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AMC10美国数学竞赛真题2002A卷


Problem 1
The ratio is closest to which of the following numbers?

Solution

Problem 2
Given that a, b, and c are non-zero real numbers, define Find . .

Solution

Problem 3
According to the standard convention for exponentiation, . If the order in which the exponentiations are performed is changed, how many other values are possible?

Solution

Problem 4
For how many positive integers ? is there at least 1 positive integer infinitely many Solution such that

Problem 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 those circles is tangent to

the large circle and to its small-circle neighbors. Find the area of the shaded region.

Solution

Problem 6
From a starting number, Cindy was supposed to subtract 3, and then divide by 9, but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be?

Solution

Problem 7
A arc of circle A is equal in length to a circle A's area and circle B's area? arc of circle B. What is the ratio of

Solution

Problem 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

Problem 9
There are 3 numbers A, B, and C, such that . What is the average of A, B, and C? Not uniquely determined Solution , and

Problem 10
What is the sum of all of the roots of ?

Solution

Problem 11
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB each, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?

Solution

Problem 12
Mr. Earl E. Bird leaves home every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?

Solution

Problem 13
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.

Solution

Problem 14
Both roots of the quadratic equation number of possible values of is are prime numbers. The

Solution

Problem 15
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?

Solution

Problem 16
Let ? . What is

Solution

Problem 17
Sarah pours 4 ounces of coffee into a cup that can hold 8 ounces. Then she pours 4 ounces of cream into a second cup that can also hold 8 ounces. She then pours half of the contents of the first cup into the second cup, completely mixes the contents of the second cup, then pours half of the contents of the second cup back into the first cup. What fraction of the contents in the first cup is cream?

Solution

Problem 18
A 3x3x3 cube is made of 27 normal dice. Each die's opposite sides sum to 7. What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?

Solution

Problem 19

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?

Solution

Problem 20
Points and lie, in that order, on segments, each of length 1. Point is not on line and point Find lies on . . The line segments , dividing it into five . Point lies on and are parallel. ,

Solution

Problem 21
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

Solution

Problem 22
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

Solution

Problem 23
Points and lie on a line, in that order, with . Point is not on the line, and is twice the perimeter of . Find . and . The perimeter of

Solution

Problem 24
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?

Solution

Problem 25

, we have to scale). The area of

, is

In trapezoid , , and

with bases and (diagram not

Solution


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