ΒΆ 2003 AMC 10A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2003 AMC 10A problems here.
Discussion Forum
Engage in discussion about the 2003 AMC 10A math contest by visiting Random Math AMC 10A 2003 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2003 AMC 10A problems, please refer below:
Problem 1: What is the difference between the sum of the first even counting numbers and the sum of the first odd counting numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: Members of the Rockham Soccer League buy socks and T-shirts. Socks cost per pair and each -shirt costs more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is , how many members are in the League?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: A solid box is by by . A new solid is formed by removing a cube on a side from each corner of this box. What percent of the original volume is removed?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: It takes Mary minutes to walk uphill from her home to school, but it takes her only minutes to walk from school to home along the same route. What is her average speed, in , for the round trip?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: Let and denote the solutions of . What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: Define to be for all real numbers and . Which of the following statements is not true?
Answer Choices:
A. for all and
B. for all and
C. for all
D. for all
E. if
Solution:
Problem 7: How many non-congruent triangles with perimeter have integer side lengths?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: What is the probability that a randomly drawn positive factor of is less than
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 9: Simplify
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: The polygon enclosed by the solid lines in the figure consists of congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: The sum of the two -digit numbers and is . What is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: A point is randomly picked from inside the rectangle with vertices , and . What is the probability that
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: The sum of three numbers is . The first is times the sum of the other two. The second is seven times the third. What is the product of all three?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: Let be the largest integer that is the product of exactly distinct prime numbers, and , where and are single digits. What is the sum of the digits of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: What is the probability that an integer in the set is divisible by and not divisible by
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: What is the units digit of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: What is the sum of the reciprocals of the roots of the equation
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: A base- three-digit number is selected at random. Which of the following is closest to the probability that the base- representation and the base- representation of are both three-digit numerals?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: In rectangle , we have is on with is on with , line intersects line at , and is on line with . Find the length .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have rows of small congruent equilateral triangles, with small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of small equilateral triangles?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: Sally has five red cards numbered through and four blue cards numbered through . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: Let be a -digit number, and let and be the quotient and remainder, respectively, when is divided by . For how many values of is divisible by
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions