Difference between revisions of "1999 AIME Problems"

Latest revision as of 12:39, 16 August 2020

1999 AIME (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

Solution

Problem 2

Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 3

Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.

Solution

Problem 4

The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\overline{AB}$ is $\frac{43}{99}$ and the area of octagon $ABCDEFGH$ is $\frac{m}{n}$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 5

For any positive integer $x_{}$ , let $S(x)$ be the sum of the digits of $x_{}$ , and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999?

Solution

Problem 6

A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k_{}.$

Solution

Problem 7

There is a set of 1000 switches, each of which has four positions, called $A, B, C$ , and $D$ . When the position of any switch changes, it is only from $A$ to $B$ , from $B$ to $C$ , from $C$ to $D$ , or from $D$ to $A$ . Initially each switch is in position $A$ . The switches are labeled with the 1000 different integers $(2^{x})(3^{y})(5^{z})$ , where $x, y$ , and $z$ take on the values $0, 1, \ldots, 9$ . At step $i$ of a 1000-step process, the $i$ -th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$ -th switch. After step 1000 has been completed, how many switches will be in position $A$ ?

Solution

Problem 8

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n.$

Solution

Problem 9

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ 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 $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

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 $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 11

Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$

Solution

Problem 12

The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P_{},$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.

Solution

Problem 13

Forty teams play a tournament in which every team plays every other( $39$ different opponents) team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $\log_2 n.$

Solution

Problem 14

Point $P_{}$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 15

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ 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?

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1999}}
 == Problem 1 ==
 Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
@@ Line 5: / Line 7: @@
 == Problem 2 ==
-Consider the parallelogram with vertices <math>\displaystyle (10,45),</math> <math>\displaystyle (10,114),</math> <math>\displaystyle (28,153),</math> and <math>\displaystyle (28,84).</math>  A line through the origin cuts this figure into two congruent polygons.  The slope of the line is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</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]]
 == Problem 3 ==
-Find the sum of all positive integers <math>\displaystyle n</math> for which <math>\displaystyle n^2-19n+99</math> is a perfect square.
+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]]
 == Problem 4 ==
-The two squares shown share the same center <math>\displaystyle O_{}</math> and have sides of length 1. The length of <math>\displaystyle \overline{AB}</math> is <math>\displaystyle 43/99</math> and the area of octagon <math>\displaystyle ABCDEFGH</math> is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</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]]
@@ Line 22: / Line 24: @@
 == Problem 5 ==
-For any positive integer <math>\displaystyle x_{}</math>, let <math>\displaystyle S(x)</math> be the sum of the digits of <math>\displaystyle x_{}</math>, and let <math>\displaystyle T(x)</math> be <math>\displaystyle |S(x+2)-S(x)|.</math>  For example, <math>\displaystyle T(199)=|S(201)-S(199)|=|3-19|=16.</math>  How many values <math>\displaystyle T(x)</math> do not exceed 1999?
+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]]
 == Problem 6 ==
-A transformation of the first quadrant of the coordinate plane maps each point <math>\displaystyle (x,y)</math> to the point <math>\displaystyle (\sqrt{x},\sqrt{y}).</math>  The vertices of quadrilateral <math>\displaystyle ABCD</math> are <math>\displaystyle A=(900,300), B=(1800,600), C=(600,1800),</math> and <math>\displaystyle D=(300,900).</math>  Let <math>\displaystyle k_{}</math> be the area of the region enclosed by the image of quadrilateral <math>\displaystyle ABCD.</math>  Find the greatest integer that does not exceed <math>\displaystyle k_{}.</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]]
 == 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]]
 == Problem 8 ==
-Let <math>\displaystyle \mathcal{T}</math> be the set of ordered triples <math>\displaystyle (x,y,z)</math> of nonnegative real numbers that lie in the plane <math>\displaystyle x+y+z=1.</math>  Let us say that <math>\displaystyle (x,y,z)</math> supports <math>\displaystyle (a,b,c)</math> when exactly two of the following are true: <math>\displaystyle x\ge a, y\ge b, z\ge c.</math>  Let <math>\displaystyle \mathcal{S}</math> consist of those triples in <math>\displaystyle \mathcal{T}</math> that support <math>\displaystyle \left(\frac 12,\frac 13,\frac 16\right).</math>  The area of <math>\displaystyle \mathcal{S}</math> divided by the area of <math>\displaystyle \mathcal{T}</math> is <math>\displaystyle m/n,</math>  where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers, find <math>\displaystyle m+n.</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]]
 == Problem 9 ==
-A function <math>\displaystyle f</math> is defined on the complex numbers by <math>\displaystyle f(z)=(a+bi)z,</math> where <math>\displaystyle a_{}</math> and <math>\displaystyle 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>\displaystyle |a+bi|=8</math> and that <math>\displaystyle b^2=m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</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]]
 == 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>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</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]]
 == Problem 11 ==
-Given that <math>\displaystyle \sum_{k=1}^{35}\sin 5k=\tan \frac mn,</math> where angles are measured in degrees, and <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers that satisfy <math>\displaystyle \frac mn<90,</math> find <math>\displaystyle m+n.</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]]
 == Problem 12 ==
-The inscribed circle of triangle <math>\displaystyle ABC</math> is tangent to <math>\displaystyle \overline{AB}</math> at <math>\displaystyle P_{},</math>  and its radius is 21.  Given that <math>\displaystyle AP=23</math> and <math>\displaystyle PB=27,</math> find the perimeter of the triangle.
+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]]
 == 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>\displaystyle 50 \%</math> chance of winning any game it plays.  The probability that no two teams win the same number of games is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle \log_2 n.</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]]
 == Problem 14 ==
-Point <math>\displaystyle P_{}</math> is located inside traingle <math>\displaystyle ABC</math> so that angles <math>\displaystyle PAB, PBC,</math> and <math>\displaystyle PCA</math> are all congruent.  The sides of the triangle have lengths <math>\displaystyle AB=13, BC=14,</math> and <math>\displaystyle CA=15,</math> and the tangent of angle <math>\displaystyle PAB</math> is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</math>
+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]]
 == 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]]
 == See also ==
+{{AIME box|year = 1999|before=[[1998 AIME Problems]]|after=[[2000 AIME I Problems]]}}
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

1999 AIME (Problems • Answer Key • Resources)
Preceded by 1998 AIME Problems	Followed by 2000 AIME I Problems
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Difference between revisions of "1999 AIME Problems"

Latest revision as of 12:39, 16 August 2020

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also