Difference between revisions of "1995 AIME Problems"
m |
(→Problem 7) |
||
(15 intermediate revisions by 7 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AIME Problems|year=1995}} | ||
+ | |||
== Problem 1 == | == Problem 1 == | ||
− | Square <math> | + | Square <math>S_{1}</math> is <math>1\times 1.</math> For <math>i\ge 1,</math> the lengths of the sides of square <math>S_{i+1}</math> are half the lengths of the sides of square <math>S_{i},</math> two adjacent sides of square <math>S_{i}</math> are perpendicular bisectors of two adjacent sides of square <math>S_{i+1},</math> and the other two sides of square <math>S_{i+1},</math> are the perpendicular bisectors of two adjacent sides of square <math>S_{i+2}.</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> |
[[Image:AIME 1995 Problem 1.png]] | [[Image:AIME 1995 Problem 1.png]] | ||
Line 13: | Line 15: | ||
== Problem 3 == | == Problem 3 == | ||
− | Starting at <math> | + | Starting at <math>(0,0),</math> an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let <math>p</math> be the probability that the object reaches <math>(2,2)</math> in six or fewer steps. Given that <math>p</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n.</math> |
[[1995 AIME Problems/Problem 3|Solution]] | [[1995 AIME Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
− | Circles of radius <math> | + | Circles of radius <math>3</math> and <math>6</math> are externally tangent to each other and are internally tangent to a circle of radius <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. |
[[1995 AIME Problems/Problem 4|Solution]] | [[1995 AIME Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
− | For certain real values of <math> | + | For certain real values of <math>a, b, c,</math> and <math>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d=0</math> has four non-real roots. The product of two of these roots is <math>13+i</math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math> |
[[1995 AIME Problems/Problem 5|Solution]] | [[1995 AIME Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | Let <math> | + | Let <math>n=2^{31}3^{19}.</math> How many positive integer divisors of <math>n^2</math> are less than <math>n_{}</math> but do not divide <math>n_{}</math>? |
[[1995 AIME Problems/Problem 6|Solution]] | [[1995 AIME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | Given that <math> | + | Given that <math>(1+\sin t)(1+\cos t)=5/4</math> and |
<center><math>(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},</math></center> | <center><math>(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},</math></center> | ||
− | where <math> | + | where <math>k, m,</math> and <math>n_{}</math> are positive integers with <math>m_{}</math> and <math>n_{}</math> relatively prime, find <math>k+m+n.</math> |
[[1995 AIME Problems/Problem 7|Solution]] | [[1995 AIME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | For how many ordered pairs of positive integers <math>(x,y),</math> with <math>y<x\le 100,</math> are both <math>\frac xy</math> and <math>\frac{x+1}{y+1}</math> integers? | ||
[[1995 AIME Problems/Problem 8|Solution]] | [[1995 AIME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | Triangle <math>ABC</math> is isosceles, with <math>AB=AC</math> and altitude <math>AM=11.</math> Suppose that there is a point <math>D</math> on <math>\overline{AM}</math> with <math>AD=10</math> and <math>\angle BDC=3\angle BAC.</math> Then the perimeter of <math>\triangle ABC</math> may be written in the form <math>a+\sqrt{b},</math> where <math>a</math> and <math>b</math> are integers. Find <math>a+b.</math> | ||
+ | |||
+ | [[Image:AIME_1995_Problem_9.png]] | ||
[[1995 AIME Problems/Problem 9|Solution]] | [[1995 AIME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
+ | What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer? | ||
[[1995 AIME Problems/Problem 10|Solution]] | [[1995 AIME Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | A right rectangular prism <math>P_{}</math> (i.e., a rectangular parallelepiped) has sides of integral length <math>a, b, c,</math> with <math>a\le b\le c.</math> A plane parallel to one of the faces of <math>P_{}</math> cuts <math>P_{}</math> into two prisms, one of which is similar to <math>P_{},</math> and both of which have nonzero volume. Given that <math>b=1995,</math> for how many ordered triples <math>(a, b, c)</math> does such a plane exist? | ||
[[1995 AIME Problems/Problem 11|Solution]] | [[1995 AIME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | Pyramid <math>OABCD</math> has square base <math>ABCD,</math> congruent edges <math>\overline{OA}, \overline{OB}, \overline{OC},</math> and <math>\overline{OD},</math> and <math>\angle AOB=45^\circ.</math> Let <math>\theta</math> be the measure of the dihedral angle formed by faces <math>OAB</math> and <math>OBC.</math> Given that <math>\cos \theta=m+\sqrt{n},</math> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> | ||
[[1995 AIME Problems/Problem 12|Solution]] | [[1995 AIME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | Let <math>f(n)</math> be the integer closest to <math>\sqrt[4]{n}.</math> Find <math>\sum_{k=1}^{1995}\frac 1{f(k)}.</math> | ||
[[1995 AIME Problems/Problem 13|Solution]] | [[1995 AIME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positive integers and <math>d_{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> | ||
[[1995 AIME Problems/Problem 14|Solution]] | [[1995 AIME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | Let <math>p_{}</math> be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that <math>p_{}</math> can be written in the form <math>m/n</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers, find <math>m+n</math>. | ||
[[1995 AIME Problems/Problem 15|Solution]] | [[1995 AIME Problems/Problem 15|Solution]] | ||
== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year=1995|before=[[1994 AIME Problems]]|after=[[1996 AIME 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 00:25, 27 March 2024
1995 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
Square is
For
the lengths of the sides of square
are half the lengths of the sides of square
two adjacent sides of square
are perpendicular bisectors of two adjacent sides of square
and the other two sides of square
are the perpendicular bisectors of two adjacent sides of square
The total area enclosed by at least one of
can be written in the form
where
and
are relatively prime positive integers. Find
Problem 2
Find the last three digits of the product of the positive roots of
Problem 3
Starting at an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let
be the probability that the object reaches
in six or fewer steps. Given that
can be written in the form
where
and
are relatively prime positive integers, find
Problem 4
Circles of radius and
are externally tangent to each other and are internally tangent to a circle of radius
. The circle of radius
has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
Problem 5
For certain real values of and
the equation
has four non-real roots. The product of two of these roots is
and the sum of the other two roots is
where
Find
Problem 6
Let How many positive integer divisors of
are less than
but do not divide
?
Problem 7
Given that and
![$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$](http://latex.artofproblemsolving.com/c/f/2/cf29d52380b6425977553a19ef91fe310d5153cc.png)
where and
are positive integers with
and
relatively prime, find
Problem 8
For how many ordered pairs of positive integers with
are both
and
integers?
Problem 9
Triangle is isosceles, with
and altitude
Suppose that there is a point
on
with
and
Then the perimeter of
may be written in the form
where
and
are integers. Find
Problem 10
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
Problem 11
A right rectangular prism (i.e., a rectangular parallelepiped) has sides of integral length
with
A plane parallel to one of the faces of
cuts
into two prisms, one of which is similar to
and both of which have nonzero volume. Given that
for how many ordered triples
does such a plane exist?
Problem 12
Pyramid has square base
congruent edges
and
and
Let
be the measure of the dihedral angle formed by faces
and
Given that
where
and
are integers, find
Problem 13
Let be the integer closest to
Find
Problem 14
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form where
and
are positive integers and
is not divisible by the square of any prime number. Find
Problem 15
Let be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that
can be written in the form
where
and
are relatively prime positive integers, find
.
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1994 AIME Problems |
Followed by 1996 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.