2021年大联盟(Math League)国际夏季四年级数学挑战活动一.docx

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 QuestionsQuestion 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 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 that 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 quarters 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.5Question 2:The pages of a book are consecutively numbered from 1 through 384. How many times does the digit 8 appear in this numbering?Question 3:Annabella is 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?Question 4:Sarah is using popsicle sticks to build some grids for her city planning project. 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?Question 5:Answer: 192Question 6The sides of the large rectangle are 20 m and 16 m, figure below, not drawn to scale. All six shaded rectangles are identical. What is the total area of all the shaded regions, in square meters?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 each 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, andClarys 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-deterministicQuestion 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 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?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?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 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.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 of the field would increase by 245 square feet. Find the area of the original square.