Difference between revisions of "2002 AMC 10P Problems"
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== Problem 3 == | == Problem 3 == | ||
− | + | Mary typed a six-digit number, but the two <math>1</math>s she typed didn't show. What appeared was <math>2002.</math> How many different six-digit numbers could she have typed? | |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }4 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }8 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }10 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }15 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }20 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 3|Solution]] |
== Problem 4 == | == Problem 4 == | ||
− | + | Which of the following numbers is a perfect square? | |
− | |||
− | |||
− | |||
− | <math> | + | <math>\text{(A) }4^4 5^5 6^6 \qquad \text{(B) }4^4 5^6 6^5 \qquad \text{(C) }4^5 5^4 6^6 \qquad \text{(D) }4^6 5^4 6^5 \qquad \text{(E) }4^6 5^5 6^4</math> |
− | \text{(A) } | ||
− | \qquad | ||
− | \text{(B) } | ||
− | \qquad | ||
− | \text{(C) } | ||
− | \qquad | ||
− | \text{(D) } | ||
− | \qquad | ||
− | \text{(E) } | ||
− | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 4|Solution]] |
== Problem 5 == | == Problem 5 == | ||
− | + | Let <math>(a_n)_{n \geq 1}</math> be a sequence such that <math>a_1 = 1</math> and <math>3a_{n+1} - 3a_n = 1</math> for all <math>n \geq 1.</math> Find <math>a_{2002}.</math> | |
− | < | ||
− | |||
− | |||
<math> | <math> | ||
− | \text{(A) | + | \text{(A) }666 |
\qquad | \qquad | ||
− | \text{(B) | + | \text{(B) }667 |
\qquad | \qquad | ||
− | \text{(C) | + | \text{(C) }668 |
\qquad | \qquad | ||
− | \text{(D) | + | \text{(D) }669 |
\qquad | \qquad | ||
− | \text{(E) | + | \text{(E) }670 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 5|Solution]] |
== Problem 6 == | == Problem 6 == | ||
− | + | The perimeter of a rectangle <math>100</math> and its diagonal has length <math>x.</math> What is the area of this rectangle? | |
− | |||
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }625-x^2 |
\qquad | \qquad | ||
− | \text{(B) }\frac{ | + | \text{(B) }625-\frac{x^2}{2} |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }1250-x^2 |
\qquad | \qquad | ||
− | \text{(D) }\frac{ | + | \text{(D) }1250-\frac{x^2}{2} |
\qquad | \qquad | ||
− | \text{(E) }\frac{ | + | \text{(E) }2500-\frac{x^2}{2} |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 6|Solution]] |
== Problem 7 == | == Problem 7 == | ||
− | + | 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> | ||
− | \text{(A) } | + | \text{(A) }36 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }38 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }42 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }44 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }92 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 7|Solution]] |
== Problem 8 == | == Problem 8 == | ||
− | + | How many ordered triples of positive integers <math>(x,y,z)</math> satisfy <math>(x^y)^z=64?</math> | |
<math> | <math> | ||
\text{(A) }5 | \text{(A) }5 | ||
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }6 |
\qquad | \qquad | ||
\text{(C) }7 | \text{(C) }7 | ||
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }8 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }9 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 8|Solution]] |
== Problem 9 == | == Problem 9 == | ||
− | + | The function <math>f</math> is given by the table | |
+ | |||
+ | <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> | ||
− | \text{(A) } | + | \text{(A) }1 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }2 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }3 |
\qquad | \qquad | ||
\text{(D) }4 | \text{(D) }4 | ||
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }5 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 2|Solution]] |
== Problem 10 == | == Problem 10 == | ||
− | Let <math> | + | 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> | ||
− | < | + | Find <math>\frac{a}{b}</math> |
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }0.4 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }0.5 |
\qquad | \qquad | ||
− | \text{(C) }6 | + | \text{(C) }0.6 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }0.7 |
\qquad | \qquad | ||
− | \text{(E) | + | \text{(E) }0.8 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 10|Solution]] |
== Problem 11 == | == Problem 11 == | ||
− | Let <math> | + | Let <math>P(x)=kx^3 + 2k^2x^2+k^3.</math> Find the sum of all real numbers <math>k</math> for which <math>x-2</math> is a factor of <math>P(x).</math> |
− | |||
− | < | ||
<math> | <math> | ||
− | \text{(A) } | + | \text{(A) }-8 |
\qquad | \qquad | ||
− | \text{(B) } | + | \text{(B) }-4 |
\qquad | \qquad | ||
− | \text{(C) } | + | \text{(C) }0 |
\qquad | \qquad | ||
− | \text{(D) } | + | \text{(D) }5 |
\qquad | \qquad | ||
− | \text{(E) } | + | \text{(E) }8 |
</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 11|Solution]] |
== Problem 12 == | == Problem 12 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 12|Solution]] |
== Problem 13 == | == Problem 13 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 13|Solution]] |
== Problem 14 == | == Problem 14 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 14|Solution]] |
== Problem 15 == | == Problem 15 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 15|Solution]] |
== Problem 16 == | == Problem 16 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 16|Solution]] |
== Problem 17 == | == Problem 17 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 17|Solution]] |
== Problem 18 == | == Problem 18 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 18|Solution]] |
== Problem 19 == | == Problem 19 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 19|Solution]] |
== Problem 20 == | == Problem 20 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 20|Solution]] |
== Problem 21 == | == Problem 21 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 21|Solution]] |
== Problem 22 == | == Problem 22 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 22|Solution]] |
== Problem 23 == | == Problem 23 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 23|Solution]] |
== Problem 24 == | == Problem 24 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 24|Solution]] |
== Problem 25 == | == Problem 25 == | ||
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</math> | </math> | ||
− | [[2002 AMC | + | [[2002 AMC 10P Problems/Problem 25|Solution]] |
== See also == | == See also == | ||
− | {{ | + | {{AMC10 box|year=2002|ab=P|before=[[2001 AMC 10 Problems]]|after=[[2002 AMC 10A Problems]]}} |
− | * [[AMC | + | * [[AMC 10]] |
− | * [[AMC | + | * [[AMC 10 Problems and Solutions]] |
− | * [[2002 AMC | + | * [[2002 AMC 10P]] |
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 02:19, 14 July 2024
2002 AMC 10P (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
The ratio equals
Problem 2
The sum of eleven consecutive integers is What is the smallest of these integers?
Problem 3
Mary typed a six-digit number, but the two s she typed didn't show. What appeared was How many different six-digit numbers could she have typed?
Problem 4
Which of the following numbers is a perfect square?
Problem 5
Let be a sequence such that and for all Find
Problem 6
The perimeter of a rectangle and its diagonal has length What is the area of this rectangle?
Problem 7
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 8
How many ordered triples of positive integers satisfy
Problem 9
The function is given by the table
If and for , find
Problem 10
Let and be distinct real numbers for which
Find
Problem 11
Let Find the sum of all real numbers for which is a factor of
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, poitns 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, the only way to obtain a score of is to have correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of can be obtained with correct answers, unanswered question, and incorrect, and also with correct answers and unanswered questions. There are (three) 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
2002 AMC 10P (Problems • Answer Key • Resources) | ||
Preceded by 2001 AMC 10 Problems |
Followed by 2002 AMC 10A 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.