Difference between revisions of "2000 AMC 12 Problems"

Latest revision as of 09:35, 16 June 2024

2000 AMC 12 (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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 $I + M + O$ ?

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

Solution

Problem 2

$2000(2000^{2000}) =$

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

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?

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

Solution

Problem 4

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?

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

Solution

Problem 5

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

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

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?

$\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 231$

Solution

Problem 7

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

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

Solution

Problem 8

Figures $0$ , $1$ , $2$ , and $3$ consist of $1$ , $5$ , $13$ , and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?

$[asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S); [/asy]$

$\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{(D)}\ 39801 \qquad\textbf{(E)}\ 40801$

Solution

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?

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

Solution

Problem 10

The point $P = (1,2,3)$ is reflected in the $xy$ -plane, then its image $Q$ is rotated $180^\circ$ about the $x$ -axis to produce $R$ , and finally, $R$ is translated $5$ units in the positive- $y$ direction to produce $S$ . What are the coordinates of $S$ ?

$\textbf {(A) } (1,7, - 3) \qquad \textbf {(B) } ( - 1,7, - 3) \qquad \textbf {(C) } ( - 1, - 2,8) \qquad \textbf {(D) } ( - 1,3,3) \qquad \textbf {(E) } (1,3,3)$

Solution

Problem 11

Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$ . Which of the following is a possible value of $\frac {a}{b} + \frac {b}{a} - ab$ ?

$\textbf{(A)} \ - 2 \qquad \textbf{(B)} \ \frac {- 1}{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2$

Solution

Problem 12

Let $A, M,$ and $C$ be nonnegative integers such that $A + M + C=12$ . What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$ ?

$\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112$

Solution

Problem 13

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?

$\textbf {(A)}\ 3 \qquad \textbf {(B)}\ 4 \qquad \textbf {(C)}\ 5 \qquad \textbf {(D)}\ 6 \qquad \textbf {(E)}\ 7$

Solution

Problem 14

When the mean, median, and mode of the list

$10,2,5,2,4,2,x$

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$ ?

$\textbf {(A)}\ 3 \qquad \textbf {(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf {(D)}\ 17 \qquad \textbf {(E)}\ 20$

Solution

Problem 15

Let $f$ be a function for which $f(x/3) = x^2 + x + 1$ . Find the sum of all values of $z$ for which $f(3z) = 7$ .

$\textbf {(A)}\ -1/3 \qquad \textbf {(B)}\ -1/9 \qquad \textbf {(C)}\ 0 \qquad \textbf {(D)}\ 5/9 \qquad \textbf {(E)}\ 5/3$

Solution

Problem 16

A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$ , the second row $18,19,\ldots,34$ , and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13$ , the second column $14,15,\ldots,26$ and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).

$\textbf {(A)}\ 222 \qquad \textbf {(B)}\ 333\qquad \textbf {(C)}\ 444 \qquad \textbf {(D)}\ 555 \qquad \textbf {(E)}\ 666$

Solution

Problem 17

A circle centered at $O$ has radius $1$ and contains the point $A$ . The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$ . If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$ , then $OC =$

$\textbf {(A)}\ \sec^2 \theta - \tan \theta \qquad \textbf {(B)}\ \frac 12 \qquad \textbf {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta}\qquad \textbf {(D)}\ \frac{1}{1+\sin\theta} \qquad \textbf {(E)}\ \frac{\sin \theta}{\cos^2 \theta}$

Solution

Problem 18

In year $N$ , the $300$ th day of the year is a Tuesday. In year $N+1$ , the $200$ th day is also a Tuesday. On what day of the week did the $100$ th day of year $N-1$ occur?

$\textbf {(A)}\ \text{Thursday} \qquad \textbf {(B)}\ \text{Friday}\qquad \textbf {(C)}\ \text{Saturday}\qquad \textbf {(D)}\ \text{Sunday}\qquad \textbf {(E)}\ \text{Monday}$

Solution

Problem 19

In triangle $ABC$ , $AB = 13$ , $BC = 14$ , $AC = 15$ . Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$ . Which of the following is closest to the area of the triangle $ADE$ ?

$\textbf {(A)}\ 2 \qquad \textbf {(B)}\ 2.5 \qquad \textbf {(C)}\ 3 \qquad \textbf {(D)}\ 3.5 \qquad \textbf {(E)}\ 4$

Solution

Problem 20

If $x,y,$ and $z$ are positive numbers satisfying $x + \frac{1}{y} = 4, y + \frac{1}{z} = 1,$ and $z + \frac{1}{x} = \frac73,$ then what is the value of $xyz$ ?

