Difference between revisions of "2000 AIME II Problems"

 
 
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{{AIME Problems|year=2000|n=II}}
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== Problem 1 ==
 
== Problem 1 ==
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The number
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<center><math>\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}</math></center>
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can be written as <math>\frac mn</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n</math>.
  
 
[[2000 AIME II Problems/Problem 1|Solution]]
 
[[2000 AIME II Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
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A point whose coordinates are both integers is called a lattice point.  How many lattice points lie on the hyperbola <math>x^2 - y^2 = 2000^2</math>?
  
 
[[2000 AIME II Problems/Problem 2|Solution]]
 
[[2000 AIME II Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
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A deck of forty cards consists of four 1's, four 2's,..., and four 10's.  A matching pair (two cards with the same number) is removed from the deck.  Given that these cards are not returned to the deck, let <math>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
  
 
[[2000 AIME II Problems/Problem 3|Solution]]
 
[[2000 AIME II Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
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What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
  
 
[[2000 AIME II Problems/Problem 4|Solution]]
 
[[2000 AIME II Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
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Given eight distinguishable rings, let <math>n</math> be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of <math>n</math>.
  
 
[[2000 AIME II Problems/Problem 5|Solution]]
 
[[2000 AIME II Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
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One base of a trapezoid is <math>100</math> units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio <math>2: 3</math>. Let <math>x</math> be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed <math>x^2/100</math>.
  
 
[[2000 AIME II Problems/Problem 6|Solution]]
 
[[2000 AIME II Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
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Given that <center><math>\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}</math></center> find the greatest integer that is less than <math>\frac N{100}</math>.
  
 
[[2000 AIME II Problems/Problem 7|Solution]]
 
[[2000 AIME II Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
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In trapezoid <math>ABCD</math>, leg <math>\overline{BC}</math> is perpendicular to bases <math>\overline{AB}</math> and <math>\overline{CD}</math>, and diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math> are perpendicular. Given that <math>AB=\sqrt{11}</math> and <math>AD=\sqrt{1001}</math>, find <math>BC^2</math>.
  
 
[[2000 AIME II Problems/Problem 8|Solution]]
 
[[2000 AIME II Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Given that <math>z</math> is a complex number such that <math>z+\frac 1z=2\cos 3^\circ</math>, find the least integer that is greater than <math>z^{2000}+\frac 1{z^{2000}}</math>.
  
 
[[2000 AIME II Problems/Problem 9|Solution]]
 
[[2000 AIME II Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
A circle is inscribed in quadrilateral <math>ABCD</math>, tangent to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <math>AP=19</math>, <math>PB=26</math>, <math>CQ=37</math>, and <math>QD=23</math>, find the square of the radius of the circle.
  
 
[[2000 AIME II Problems/Problem 10|Solution]]
 
[[2000 AIME II Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
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The coordinates of the vertices of isosceles trapezoid <math>ABCD</math> are all integers, with <math>A=(20,100)</math> and <math>D=(21,107)</math>. The trapezoid has no horizontal or vertical sides, and <math>\overline{AB}</math> and <math>\overline{CD}</math> are the only parallel sides. The sum of the absolute values of all possible slopes for <math>\overline{AB}</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2000 AIME II Problems/Problem 11|Solution]]
 
[[2000 AIME II Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
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The points <math>A</math>, <math>B</math> and <math>C</math> lie on the surface of a sphere with center <math>O</math> and radius <math>20</math>. It is given that <math>AB=13</math>, <math>BC=14</math>, <math>CA=15</math>, and that the distance from <math>O</math> to triangle <math>ABC</math> is <math>\frac{m\sqrt{n}}k</math>, where <math>m</math>, <math>n</math>, and <math>k</math> are positive integers, <math>m</math> and <math>k</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>.
  
 
[[2000 AIME II Problems/Problem 12|Solution]]
 
[[2000 AIME II Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
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The equation <math>2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n}}r</math>, where <math>m</math>, <math>n</math> and <math>r</math> are integers, <math>m</math> and <math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>.
  
 
[[2000 AIME II Problems/Problem 13|Solution]]
 
[[2000 AIME II Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
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Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math> is an integer, <math>0\le f_i\le i</math>, and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>.
  
 
[[2000 AIME II Problems/Problem 14|Solution]]
 
[[2000 AIME II Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
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Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center>
  
 
[[2000 AIME II Problems/Problem 15|Solution]]
 
[[2000 AIME II Problems/Problem 15|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AIME box|year = 2000|n=II|before=[[2000 AIME I Problems]]|after=[[2001 AIME I Problems]]}}
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* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
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{{MAA Notice}}

Latest revision as of 20:11, 19 February 2019

2000 AIME II (Answer Key)
Printable version | AoPS Contest Collections

Instructions

  1. 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.
  2. 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

The number

$\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}$

can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 2

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$?

Solution

Problem 3

A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 4

What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?

Solution

Problem 5

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.

Solution

Problem 6

One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$.

Solution

Problem 7

Given that

$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$

find the greatest integer that is less than $\frac N{100}$.

Solution

Problem 8

In trapezoid $ABCD$, leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD}$, and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001}$, find $BC^2$.

Solution

Problem 9

Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$.

Solution

Problem 10

A circle is inscribed in quadrilateral $ABCD$, tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle.

Solution

Problem 11

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 12

The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k$.

Solution

Problem 13

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$.

Solution

Problem 14

Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m)$, meaning that $k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m$, where each $f_i$ is an integer, $0\le f_i\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\cdots+1968!-1984!+2000!$, find the value of $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j$.

Solution

Problem 15

Find the least positive integer $n$ such that

$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$

Solution

See also

2000 AIME II (ProblemsAnswer KeyResources)
Preceded by
2000 AIME I Problems
Followed by
2001 AIME I Problems
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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