2024年AMC 10A 真题:
Problem 1
What is the value of
Problem 2
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form where and are constants, is the time in minutes, is the length of the trail in miles, and is the altitude gain in feet. The model estimates that it will take minutes to hike to the top if a trail is miles long and ascends feet, as well as if a trail is miles long and ascends feet. How many minutes does the model estimates it will take to hike to the top if the trail is miles long and ascends feet?
Problem 3
What is the sum of the digits of the smallest prime that can be written as a sum of distinct primes?
Problem 4
The number is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
Problem 5
What is the least value of such that is a multiple of ?
Problem 6
What is the minimum number of successive swaps of adjacent letters in the string that are needed to change the string to (For example, swaps are required to change to one such sequence of swaps is )
Problem 7
The product of three integers is . What is the least possible positive sum of the three integers?
Problem 8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at and were able to pack , , and packages, respectively, every minutes. At some later time, Daria joined the group, and Daria was able to pack packages every minutes. Together, they finished packing packages at exactly . At what time did Daria join the group?
Problem 9
In how many ways can juniors and seniors form disjoint teams of people so that each team has juniors and seniors?
Problem 10
Consider the following operation. Given a positive integer , if is a multiple of , then you replace by . If is not a multiple of , then you replace by . For example, beginning with , this procedure gives . Suppose you start with . What value results if you perform this operation exactly times?
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