2021年大联盟(Math League)国际夏季四年级数学挑战活动一

2021 美国"大联盟"(Math League)国际夏季数学挑战活动2021 Math League International Summer Challenge Grade 4, Individual Questions & Solutions Question 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. Answer: ( 8 ) 学科网（北京）股份有限公司 20 80 Solution: You can win this game with just one elegant move: Take any twenty quarters from the hundred, 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 separate 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 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 pa