Difference between revisions of "1999 AIME Problems"
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== Problem 1 == | == Problem 1 == | ||
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. | Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. | ||
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== Problem 2 == | == Problem 2 == | ||
− | Consider the parallelogram with vertices <math> | + | Consider the parallelogram with vertices <math>(10,45),</math> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</math> A line through the origin cuts this figure into two congruent polygons. The slope of the line is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> |
[[1999 AIME Problems/Problem 2|Solution]] | [[1999 AIME Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
− | Find the sum of all positive integers <math> | + | Find the sum of all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square. |
[[1999 AIME Problems/Problem 3|Solution]] | [[1999 AIME Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
− | The two squares shown share the same center <math> | + | The two squares shown share the same center <math>O_{}</math> and have sides of length 1. The length of <math>\overline{AB}</math> is <math>\frac{43}{99}</math> and the area of octagon <math>ABCDEFGH</math> is <math>\frac{m}{n}</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> |
[[Image:AIME_1999_Problem_4.png]] | [[Image:AIME_1999_Problem_4.png]] | ||
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== Problem 5 == | == Problem 5 == | ||
− | For any positive integer <math> | + | For any positive integer <math>x_{}</math>, let <math>S(x)</math> be the sum of the digits of <math>x_{}</math>, and let <math>T(x)</math> be <math>|S(x+2)-S(x)|.</math> For example, <math>T(199)=|S(201)-S(199)|=|3-19|=16.</math> How many values of <math>T(x)</math> do not exceed 1999? |
[[1999 AIME Problems/Problem 5|Solution]] | [[1999 AIME Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | A transformation of the first quadrant of the coordinate plane maps each point <math> | + | A transformation of the first quadrant of the coordinate plane maps each point <math>(x,y)</math> to the point <math>(\sqrt{x},\sqrt{y}).</math> The vertices of quadrilateral <math>ABCD</math> are <math>A=(900,300), B=(1800,600), C=(600,1800),</math> and <math>D=(300,900).</math> Let <math>k_{}</math> be the area of the region enclosed by the image of quadrilateral <math>ABCD.</math> Find the greatest integer that does not exceed <math>k_{}.</math> |
[[1999 AIME Problems/Problem 6|Solution]] | [[1999 AIME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</math>, and <math>D</math>. When the position of any switch changes, it is only from <math>A</math> to <math>B</math>, from <math>B</math> to <math>C</math>, from <math>C</math> to <math>D</math>, or from <math>D</math> to <math>A</math>. Initially each switch is in position <math>A</math>. The switches are labeled with the 1000 different integers <math>(2^{x})(3^{y})(5^{z})</math>, where <math>x, y</math>, and <math>z</math> take on the values <math>0, 1, \ldots, 9</math>. At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are all the other switches whose labels divide the label on the <math>i</math>-th switch. After step 1000 has been completed, how many switches will be in position <math>A</math>? | + | There is a set of 1000 switches, each of which has four positions, called <math>A, B, C</math>, and <math>D</math>. When the position of any switch changes, it is only from <math>A</math> to <math>B</math>, from <math>B</math> to <math>C</math>, from <math>C</math> to <math>D</math>, or from <math>D</math> to <math>A</math>. Initially each switch is in position <math>A</math>. The switches are labeled with the 1000 different integers <math>(2^{x})(3^{y})(5^{z})</math>, where <math>x, y</math>, and <math>z</math> take on the values <math>0, 1, \ldots, 9</math>. At step <math>i</math> of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are all the other switches whose labels divide the label on the <math>i</math>-th switch. After step 1000 has been completed, how many switches will be in position <math>A</math>? |
[[1999 AIME Problems/Problem 7|Solution]] | [[1999 AIME Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
− | Let <math> | + | Let <math>\mathcal{T}</math> be the set of ordered triples <math>(x,y,z)</math> of nonnegative real numbers that lie in the plane <math>x+y+z=1.</math> Let us say that <math>(x,y,z)</math> supports <math>(a,b,c)</math> when exactly two of the following are true: <math>x\ge a, y\ge b, z\ge c.</math> Let <math>\mathcal{S}</math> consist of those triples in <math>\mathcal{T}</math> that support <math>\left(\frac 12,\frac 13,\frac 16\right).</math> The area of <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers, find <math>m+n.</math> |
[[1999 AIME Problems/Problem 8|Solution]] | [[1999 AIME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
− | A function <math> | + | A function <math>f</math> is defined on the complex numbers by <math>f(z)=(a+bi)z,</math> where <math>a_{}</math> and <math>b_{}</math> are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> |
