Problem 1
Open on its own pageThe ratio is closest to which of the following numbers?
2002 AMC 10A
All 25 problems from the 2002 AMC 10A. Take it as a timed mock test — 1:15:00 on the clock, scored the way the real contest is — or read through the problems and open any of them for hints and a worked solution.
25 problems · 6 points correct, 1.5 for blank, 0 for wrong
Problem 2
Open on its own pageFor the nonzero numbers and define FindProblem 3
Open on its own pageAccording 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
Open on its own pageFor how many positive integers does there exist at least one positive integer such thatProblem 5
Open on its own pageEach 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
Open on its own pageCindy 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
Open on its own pageIf 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 isProblem 8
Open on its own pageBetsy 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
Open on its own pageSuppose and are three numbers for which and The average of the three numbers and isProblem 10
Open on its own pageCompute the sum of all the roots ofProblem 11
Open on its own pageJamal 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
Open on its own pageMr. Earl E. Bird leaves his house for work at exactly 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
Open on its own pageThe sides of a triangle have lengths of and Find the length of the shortest altitude.Problem 14
Open on its own pageBoth roots of the quadratic equation are prime numbers. The number of possible values of isProblem 15
Open on its own pageThe 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
Open on its own pageIf then isProblem 17
Open on its own pageSarah 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
Open on its own pageA cube is formed by gluing together standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is ) The smallest possible sum of all the numbers showing on the surface of the cube isProblem 19
Open on its own pageSpot’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 the doghouse that Spot can reach?Problem 20
Open on its own pagePoints and lie, in that order, on dividing it into five segments, each of length Point is not on line Point lies on and point lies on The line segments and are parallel. Find
Problem 21
Open on its own pageThe mean, median, unique mode, and range of a collection of eight integers are all equal to The largest integer that can be an element of this collection isProblem 22
Open on its own pageA set of tiles numbered through is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with How many times must the operation be performed to reduce the number of tiles in the set to one?Problem 23
Open on its own pagePoints 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
Open on its own pageTina 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 isProblem 25
Open on its own pageIn trapezoid with bases and we have and The area of is
0 of 25 answered. Each problem left blank still scores 1.5 points.
Practise one problem at a time, with hints and solutions, on the practice page. Or work through the same ideas across every year by topic, or browse the full AMC 10 archive.