$\textbf {(A)}\ 2/3 \qquad \textbf {(B)}\ 1 \qquad \textbf {(C)}\ 4/3 \qquad \textbf {(D)}\ 2 \qquad \textbf {(E)}\ 7/3$

Solution

Problem 21

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 $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is

$\textbf {(A)}\ \frac{1}{2m+1} \qquad \textbf {(B)}\ m \qquad \textbf {(C)}\ 1-m \qquad \textbf {(D)}\ \frac{1}{4m} \qquad \textbf {(E)}\ \frac{1}{8m^2}$

Solution

Problem 22

The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$ . Which of the following is the smallest?

$\textbf{(A)}\ P(-1)\\ \textbf{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ \textbf{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ \textbf{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\ \textbf{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P$

Solution

Problem 23

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$ , inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?

$\textbf {(A)}\ 1/5 \qquad \textbf {(B)}\ 1/4 \qquad \textbf {(C)}\ 1/3 \qquad \textbf {(D)}\ 1/2 \qquad \textbf {(E)}\ 1$

Solution

Problem 24

If circular arcs $AC$ and $BC$ have centers at $B$ and $A$ , respectively, then there exists a circle tangent to both $\stackrel{\frown}{AC}$ and $\stackrel{\frown}{BC}$ , and to $\overline{AB}$ . If the length of $\stackrel{\frown}{BC}$ is $12$ , then the circumference of the circle is

$\textbf {(A)}\ 24 \qquad \textbf {(B)}\ 25 \qquad \textbf {(C)}\ 26 \qquad \textbf {(D)}\ 27 \qquad \textbf {(E)}\ 28$

Solution

Problem 25

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

$\textbf {(A)}\ 210 \qquad \textbf {(B)}\ 560 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 1260 \qquad \textbf {(E)}\ 1680$

$[asy] import three; import math; unitsize(1.5cm); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7)); [/asy]$

Solution

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by 2000 AMC 12 Problems	Followed by 2001 AMC 12 Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "2000 AMC 12 Problems"

