2021年必威体育精装版美国数学竞赛2021 AMC12B（解析版）.pdf

2021 AMC 12B Problem 1 How many integer values of satisfy ? Solution 1 Since is about , we multiply 9 by 2 for the numbers from to and the numbers from to and add 1 to account for the zero to get ~smarty101 and edited by Tony_Li2007 Problem 2 At a math contest, students are wearing blue shirts, and another students are wearing yellow shirts. The students are assigned into pairs. In exactly of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? Solution There are students paired with a blue partner. The other students wearing blue shirts must each be paired with a partner wearing a shirt of the opposite color. There are students remaining. Therefore the requested number of pairs is ~Punxsutawney Phil Problem 3 Suppose What is the value of Solution 1 / 21 Subtracting from both sides and taking reciprocals gives . Subtracting from both sides and taking reciprocals again gives . Subtracting from both sides and taking reciprocals for the final time gives or . Problem 4 Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is , and the afternoon classs mean score is . The ratio of the number of students in the morning class to the number of students in the afternoon class is . What is the mean of the scores of all the students? Solution 1 WLOG, assume there are students in the morning class and in the afternoon class. Then the average is Solution 2 Let there be students in the morning class and students in the afternoon class. The total number of students is . The average is . Therefore, the answer is . Problem 5