2002 AMC 10A
1、The ratio is clost to which of the following numbers?
Solution
We factor as . As , our answer is .
2、For the nonzero numbers , , 档案管理总结, define
.
Find .
Solution
. Our answer is then .
Alternate solution for the lazy: Without computing the answer exactly, we e that , , and . The sum is , and as all the options are integers, the correct one is obviously .
3、According to the standard convention for exponentiation, 鸡肉怎么画
.
If the order in which the exponentiations are performed is changed, how many other values are possible?
Solution
The best way to solve this problem is by simple brute force.
It is convenient to drop the usual way how exponentiation is denoted, and to write the formula as , where denotes exponentiation. We are now examining all ways to add parenthes to this expression. There are 5 ways to do so:
1.
2.
3.
4.
5.
We can note that . Therefore options 1 and 2 are equal, and options 3 and 4 are equal. Option 1 is the one given in the problem statement. Thus we only need to evaluate options 3 and 5.
Thus the only other result is , and our answer is .
4、For how many positive integers does there exist at least one positive integer such that ?
infinitely many
Solution
Solution 1
For any we can pick , we get , therefore the answer is .
Solution 2
Another solution, slightly similar to this first one would be using Simon's Favorite Factoring Trick.
Let , then 浪漫的爱情诗句
This means that there are infinitely many numbers that can satisfy the inequality. So the answer is .
5、Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of tho circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
Solution
The outer circle has radius , and thus area esp是什么. The little circles have area each; since there are 7, their total area is 非主流妆. Thus, our answer is .
6、Cindy was asked by her teacher to subtract from a certain number and then divide the result by . Instead, she subtracted and then divided the result by , giving an answer of . What would her answer have been had she worked the problem correctly?
领导升职调走了祝福语Solution
We work backwards; the number that Cindy started with is . Now, the correct result is . Our answer is .
7、If an arc of on circle has the same length as an arc of on circle , then the ratio of the area of circle to the area of circle is
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Solution困难出办法
Let and be the radii of circles A and B, respectively.
It is well known that in a circle with radius r, a subtended arc opposite an angle of degrees has length .
Using that here, the arc of circle A has length . The arc of circle B has length . We know that they are equal, so , so we multiply through and simplify to get . As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is .
8、Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let be the total area of the blue triangles, the total area of the white squares, and the area of the red square. Which of the following is correct?
Solution
The blue that's touching the center red square makes up 8 triangles, or 4 squares. Each of the corners is 2 squares and each of the edges is 1, totaling 12 squares. There are 12 white squares, thus we have .
9、There are 3 numbers A, B, and C, such that , and . What is the average of A, B, and C?
More than 1
Solution
Notice that we don't need to find what A, B, and C actually are, just their average. In other words, if we can find A+B+C, we will be done.
Adding up the equations gives so and the average is . Our answer is .
10、Compute the sum of all the roots of .
Solution
Solution 1
We expand to get which is after combining like terms. Using the quadratic part of Vieta's Formulas, we find the sum of the roots is .
Solution 2
Combine terms to get , hence the roots are and , thus our answer is .
11、Jamal wants to store computer files on floppy disks, each of which has a capacity of megabytes (MB). Three of his files require MB of memory each, more require MB each, and the remaining require MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
