Difference between revisions of "2000 AMC 12 Problems"
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<math> \mathrm{(A) \ 23 } \qquad \mathrm{(B) \ 55 } \qquad \mathrm{(C) \ 99 } \qquad \mathrm{(D) \ 111 } \qquad \mathrm{(E) \ 671 } </math> | <math> \mathrm{(A) \ 23 } \qquad \mathrm{(B) \ 55 } \qquad \mathrm{(C) \ 99 } \qquad \mathrm{(D) \ 111 } \qquad \mathrm{(E) \ 671 } </math> | ||
− | [[2000 AMC 12 Problem 1|Solution]] | + | [[2000 AMC 12/Problem 1|Solution]] |
== Problem 2 == | == Problem 2 == | ||
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− | [[2000 AMC 12 Problem 2|Solution]] | + | [[2000 AMC 12/Problem 2|Solution]] |
== Problem 3 == | == Problem 3 == | ||
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<center><math> \mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 } </math></center> | <center><math> \mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 } </math></center> | ||
− | [[2000 AMC 12 Problem 3|Solution]] | + | [[2000 AMC 12/Problem 3|Solution]] |
== Problem 4 == | == Problem 4 == | ||
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<center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 } </math></center> | <center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 } </math></center> | ||
− | [[2000 AMC 12 Problem 4|Solution]] | + | [[2000 AMC 12/Problem 4|Solution]] |
== Problem 5 == | == Problem 5 == | ||
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− | [[2000 AMC 12 Problem 5|Solution]] | + | [[2000 AMC 12/Problem 5|Solution]] |
== Problem 6 == | == Problem 6 == | ||
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<center><math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math></center> | <center><math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math></center> | ||
− | [[2000 AMC 12 Problem 6|Solution]] | + | [[2000 AMC 12/Problem 6|Solution]] |
== Problem 7 == | == Problem 7 == | ||
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<center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math></center> | <center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math></center> | ||
− | [[2000 AMC 12 Problem 7|Solution]] | + | [[2000 AMC 12/Problem 7|Solution]] |
== Problem 8 == | == Problem 8 == | ||
− | [[2000 AMC 12 Problem 8|Solution]] | + | [[2000 AMC 12/Problem 8|Solution]] |
== Problem 9 == | == Problem 9 == | ||
− | [[2000 AMC 12 Problem 9|Solution]] | + | [[2000 AMC 12/Problem 9|Solution]] |
== Problem 10 == | == Problem 10 == | ||
− | [[2000 AMC 12 Problem 10|Solution]] | + | [[2000 AMC 12/Problem 10|Solution]] |
== Problem 11 == | == Problem 11 == | ||
− | [[2000 AMC 12 Problem 11|Solution]] | + | [[2000 AMC 12/Problem 11|Solution]] |
== Problem 12 == | == Problem 12 == | ||
− | [[2000 AMC 12 Problem 12|Solution]] | + | [[2000 AMC 12/Problem 12|Solution]] |
== Problem 13 == | == Problem 13 == | ||
− | [[2000 AMC 12 Problem 13|Solution]] | + | [[2000 AMC 12/Problem 13|Solution]] |
== Problem 14 == | == Problem 14 == | ||
− | [[2000 AMC 12 Problem 14|Solution]] | + | [[2000 AMC 12/Problem 14|Solution]] |
== Problem 15 == | == Problem 15 == | ||
− | [[2000 AMC 12 Problem 15|Solution]] | + | [[2000 AMC 12/Problem 15|Solution]] |
== Problem 16 == | == Problem 16 == | ||
− | [[2000 AMC 12 Problem 16|Solution]] | + | [[2000 AMC 12/Problem 16|Solution]] |
== Problem 17 == | == Problem 17 == | ||
− | [[2000 AMC 12 Problem 17|Solution]] | + | [[2000 AMC 12/Problem 17|Solution]] |
== Problem 18 == | == Problem 18 == | ||
− | [[2000 AMC 12 Problem 18|Solution]] | + | [[2000 AMC 12/Problem 18|Solution]] |
== Problem 19 == | == Problem 19 == | ||
− | [[2000 AMC 12 Problem 19|Solution]] | + | [[2000 AMC 12/Problem 19|Solution]] |
== Problem 20 == | == Problem 20 == | ||
− | [[2000 AMC 12 Problem 20|Solution]] | + | [[2000 AMC 12/Problem 20|Solution]] |
== Problem 21 == | == Problem 21 == | ||
− | [[2000 AMC 12 Problem 21|Solution]] | + | [[2000 AMC 12/Problem 21|Solution]] |
== Problem 22 == | == Problem 22 == | ||
− | [[2000 AMC 12 Problem 22|Solution]] | + | [[2000 AMC 12/Problem 22|Solution]] |
== Problem 23 == | == Problem 23 == | ||
− | [[2000 AMC 12 Problem 23|Solution]] | + | [[2000 AMC 12/Problem 23|Solution]] |
== Problem 24 == | == Problem 24 == | ||
− | [[2000 AMC 12 Problem 24|Solution]] | + | [[2000 AMC 12/Problem 24|Solution]] |
== Problem 25 == | == Problem 25 == | ||
− | [[2000 AMC 12 Problem 25|Solution]] | + | [[2000 AMC 12/Problem 25|Solution]] |
== See also == | == See also == |
Revision as of 21:43, 11 July 2006
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 , 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
![$\mathrm{(A) \ 2000^{2001} } \qquad \mathrm{(B) \ 4000^{2000} } \qquad \mathrm{(C) \ 2000^{4000} } \qquad \mathrm{(D) \ 4,000,000^{2000} } \qquad \mathrm{(E) \ 2000^{4,000,000} }$](http://latex.artofproblemsolving.com/0/3/1/031037e9f6203a6576d24167fd8b6b9f94c6d7c9.png)
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?
![$\mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 }$](http://latex.artofproblemsolving.com/c/9/6/c96a5445db4357effee954e6a74057236f09f952.png)
Problem 4
The Fibonacci sequence 1,1,2,3,5,8,13,21,... starts with two 1s, 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?
![$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }$](http://latex.artofproblemsolving.com/4/0/8/408fa8806f5e694766a9cc0913fb1e3ca60cff0d.png)
Problem 5
If where
then
![$\mathrm{(A) \ -2 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 2-2p } \qquad \mathrm{(D) \ 2p-2 } \qquad \mathrm{(E) \ |2p-2| }$](http://latex.artofproblemsolving.com/3/0/4/304ed31d5450f185d35e8ef6342aa85f233e3681.png)
Problem 6
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?
![$\mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$](http://latex.artofproblemsolving.com/3/8/b/38b11bf36eef49ea994e3758c15cd29e957a9795.png)
Problem 7
How many positive integers have the property that
is a positive integer?
![$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }$](http://latex.artofproblemsolving.com/1/1/2/112e29de497e42cd4612addaf17d301844a116ed.png)