Difference between revisions of "2000 AMC 12 Problems"
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== Problem 1 == | == Problem 1 == | ||
− | In the year <math>2001</math>, the United States will host the International Mathematical Olympiad. Let <math> | + | In the year <math>2001</math>, the United States will host the International Mathematical Olympiad. Let <math>I,M,</math> and <math>O</math> be distinct positive integers such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? |
<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> | ||
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== Problem 2 == | == Problem 2 == | ||
− | <math> | + | <math>2000(2000^{2000}) =</math> |
<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> | <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> | ||
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== Problem 5 == | == Problem 5 == | ||
− | If <math> | + | If <math>|x - 2| = p,</math> where <math>x < 2,</math> then <math>x - p =</math> |
<math> \mathrm{(A) \ -2 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 2-2p } \qquad \mathrm{(D) \ 2p-2 } \qquad \mathrm{(E) \ |2p-2| } </math> | <math> \mathrm{(A) \ -2 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 2-2p } \qquad \mathrm{(D) \ 2p-2 } \qquad \mathrm{(E) \ |2p-2| } </math> | ||
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== Problem 7 == | == Problem 7 == | ||
− | How many positive integers <math> | + | How many positive integers <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer? |
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math> | <math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math> | ||
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<math>\text {(A)}10401 \qquad \text {(B)}19801 \qquad \text {(C)} 20201 \qquad \text {(D)} 39801 \qquad \text {(E)}40801</math> | <math>\text {(A)}10401 \qquad \text {(B)}19801 \qquad \text {(C)} 20201 \qquad \text {(D)} 39801 \qquad \text {(E)}40801</math> | ||
− | + | [[Image:2000 AHSME number 8.png]] | |
[[2000 AMC 12 Problems/Problem 8|Solution]] | [[2000 AMC 12 Problems/Problem 8|Solution]] |
Revision as of 19:37, 1 March 2008
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
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
The Fibonacci sequence 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?
Problem 5
If where
then
Problem 6
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 7
How many positive integers have the property that
is a positive integer?
Problem 8
Figures ,
,
, and
consist of
,
,
, and
non-overlapping squares. If the pattern continued, how many non-overlapping squares would there be in figure
?
Problem 9
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. Walters entered?
Problem 10
The point is reflected in the
-plane, then its image
is rotated by
about the
-axis to produce
, and finally,
is translated by 5 units in the positive-
direction to produce
. What are the coordinates of
?
Problem 11
Two non-zero real numbers, and
satisfy
. Which of the following is a possible value of
?
Problem 12
Let A, M, and C be nonnegative integers such that . What is the maximum value of
+
+
+
?