Difference between revisions of "1995 AIME Problems"
m |
|||
(7 intermediate revisions by 5 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> | + | 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> | + | 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]] | [[Image:AIME_1995_Problem_9.png]] | ||
Line 57: | Line 59: | ||
== Problem 11 == | == Problem 11 == | ||
− | A right rectangular prism <math> | + | 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> | + | 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 07:37, 7 September 2018
1995 AIME (Answer Key) | AoPS Contest Collections | ||
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
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.