Latest revision as of 09:35, 16 June 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 1: / Line 1: @@
+{{AMC12 Problems|year=2000|ab=}}
 == Problem 1 ==
-In the year <math>2001</math>, the United States will host the International Mathematical Olympiad. Let <math> \displaystyle I,M,</math> and <math>\displaystyle 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>\displaystyle I + M + O</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>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 671</math>
 [[2000 AMC 12 Problems/Problem 1|Solution]]
@@ Line 9: / Line 10: @@
 == Problem 2 ==
-<math>\displaystyle 2000(2000^{2000}) =</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>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000} \qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000}</math>
 [[2000 AMC 12 Problems/Problem 2|Solution]]
@@ Line 20: / Line 20: @@
 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?
-<math> \mathrm{(A) \ 40 } \qquad \mathrm{(B) \ 50 } \qquad \mathrm{(C) \ 55 } \qquad \mathrm{(D) \ 60 } \qquad \mathrm{(E) \ 75 }  </math>
+<math>\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math>
 [[2000 AMC 12 Problems/Problem 3|Solution]]
@@ Line 26: / Line 26: @@
 == Problem 4 ==
-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?
+The Fibonacci sequence <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two <math>1</math>'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?
-<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }  </math>
+<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9</math>
 [[2000 AMC 12 Problems/Problem 4|Solution]]
@@ Line 34: / Line 34: @@
 == Problem 5 ==
-If <math>\displaystyle |x - 2| = p,</math> where <math>\displaystyle x < 2,</math> then <math>\displaystyle x - p =</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>\textbf{(A)}\ -2 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 2-2p \qquad \textbf{(D)}\ 2p-2 \qquad \textbf{(E)}\ |2p-2|</math>
 [[2000 AMC 12 Problems/Problem 5|Solution]]
@@ Line 44: / Line 44: @@
 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?
-<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }  </math>
+<math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 231</math>
 [[2000 AMC 12 Problems/Problem 6|Solution]]
@@ Line 50: / Line 50: @@
 == Problem 7 ==
-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>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>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
 [[2000 AMC 12 Problems/Problem 7|Solution]]
 == Problem 8 ==
-Figures <math>0</math>, <math>1</math>, <math>2</math>, and <math>3</math> consist of <math>1</math>, <math>5</math>, <math>13</math>, and <math>25</math> non-overlapping squares. If the pattern continued, how many non-overlapping squares would there be in figure <math>100</math>?
-<math>\text {(A)}10401 \qquad \text {(B)}19801 \qquad \text {(C)} 20201 \qquad \text {(D)} 39801 \qquad \text {(E)}40801</math>
+Figures <math>0</math>, <math>1</math>, <math>2</math>, and <math>3</math> consist of <math>1</math>, <math>5</math>, <math>13</math>, and <math>25</math> nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?
+<asy>
+unitsize(8);
+draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
+draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
+draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
+draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
+draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
+draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
+draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
+draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
+draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
+draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
+label("Figure",(0.5,-1),S);
+label("$0$",(0.5,-2.5),S);
+label("Figure",(9.5,-1),S);
+label("$1$",(9.5,-2.5),S);
+label("Figure",(19.5,-1),S);
+label("$2$",(19.5,-2.5),S);
+label("Figure",(32.5,-1),S);
+label("$3$",(32.5,-2.5),S);
+</asy>
-{{image}}
+<math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{(D)}\ 39801 \qquad\textbf{(E)}\ 40801</math>
 [[2000 AMC 12 Problems/Problem 8|Solution]]
 == 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?
+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>71,76,80,82,</math> and <math>91</math>. What was the last score Mrs. Walters entered?
-<math>\text{(A)} \ 71 \qquad \text{(B)} \ 76 \qquad \text{(C)} \ 80 \qquad \text{(D)} \ 82 \qquad \text{(E)} \ 91</math>
+<math>\textbf{(A)} \ 71 \qquad \textbf{(B)} \ 76 \qquad \textbf{(C)} \ 80 \qquad \textbf{(D)} \ 82 \qquad \textbf{(E)} \ 91</math>
 [[2000 AMC 12 Problems/Problem 9|Solution]]
