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AMC 10 2000

Question 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 ?

Solution

  
  2020-07-09 06:36:05

Question 2

Solution

  
  2020-07-09 06:36:05

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 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

Solution

  
  2020-07-09 06:36:05

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 7

In rectangle , , is on , and and trisect . What is the perimeter of ?

Solution

  
  2020-07-09 06:36:05

Question 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.

Solution

  
  2020-07-09 06:36:05

Question 9

If , where , then

Solution

  
  2020-07-09 06:36:05

Question 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 ?

Solution

  
  2020-07-09 06:36:05

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 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?


Solution

The nth item for the sequence is:

An=An-1+4n

We add increasing multiples of 4 each time we go up a figure.

So, to go from Figure 0 to 100, we add

4 *1+4*2+...+4*99+4*100=4*5050=20200

We then add 20200 to the number of squares in Figure 0 to get 20201
  
  2017-01-21 15:41:52

Question 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?


Solution

In each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.

In the second column from the right, there are two spaces for pegs. One of them is in the same row as the rightmost corner peg, so there is only one remaining choice left for the yellow peg in this column.

By similar logic, we can fill in the yellow pegs at top of each column.

By similar logic, we can fill in the red pegs at 2nd top position of each column (not including the rightmost column)

By similar logic, we can fill in the green pegs at 3rd top position of each remaining column.

......

So we only have 1 way of placing the pegs!
  
  2017-01-21 15:48:51

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 15

Two non-zero real numbers, and , satisfy . Find a possible value of .

Solution

a/b + b/a - ab = (a^2 + b^2)/ab - (a-b) =
(a^2 + b^2)/ab - (a-b)^2/(a-b) =
(a^2 + b^2)/ab - (a-b)^2/ab =
2ab/ab = 2
  
  2017-01-21 15:44:36

Question 16

The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .

Solution

  
  2020-07-09 06:36:05

Question 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?

Solution

First, change all values to pennies. Then, from the conditions, you know the input of 1 penny will get the output of 124 pennies. And the other 2 types of transactions will not change the amount Boris has at all. Then the conclusion is that the amount Boris has with multiple transactions will be 124*n+1.
  
  2017-01-03 02:10:42

Question 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?

Solution

  
  2020-07-09 06:36:05

Question 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

Solution

  
  2020-07-09 06:36:05

Question 20

Let , , and be nonnegative integers such that . What is the maximum value of ?

Solution

  
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Question 21

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?

Solution

  
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Question 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?

Solution

  
  2020-07-09 06:36:05

Question 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 ?

Solution

  
  2020-07-09 06:36:05

Question 24

Let be a function for which . Find the sum of all values of for which .

Solution

if f(x/3) = x^2 + x + 1
then
f(x) = 9x^2+3x+1
f(3x) = 81x^2+9x+1
f(3x) - 7 = 81x^2+9x+1-7 = 81x^2+9x-6

Equation 81x^2+9x-6 = 0 -> x^2 + 1/9 * x - 6/81 = 0
From Vieta's formula, sum of the 2 roots is -1/9.

  
  2017-01-07 16:00:20

Question 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?

Solution

  
  2020-07-09 06:36:05

Answer Keys


Question 1: E
Question 2: A
Question 3: B
Question 4: D
Question 5: B
Question 6: C
Question 7: B
Question 8: D
Question 9: C
Question 10: D
Question 11: C
Question 12: C
Question 13: B
Question 14: C
Question 15: E
Question 16: B
Question 17: D
Question 18: C
Question 19: D
Question 20: C
Question 21: B
Question 22: C
Question 23: E
Question 24: B
Question 25: A