2021年必威体育精装版美国数学竞赛2021 AMC 12A （解析版）.pdf

批注[c1]: 2021 AMC 12A Problem 1 What is the value of Solution We evaluate the given expression to get that Problem 2 Under what conditions does hold, where and are real numbers? It is never true. It is true if and only if . It is true if and only if . It is true if and only if and . It is always true. Solution 1 Square both sides to get . Then, . Also, it is clear that both sides of the equation must be nonnegative. The answer is . Problem 3 The sum of two natural numbers is . One of the two numbers is divisible by . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? Solution 1 The units digit of a multiple of will always be . We add a whenever we multiply by . So, removing the units digit is equal to dividing by . 1 / 22 Let the smaller number (the one we get after removing the units digit) be . This means the bigger number would be . We know the sum is so . So . The difference is . So, the answer is . Problem 4 Tom has a collection of snakes, of which are purple and of which are happy. He observes that all of his happy snakes can add, none of his purple snakes can subtract, and all of his snakes that cant subtract also cant add. Which of these conclusions can be drawn about Toms snakes? Purple snakes can add. Purple snakes are happy. Snakes that can add are purple. Happy snakes are not purple. Happy snakes cant subtract. Solution 1 We know that purple snakes cannot subtract, thus they canno