2000 AMC 10 Problems
2000 AMC 10 (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
In the year 2001, 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 ?
Problem 2
Problem 3
Each day, Jenny ate of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, remained. How many jellybeans were in the jar originally?
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?
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?
(a) the length of the segment
(b) the perimeter of
(c) the area of
(d) the area of trapezoid
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?
Problem 7
In rectangle , , is on , and and trisect . What is the perimeter of ?
Problem 8
At Olympic High School, of the freshmen and of the sophomores took the AMC 10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?
There are five times as many sophomores as freshmen.
There are twice as many sophomores as freshmen.
There are as many freshmen as sophomores.
There are twice as many freshmen as sophomores.
There are five times as many freshmen as sophomores.
Problem 9
If , where , then
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 ?
Problem 11
Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
Problem 12
Figures , , , and consist of , , , and nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?
Problem 13
There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 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?
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 score Mrs. Walter entered?
Problem 15
Two non-zero real numbers, and , satisfy . Find a possible value of .
Problem 16
The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .
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?
Problem 18
Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can see exactly km 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?
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 triangles 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
Problem 20
Let , , and be nonnegative integers such that . What is the maximum value of ?
Problem 21
If all alligators are ferocious creatures and some ugly crawlers are alligators, which statement(s) must be true?
Problem 22
One morning each member of Angela's family drank an 8-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?
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 values of ?
-Error fixed(BIGFOOT09) Changes Copyrighted©
Problem 24
Let be a function for which . Find the sum of all values of for which .
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?
See also
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by First AMC 10 |
Followed by 2001 AMC 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.