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Difference between revisions of "2000 AMC 10 Problems"

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== Problem 14 ==
 
== Problem 14 ==
  
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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 71, 76, 80, 82, and 91.  What was the last score Mrs. Walter entered?
+
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 <math>71</math>, <math>76</math>, <math>80</math>, <math>82</math>, and <math>91</math>.  What was the last score Mrs. Walter entered?
  
 
<math>\mathrm{(A)}\ 71 \qquad\mathrm{(B)}\ 76 \qquad\mathrm{(C)}\ 80 \qquad\mathrm{(D)}\ 82 \qquad\mathrm{(E)}\ 91</math>
 
<math>\mathrm{(A)}\ 71 \qquad\mathrm{(B)}\ 76 \qquad\mathrm{(C)}\ 80 \qquad\mathrm{(D)}\ 82 \qquad\mathrm{(E)}\ 91</math>

Revision as of 21:42, 6 January 2009

Problem 1

In the year 2001, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?

$\mathrm{(A)}\ 23 \qquad\mathrm{(B)}\ 55 \qquad\mathrm{(C)}\ 99 \qquad\mathrm{(D)}\ 111 \qquad\mathrm{(E)}\ 671$

Solution

Problem 2

$2000({2000}^{2000}) =$

$\mathrm{(A)}\ {2000}^{2001} \qquad \mathrm{(B)}\ {4000}^{2000} \qquad \mathrm{(C)}\ {2000}^{4000} \qquad \mathrm{(D)}\ {4,000,000}^{2000} \qquad\mathrm{(E)}\ {2000}^{4,000,000}$

Solution

Problem 3

Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, $32$ remained. How many jellybeans were in the jar originally?

$\mathrm{(A)}\ 40 \qquad\mathrm{(B)}\ 50 \qquad\mathrm{(C)}\ 55 \qquad\mathrm{(D)}\ 60 \qquad\mathrm{(E)}\ 75$

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 $$$12.48$, but in January her bill was $$$17.54$ because she used twice as much connect time as in December. What is the fixed monthly fee?

$\mathrm{(A)}\ $ $2.53 \qquad\mathrm{(B)}\ $ $5.06 \qquad\mathrm{(C)}\ $ $6.24 \qquad\mathrm{(D)}\ $ $7.42 \qquad\mathrm{(E)}\ $ $8.77$

Solution

Problem 5

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?

(a) the length of the segment $MN$

(b) the perimeter of $\triangle PAB$

(c) the area of $\triangle PAB$

(d) the area of trapezoid $ABNM$

[asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N); [/asy]

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 3 \qquad\mathrm{(E)}\ 4$

Solution

Problem 6

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ starts with two $1$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?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 4 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 9$

Solution

Problem 7

In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. What is the perimeter of $\triangle BDP$?

[asy] draw((0,2)--(3.4,2)--(3.4,0)--(0,0)--cycle); draw((0,0)--(1.3,2)); draw((0,0)--(3.4,2)); dot((0,0)); dot((0,2)); dot((3.4,2)); dot((3.4,0)); dot((1.3,2)); label("$A$",(0,2),NW); label("$B$",(3.4,2),NE); label("$C$",(3.4,0),SE); label("$D$",(0,0),SW); label("$P$",(1.3,2),N); [/asy]

$\mathrm{(A)}\ 3+\frac{\sqrt{3}}{3} \qquad\mathrm{(B)}\ 2+\frac{4\sqrt{3}}{3} \qquad\mathrm{(C)}\ 2+2\sqrt{2} \qquad\mathrm{(D)}\ \frac{3+3\sqrt{5}}{2} \qquad\mathrm{(E)}\ 2+\frac{5\sqrt{3}}{3}$

Solution

Problem 8

At Olympic High School, $\frac{2}{5}$ of the freshmen and $\frac{4}{5}$ 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?

$\mathrm{(A)}$ There are five times as many sophomores as freshmen.

$\mathrm{(B)}$ There are twice as many sophomores as freshmen.

$\mathrm{(C)}$ There are as many freshmen as sophomores.

$\mathrm{(D)}$ There are twice as many freshmen as sophomores.

$\mathrm{(E)}$ There are five times as many freshmen as sophomores.

Solution

Problem 9

If $|x-2|=p$, where $x<2$, then $x-p=$

$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 2-2p \qquad\mathrm{(D)}\ 2p-2 \qquad\mathrm{(E)}\ |2p-2|$

Solution

Problem 10

The sides of a triangle with positive area have lengths $4$, $6$, and $x$. The sides of a second triangle with positive area have lengths $4$, $6$, and $y$. What is the smallest positive number that is not a possible value of $|x-y|$?

$\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 4 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 8 \qquad\mathrm{(E)}\ 10$

Solution

Problem 11

Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$\mathrm{(A)}\ 21 \qquad\mathrm{(B)}\ 60 \qquad\mathrm{(C)}\ 119 \qquad\mathrm{(D)}\ 180 \qquad\mathrm{(E)}\ 231$

Solution

Problem 12

Solution

Problem 13

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 $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?

$\mathrm{(A)}\ 71 \qquad\mathrm{(B)}\ 76 \qquad\mathrm{(C)}\ 80 \qquad\mathrm{(D)}\ 82 \qquad\mathrm{(E)}\ 91$

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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