Difference between revisions of "1989 AIME Problems"
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== Problem 4 == | == Problem 4 == | ||
− | If <math>a<b<c<d<e^{}_{}</math> are consecutive positive integers such that <math>b+c+d | + | If <math>a<b<c<d<e^{}_{}</math> are consecutive positive integers such that <math>b+c+d</math> is a perfect square and <math>a+b+c+d+e^{}_{}</math> is a perfect cube, what is the smallest possible value of <math>c^{}_{}</math>? |
[[1989 AIME Problems/Problem 4|Solution]] | [[1989 AIME Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
− | When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to <math>0 | + | When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to <math>0</math> and is the same as that of getting heads exactly twice. Let <math>\frac ij^{}_{}</math>, in lowest terms, be the probability that the coin comes up heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j^{}_{}</math>. |
[[1989 AIME Problems/Problem 5|Solution]] | [[1989 AIME Problems/Problem 5|Solution]] | ||
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== Problem 6 == | == Problem 6 == | ||
Two skaters, Allie and Billie, are at points <math>A^{}_{}</math> and <math>B^{}_{}</math>, respectively, on a flat, frozen lake. The distance between <math>A^{}_{}</math> and <math>B^{}_{}</math> is <math>100^{}_{}</math> meters. Allie leaves <math>A^{}_{}</math> and skates at a speed of <math>8^{}_{}</math> meters per second on a straight line that makes a <math>60^\circ</math> angle with <math>AB^{}_{}</math>. At the same time Allie leaves <math>A^{}_{}</math>, Billie leaves <math>B^{}_{}</math> at a speed of <math>7^{}_{}</math> meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie? | Two skaters, Allie and Billie, are at points <math>A^{}_{}</math> and <math>B^{}_{}</math>, respectively, on a flat, frozen lake. The distance between <math>A^{}_{}</math> and <math>B^{}_{}</math> is <math>100^{}_{}</math> meters. Allie leaves <math>A^{}_{}</math> and skates at a speed of <math>8^{}_{}</math> meters per second on a straight line that makes a <math>60^\circ</math> angle with <math>AB^{}_{}</math>. At the same time Allie leaves <math>A^{}_{}</math>, Billie leaves <math>B^{}_{}</math> at a speed of <math>7^{}_{}</math> meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie? | ||
− | + | <center><asy> | |
− | [[Image:AIME_1989_Problem_6.png]] | + | pointpen=black; pathpen=black+linewidth(0.7); |
+ | pair A=(0,0),B=(10,0),C=6*expi(pi/3); | ||
+ | D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); | ||
+ | </asy></center><!-- Minsoen's image: [[Image:AIME_1989_Problem_6.png]] --> | ||
[[1989 AIME Problems/Problem 6|Solution]] | [[1989 AIME Problems/Problem 6|Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
− | One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer | + | One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>. |
[[1989 AIME Problems/Problem 9|Solution]] | [[1989 AIME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | Let <math> | + | Let <math>a</math>, <math>b</math>, <math>c</math> be the three sides of a triangle, and let <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find |
<center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center> | <center><math>\frac{\cot \gamma}{\cot \alpha+\cot \beta}</math></center> | ||
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Let <math>ABCD^{}_{}</math> be a tetrahedron with <math>AB=41^{}_{}</math>, <math>AC=7^{}_{}</math>, <math>AD=18^{}_{}</math>, <math>BC=36^{}_{}</math>, <math>BD=27^{}_{}</math>, and <math>CD=13^{}_{}</math>, as shown in the figure. Let <math>d^{}_{}</math> be the distance between the midpoints of edges <math>AB^{}_{}</math> and <math>CD^{}_{}</math>. Find <math>d^{2}_{}</math>. | Let <math>ABCD^{}_{}</math> be a tetrahedron with <math>AB=41^{}_{}</math>, <math>AC=7^{}_{}</math>, <math>AD=18^{}_{}</math>, <math>BC=36^{}_{}</math>, <math>BD=27^{}_{}</math>, and <math>CD=13^{}_{}</math>, as shown in the figure. Let <math>d^{}_{}</math> be the distance between the midpoints of edges <math>AB^{}_{}</math> and <math>CD^{}_{}</math>. Find <math>d^{2}_{}</math>. | ||
− | [[Image:AIME_1989_Problem_12.png]] | + | [[Image:AIME_1989_Problem_12.png|center]] |
[[1989 AIME Problems/Problem 12|Solution]] | [[1989 AIME Problems/Problem 12|Solution]] | ||
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== Problem 14 == | == Problem 14 == | ||
− | Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation | + | Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>0,1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation |
<center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> | <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> | ||
is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write | is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write |
Latest revision as of 20:47, 14 December 2023
1989 AIME (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
Compute .
