Difference between revisions of "2000 AMC 12 Problems"

Revision as of 23:36, 26 November 2011

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?

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

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

@@ Line 3: / Line 3: @@
 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 11: / Line 11: @@
 <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 19: @@
 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 25: @@
 == 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 36: / Line 35: @@
 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 43: @@
 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)}\ 21 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180 \qquad \textbf{(E)}\ 231</math>
 [[2000 AMC 12 Problems/Problem 6|Solution]]
@@ Line 52: / Line 51: @@
 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]]
@@ Line 82: / Line 81: @@
 </asy>
+<math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{(D)}\ 39801 \qquad\textbf{(E)}\ 40801</math>
-<math>\mathrm{(A)}\ 10401 \qquad\mathrm{(B)}\ 19801 \qquad\mathrm{(C)}\ 20201 \qquad\mathrm{(D)}\ 39801 \qquad\mathrm{(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 106: / Line 102: @@
 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 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?
+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>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \text {(E)}\ 7</math>
+<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]]
@@ Line 131: / Line 127: @@
 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>\text {(A)}\ 3 \qquad \text {(B)}\ 6 \qquad \text {(C)}\ 9 \qquad \text {(D)}\ 17 \qquad \text {(E)}\ 20</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]]
@@ Line 138: / Line 134: @@
 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>\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\ 5/9 \qquad \text {(E)}\ 5/3</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).
+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>\text {(A)}\ 222 \qquad \text {(B)}\ 333\qquad \text {(C)}\ 444 \qquad \text {(D)}\ 555 \qquad \text {(E)}\ 666</math>
+<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]]
@@ Line 155: / Line 152: @@
 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>\text {(A)}\ \sec^2 \theta - \tan \theta \qquad \text {(B)}\ \frac 12 \qquad \text {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta}\qquad \text {(D)}\ \frac{1}{1+\sin\theta} \qquad \text {(E)}\ \frac{\sin \theta}{\cos^2 \theta}</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]]
@@ Line 161: / Line 158: @@
 == Problem 18 ==
-In year <math>N</math>, the <math>300^{\text{th}}</math> day of the year is a Tuesday. In year <math>N+1</math>, the <math>200^{\text{th}}</math> day is also a Tuesday. On what day of the week did the <math>100</math><sup>th</sup> day of year <math>N-1</math> occur?
+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>\text {(A)}\ \text{Thursday} \qquad \text {(B)}\ \text{Friday}\qquad \text {(C)}\ \text{Saturday}\qquad \text {(D)}\ \text{Sunday}\qquad \text {(E)}\ \text{Monday}</math>
+<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]]
@@ Line 171: / Line 168: @@
 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>\text {(A)}\ 2 \qquad \text {(B)}\ 2.5 \qquad \text {(C)}\ 3 \qquad \text {(D)}\ 3.5 \qquad \text {(E)}\ 4</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]]
@@ Line 177: / Line 174: @@
 == Problem 20 ==
-If <math>x,y,</math> and <math>z</math> are positive numbers satisfying
+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> ?
-<cmath>x + 1/y = 4,\qquad y + 1/z = 1, \qquad \text{and} \qquad z + 1/x = 7/3</cmath>
-Then what is the value of <math>xyz</math> ?
-<math>\text {(A)}\ 2/3 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 4/3 \qquad \text {(D)}\ 2 \qquad \text {(E)}\ 7/3</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]]
@@ Line 190: / Line 183: @@
 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>\text {(A)}\ \frac{1}{2m+1} \qquad \text {(B)}\ m \qquad \text {(C)}\ 1-m \qquad \text {(D)}\ \frac{1}{4m} \qquad \text {(E)}\ \frac{1}{8m^2}</math>
+<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]]
@@ Line 197: / Line 190: @@
 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>\text{(A)}\ P(-1)\\
+<math>\textbf{(A)}\ P(-1)\\
-\text{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\
+\textbf{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\
-\text{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\
+\textbf{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\
-\text{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
+\textbf{(D)}\ \text{The\ sum\ of\ the\ coefficients\ of\ } P \\
-\text{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math>
+\textbf{(E)}\ \text{The\ sum\ of\ the\ real\ zeros\ of\ } P</math>
-[[Image:2000_12_AMC-22.png]]
+[[Image:2000_12_AMC-22.png|center]]
 [[2000 AMC 12 Problems/Problem 22|Solution]]
@@ Line 210: / Line 203: @@
 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>\text {(A)}\ 1/5 \qquad \text {(B)}\ 1/4 \qquad \text {(C)}\ 1/3 \qquad \text {(D)}\ 1/2 \qquad \text {(E)}\ 1 </math>
+<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]]
@@ Line 219: / Line 212: @@
 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>\text {(A)}\ 24 \qquad \text {(B)}\ 25 \qquad \text {(C)}\ 26 \qquad \text {(D)}\ 27 \qquad \text {(E)}\ 28</math>
+<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]]
@@ Line 227: / Line 220: @@
 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>\text {(A)}\ 210 \qquad \text {(B)}\ 560 \qquad \text {(C)}\ 840 \qquad \text {(D)}\ 1260 \qquad \text {(E)}\ 1680</math>
+<math>\textbf {(A)}\ 210 \qquad \textbf {(B)}\ 560 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 1260 \qquad \textbf {(E)}\ 1680</math>
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Difference between revisions of "2000 AMC 12 Problems"

Revision as of 23:36, 26 November 2011

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