Difference between revisions of "2002 AIME II Problems"
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+ | {{AIME Problems|year=2002|n=II}} | ||
+ | |||
== Problem 1 == | == Problem 1 == | ||
− | + | Given that <math>x</math> and <math>y</math> are both integers between <math>100</math> and <math>999</math>, inclusive; <math>y</math> is the number formed by reversing the digits of <math>x</math>; and <math>z=|x-y|</math>. How many distinct values of <math>z</math> are possible? | |
[[2002 AIME II Problems/Problem 1|Solution]] | [[2002 AIME II Problems/Problem 1|Solution]] | ||
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== Problem 3 == | == Problem 3 == | ||
− | It is given that <math>\log_{6}a + \log_{6}b + \log_{6}c = 6 | + | It is given that <math>\log_{6}a + \log_{6}b + \log_{6}c = 6</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are [[positive]] [[integer]]s that form an increasing [[geometric sequence]] and <math>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>. |
[[2002 AIME II Problems/Problem 3|Solution]] | [[2002 AIME II Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
+ | Patio blocks that are hexagons <math>1</math> unit on a side are used to outline a garden by placing the blocks edge to edge with <math>n</math> on each side. The diagram indicates the path of blocks around the garden when <math>n=5</math>. | ||
+ | |||
+ | [[Image:AIME 2002 II Problem 4.gif]] | ||
+ | |||
+ | If <math>n=202</math>, then the area of the garden enclosed by the path, not including the path itself, is <math>m\left(\sqrt3/2\right)</math> square units, where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. | ||
[[2002 AIME II Problems/Problem 4|Solution]] | [[2002 AIME II Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
+ | Find the sum of all positive integers <math>a=2^n3^m</math> where <math>n</math> and <math>m</math> are non-negative integers, for which <math>a^6</math> is not a divisor of <math>6^a</math>. | ||
[[2002 AIME II Problems/Problem 5|Solution]] | [[2002 AIME II Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
+ | Find the integer that is closest to <math>1000\sum_{n=3}^{10000}\frac1{n^2-4}</math>. | ||
[[2002 AIME II Problems/Problem 6|Solution]] | [[2002 AIME II Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | It is known that, for all positive integers <math>k</math>, | ||
+ | |||
+ | <center><math>1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6</math>.</center> | ||
+ | |||
+ | Find the smallest positive integer <math>k</math> such that <math>1^2+2^2+3^2+\ldots+k^2</math> is a multiple of <math>200</math>. | ||
[[2002 AIME II Problems/Problem 7|Solution]] | [[2002 AIME II Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | Find the least positive integer <math>k</math> for which the equation <math>\left\lfloor\frac{2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math>. (The notation <math>\lfloor x\rfloor</math> means the greatest integer less than or equal to <math>x</math>.) | ||
[[2002 AIME II Problems/Problem 8|Solution]] | [[2002 AIME II Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | Let <math>\mathcal{S}</math> be the set <math>\lbrace1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</math> is divided by <math>1000</math>. | ||
[[2002 AIME II Problems/Problem 9|Solution]] | [[2002 AIME II Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
+ | While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of <math>x</math> for which the sine of <math>x</math> degrees is the same as the sine of <math>x</math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>. | ||
[[2002 AIME II Problems/Problem 10|Solution]] | [[2002 AIME II Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | Two distinct, real, infinite geometric series each have a sum of <math>1</math> and have the same second term. The third term of one of the series is <math>1/8</math>, and the second term of both series can be written in the form <math>\frac{\sqrt{m}-n}p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</math>. | ||
[[2002 AIME II Problems/Problem 11|Solution]] | [[2002 AIME II Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
− | A basketball player has a constant probability of <math>.4</math> of making any given shot, independent of previous shots. Let <math>a_n</math> be the ratio of shots made to shots attempted after <math>n</math> shots. The probability that <math>a_{10} | + | A basketball player has a constant probability of <math>.4</math> of making any given shot, independent of previous shots. Let <math>a_n</math> be the ratio of shots made to shots attempted after <math>n</math> shots. The probability that <math>a_{10}=.4</math> and <math>a_n\le.4</math> for all <math>n</math> such that <math>1\le n\le9</math> is given to be <math>p^aq^br/\left(s^c\right)</math> where <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> are primes, and <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. |
