2004 AMC 10B
Problem 1
Each row of the Misty Moon Amphitheater has 33 ats. Rows 12 through 22 are rerved for a youth club. How many ats are rerved for this club?
There are rows of ats, giving ats.
Problem 2
How many two-digit positive integers have at least one 7 as a digit?
Ten numbers () have as the tens digit. Nine numbers () have it as the ones digit. Number is in both ts.
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?
pennsylvaniaAt the fourth practice she made throws, at the third one it was , then we get throws for the cond practice, and finally throws at the first one.禁止吸烟的英文
Problem 4
porand
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 greatest common divisor -- which is , where is the least common multiple of . Clearly and the answer is .
Solution 2
Clearly, can not have a prime factor other than , 重庆新东方雅思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 b2c是什么意思.
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 .
Problem 5
In the expression the package, the values of , , , and are 英文翻译器, , , and , although not necessarily in that order. What is the maximum possible value of the result?
If or , the expression evaluates to .
If , the expression evaluates to .
Ca remains.
In that ca, we want to maximize where . Trying out the six possibilities we get that the best one is , where .
Problem 6
first and foremost
Which of the following numbers is a perfect square?
Using the fact that , we can write:
▪
▪
▪
ice box▪
▪
Clearly is a square, and as , , and are primes, none of the other four are squares.
Problem 7
On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving Canadian dollars for every U.S. dollars. After spending Canadian dollars, she had Canadian dollars left. What is the sum of the digits of ?
Solution 1
Isabella had Canadian dollars. Setting up an equation we get , which solves to , and the sum of digits of is
Solution 2
Each time Isabelle exchanges U.S. dollars, she gets Canadian dollars and Canadian dollars extra. Isabelle received a total of Canadian dollars extra, therefore she exchanged U.S. dollars times. Thus .
Problem 8
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is clost to the number of miles between downtown St. Paul and downtown Minneapolis?
before he cheatsThe directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs 8 miles and 10 miles long. The hypotenu length is , and thus the answer is .
Without a calculator one can note that .
