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Difference between revisions of "2001 AIME I Problems"

 
Line 1: Line 1:
 
== Problem 1 ==
 
== Problem 1 ==
 +
Find the sum of all positive two-digit integers that are divisible by each of their digits.
  
 
[[2001 AIME I Problems/Problem 1|Solution]]
 
[[2001 AIME I Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
A finite set <math>\mathcal{S}</math> of distinct real numbers has the following properties: the mean of <math>\mathcal{S}\cup\{1\}</math> is <math>13</math> less than the mean of <math>\mathcal{S}</math>, and the mean of <math>\mathcal{S}\cup\{2001\}</math> is <math>27</math> more than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>.
  
 
[[2001 AIME I Problems/Problem 2|Solution]]
 
[[2001 AIME I Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
Find the sum of the roots, real and non-real, of the equation <math>x^{2001}+\left(\frac 12-x\right)^{2001}=0</math>, given that there are no multiple roots.
  
 
[[2001 AIME I Problems/Problem 3|Solution]]
 
[[2001 AIME I Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
In triangle <math>ABC</math>, angles <math>A</math> and <math>B</math> measure <math>60</math> degrees and <math>45</math> degrees, respectively. The bisector of angle <math>A</math> intersects <math>\overline{BC}</math> at <math>T</math>, and <math>AT=24</math>. The area of triangle <math>ABC</math> can be written in the form <math>a+b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.
  
 
[[2001 AIME I Problems/Problem 4|Solution]]
 
[[2001 AIME I Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
An equilateral triangle is inscribed in the ellipse whose equation is <math>x^2+4y^2=4</math>. One vertex of the triangle is <math>(0,1)</math>, one altitude is contained in the y-axis, and the length of each side is <math>\sqrt{\frac mn}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2001 AIME I Problems/Problem 5|Solution]]
 
[[2001 AIME I Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed 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>.
  
 
[[2001 AIME I Problems/Problem 6|Solution]]
 
[[2001 AIME I Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
Triangle <math>ABC</math> has <math>AB=21</math>, <math>AC=22</math> and <math>BC=20</math>. Points <math>D</math> and <math>E</math> are located on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>\overline{DE}</math> is parallel to <math>\overline{BC}</math> and contains the center of the inscribed circle of triangle <math>ABC</math>. Then <math>DE=m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2001 AIME I Problems/Problem 7|Solution]]
 
[[2001 AIME I Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Call a positive integer <math>N</math> a <math>\textit{7-10 double}</math> if the digits of the base-7 representation of <math>N</math> form a base-10 number that is twice <math>N</math>. For example, <math>51</math> is a 7-10 double because its base-7 representation is <math>102</math>. What is the largest 7-10 double?
  
 
[[2001 AIME I Problems/Problem 8|Solution]]
 
[[2001 AIME I Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
In triangle <math>ABC</math>, <math>AB=13</math>, <math>BC=15</math> and <math>CA=17</math>. Point <math>D</math> is on <math>\overline{AB}</math>, <math>E</math> is on <math>\overline{BC}</math>, and <math>F</math> is on <math>\overline{CA}</math>. Let <math>AD=p\cdot AB</math>, <math>BE=q\cdot BC</math>, and <math>CF=r\cdot CA</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are positive and satisfy <math>p+q+r=2/3</math> and <math>p^2+q^2+r^2=2/5</math>. The ratio of the area of triangle <math>DEF</math> to the area of triangle <math>ABC</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>.
  
 
[[2001 AIME I Problems/Problem 9|Solution]]
 
[[2001 AIME I Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
Let <math>S</math> be the set of points whose coordinates <math>x,</math> <math>y,</math> and <math>z</math> are integers that satisfy <math>0\le x\le2,</math> <math>0\le y\le3,</math> and <math>0\le z\le4.</math>  Two distinct points are randomly chosen from <math>S.</math>  The probability that the midpoint of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
[[2001 AIME I Problems/Problem 10|Solution]]
 
[[2001 AIME I Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
In a rectangular array of points, with 5 rows and <math>N</math> columns, the points are numbered consecutively from left to right beginning with the top row.  Thus the top row is numbered 1 through <math>N,</math> the second row is numbered <math>N + 1</math> through <math>2N,</math> and so forth.  Five points, <math>P_1, P_2, P_3, P_4,</math> and <math>P_5,</math> are selected so that each <math>P_i</math> is in row <math>i.</math>  Let <math>x_i</math> be the number associated with <math>P_i.</math>  Now renumber the array consecutively from top to bottom, beginning with the first column.  Let <math>y_i</math> be the number associated with <math>P_i</math> after the renumbering.  It is found that <math>x_1 = y_2,</math> <math>x_2 = y_1,</math> <math>x_3 = y_4,</math> <math>x_4 = y_5,</math> and <math>x_5 = y_3.</math>  Find the smallest possible value of <math>N.</math>
  
 
[[2001 AIME I Problems/Problem 11|Solution]]
 
[[2001 AIME I Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
A sphere is inscribed in the tetrahedron whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math>  The radius of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
[[2001 AIME I Problems/Problem 12|Solution]]
 
[[2001 AIME I Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
In a certain circle, the chord of a <math>d</math>-degree arc is 22 centimeters long, and the chord of a <math>2d</math>-degree arc is 20 centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math>  The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers.  Find <math>m + n.</math>
  
 
[[2001 AIME I Problems/Problem 13|Solution]]
 
[[2001 AIME I Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street.  The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day.  How many different patterns of mail delivery are possible?
  
 
[[2001 AIME I Problems/Problem 14|Solution]]
 
[[2001 AIME I Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number.  The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
[[2001 AIME I Problems/Problem 15|Solution]]
 
[[2001 AIME I Problems/Problem 15|Solution]]

Revision as of 23:15, 19 November 2007

Problem 1

Find the sum of all positive two-digit integers that are divisible by each of their digits.

Solution

Problem 2

A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$. Find the mean of $\mathcal{S}$.

Solution

Problem 3

Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots.

Solution

Problem 4

In triangle $ABC$, angles $A$ and $B$ measure $60$ degrees and $45$ degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24$. The area of triangle $ABC$ can be written in the form $a+b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c$.

Solution

Problem 5

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the length of each side is $\sqrt{\frac mn}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 6

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 8

Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N$. For example, $51$ is a 7-10 double because its base-7 representation is $102$. What is the largest 7-10 double?

Solution

Problem 9

In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 10

Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 11

In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$

Solution

Problem 12

A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 13

In a certain circle, the chord of a $d$-degree arc is 22 centimeters long, and the chord of a $2d$-degree arc is 20 centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of the chord of a $3d$-degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integers. Find $m + n.$

Solution

Problem 14

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

Solution

Problem 15

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

See also

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