Problem 2
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?
Problem 3
Suppose is a positive integer and
is a single digit in base 10. Find
if
![$\frac{n}{810}=0.d25d25d25\ldots$](http://latex.artofproblemsolving.com/0/d/2/0d2ffb8013a922ca8e69ae1b3c9575aa9b68e7bf.png)
Problem 4
If are consecutive positive integers such that
is a perfect square and
is a perfect cube, what is the smallest possible value of
?
Problem 5
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to and is the same as that of getting heads exactly twice. Let
, in lowest terms, be the probability that the coin comes up heads in exactly
out of
flips. Find
.
Problem 6
Two skaters, Allie and Billie, are at points and
, respectively, on a flat, frozen lake. The distance between
and
is
meters. Allie leaves
and skates at a speed of
meters per second on a straight line that makes a
angle with
. At the same time Allie leaves
, Billie leaves
at a speed of
meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
![[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=6*expi(pi/3); D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); [/asy]](http://latex.artofproblemsolving.com/a/3/0/a30f95a4bc471ad519d35b683f2f319118167660.png)
Problem 7
If the integer is added to each of the numbers
,
, and
, one obtains the squares of three consecutive terms of an arithmetic series. Find
.
Problem 8
Assume that are real numbers such that
Find the value of
.
Problem 9
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that Find the value of
.
Problem 10
Let ,
,
be the three sides of a triangle, and let
,
,
, be the angles opposite them. If
, find
![$\frac{\cot \gamma}{\cot \alpha+\cot \beta}$](http://latex.artofproblemsolving.com/8/5/8/858c49b569dc76c503dbdbfa8d0b78fd2c04fe1d.png)
Problem 11
A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of
? (For real
,
is the greatest integer less than or equal to
.)
Problem 12
Let be a tetrahedron with
,
,
,
,
, and
, as shown in the figure. Let
be the distance between the midpoints of edges
and
. Find
.
Problem 13
Let be a subset of
such that no two members of
differ by
or
. What is the largest number of elements
can have?
Problem 14
Given a positive integer , it can be shown that every complex number of the form
, where
and
are integers, can be uniquely expressed in the base
using the integers
as digits. That is, the equation
![$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$](http://latex.artofproblemsolving.com/f/6/4/f6491a5fef34ba26ef6657aae299c94700b40a0e.png)
is true for a unique choice of non-negative integer and digits
chosen from the set
, with
. We write
![$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$](http://latex.artofproblemsolving.com/5/6/a/56aa9ca7c641491fbd454fff558b7a8fb8918ced.png)
to denote the base expansion of
. There are only finitely many integers
that have four-digit expansions
![$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$](http://latex.artofproblemsolving.com/6/0/f/60f64fe15a0ae3ffb0c11f59e745a2bd2b665a3d.png)
Find the sum of all such .
Problem 15
Point is inside
. Line segments
,
, and
are drawn with
on
,
on
, and
on
(see the figure below). Given that
,
,
,
, and
, find the area of
.
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1988 AIME Problems |
Followed by 1990 AIME 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.