Solution:To find the heavier weight of each color, you only need to use the balance scale twice. First, you must weigh a red and an orang
23、e weight against a blue and an orange weight.If the pans balance: You can be sure that there is a heavy and a light weight on each pan. Take both the red and blue weights off the scale and leave the orange weights oneach side. This will show you which of the orange weights is the heavier one, which
24、will then show you whether the red or blue weight that was just on the pan was the heavier one, which will then allow you to, by process of elimination, determine the weight of the red and blue weight that were never on the pan.If the pans do not balance: You can be sure that the orange weight on th
25、e side of the scale that went down is the heavier of the two orange weights. To find the heavier weights of the red and blue pair, you must take the red weight that you just weighed and weigh it against the blue weight that has not yet been on the scale. Seeing what happens here, plus remembering wh
26、at happened when you made the first weighing, will allow you to correctly label each weight.Question 8: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?Answer: 3
27、0 Solution:5 have both cats and dogs. 15 5 = 10 have cats only.12 5 = 7 have dogs only. 8 have neither cats or dogs. 5 + 10 + 7 + 8 = 30.Question 9: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
28、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
29、 of the fourth digit.Answer: 16842 Solution:“The first digit is one-eighth of the third digit.”: the first digit has to be 1, and the third digit has to be 8.“The first digit is one-fourth of the fourth digit.”: the fourth digit is 4.“The sum of the first two digits is one smaller than the third dig
30、it.”: the second digit is 6.“The fourth digit is double the last digit.”: the last (fifth) digit is 2. So the answer is 16842.Please note some conditions are redundant.Question 10:If two sides of a square field were increased by five feet, as seen in the diagram below, not drawn to scale, the area o
31、f the field would increase by 245 square feet. Find the area of the original square.Answer: 484 Solution:The area of the shaded square is 5 5 = 25, figure below. The area of each of the two “new” rectangles is (245 25) 2 = 110. Thus the length of the “new” rectangle, or the side length of the original square, is 110 5 = 22. So the area of the original square is 22 22 = 484.