Difference between revisions of "2002 AMC 12P Problems"
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</math> | </math> | ||
− | + | [[2002 AMC 12P Problems/Problem 1|Solution]] | |
− | [[2002 AMC 12P Problems/Problem | ||
== Problem 2 == | == Problem 2 == | ||
The function <math>f</math> is given by the table | The function <math>f</math> is given by the table | ||
− | If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n | + | <cmath> |
+ | \begin{tabular}{|c||c|c|c|c|c|} | ||
+ | \hline | ||
+ | x & 1 & 2 & 3 & 4 & 5 \\ | ||
+ | \hline | ||
+ | f(x) & 4 & 1 & 3 & 5 & 2 \\ | ||
+ | \hline | ||
+ | \end{tabular} | ||
+ | </cmath> | ||
+ | |||
+ | If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n \ge 0</math>, find <math>u_{2002}</math> | ||
<math> | <math> | ||
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\text{(E) }5 | \text{(E) }5 | ||
</math> | </math> | ||
+ | |||
[[2002 AMC 12P Problems/Problem 2|Solution]] | [[2002 AMC 12P Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
− | The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in^3. Find the minimum possible sum of the three dimensions. | + | The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in<math>^3</math>. Find the minimum possible sum of the three dimensions. |
<math> | <math> | ||
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Let <math>a</math> and <math>b</math> be distinct real numbers for which | Let <math>a</math> and <math>b</math> be distinct real numbers for which | ||
<cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath> | <cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath> | ||
+ | |||
Find <math>\frac{a}{b}</math> | Find <math>\frac{a}{b}</math> | ||
− | <math>\text{(A)} | + | <math> |
+ | \text{(A) }0.4 | ||
+ | \qquad | ||
+ | \text{(B) }0.5 | ||
+ | \qquad | ||
+ | \text{(C) }0.6 | ||
+ | \qquad | ||
+ | \text{(D) }0.7 | ||
+ | \qquad | ||
+ | \text{(E) }0.8 | ||
+ | </math> | ||
[[2002 AMC 12P Problems/Problem 4|Solution]] | [[2002 AMC 12P Problems/Problem 4|Solution]] | ||
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<cmath>\frac{2002}{m^2 -2}</cmath> | <cmath>\frac{2002}{m^2 -2}</cmath> | ||
− | <math>\text{(A)} | + | a positive integer? |
− | \text{(C)} | + | |
− | \text{(E)} | + | <math> |
+ | \text{(A) one} | ||
+ | \qquad | ||
+ | \text{(B) two} | ||
+ | \qquad | ||
+ | \text{(C) three} | ||
+ | \qquad | ||
+ | \text{(D) four} | ||
+ | \qquad | ||
+ | \text{(E) more than four} | ||
+ | </math> | ||
[[2002 AMC 12P Problems/Problem 5|Solution]] | [[2002 AMC 12P Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | Participation in the local soccer league this year is <math>10%</math> higher than last year. The number of males increased by <math>5%</math> and the number of females increased by <math>20%</math>. What fraction of the soccer league is now female? | + | Participation in the local soccer league this year is <math>10\%</math> higher than last year. The number of males increased by <math>5\%</math> and the number of females increased by <math>20\%</math>. What fraction of the soccer league is now female? |
<math> | <math> | ||
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\text{(E) }\frac{1}{2} | \text{(E) }\frac{1}{2} | ||
</math> | </math> | ||
− | |||
[[2002 AMC 12P Problems/Problem 6|Solution]] | [[2002 AMC 12P Problems/Problem 6|Solution]] | ||
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== Problem 7 == | == Problem 7 == | ||
− | How many three-digit numbers have at least one 2 and at least one 3? | + | How many three-digit numbers have at least one <math>2</math> and at least one <math>3</math>? |
<math> | <math> | ||
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== Problem 9 == | == Problem 9 == | ||
− | Two walls and the ceiling of a room meet at right angles at point <math>P.</math> A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point <math>P</math>. How many meters is the fly from the ceiling? | + | Two walls and the ceiling of a room meet at right angles at point <math>P.</math> A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point <math>P</math>. How many meters is the fly from the ceiling? |
− | <math>\text{(A)} | + | <math> |
+ | \text{(A) }\sqrt{13} | ||
+ | \qquad | ||
+ | \text{(B) }\sqrt{14} | ||
+ | \qquad | ||
+ | \text{(C) }\sqrt{15} | ||
+ | \qquad | ||
+ | \text{(D) }4 | ||
+ | \qquad | ||
+ | \text{(E) }\sqrt{17} | ||
+ | </math> | ||
[[2002 AMC 12P Problems/Problem 9|Solution]] | [[2002 AMC 12P Problems/Problem 9|Solution]] | ||
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== Problem 10 == | == Problem 10 == | ||
− | Let <math>f_n (x) = sin^n x + cos^n x.</math> For how many <math>x</math> in | + | Let <math>f_n (x) = \text{sin}^n x + \text{cos}^n x.</math> For how many <math>x</math> in <math>[0,\pi]</math> is it true that |
+ | |||
+ | <cmath>6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?</cmath> | ||
<math> | <math> | ||
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\text{(D) }8 | \text{(D) }8 | ||
\qquad | \qquad | ||
− | \text{(E) | + | \text{(E) more than }8 |
</math> | </math> | ||
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Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find | Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find | ||
− | <cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{ | + | <cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}</cmath> |
<math> | <math> | ||
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<math> | <math> | ||
− | \text{(A) } | + | \text{(A) one} |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) two} |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) three} |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) four} |
