Difference between revisions of "2000 AMC 12 Problems"
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+ | {{AMC12 Problems|year=2000|ab=}} | ||
== Problem 1 == | == Problem 1 == | ||
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>? | 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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 1|Solution]] | ||
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<math>2000(2000^{2000}) =</math> | <math>2000(2000^{2000}) =</math> | ||
− | <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]] | [[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? | 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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 3|Solution]] | ||
Line 26: | Line 26: | ||
== Problem 4 == | == Problem 4 == | ||
− | The Fibonacci sequence <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two | + | 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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 4|Solution]] | ||
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If <math>|x - 2| = p,</math> where <math>x < 2,</math> then <math>x - p =</math> | If <math>|x - 2| = p,</math> where <math>x < 2,</math> then <math>x - p =</math> | ||
− | <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]] | [[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? | 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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 6|Solution]] | ||
Line 52: | Line 52: | ||
How many positive integers <math>b</math> have the property that <math>\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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
− | |||
− | <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> | ||
+ | |||
+ | <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]] | [[2000 AMC 12 Problems/Problem 8|Solution]] | ||
== Problem 9 == | == 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 9|Solution]] | ||
== Problem 10 == | == 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 | + | 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> |
− | \ | ||
− | </math> | ||
[[2000 AMC 12 Problems/Problem 10|Solution]] | [[2000 AMC 12 Problems/Problem 10|Solution]] | ||
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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>? | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 11|Solution]] | ||
== Problem 12 == | == 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 | + | 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> \ | + | <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]] | [[2000 AMC 12 Problems/Problem 12|Solution]] | ||
== Problem 13 == | == 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 13|Solution]] | ||
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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>? | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 14|Solution]] | ||
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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>. | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 15|Solution]] | ||
== Problem 16 == | == 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]] | [[2000 AMC 12 Problems/Problem 16|Solution]] | ||
== Problem 17 == | == 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]] | [[2000 AMC 12 Problems/Problem 17|Solution]] | ||
== Problem 18 == | == 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]] | [[2000 AMC 12 Problems/Problem 18|Solution]] | ||
== Problem 19 == | == 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]] | [[2000 AMC 12 Problems/Problem 19|Solution]] | ||
== Problem 20 == | == 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]] | [[2000 AMC 12 Problems/Problem 20|Solution]] | ||
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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 | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 21|Solution]] | ||
== Problem 22 == | == 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]] | [[2000 AMC 12 Problems/Problem 22|Solution]] | ||
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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? | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 23|Solution]] | ||
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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 | 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>\ | + | <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]] | [[2000 AMC 12 Problems/Problem 24|Solution]] | ||
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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.) | 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>\ | + | <math>\textbf {(A)}\ 210 \qquad \textbf {(B)}\ 560 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 1260 \qquad \textbf {(E)}\ 1680</math> |
<center><asy> | <center><asy> | ||
Line 192: | Line 244: | ||
== See also == | == See also == | ||
+ | |||
+ | {{AMC12 box|year=2000|before=[[2000 AMC 12 Problems]]|after=[[2001 AMC 12 Problems]]}} | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[2000 AMC 12]] | * [[2000 AMC 12]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 10:35, 16 June 2024
2000 AMC 12 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
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
'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?
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
nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?
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 and
. What was the last score Mrs. Walters entered?
Problem 10
The point is reflected in the
-plane, then its image
is rotated
about the
-axis to produce
, and finally,
is translated
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 and
be nonnegative integers such that
. What is the maximum value of
?
Problem 13
One morning each member of Angela’s family drank an -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?
Problem 14
When the mean, median, and mode of the list
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?
Problem 15
Let be a function for which
. Find the sum of all values of
for which
.
Problem 16
A checkerboard of rows and
columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered
, the second row
, and so on down the board. If the board is renumbered so that the left column, top to bottom, is
, the second column
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).
Problem 17
A circle centered at has radius
and contains the point
. The segment
is tangent to the circle at
and
. If point
lies on
and
bisects
, then
Problem 18
In year , the
th day of the year is a Tuesday. In year
, the
th day is also a Tuesday. On what day of the week did the
th day of year
occur?
Problem 19
In triangle ,
,
,
. Let
denote the midpoint of
and let
denote the intersection of
with the bisector of angle
. Which of the following is closest to the area of the triangle
?
Problem 20
If and
are positive numbers satisfying
and
then what is the value of
?
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 times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
Problem 22
The graph below shows a portion of the curve defined by the quartic polynomial . Which of the following is the smallest?
Problem 23
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from through
, 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?
Problem 24
If circular arcs and
have centers at
and
, respectively, then there exists a circle tangent to both
and
, and to
. If the length of
is
, then the circumference of the circle is
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.)
![[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]](http://latex.artofproblemsolving.com/7/d/1/7d13f21ea69c61ec44e671434db41a0bdedb8189.png)
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
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