[[1999 AIME Problems/Problem 9|Solution]] | [[1999 AIME Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is <math> | + | Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> |
[[1999 AIME Problems/Problem 10|Solution]] | [[1999 AIME Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
− | Given that <math> | + | Given that <math>\sum_{k=1}^{35}\sin 5k=\tan \frac mn,</math> where angles are measured in degrees, and <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers that satisfy <math>\frac mn<90,</math> find <math>m+n.</math> |
[[1999 AIME Problems/Problem 11|Solution]] | [[1999 AIME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
− | The inscribed circle of triangle <math> | + | The inscribed circle of triangle <math>ABC</math> is tangent to <math>\overline{AB}</math> at <math>P_{},</math> and its radius is 21. Given that <math>AP=23</math> and <math>PB=27,</math> find the perimeter of the triangle. |
[[1999 AIME Problems/Problem 12|Solution]] | [[1999 AIME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
− | Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a <math> | + | Forty teams play a tournament in which every team plays every other(<math>39</math> different opponents) team exactly once. No ties occur, and each team has a <math>50 \%</math> chance of winning any game it plays. The probability that no two teams win the same number of games is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>\log_2 n.</math> |
[[1999 AIME Problems/Problem 13|Solution]] | [[1999 AIME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | Point <math>P_{}</math> is located inside triangle <math>ABC</math> so that angles <math>PAB, PBC,</math> and <math>PCA</math> are all congruent. The sides of the triangle have lengths <math>AB=13, BC=14,</math> and <math>CA=15,</math> and the tangent of angle <math>PAB</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
[[1999 AIME Problems/Problem 14|Solution]] | [[1999 AIME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | Consider the paper triangle whose vertices are <math>(0,0), (34,0),</math> and <math>(16,24).</math> The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | ||
[[1999 AIME Problems/Problem 15|Solution]] | [[1999 AIME Problems/Problem 15|Solution]] | ||
== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 1999|before=[[1998 AIME Problems]]|after=[[2000 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 12:39, 16 August 2020
1999 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
Problem 2
Consider the parallelogram with vertices and A line through the origin cuts this figure into two congruent polygons. The slope of the line is where and are relatively prime positive integers. Find
Problem 3
Find the sum of all positive integers for which is a perfect square.
Problem 4
The two squares shown share the same center and have sides of length 1. The length of is and the area of octagon is where and are relatively prime positive integers. Find
Problem 5
For any positive integer , let be the sum of the digits of , and let be For example, How many values of do not exceed 1999?
Problem 6
A transformation of the first quadrant of the coordinate plane maps each point to the point The vertices of quadrilateral are and Let be the area of the region enclosed by the image of quadrilateral Find the greatest integer that does not exceed
Problem 7
There is a set of 1000 switches, each of which has four positions, called , and . When the position of any switch changes, it is only from to , from to , from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers , where , and take on the values . At step of a 1000-step process, the -th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?
Problem 8
Let be the set of ordered triples of nonnegative real numbers that lie in the plane Let us say that supports when exactly two of the following are true: Let consist of those triples in that support The area of divided by the area of is where and are relatively prime positive integers, find
Problem 9
A function is defined on the complex numbers by where and are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that and that where and are relatively prime positive integers. Find
Problem 10
Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is where and are relatively prime positive integers. Find
Problem 11
Given that where angles are measured in degrees, and and are relatively prime positive integers that satisfy find
Problem 12
The inscribed circle of triangle is tangent to at and its radius is 21. Given that and find the perimeter of the triangle.
Problem 13
Forty teams play a tournament in which every team plays every other( different opponents) team exactly once. No ties occur, and each team has a chance of winning any game it plays. The probability that no two teams win the same number of games is where and are relatively prime positive integers. Find
Problem 14
Point is located inside triangle so that angles and are all congruent. The sides of the triangle have lengths and and the tangent of angle is where and are relatively prime positive integers. Find
Problem 15
Consider the paper triangle whose vertices are and The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1998 AIME Problems |
Followed by 2000 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.