\qquad | \qquad | ||
− | \text{(E) | + | \text{(E) more than four} |
</math> | </math> | ||
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What is the maximum value of <math>n</math> for which there is a set of distinct positive integers <math>k_1, k_2, ... k_n</math> for which | What is the maximum value of <math>n</math> for which there is a set of distinct positive integers <math>k_1, k_2, ... k_n</math> for which | ||
− | < | + | <cmath>k^2_1 + k^2_2 + ... + k^2_n = 2002?</cmath> |
<math> | <math> | ||
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== Problem 14 == | == Problem 14 == | ||
− | Find <math>i + 2i^2 +3i^3 + ... + 2002i^2002.</math> | + | Find <math>i + 2i^2 +3i^3 + . . . + 2002i^{2002}.</math> |
<math> | <math> | ||
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== Problem 15 == | == Problem 15 == | ||
− | There are <math>1001 red marbles and </math> | + | There are <math>1001</math> red marbles and <math>1001</math> black marbles in a box. Let <math>P_s</math> be the probability that two marbles drawn at random from the box are the same color, and let <math>P_d</math> be the probability that they are different colors. Find <math>|P_s-P_d|.</math> |
<math> | <math> | ||
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<math> | <math> | ||
− | \text{(A) }72^ | + | \text{(A) }72^\circ |
\qquad | \qquad | ||
− | \text{(B) }75^ | + | \text{(B) }75^\circ |
\qquad | \qquad | ||
− | \text{(C) }90^ | + | \text{(C) }90^\circ |
\qquad | \qquad | ||
− | \text{(D) }108^ | + | \text{(D) }108^\circ |
\qquad | \qquad | ||
− | \text{(E) }120^ | + | \text{(E) }120^\circ |
</math> | </math> | ||
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== Problem 17 == | == Problem 17 == | ||
− | Let <math>f(x) = | + | Let <math>f(x) = \sqrt{\sin^4{x} + 4 \cos^2{x}} - \sqrt{\cos^4{x} + 4 \sin^2{x}}.</math> An equivalent form of <math>f(x)</math> is |
− | < | + | |
− | \text{(A) }\ | + | <math> |
+ | \text{(A) }1-\sqrt{2}\sin{x} | ||
\qquad | \qquad | ||
− | \text{(B) }\ | + | \text{(B) }-1+\sqrt{2}\cos{x} |
\qquad | \qquad | ||
− | \text{(C) }\frac { | + | \text{(C) }\cos{\frac{x}{2}} - \sin{\frac{x}{2}} |
\qquad | \qquad | ||
− | \text{(D) }\ | + | \text{(D) }\cos{x} - \sin{x} |
\qquad | \qquad | ||
− | \text{(E) }\ | + | \text{(E) }\cos{2x} |
− | <math> | + | </math> |
[[2002 AMC 12P Problems/Problem 17|Solution]] | [[2002 AMC 12P Problems/Problem 17|Solution]] | ||
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== Problem 18 == | == Problem 18 == | ||
− | + | If <math>a,b,c</math> are real numbers such that <math>a^2 + 2b =7</math>, <math>b^2 + 4c= -7,</math> and <math>c^2 + 6a= -14</math>, find <math>a^2 + b^2 + c^2.</math> | |
− | |||
− | < | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | </ | ||
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }14 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }21 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }28 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }35 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }49 |
</math> | </math> | ||
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== Problem 19 == | == Problem 19 == | ||
− | + | In quadrilateral <math>ABCD</math>, <math>m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,</math> and <math>CD=5.</math> Find the area of <math>ABCD.</math> | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }15 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }9 \sqrt{3} |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }\frac{45 \sqrt{3}}{4} |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }\frac{47 \sqrt{3}}{4} |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }15 \sqrt{3} |
</math> | </math> | ||
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== Problem 20 == | == Problem 20 == | ||
− | + | Let <math>f</math> be a real-valued function such that | |
+ | |||
+ | <cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath> | ||
+ | |||
+ | for all <math>x>0.</math> Find <math>f(2).</math> | ||
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }1000 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }2000 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }3000 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }4000 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }6000 |
</math> | </math> | ||
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== Problem 21 == | == Problem 21 == | ||
− | + | Let <math>a</math> and <math>b</math> be real numbers greater than <math>1</math> for which there exists a positive real number <math>c,</math> different from <math>1</math>, such that | |
− | <cmath> | + | <cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath> |
− | \ | ||
− | |||
− | \ | ||
− | |||
− | \ | ||
− | |||
− | |||
− | </cmath> | ||
− | + | Find the largest possible value of <math>\log_a b.</math> | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }\sqrt{2} |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }\sqrt{3} |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }2 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }\sqrt{6} |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }3 |
</math> | </math> | ||
− | [[ | + | [[2002 AMC 12P Problems/Problem 21|Solution]] |
== Problem 22 == | == Problem 22 == | ||
