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 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 be
2、low. 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
3、 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, Danas six arrows land on the target shown. Each arrow is inside one of the regions of the ta
4、rget. 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
5、 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
6、 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 s
7、hook 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 doesnt shake his/her own hand.(2) One doesnt s
8、hake the same persons 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 j
9、ust 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 tel
10、ls Bernard the day of her birthday.)Conversation:Albert: I dont know when Cheryls birthday is, but I know that Bernard also doesnt know.Bernard: At first I didnt know when Cheryls birthday is, but I know now.Albert: Then I also know when Cheryls birthday is. When is Cheryls birthday?Please enter the
11、 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
12、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
13、 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 p
14、ossible permutations, how many permutations are impossible to solve? That is no matter how you pour the juices, you cant 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! p
15、ossible 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 cant arrange the juices in order.(4) What is
16、 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-file line, positions determined randomly, all facing the same directi
17、on. The person in the back of the line, who will be referred to as the twelfth person, can see the eleven people standing in front of him; the eleventh person can see the ten people standing in front of him, but not the twelfth person who is standing behind him; so on and so forth until you get to t
18、he first person in line who cannot see anyone. The warden of this prison likes to play games with his prisoners and has lined everyone up in this fashion so that he can give them an opportunity to gain their freedom.The warden will place, at random, a black or white hat on each prisoners head. The p
19、risoners cannot see the color of the hat they are wearing, but they can see the color of the hats of each prisoner standing in front of them in line. After all of the hats have been placed on the prisoners heads, the warden asks the person in the back of the line, the twelfth person, “what is the co
20、lor of the hat on your head?” The prisoner can reply by saying either “black” or “white”, but nothing else. When the prisoner answers the warden, all the prisoners in line can hear the reply. If the prisoner replied with the correct color of his hat, he will be freed from the prison. If he is wrong,
21、 he will be executed that night. This game continues until all twelve of the prisoners in line have been asked about the color of their hat. While the game is in progress, no prisoner is allowed to speak, move, or do anything until it is their turn to answer the question. The warden allows the priso
22、ners to get together in the courtyard the day before this game to come up with a plan to maximize the number of lives they can save. The prisoners come up with a plan that guarantees x prisoner(s) will survive, and 12 x prisoner(s) will each have a 50% chance to survive. What is the largest possible
23、 value of x?Question 10:Somewhere in the world there is a small neighborhood full of very eccentric and intelligent people. There are five houses that are each a different color. There is a person of a different nationality who lives in each house. Each of these five people drinks their own special
24、drink, smokes their own special brand of cigarettes, and has their own special pet. No one has the same pet, smokes the same brand of cigarettes, or drinks the same drink. One of these five people owns a pet fish. With just the following information, figure out who owns the cat and who owns the fish
25、:(1) The British man lives in the red house.(2) The Swedish man has a dog for a pet.(3) The Danish man drinks tea.(4) The green house and the white house are next to each other, and the green house is to the left of the white house.(5) The owner of the green house drinks coffee.(6) The person that s
26、mokes Pall Mall cigarettes has a bird for a pet.(7) The owner of the yellow house smokes Dunhill cigarettes.(8) The person who lives in the middle house drinks milk.(9) The Norwegian lives in the first house, counting from left to right.(10) The person who smokes Blend cigarettes lives next to the o
27、ne that has a cat for a pet.(11) The person who has a horse for a pet lives next to the one who smokes Dunhill cigarettes.(12) The one who smokes Bluemaster cigarettes drinks beer.(13) The German smokes Prince cigarettes.(14) The Norwegian lives next to a blue house.(15) The person who smokes Blend
28、cigarettes has a neighbor who drinks water.(16) The five houses are positioned in a single-file line.Question (1): Who owns the cat? (Please enter 1 if your answer is Norwegian, 2 if your answer is Swedish, 3 if your answer is German, 4 if your answer is British, and 5 if your answer is Danish.)Question (2): Who owns the fish? (Please enter 1 if your answer is Norwegian, 2 if your answer is Swedish, 3 if your answer is German, 4 if your answer is British, and 5 if your answer is Danish.)