[[2002 AIME II Problems/Problem 12|Solution]] | [[2002 AIME II Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
− | In triangle <math>ABC</math>, point <math>D</math> is on <math>\overline{BC}</math> with <math>CD=2</math> and <math>DB=5</math>, point <math>E</math> is on <math>\overline{AC}</math> with <math>CE=1</math> and <math>EA= | + | In triangle <math>ABC</math>, point <math>D</math> is on <math>\overline{BC}</math> with <math>CD=2</math> and <math>DB=5</math>, point <math>E</math> is on <math>\overline{AC}</math> with <math>CE=1</math> and <math>EA=3</math>, <math>AB=8</math>, and <math>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>P</math>. Points <math>Q</math> and <math>R</math> lie on <math>\overline{AB}</math> so that <math>\overline{PQ}</math> is parallel to <math>\overline{CA}</math> and <math>\overline{PR}</math> is parallel to <math>\overline{CB}</math>. It is given that the ratio of the area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[2002 AIME II Problems/Problem 13|Solution]] | [[2002 AIME II Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
− | The perimeter of triangle <math>APM</math> is <math>152</math> and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM} | + | The perimeter of triangle <math>APM</math> is <math>152</math>, and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM}</math>. Given that <math>OP=m/n</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. |
[[2002 AIME II Problems/Problem 14|Solution]] | [[2002 AIME II Problems/Problem 14|Solution]] | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2002|n=II|before=[[2002 AIME I Problems]]|after=[[2003 AIME I Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 15:42, 11 August 2023
2002 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Given that and
are both integers between
and
, inclusive;
is the number formed by reversing the digits of
; and
. How many distinct values of
are possible?
Problem 2
Three vertices of a cube are ,
, and
. What is the surface area of the cube?
Problem 3
It is given that , where
,
, and
are positive integers that form an increasing geometric sequence and
is the square of an integer. Find
.
Problem 4
Patio blocks that are hexagons unit on a side are used to outline a garden by placing the blocks edge to edge with
on each side. The diagram indicates the path of blocks around the garden when
.
If , then the area of the garden enclosed by the path, not including the path itself, is
square units, where
is a positive integer. Find the remainder when
is divided by
.
Problem 5
Find the sum of all positive integers where
and
are non-negative integers, for which
is not a divisor of
.
Problem 6
Find the integer that is closest to .
Problem 7
It is known that, for all positive integers ,
![$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$](http://latex.artofproblemsolving.com/4/f/c/4fc69b2d25fe9c2e53533b462d9b0cce6f4ceed0.png)
Find the smallest positive integer such that
is a multiple of
.
Problem 8
Find the least positive integer for which the equation
has no integer solutions for
. (The notation
means the greatest integer less than or equal to
.)
Problem 9
Let be the set
Let
be the number of sets of two non-empty disjoint subsets of
. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when
is divided by
.
Problem 10
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of for which the sine of
degrees is the same as the sine of
radians are
and
, where
,
,
, and
are positive integers. Find
.
Problem 11
Two distinct, real, infinite geometric series each have a sum of and have the same second term. The third term of one of the series is
, and the second term of both series can be written in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 12
A basketball player has a constant probability of of making any given shot, independent of previous shots. Let
be the ratio of shots made to shots attempted after
shots. The probability that
and
for all
such that
is given to be
where
,
,
, and
are primes, and
,
, and
are positive integers. Find
.
Problem 13
In triangle , point
is on
with
and
, point
is on
with
and
,
, and
and
intersect at
. Points
and
lie on
so that
is parallel to
and
is parallel to
. It is given that the ratio of the area of triangle
to the area of triangle
is
, where
and
are relatively prime positive integers. Find
.
Problem 14
The perimeter of triangle is
, and the angle
is a right angle. A circle of radius
with center
on
is drawn so that it is tangent to
and
. Given that
where
and
are relatively prime positive integers, find
.
Problem 15
Circles and
intersect at two points, one of which is
, and the product of the radii is
. The x-axis and the line
, where
, are tangent to both circles. It is given that
can be written in the form
, where
,
, and
are positive integers,
is not divisible by the square of any prime, and
and
are relatively prime. Find
.
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2002 AIME I Problems |
Followed by 2003 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.