− | + | Under the new AMC <math>10, 12</math> scoring method, <math>6</math> points are given for each correct answer, <math>2.5</math> points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between <math>0</math> and <math>150</math> can be obtained in only one way, for example, a score of <math>104.5</math> can be obtained with <math>17</math> correct answers, <math>1</math> unanswered question, and <math>7</math> incorrect answers, and also with <math>12</math> correct answers and <math>13</math> unanswered questions. There are scores that can be obtained in exactly three ways. What is their sum? | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }175 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }179.5 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }182 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }188.5 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }201 |
</math> | </math> | ||
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== Problem 23 == | == Problem 23 == | ||
− | + | The equation <math>z(z+i)(z+3i)=2002i</math> has a zero of the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are positive real numbers. Find <math>a.</math> | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }\sqrt{118} |
\qquad | \qquad | ||
− | \text{(B) }\ | + | \text{(B) }\sqrt{210} |
\qquad | \qquad | ||
− | \text{(C) }\ | + | \text{(C) }2 \sqrt{210} |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }\sqrt{2002} |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }100 \sqrt{2} |
</math> | </math> | ||
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== Problem 24 == | == Problem 24 == | ||
− | + | Let <math>ABCD</math> be a regular tetrahedron and Let <math>E</math> be a point inside the face <math>ABC.</math> Denote by <math>s</math> the sum of the distances from <math>E</math> to the faces <math>DAB, DBC, DCA,</math> and by <math>S</math> the sum of the distances from <math>E</math> to the edges <math>AB, BC, CA.</math> Then <math>\frac{s}{S}</math> equals | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }\sqrt{2} |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }\frac{2 \sqrt{2}}{3} |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }\frac{\sqrt{6}}{2} |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }2 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }3 |
</math> | </math> | ||
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== Problem 25 == | == Problem 25 == | ||
− | + | Let <math>a</math> and <math>b</math> be real numbers such that <math>\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}</math> and <math>\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.</math> Find <math>\sin{(a+b)}.</math> | |
<math> | <math> | ||
− | \text{(A) }1 | + | \text{(A) }\frac{1}{2} |
\qquad | \qquad | ||
− | \text{(B) }2 | + | \text{(B) }\frac{\sqrt{2}}{2} |
\qquad | \qquad | ||
− | \text{(C) }3 | + | \text{(C) }\frac{\sqrt{3}}{2} |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }\frac{\sqrt{6}}{2} |
\qquad | \qquad | ||
− | \text{(E) | + | \text{(E) }1 |
</math> | </math> | ||
Latest revision as of 01:46, 31 December 2023
2002 AMC 12P (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
Which of the following numbers is a perfect square?
Problem 2
The function is given by the table
If and for , find
Problem 3
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is in. Find the minimum possible sum of the three dimensions.
Problem 4
Let and be distinct real numbers for which
Find
Problem 5
For how many positive integers is
a positive integer?
Problem 6
Participation in the local soccer league this year is higher than last year. The number of males increased by and the number of females increased by . What fraction of the soccer league is now female?
Problem 7
How many three-digit numbers have at least one and at least one ?
Problem 8
Let be a segment of length , and let points and be located on such that and . Let and be points on one of the semicircles with diameter for which and are perpendicular to . Find
Problem 9
Two walls and the ceiling of a room meet at right angles at point A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point . How many meters is the fly from the ceiling?
Problem 10
Let For how many in is it true that
Problem 11
Let be the th triangular number. Find
Problem 12
For how many positive integers is a prime number?
Problem 13
What is the maximum value of for which there is a set of distinct positive integers for which
Problem 14
Find
Problem 15
There are red marbles and black marbles in a box. Let be the probability that two marbles drawn at random from the box are the same color, and let be the probability that they are different colors. Find
Problem 16
The altitudes of a triangle are and The largest angle in this triangle is
Problem 17
Let An equivalent form of is
Problem 18
If are real numbers such that , and , find
Problem 19
In quadrilateral , and Find the area of
Problem 20
Let be a real-valued function such that
for all Find
Problem 21
Let and be real numbers greater than for which there exists a positive real number different from , such that
Find the largest possible value of
Problem 22
Under the new AMC scoring method, points are given for each correct answer, points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between and can be obtained in only one way, for example, a score of can be obtained with correct answers, unanswered question, and incorrect answers, and also with correct answers and unanswered questions. There are scores that can be obtained in exactly three ways. What is their sum?
Problem 23
The equation has a zero of the form , where and are positive real numbers. Find
Problem 24
Let be a regular tetrahedron and Let be a point inside the face Denote by the sum of the distances from to the faces and by the sum of the distances from to the edges Then equals
Problem 25
Let and be real numbers such that and Find
See also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by 2000 AMC 12 Problems |
Followed by 2002 AMC 12A 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.