Difference between revisions of "2006 AIME II Problems"
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Equilateral <math> \triangle ABC </math> is inscribed in a circle of radius 2. Extend <math> \overline{AB} </math> through <math> B </math> to point <math> D </math> so that <math> AD=13, </math> and extend <math> \overline{AC} </math> through <math> C </math> to point <math> E </math> so that <math> AE = 11. </math> Through <math> D, </math> draw a line <math> l_1 </math> parallel to <math> \overline{AE}, </math> and through <math> E, </math> draw a line <math> l_2 </math> parallel to <math> \overline{AD}. </math> Let <math> F </math> be the intersection of <math> l_1 </math> and <math> l_2. </math> Let <math> G </math> be the point on the circle that is collinear with <math> A </math> and <math> F </math> and distinct from <math> A. </math> Given that the area of <math> \triangle CBG </math> can be expressed in the form <math> \frac{p\sqrt{q}}{r}, </math> where <math> p, q, </math> and <math> r </math> are positive integers, <math> p </math> and <math> r</math> are relatively prime, and <math> q </math> is not divisible by the square of any prime, find <math> p+q+r. </math> | Equilateral <math> \triangle ABC </math> is inscribed in a circle of radius 2. Extend <math> \overline{AB} </math> through <math> B </math> to point <math> D </math> so that <math> AD=13, </math> and extend <math> \overline{AC} </math> through <math> C </math> to point <math> E </math> so that <math> AE = 11. </math> Through <math> D, </math> draw a line <math> l_1 </math> parallel to <math> \overline{AE}, </math> and through <math> E, </math> draw a line <math> l_2 </math> parallel to <math> \overline{AD}. </math> Let <math> F </math> be the intersection of <math> l_1 </math> and <math> l_2. </math> Let <math> G </math> be the point on the circle that is collinear with <math> A </math> and <math> F </math> and distinct from <math> A. </math> Given that the area of <math> \triangle CBG </math> can be expressed in the form <math> \frac{p\sqrt{q}}{r}, </math> where <math> p, q, </math> and <math> r </math> are positive integers, <math> p </math> and <math> r</math> are relatively prime, and <math> q </math> is not divisible by the square of any prime, find <math> p+q+r. </math> | ||
− | < | + | <asy> |
+ | size(250); pointpen = black; pathpen = black + linewidth(0.65); pen s = fontsize(8); | ||
+ | pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D; | ||
+ | path O = CP((0,-2),A); pair G = OP(A--F,O); | ||
+ | D(MP("A",A,N,s)--MP("B",B,W,s)--MP("C",C,E,s)--cycle);D(O); | ||
+ | D(B--MP("D",D,W,s)--MP("F",F,s)--MP("E",E1,E,s)--C); | ||
+ | D(A--F);D(B--MP("G",G,SW,s)--C); | ||
+ | MP("11",(A+E1)/2,NE);MP("13",(A+D)/2,NW);MP("l_1",(D+F)/2,SW);MP("l_2",(E1+F)/2,SE); | ||
+ | </asy> | ||
[[2006 AIME II Problems/Problem 12|Solution]] | [[2006 AIME II Problems/Problem 12|Solution]] |
Revision as of 03:19, 2 July 2014
2006 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
In convex hexagon , all six sides are congruent,
and
are right angles, and
and
are congruent. The area of the hexagonal region is
Find
.
Problem 2
The lengths of the sides of a triangle with positive area are ,
, and
, where
is a positive integer. Find the number of possible values for
.
Problem 3
Let be the product of the first 100 positive odd integers. Find the largest integer
such that
is divisible by
.
Problem 4
Let be a permutation of
for which
![$a_1>a_2>a_3>a_4>a_5>a_6 \mathrm{\ and \ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$](http://latex.artofproblemsolving.com/7/1/4/714f8aaced4000d779337cb9bddced3aff9ed3e7.png)
An example of such a permutation is Find the number of such permutations.
Problem 5
When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face is greater than 1/6, the probability of obtaining the face opposite is less than 1/6, the probability of obtaining any one of the other four faces is 1/6, and the sum of the numbers on opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is 47/288. Given that the probability of obtaining face
is
where
and
are relatively prime positive integers, find
Problem 6
Square has sides of length 1. Points
and
are on
and
respectively, so that
is equilateral. A square with vertex
has sides that are parallel to those of
and a vertex on
The length of a side of this smaller square is
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 7
Find the number of ordered pairs of positive integers such that
and neither
nor
has a zero digit.
Problem 8
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
Problem 9
Circles and
have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line
is a common internal tangent to
and
and has a positive slope, and line
is a common internal tangent to
and
and has a negative slope. Given that lines
and
intersect at
and that
where
and
are positive integers and
is not divisible by the square of any prime, find
Problem 10
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team
beats team
The probability that team
finishes with more points than team
is
where
and
are relatively prime positive integers. Find
Problem 11
A sequence is defined as follows and, for all positive integers
Given that
and
find the remainder when
is divided by 1000.
Problem 12
Equilateral is inscribed in a circle of radius 2. Extend
through
to point
so that
and extend
through
to point
so that
Through
draw a line
parallel to
and through
draw a line
parallel to
Let
be the intersection of
and
Let
be the point on the circle that is collinear with
and
and distinct from
Given that the area of
can be expressed in the form
where
and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime, find
Problem 13
How many integers less than 1000 can be written as the sum of
consecutive positive odd integers from exactly 5 values of
?
Problem 14
Let be the sum of the reciprocals of the non-zero digits of the integers from
to
inclusive. Find the smallest positive integer n for which
is an integer.
Problem 15
Given that and
are real numbers that satisfy:
![$x = \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}$](http://latex.artofproblemsolving.com/4/8/2/482ec834e100a5c445119023117da55b0a08dae6.png)
![$y = \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}$](http://latex.artofproblemsolving.com/0/7/e/07e3a5bfe68da11f9902c4b34d8381e913e31859.png)
![$z = \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}$](http://latex.artofproblemsolving.com/0/0/b/00b6096d5f7c9f1584a2efea64e687039b0c28ff.png)
and that where
and
are positive integers and
is not divisible by the square of any prime, find
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- 2006 AIME I Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.