Difference between revisions of "Mock Geometry AIME 2011 Problems"
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Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles? | Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles? | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
In triangle <math>ABC,</math> <math>AB=6, BC=9, \angle ABC=120^{\circ}</math> Let <math>P</math> and <math>Q</math> be points on <math>AC</math> such that <math>BPQ</math> is equilateral. The perimeter of <math>BPQ</math> can be expressed in the form <math>\frac{m} {\sqrt{n}},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | In triangle <math>ABC,</math> <math>AB=6, BC=9, \angle ABC=120^{\circ}</math> Let <math>P</math> and <math>Q</math> be points on <math>AC</math> such that <math>BPQ</math> is equilateral. The perimeter of <math>BPQ</math> can be expressed in the form <math>\frac{m} {\sqrt{n}},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
In triangle <math>ABC,</math> <math>AB=36,BC=40,CA=44.</math> The bisector of angle <math>A</math> meet <math>BC</math> at <math>D</math> and the circumcircle at <math>E</math> different from <math>A</math>. Calculate the value of <math>DE^2</math> | In triangle <math>ABC,</math> <math>AB=36,BC=40,CA=44.</math> The bisector of angle <math>A</math> meet <math>BC</math> at <math>D</math> and the circumcircle at <math>E</math> different from <math>A</math>. Calculate the value of <math>DE^2</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
Three points <math>A,B,C</math> are chosen at random on a circle. The probability that there exists a point <math>P</math> inside an equilateral triangle <math>A_1B_1C_1</math> such that <math>PA_1=BC,PB_1=AC,PC_1=AB</math> can be expressed in the form <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | Three points <math>A,B,C</math> are chosen at random on a circle. The probability that there exists a point <math>P</math> inside an equilateral triangle <math>A_1B_1C_1</math> such that <math>PA_1=BC,PB_1=AC,PC_1=AB</math> can be expressed in the form <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
In trapezoid <math>ABCD,</math> <math>AB||CD,</math> and <math>AB\perp BC.</math> There is a point <math>P</math> on side <math>AD</math> such that the circumcircle of triangle <math>BPC</math> is tangent to <math>AD.</math> If <math>AB=3, AD=78, CD=75,</math> <math>CP-BP</math> can be expressed in the form <math>\frac{a\sqrt{b}} {c},</math> where <math>a,b,c</math> are positive integers and <math>a,c</math> are relatively prime. Find <math>a+b+c.</math> | In trapezoid <math>ABCD,</math> <math>AB||CD,</math> and <math>AB\perp BC.</math> There is a point <math>P</math> on side <math>AD</math> such that the circumcircle of triangle <math>BPC</math> is tangent to <math>AD.</math> If <math>AB=3, AD=78, CD=75,</math> <math>CP-BP</math> can be expressed in the form <math>\frac{a\sqrt{b}} {c},</math> where <math>a,b,c</math> are positive integers and <math>a,c</math> are relatively prime. Find <math>a+b+c.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
Two circles <math>\omega_1,\omega_2</math> have center <math>O_1,O_2</math> and radius <math>25,39</math> respectively. The smallest distance between a point on <math>\omega_1</math> with a point on <math>\omega_2</math> is <math>1</math>. Tangents from <math>O_2</math> to <math>\omega_1</math> meet <math>\omega_1</math> at <math>S_1,T_1,</math> and tangents from <math>O_1</math> to <math>\omega_2</math> meet <math>\omega_2</math> at <math>S_2,T_2,</math> such that <math>S_1,S_2</math> are on the same side of line <math>O_1O_2.</math> <math>O_1S_1</math> meets <math>O_2S_2</math> at <math>P</math> and <math>O_1T_1</math> meets <math>O_2T_2</math> at Q. The length of <math>PQ</math> can be expressed in the form <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | Two circles <math>\omega_1,\omega_2</math> have center <math>O_1,O_2</math> and radius <math>25,39</math> respectively. The smallest distance between a point on <math>\omega_1</math> with a point on <math>\omega_2</math> is <math>1</math>. Tangents from <math>O_2</math> to <math>\omega_1</math> meet <math>\omega_1</math> at <math>S_1,T_1,</math> and tangents from <math>O_1</math> to <math>\omega_2</math> meet <math>\omega_2</math> at <math>S_2,T_2,</math> such that <math>S_1,S_2</math> are on the same side of line <math>O_1O_2.</math> <math>O_1S_1</math> meets <math>O_2S_2</math> at <math>P</math> and <math>O_1T_1</math> meets <math>O_2T_2</math> at Q. The length of <math>PQ</math> can be expressed in the form <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
<math>P-ABCD</math> is a right pyramid with square base <math>ABCD</math> edge length 6, and <math>PA=PB=PC=PD=6\sqrt{2}.</math> The probability that a randomly selected point inside the pyramid is at least <math>\frac{\sqrt{6}} {3}</math> units away from each face can be expressed in the form <math>\frac{m}{n}</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | <math>P-ABCD</math> is a right pyramid with square base <math>ABCD</math> edge length 6, and <math>PA=PB=PC=PD=6\sqrt{2}.</math> The probability that a randomly selected point inside the pyramid is at least <math>\frac{\sqrt{6}} {3}</math> units away from each face can be expressed in the form <math>\frac{m}{n}</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
Circle <math>\omega_1</math> is defined by the equation <math>(x-7)^2+(y-1)^2=k,</math> where <math>k</math> is a positive real number. Circle <math>\omega_2</math> passes through the center of <math>\omega_1</math> and its center lies on the line <math>7x+y=28.</math> Suppose that one of the tangent lines from the origin to circles <math>\omega_1</math> and <math>\omega_2</math> meets <math>\omega_1</math> and <math>\omega_2</math> at <math>A_1,A_2</math> respectively, that <math>OA_1=OA_2,</math> where <math>O</math> is the origin, and that the radius of <math>\omega_2</math> is <math>\frac{2011} {211}</math>. What is <math>k</math>? | Circle <math>\omega_1</math> is defined by the equation <math>(x-7)^2+(y-1)^2=k,</math> where <math>k</math> is a positive real number. Circle <math>\omega_2</math> passes through the center of <math>\omega_1</math> and its center lies on the line <math>7x+y=28.</math> Suppose that one of the tangent lines from the origin to circles <math>\omega_1</math> and <math>\omega_2</math> meets <math>\omega_1</math> and <math>\omega_2</math> at <math>A_1,A_2</math> respectively, that <math>OA_1=OA_2,</math> where <math>O</math> is the origin, and that the radius of <math>\omega_2</math> is <math>\frac{2011} {211}</math>. What is <math>k</math>? | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
<math>C</math> is on a semicircle with diameter <math>AB</math> and center <math>O.</math> Circle radius <math>r_1</math> is tangent to <math>OA,OC,</math> and arc <math>AC,</math> and circle radius <math>r_2</math> is tangent to <math>OB,OC,</math> and arc <math>BC</math>. It is known that <math>\tan AOC=\frac{24}{7}</math>. The ratio <math>\frac{r_2} {r_1}</math> can be expressed <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | <math>C</math> is on a semicircle with diameter <math>AB</math> and center <math>O.</math> Circle radius <math>r_1</math> is tangent to <math>OA,OC,</math> and arc <math>AC,</math> and circle radius <math>r_2</math> is tangent to <math>OB,OC,</math> and arc <math>BC</math>. It is known that <math>\tan AOC=\frac{24}{7}</math>. The ratio <math>\frac{r_2} {r_1}</math> can be expressed <math>\frac{m} {n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as <math>\frac{a+\sqrt{b}} {c}</math>, where <math>a,b,c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a+b+c</math>. | A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as <math>\frac{a+\sqrt{b}} {c}</math>, where <math>a,b,c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a+b+c</math>. | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
In acute triangle <math>ABC,</math> <math>\ell</math> is the bisector of <math>\angle BAC</math>. <math>M</math> is the midpoint of <math>BC</math>. a line through <math>M</math> parallel to <math>\ell</math> meets <math>AC,AB</math> at <math>E,F,</math> respectively. Given that <math>AE=1,EF=\sqrt{3}, AB=21,</math> the sum of all possible values of <math>BC</math> can be expressed as <math>\sqrt{a}+\sqrt{b},</math> where <math>a,b</math> are positive integers. What is <math>a+b</math>? | In acute triangle <math>ABC,</math> <math>\ell</math> is the bisector of <math>\angle BAC</math>. <math>M</math> is the midpoint of <math>BC</math>. a line through <math>M</math> parallel to <math>\ell</math> meets <math>AC,AB</math> at <math>E,F,</math> respectively. Given that <math>AE=1,EF=\sqrt{3}, AB=21,</math> the sum of all possible values of <math>BC</math> can be expressed as <math>\sqrt{a}+\sqrt{b},</math> where <math>a,b</math> are positive integers. What is <math>a+b</math>? | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
The point <math>(10,26)</math> is a focus of a non-degenerate ellipse tangent to the positive <math>x</math> and <math>y</math> axes. the locus of the center of the ellipse lies along graph of, <math>ax-by+c=0,</math> where <math>a,b,c</math> are positive integers with no common factor other than <math>1</math>. Find <math>a+b+c.</math> | The point <math>(10,26)</math> is a focus of a non-degenerate ellipse tangent to the positive <math>x</math> and <math>y</math> axes. the locus of the center of the ellipse lies along graph of, <math>ax-by+c=0,</math> where <math>a,b,c</math> are positive integers with no common factor other than <math>1</math>. Find <math>a+b+c.</math> | ||
