2002 AMC 10A Problems
2002 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
The ratio is closest to which of the following numbers?
Problem 2
For the nonzero numbers , , , define . Find .
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?
Problem 4
For how many positive integers does there exist at least one positive integer such that ?
infinitely many
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.
Problem 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?
Problem 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
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?
Problem 9
There are 3 numbers A, B, and C, such that , and . What is the average of A, B, and C?
Problem 10
Compute the sum of all the roots of .
Problem 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?
Problem 12
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages miles per hour, he arrives at his workplace three minutes late. When he averages miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
Problem 13
The sides of a triangle have lengths 15, 20, and 25. Find the length of the shortest altitude.
Problem 14
Both roots of the quadratic equation are prime numbers. The number of possible values of is
Problem 15
The digits , , , , , , , and are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
Problem 16
If , then is
Problem 17
Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
Problem 18
A cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the cube is
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?
Problem 20
Points and lie, in that order, on , dividing it into five segments, each of length 1. Point is not on line . Point lies on , and point lies on . The line segments and are parallel. Find .
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
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?
Problem 23
Points and lie on a line, in that order, with and . Point is not on the line, and . The perimeter of is twice the perimeter of . Find .
Problem 24
Tina randomly selects two distinct numbers from the set , and Sergio randomly selects a number from the set . The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
Problem 25
In trapezoid with bases and , we have , , , and . The area of is
See also
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2001 AMC 10 Problems |
Followed by 2002 AMC 10B Problems | |
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All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.