¶ 2000 AMC 10 Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2000 AMC 10 problems here.
Discussion Forum
Engage in discussion about the 2000 AMC 10 math contest by visiting Random Math AMC 10 2000 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2000 AMC 10 problems, please refer below:
Problem 1: In the year , the United States will host the International Mathematical Olympiad. Let , and be distinct positive integers such that the product . What is the largest possible value of the sum
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: Each day, Jenny ate of the jellybeans that were in her jar at the beginning of that day. At the end of second day, remained. How many jellybeans were in the jar originally?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixed monthly fee?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: Points and are the midpoints of sides and of . As moves along a line that is parallel to side , how many of the four quantities listed below change?
the length of the segment
the perimeter of
the area of
the area of trapezoid
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: The Fibonacci sequence starts with two s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: In rectangle is on , and and trisect . What is the perimeter of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: AT Olympic High School, of the freshmen and of the sophomores took the . Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?
Answer Choices:
A. There are five times as many sophomores as freshmen.
B. There are twice as many sophomores as freshmen.
C. There are as many freshmen as sophomores.
D. There are twice as many freshmen as sophomores.
E. There are five times as many freshmen as sophomores.
Solution:
Problem 9: If , where , then
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: The sides of a triangle with positive area have lengths , and . The sides of a second triangle with positive area have lengths , and . What is the smallest positive number that is not a possible value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following number could be obtained?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: Figure , and consist of , and nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure
.jpg)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: There are yellow pegs, red pegs, green pegs, blue pegs, and orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: Mrs.Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , and . What was the last scores Mrs.Walter entered?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: Two non-zero real numbers, and , satisfy . Find a possible value of .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: Charlyn walks completely around the boundary of a square whose sides are each long. From any point on her path she can see exactly horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangle is times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: Let , and be nonnegative integers such that . What is the maximum value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?
. All alligators are creepy crawlers.
. Some ferocious creatures are creepy crawlers.
. Some alligators are not creepy crawlers.
Answer Choices:
A. only
B. only
C. only
D. and only
E. None must be true
Solution:
Problem 22: One morning each member of Angela's family drank an -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 23: When the mean, median, and mode of the list
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: Let be a function for which . Find the sum of all values of for which .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: In year , the day of the year is a Tuesday. In year , the day is also a Tuesday. On what day of the week did the day of year occur?
Answer Choices:
A. Thursday
B. Friday
C. Saturday
D. Sunday
E. Monday
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions