2021年大联盟(Math League)国际夏季五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 5, Individual QuestionsQuestion 1: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 2:The 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?8Question 3: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.Question 4: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.)(a) 44(b) 31(c) 26(d) 16Question 5:There are 1000 people in a room. Each pair either shakes hands or does not.(1) Is it always true that some two people shook the same number of hands? (Please enter 1 if your answer is Yes, and 0 if your answer is No.)(2) Is it always true that some three people shook the same number of hands? (Please enter 1 if your answer is Yes, and 0 if your answer is No.)(3) Is it always true that some four people shook the same number of hands? (Please enter 1 if your answer is Yes, and 0 if your answer is No.)(4) Is it always true that some five people shook the same number of hands? (Please enter 1 if your answer is Yes, and 0 if your answer is No.)Note:(1) One doesn’t shake his/her own hand.(2) One doesn’t shake the same person’s hand more than once.Question 6:The numbers 1 through 10 are written in a row. Can the signs “+” and “–” be placed between them, so that the value of the resulting expression is 0? (Note: There are 10 numbers, and there are 9 places to place signs.)Question 7:Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates, figure below. Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively. (That is Cheryl tells Albert the month of her birthday, and tells Bernard the day of her birthday.)Conversation:Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard also doesn’t know.Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.Albert: Then I also know when Cheryl’s birthday is. When is Cheryl’s birthday?Please enter the month first, followed by the day. When you enter the month, please enter 5 for May, 6 for June, 7 for July, and 8 for August.Question 8:Goal:1. Arrange the six different juices into the following order.2. The order is: 1 – red, 2 – orange, 3 – yellow, 4 – green, 5 – blue, 6 – violet.Rules:1. You may only pour a filled cup into an empty cup.2. You may not switch the positions of any cups.3. The empty cup must end up on the right.Examples:2 Pours are necessary to solve the following example puzzle, figure below.Pour One, figure below.Pour Two, figure below:For another example below, this example needs four pours.Pour One, figure below.Pour Two, figure below.Pour Three, figure below.Pour Four, figure below.For 6 different juices and one empty cup, there are 5040 (= 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1) possible permutations. Two of them were listed above. One needs 2 pours. The other needs 4 pours.(1) Of all the 5040 possible permutations, how many permutations are impossible to solve? That is no matter how you pour the juices, you can’t arrange the juices in order.(2) What is the largest number of pours you need to solve any solvable permutation?For 1000001 different juices and one empty cup, there are 1000002! possible permutations. There is a pre-defined order of the 1000001 juices, just as the order for the 6 juices above.(3) Of all the 1000002! possible permutations, how many permutations are impossible to solve? That is no matter how you pour the juices, you can’t arrange the juices in order.(4) What is the largest number of pours you need to solve any solvable permutation (for 1000001 different juices and one empty cup)?Question 9:Twelve straightjacketed prisoners are on the death row. Tomorrow they will be arranged in a single-fil。