Difference between revisions of "2000 AIME II Problems"
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== Problem 6 == | == Problem 6 == | ||
− | 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 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 <math>x^2/100</math>. | + | 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]] | ||
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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>. | 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 | + | [[2000 AIME II Problems/Problem 11|Solution]] |
== Problem 12 == | == Problem 12 == | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2000|n=II|before=[[2000 AIME I Problems]]|after=[[2001 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 21:11, 19 February 2019
2000 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
The number
![$\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}$](http://latex.artofproblemsolving.com/1/c/a/1ca16383129aebb061bb0e345327dff7fbc71548.png)
can be written as where
and
are relatively prime positive integers. Find
.
Problem 2
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola ?
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 be the probability that two randomly selected cards also form a pair, where
and
are relatively prime positive integers. Find
Problem 4
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
Problem 5
Given eight distinguishable rings, let 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
.
Problem 6
One base of a trapezoid is 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
. Let
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
.
Problem 7
Given that
![$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$](http://latex.artofproblemsolving.com/1/6/d/16d26af464ce4b8d84de3933589ddc93c5d5b57e.png)
find the greatest integer that is less than .
Problem 8
In trapezoid , leg
is perpendicular to bases
and
, and diagonals
and
are perpendicular. Given that
and
, find
.
Problem 9
Given that is a complex number such that
, find the least integer that is greater than
.
Problem 10
A circle is inscribed in quadrilateral , tangent to
at
and to
at
. Given that
,
,
, and
, find the square of the radius of the circle.
Problem 11
The coordinates of the vertices of isosceles trapezoid are all integers, with
and
. The trapezoid has no horizontal or vertical sides, and
and
are the only parallel sides. The sum of the absolute values of all possible slopes for
is
, where
and
are relatively prime positive integers. Find
.
Problem 12
The points ,
and
lie on the surface of a sphere with center
and radius
. It is given that
,
,
, and that the distance from
to triangle
is
, where
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 13
The equation has exactly two real roots, one of which is
, where
,
and
are integers,
and
are relatively prime, and
. Find
.
Problem 14
Every positive integer has a unique factorial base expansion
, meaning that
, where each
is an integer,
, and
. Given that
is the factorial base expansion of
, find the value of
.
Problem 15
Find the least positive integer such that
![$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$](http://latex.artofproblemsolving.com/4/c/9/4c9c990a7b70753fe475b450cb0915460af6cf64.png)
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
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 |
- 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.