AMC-美国数学竞赛-2004-AMC-10B-试题及答案解析.doc

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Problem 1
Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
Solution
There are rows of seats, giving seats.
Problem 2
How many two-digit positive integers have at least one 7 as a digit?
Solution
Ten numbers () have as the tens digit. Nine numbers () have it as the ones digit. Number is in both sets.
Thus the result is .
Problem 3
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
Solution
At the fourth practice she made throws, at the third one it was , then we get throws for the second practice, and finally throws at the first one.
Problem 4
A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P?
Solution 1
The product of all six numbers is . The products of numbers that can be visible are , , ..., . The answer to this problem is their mon divisor -- which is , where is the mon multiple of . Clearly and the answer is .
Solution 2
Clearly, can not have a prime factor other than , and .
We can not guarantee that the product will be divisible by , as the number can end on the bottom.
We can guarantee that the product will be divisible by (one of and will always be visible), but not by .
Finally, there are three even numbers, hence two of them are always visible and thus the product is divisible by . This is the most we can guarantee, as when the is on the bottom side, the two visible even numbers are and , and their product is not divisible by .
Hence .
Solution
Problem 5
In the expression , the values of , , , and are , , , and , although not necessarily in that order. What is the maximum possible value of the result?
Solution
If or , the expression e