 == Problem 10 ==
-The point <math>P = (1,2,3)</math> is reflected in the <math>xy</math>-plane, then its image <math>Q</math> is rotated by <math>180^\circ</math> about the <math>x</math>-axis to produce <math>R</math>, and finally, <math>R</math> is translated by 5 units in the positive-<math>y</math> direction to produce <math>S</math>. What are the coordinates of <math>S</math>?
+The point <math>P = (1,2,3)</math> is reflected in the <math>xy</math>-plane, then its image <math>Q</math> is rotated <math>180^\circ</math> about the <math>x</math>-axis to produce <math>R</math>, and finally, <math>R</math> is translated  <math>5</math> units in the positive-<math>y</math> direction to produce <math>S</math>. What are the coordinates of <math>S</math>?
-<math>
+<math>\textbf {(A) } (1,7, - 3) \qquad \textbf {(B) } ( - 1,7, - 3) \qquad \textbf {(C) } ( - 1, - 2,8) \qquad \textbf {(D) } ( - 1,3,3) \qquad \textbf {(E) } (1,3,3)</math>
-\text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1, - 2,8) \qquad \text {(D) } ( - 1,3,3) \qquad \text {(E) } (1,3,3)
-</math>
 [[2000 AMC 12 Problems/Problem 10|Solution]]
@@ Line 84: / Line 103: @@
 Two non-zero real numbers, <math>a</math> and <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>?
-<math>\text{(A)} \ - 2 \qquad \text{(B)} \ \frac { - 1}{2} \qquad \text{(C)} \ \frac {1}{3} \qquad \text{(D)} \ \frac {1}{2} \qquad \text{(E)} \ 2</math>
+<math>\textbf{(A)} \ - 2 \qquad \textbf{(B)} \ \frac {- 1}{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2</math>
 [[2000 AMC 12 Problems/Problem 11|Solution]]
 == Problem 12 ==
-Let A, M, and C be nonnegative integers such that <math>A + M + C=12</math>. What is the maximum value of <math>A \cdot M \cdot C</math>+<math>A \cdot M</math>+<math>M \cdot C</math>+<math>A\cdot C</math>?
+Let <math>A, M,</math> and <math>C</math> be nonnegative integers such that <math>A + M + C=12</math>. What is the maximum value of <math>A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C</math>?
-<math> \mathrm{(A) \ 62 } \qquad \mathrm{(B) \ 72 } \qquad \mathrm{(C) \ 92 } \qquad \mathrm{(D) \ 102 } \qquad \mathrm{(E) \ 112 }  </math>
+<math>\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112</math>
 [[2000 AMC 12 Problems/Problem 12|Solution]]
 == Problem 13 ==
+One morning each member of Angela’s family drank an <math>8</math>-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?
+<math>\textbf {(A)}\ 3 \qquad \textbf {(B)}\ 4 \qquad \textbf {(C)}\ 5 \qquad \textbf {(D)}\ 6 \qquad \textbf {(E)}\ 7</math>
 [[2000 AMC 12 Problems/Problem 13|Solution]]
 == Problem 14 ==
+When the [[mean]], [[median]], and [[mode]] of the list
+<cmath>10,2,5,2,4,2,x</cmath>
+are arranged in increasing order, they form a non-constant [[arithmetic progression]]. What is the sum of all possible real values of <math>x</math>?
+<math>\textbf {(A)}\ 3 \qquad \textbf {(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf {(D)}\ 17 \qquad \textbf {(E)}\ 20</math>
 [[2000 AMC 12 Problems/Problem 14|Solution]]
 == Problem 15 ==
+Let <math>f</math> be a [[function]] for which <math>f(x/3) = x^2 + x + 1</math>. Find the sum of all values of <math>z</math> for which <math>f(3z) = 7</math>.
+<math>\textbf {(A)}\ -1/3 \qquad \textbf {(B)}\ -1/9 \qquad \textbf {(C)}\ 0 \qquad \textbf {(D)}\ 5/9 \qquad \textbf {(E)}\ 5/3</math>
 [[2000 AMC 12 Problems/Problem 15|Solution]]
 == Problem 16 ==
+A checkerboard of <math>13</math> rows and <math>17</math> columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered <math>1,2,\ldots,17</math>, the second row <math>18,19,\ldots,34</math>, and so on down the board. If the board is renumbered so that the left column, top to bottom, is <math>1,2,\ldots,13</math>, the second column <math>14,15,\ldots,26</math> and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
+<math>\textbf {(A)}\ 222 \qquad \textbf {(B)}\ 333\qquad \textbf {(C)}\ 444 \qquad \textbf {(D)}\ 555 \qquad \textbf {(E)}\ 666</math>
 [[2000 AMC 12 Problems/Problem 16|Solution]]
 == Problem 17 ==
+[[Image:2000_12_AMC-17.png|right]]
+A [[circle]] centered at <math>O</math> has [[radius]] <math>1</math> and contains the point <math>A</math>. The segment <math>AB</math> is [[tangent (geometry)|tangent]] to the circle at <math>A</math> and <math>\angle AOB = \theta</math>. If point <math>C</math> lies on <math>\overline{OA}</math> and <math>\overline{BC}</math> bisects <math>\angle ABO</math>, then <math>OC =</math>
+<math>\textbf {(A)}\ \sec^2 \theta - \tan \theta \qquad \textbf {(B)}\ \frac 12 \qquad \textbf {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta}\qquad \textbf {(D)}\ \frac{1}{1+\sin\theta} \qquad \textbf {(E)}\ \frac{\sin \theta}{\cos^2 \theta}</math>
