Difference between revisions of "1991 AIME Problems"
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== Problem 1 == | == Problem 1 == | ||
Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such that | Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such that | ||
− | < | + | <cmath>\begin{align*} |
− | + | xy+x+y&=71, \\ | |
+ | x^2y+xy^2&=880. | ||
+ | \end{align*}</cmath> | ||
[[1991 AIME Problems/Problem 1|Solution]] | [[1991 AIME Problems/Problem 1|Solution]] | ||
Line 27: | Line 29: | ||
== Problem 5 == | == Problem 5 == | ||
− | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will | + | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will <math>20_{}^{}!</math> be the resulting product? |
[[1991 AIME Problems/Problem 5|Solution]] | [[1991 AIME Problems/Problem 5|Solution]] | ||
Line 65: | Line 67: | ||
Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | ||
− | + | <asy> | |
+ | unitsize(100); | ||
+ | draw(Circle((0,0),1)); | ||
+ | dot((0,0)); | ||
+ | draw((0,0)--(1,0)); | ||
+ | label("$1$", (0.5,0), S); | ||
+ | |||
+ | for (int i=0; i<12; ++i) | ||
+ | { | ||
+ | dot((cos(i*pi/6), sin(i*pi/6))); | ||
+ | } | ||
+ | |||
+ | for (int a=1; a<24; a+=2) | ||
+ | { | ||
+ | dot(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12))); | ||
+ | draw(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12))--((1/cos(pi/12))*cos((a+2)*pi/12), (1/cos(pi/12))*sin((a+2)*pi/12))); | ||
+ | draw(Circle(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12)), tan(pi/12))); | ||
+ | } | ||
+ | </asy> | ||
[[1991 AIME Problems/Problem 11|Solution]] | [[1991 AIME Problems/Problem 11|Solution]] | ||
Line 92: | Line 112: | ||
== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year=1991|before=[[1990 AIME Problems]]|after=[[1992 AIME Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] |
Latest revision as of 22:05, 7 June 2021
1991 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
Find if
and
are positive integers such that
Problem 2
Rectangle has sides
of length 4 and
of length 3. Divide
into 168 congruent segments with points
, and divide
into 168 congruent segments with points
. For
, draw the segments
. Repeat this construction on the sides
and
, and then draw the diagonal
. Find the sum of the lengths of the 335 parallel segments drawn.
Problem 3
Expanding by the binomial theorem and doing no further manipulation gives
![${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$](http://latex.artofproblemsolving.com/6/3/c/63c14bf916c02c40e694eaa2f957cb61f108cf07.png)
![$= A_0 + A_1 + A_2 + \cdots + A_{1000},$](http://latex.artofproblemsolving.com/7/6/1/7614a86a13eb038383bc2726776c8bf9cc6fbd24.png)
where for
. For which
is
the largest?
Problem 4
How many real numbers satisfy the equation
?
Problem 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be the resulting product?
Problem 6
Suppose is a real number for which
![$\left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546.$](http://latex.artofproblemsolving.com/d/4/9/d49bf9d61cafb77fc2043857f815732ddef15f57.png)
Find . (For real
,
is the greatest integer less than or equal to
.)
Problem 7
Find , where
is the sum of the absolute values of all roots of the following equation:
![$x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}}$](http://latex.artofproblemsolving.com/b/0/2/b02f23aa1a41aa56a1e00e986776128fffe43321.png)
Problem 8
For how many real numbers does the quadratic equation
have only integer roots for
?
Problem 9
Suppose that and that
where
is in lowest terms. Find
Problem 10
Two three-letter strings, and
, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an
when it should have been a
, or as a
when it should be an
. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let
be the three-letter string received when
is transmitted and let
be the three-letter string received when
is transmitted. Let
be the probability that
comes before
in alphabetical order. When
is written as a fraction in lowest terms, what is its numerator?
Problem 11
Twelve congruent disks are placed on a circle of radius 1 in such a way that the twelve disks cover
, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form
, where
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 12
Rhombus is inscribed in rectangle
so that vertices
,
,
, and
are interior points on sides
,
,
, and
, respectively. It is given that
,
,
, and
. Let
, in lowest terms, denote the perimeter of
. Find
.
Problem 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Problem 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from
.
Problem 15
For positive integer , define
to be the minimum value of the sum
![$\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},$](http://latex.artofproblemsolving.com/1/e/9/1e96fb544379cfe35283e8fa9f7d5367624e4676.png)
where are positive real numbers whose sum is 17. There is a unique positive integer
for which
is also an integer. Find this
.
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1990 AIME Problems |
Followed by 1992 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.