+ | [[Mock Geometry AIME 2011 Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
Two circles <math>\omega_1,\omega_2</math> radius <math>28,112</math> respectively intersect at <math>P,Q</math>. <math>A</math> is on <math>\omega_1</math> and <math>B</math> on <math>\omega_2</math> such that <math>A,P,B</math> are collinear. Tangents to <math>\omega_1,\omega_2</math> at <math>A,B</math> respectively meet at <math>T</math>. Suppose <math>\angle AQT=\angle BQT=60^{\circ}.</math> The length of <math>TQ</math> can be expressed in the form <math>a\sqrt{b}</math> where <math>a,b</math> are positive integers and <math>b</math> is not divisible by the square of any prime. Find <math>a+b.</math> | Two circles <math>\omega_1,\omega_2</math> radius <math>28,112</math> respectively intersect at <math>P,Q</math>. <math>A</math> is on <math>\omega_1</math> and <math>B</math> on <math>\omega_2</math> such that <math>A,P,B</math> are collinear. Tangents to <math>\omega_1,\omega_2</math> at <math>A,B</math> respectively meet at <math>T</math>. Suppose <math>\angle AQT=\angle BQT=60^{\circ}.</math> The length of <math>TQ</math> can be expressed in the form <math>a\sqrt{b}</math> where <math>a,b</math> are positive integers and <math>b</math> is not divisible by the square of any prime. Find <math>a+b.</math> | ||
+ | |||
+ | [[Mock Geometry AIME 2011 Problems/Problem 15|Solution]] |
Revision as of 15:57, 30 December 2011
Contents
Problem 1
Let be a unit square, and let
be its image after a
degree rotation about point
The area of the region consisting of all points inside at least one of
and
can be expressed in the form
where
are positive integers, and
shares no perfect square common factor with
. Find
Problem 2
Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?
Problem 3
In triangle
Points
and
are located on
such that
The circumcircle of
cuts
at
respectively. If
then the ratio
can be expressed in the form
where
are relatively prime positive integers. Find
Problem 4
In triangle
Let
and
be points on
such that
is equilateral. The perimeter of
can be expressed in the form
where
are relatively prime positive integers. Find
Problem 5
In triangle
The bisector of angle
meet
at
and the circumcircle at
different from
. Calculate the value of
Problem 6
Three points are chosen at random on a circle. The probability that there exists a point
inside an equilateral triangle
such that
can be expressed in the form
where
are relatively prime positive integers. Find
Problem 7
In trapezoid
and
There is a point
on side
such that the circumcircle of triangle
is tangent to
If
can be expressed in the form
where
are positive integers and
are relatively prime. Find
Problem 8
Two circles have center
and radius
respectively. The smallest distance between a point on
with a point on
is
. Tangents from
to
meet
at
and tangents from
to
meet
at
such that
are on the same side of line
meets
at
and
meets
at Q. The length of
can be expressed in the form
where
are relatively prime positive integers. Find
Problem 9
is a right pyramid with square base
edge length 6, and
The probability that a randomly selected point inside the pyramid is at least
units away from each face can be expressed in the form
where
are relatively prime positive integers. Find
Problem 10
Circle is defined by the equation
where
is a positive real number. Circle
passes through the center of
and its center lies on the line
Suppose that one of the tangent lines from the origin to circles
and
meets
and
at
respectively, that
where
is the origin, and that the radius of
is
. What is
?
Problem 11
is on a semicircle with diameter
and center
Circle radius
is tangent to
and arc
and circle radius
is tangent to
and arc
. It is known that
. The ratio
can be expressed
where
are relatively prime positive integers. Find
Problem 12
A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as , where
are positive integers,
is not divisible by the square of any prime, and
and
are relatively prime. Find
.
Problem 13
In acute triangle
is the bisector of
.
is the midpoint of
. a line through
parallel to
meets
at
respectively. Given that
the sum of all possible values of
can be expressed as
where
are positive integers. What is
?
Problem 14
The point is a focus of a non-degenerate ellipse tangent to the positive
and
axes. the locus of the center of the ellipse lies along graph of,
where
are positive integers with no common factor other than
. Find
Problem 15
Two circles radius
respectively intersect at
.
is on
and
on
such that
are collinear. Tangents to
at
respectively meet at
. Suppose
The length of
can be expressed in the form
where
are positive integers and
is not divisible by the square of any prime. Find