 [[2000 AMC 12 Problems/Problem 17|Solution]]
 == Problem 18 ==
+In year <math>N</math>, the <math>300</math>th day of the year is a Tuesday. In year <math>N+1</math>, the <math>200</math>th day is also a Tuesday. On what day of the week did the <math>100</math>th day of year <math>N-1</math> occur?
+<math>\textbf {(A)}\ \text{Thursday} \qquad \textbf {(B)}\ \text{Friday}\qquad \textbf {(C)}\ \text{Saturday}\qquad \textbf {(D)}\ \text{Sunday}\qquad \textbf {(E)}\ \text{Monday}</math>
 [[2000 AMC 12 Problems/Problem 18|Solution]]
 == Problem 19 ==
+In [[triangle]] <math>ABC</math>, <math>AB = 13</math>, <math>BC = 14</math>, <math>AC = 15</math>. Let <math>D</math> denote the [[midpoint]] of <math>\overline{BC}</math> and let <math>E</math> denote the intersection of <math>\overline{BC}</math> with the [[angle bisector|bisector]] of angle <math>BAC</math>. Which of the following is closest to the area of the triangle <math>ADE</math>?
+<math>\textbf {(A)}\ 2 \qquad \textbf {(B)}\ 2.5 \qquad \textbf {(C)}\ 3 \qquad \textbf {(D)}\ 3.5 \qquad \textbf {(E)}\ 4</math>
 [[2000 AMC 12 Problems/Problem 19|Solution]]
 == Problem 20 ==
+If <math>x,y,</math> and <math>z</math> are positive numbers satisfying <math>x + \frac{1}{y} = 4, y + \frac{1}{z} = 1,</math> and <math>z + \frac{1}{x} = \frac73,</math> then what is the value of <math>xyz</math> ?
+<math>\textbf {(A)}\ 2/3 \qquad \textbf {(B)}\ 1 \qquad \textbf {(C)}\ 4/3 \qquad \textbf {(D)}\ 2 \qquad \textbf {(E)}\ 7/3</math>
 [[2000 AMC 12 Problems/Problem 20|Solution]]
 == Problem 21 ==
+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 <math>m</math> times the area of the square. The [[ratio]] of the area of the other small right triangle to the area of the square is
+<math>\textbf {(A)}\ \frac{1}{2m+1} \qquad \textbf {(B)}\ m \qquad \textbf {(C)}\ 1-m \qquad \textbf {(D)}\ \frac{1}{4m} \qquad \textbf {(E)}\ \frac{1}{8m^2}</math>
 [[2000 AMC 12 Problems/Problem 21|Solution]]
 == Problem 22 ==
+The [[graph]] below shows a portion of the [[curve]] defined by the quartic [[polynomial]] <math>P(x) = x^4 + ax^3 + bx^2 + cx + d</math>. Which of the following is the smallest?
+<math>\textbf{(A)}\ P(-1)\\
+\textbf{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\
+\textbf{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\
+\textbf{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
+\textbf{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math>
+[[Image:2000_12_AMC-22.png|center]]
 [[2000 AMC 12 Problems/Problem 22|Solution]]
 == Problem 23 ==
+Professor Gamble buys a lottery ticket, which requires that he pick six different integers from <math>1</math>  through <math>46</math>, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?
+<math>\textbf {(A)}\ 1/5 \qquad \textbf {(B)}\ 1/4 \qquad \textbf {(C)}\ 1/3 \qquad \textbf {(D)}\ 1/2 \qquad \textbf {(E)}\ 1 </math>
 [[2000 AMC 12 Problems/Problem 23|Solution]]
 == Problem 24 ==
+[[Image:2000_12_AMC-24.png|right]]
+If circular [[arc]]s <math>AC</math> and <math>BC</math> have [[center]]s at <math>B</math> and <math>A</math>, respectively, then there exists a [[circle]] [[tangent (geometry)|tangent]] to both <math>\stackrel{\frown}{AC}</math> and <math>\stackrel{\frown}{BC}</math>, and to <math>\overline{AB}</math>. If the length of <math>\stackrel{\frown}{BC}</math> is <math>12</math>, then the [[circumference]] of the circle is
+<math>\textbf {(A)}\ 24 \qquad \textbf {(B)}\ 25 \qquad \textbf {(C)}\ 26 \qquad \textbf {(D)}\ 27 \qquad \textbf {(E)}\ 28</math>
 [[2000 AMC 12 Problems/Problem 24|Solution]]
 == Problem 25 ==
+Eight congruent [[equilateral triangle]]s, each of a different color, are used to construct a regular [[octahedron]]. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
+<math>\textbf {(A)}\ 210 \qquad \textbf {(B)}\ 560 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 1260 \qquad \textbf {(E)}\ 1680</math>
+<center><asy>
+import three;
+import math;
+unitsize(1.5cm);
+currentprojection=orthographic(2,0.2,1);
+triple A=(0,0,1);
+triple B=(sqrt(2)/2,sqrt(2)/2,0);
+triple C=(sqrt(2)/2,-sqrt(2)/2,0);
+triple D=(-sqrt(2)/2,-sqrt(2)/2,0);
+triple E=(-sqrt(2)/2,sqrt(2)/2,0);
+triple F=(0,0,-1);
+draw(A--B--E--cycle);
+draw(A--C--D--cycle);
+draw(F--C--B--cycle);
+draw(F--D--E--cycle,dotted+linewidth(0.7));
+</asy></center>
 [[2000 AMC 12 Problems/Problem 25|Solution]]
 == See also ==
+{{AMC12 box|year=2000|before=[[2000 AMC 12 Problems]]|after=[[2001 AMC 12 Problems]]}}
 * [[AMC 12]]
 * [[AMC 12 Problems and Solutions]]
 * [[2000 AMC 12]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}