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Difference between revisions of "2000 AMC 12 Problems"
(Formatting, Latex)
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<math>\displaystyle 2000(2000^{2000}) =</math>
 
<math>\displaystyle 2000(2000^{2000}) =</math>
  
−
<center><math> \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} }  </math></center>
+
<math> \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} }  </math>
  
  
Line 18: Line 18:
 
== Problem 3 ==
 
== 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?
+
Each day, Jenny ate <math>20%</math> of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, <math>32</math> remained. How many jellybeans were in the jar originally?
  
−
<center><math> \mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 }  </math></center>
+
<math> \mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 }  </math>
  
 
[[2000 AMC 12/Problem 3|Solution]]
 
[[2000 AMC 12/Problem 3|Solution]]
Line 26: Line 26:
 
== Problem 4 ==
 
== 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?
+
The Fibonacci sequence <math>1,1,2,3,5,8,13,21,\ldots </math> 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?
  
−
<center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }  </math></center>
+
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }  </math>
  
 
[[2000 AMC 12/Problem 4|Solution]]
 
[[2000 AMC 12/Problem 4|Solution]]
Line 36: Line 36:
 
If <math>\displaystyle |x - 2| = p,</math> where <math>\displaystyle x < 2,</math> then <math>\displaystyle x - p =</math>
 
If <math>\displaystyle |x - 2| = p,</math> where <math>\displaystyle x < 2,</math> then <math>\displaystyle x - p =</math>
  
−
<center><math> \mathrm{(A) \ -2 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 2-2p } \qquad \mathrm{(D) \ 2p-2 } \qquad \mathrm{(E) \ |2p-2| }  </math></center>
+
<math> \mathrm{(A) \ -2 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 2-2p } \qquad \mathrm{(D) \ 2p-2 } \qquad \mathrm{(E) \ |2p-2| }  </math>
−
 
 
  
 
[[2000 AMC 12/Problem 5|Solution]]
 
[[2000 AMC 12/Problem 5|Solution]]
Line 43: Line 42:
 
== Problem 6 ==
 
== 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?
+
Two different prime numbers between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
  
−
<center><math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math></center>
+
<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math>
  
 
[[2000 AMC 12/Problem 6|Solution]]
 
[[2000 AMC 12/Problem 6|Solution]]
Line 53: Line 52:
 
How many positive integers <math>\displaystyle b</math> have the property that <math>\displaystyle \log_{b} 729</math> is a positive integer?
 
How many positive integers <math>\displaystyle b</math> have the property that <math>\displaystyle \log_{b} 729</math> is a positive integer?
  
−
<center><math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }  </math></center>
+
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }  </math>
  
 
[[2000 AMC 12/Problem 7|Solution]]
 
[[2000 AMC 12/Problem 7|Solution]]

Revision as of 21:46, 11 July 2006

Problem 1

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

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

Solution

Problem 2

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

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


Solution

Problem 3

Each day, Jenny ate $20%$ (Error compiling LaTeX. Unknown error_msg) 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

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

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

Solution

Problem 5

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

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

Solution

Problem 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?

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

Solution

Problem 7

How many positive integers $\displaystyle b$ have the property that $\displaystyle \log_{b} 729$ is a positive integer?

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

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

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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