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Difference between revisions of "Mock Geometry AIME 2011 Problems"

(Created page with '==Problem 1== Let <math>ABCD</math> be a unit square, and let <math>AB_1C_1D_1</math> be its image after a <math>30</math> degree rotation about point <math>A.</math> The area o…')
 
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==Problem 1==  
 
==Problem 1==  
−
Let <math>ABCD</math> be a unit square, and let <math>AB_1C_1D_1</math> be its image after a <math>30</math> degree rotation about point <math>A.</math> The area of the region consisting of all points inside at least one of <math>ABCD</math> and <math>AB_1C_1D_1</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>b</math> shares no perfect square common factor with <math>c</math>. Find <math>a+b+c.</math>  
+
Let <math>ABCD</math> be a unit square, and let <math>AB_1C_1D_1</math> be its image after a <math>30</math> degree rotation about point <math>A.</math> The area of the region consisting of all points inside at least one of <math>ABCD</math> and <math>AB_1C_1D_1</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>b</math> shares no perfect square common factor with <math>c</math>. Find <math>a+b+c.</math>
 +
 
 +
[[Mock Geometry AIME 2011 Problems/Problem 1|Solution]]
  
 
==Problem 2==  
 
==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?  
 
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>PABCD</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]]

Latest revision as of 18:10, 14 June 2022

Problem 1

Let $ABCD$ be a unit square, and let $AB_1C_1D_1$ be its image after a $30$ degree rotation about point $A.$ The area of the region consisting of all points inside at least one of $ABCD$ and $AB_1C_1D_1$ can be expressed in the form $\frac{a-\sqrt{b}} {c},$ where $a,b,c$ are positive integers, and $b$ shares no perfect square common factor with $c$. Find $a+b+c.$

Solution

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?

Solution

Problem 3

In triangle $ABC,$ $BC=9.$ Points $P$ and $Q$ are located on $BC$ such that $BP=PQ=2,$ $QC=5.$ The circumcircle of $APQ$ cuts $AB,AC$ at $D,E$ respectively. If $BD=CE,$ then the ratio $\frac{AB}{AC}$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 4

In triangle $ABC,$ $AB=6, BC=9, \angle ABC=120^{\circ}$ Let $P$ and $Q$ be points on $AC$ such that $BPQ$ is equilateral. The perimeter of $BPQ$ can be expressed in the form $\frac{m} {\sqrt{n}},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 5

In triangle $ABC,$ $AB=36,BC=40,CA=44.$ The bisector of angle $A$ meet $BC$ at $D$ and the circumcircle at $E$ different from $A$. Calculate the value of $DE^2$

Solution

Problem 6

Three points $A,B,C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC,PB_1=AC,PC_1=AB$ can be expressed in the form $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 7

In trapezoid $ABCD,$ $AB||CD,$ and $AB\perp BC.$ There is a point $P$ on side $AD$ such that the circumcircle of triangle $BPC$ is tangent to $AD.$ If $AB=3, AD=78, CD=75,$ $CP-BP$ can be expressed in the form $\frac{a\sqrt{b}} {c},$ where $a,b,c$ are positive integers and $a,c$ are relatively prime. Find $a+b+c.$

Solution

Problem 8

Two circles $\omega_1,\omega_2$ have center $O_1,O_2$ and radius $25,39$ respectively. The smallest distance between a point on $\omega_1$ with a point on $\omega_2$ is $1$. Tangents from $O_2$ to $\omega_1$ meet $\omega_1$ at $S_1,T_1,$ and tangents from $O_1$ to $\omega_2$ meet $\omega_2$ at $S_2,T_2,$ such that $S_1,S_2$ are on the same side of line $O_1O_2.$ $O_1S_1$ meets $O_2S_2$ at $P$ and $O_1T_1$ meets $O_2T_2$ at Q. The length of $PQ$ can be expressed in the form $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 9

$PABCD$ is a right pyramid with square base $ABCD$ edge length 6, and $PA=PB=PC=PD=6\sqrt{2}.$ The probability that a randomly selected point inside the pyramid is at least $\frac{\sqrt{6}} {3}$ units away from each face can be expressed in the form $\frac{m}{n}$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 10

Circle $\omega_1$ is defined by the equation $(x-7)^2+(y-1)^2=k,$ where $k$ is a positive real number. Circle $\omega_2$ passes through the center of $\omega_1$ and its center lies on the line $7x+y=28.$ Suppose that one of the tangent lines from the origin to circles $\omega_1$ and $\omega_2$ meets $\omega_1$ and $\omega_2$ at $A_1,A_2$ respectively, that $OA_1=OA_2,$ where $O$ is the origin, and that the radius of $\omega_2$ is $\frac{2011} {211}$. What is $k$?

Solution

Problem 11

$C$ is on a semicircle with diameter $AB$ and center $O.$ Circle radius $r_1$ is tangent to $OA,OC,$ and arc $AC,$ and circle radius $r_2$ is tangent to $OB,OC,$ and arc $BC$. It is known that $\tan AOC=\frac{24}{7}$. The ratio $\frac{r_2} {r_1}$ can be expressed $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

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 $\frac{a+\sqrt{b}} {c}$, where $a,b,c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c$.

Solution

Problem 13

In acute triangle $ABC,$ $\ell$ is the bisector of $\angle BAC$. $M$ is the midpoint of $BC$. a line through $M$ parallel to $\ell$ meets $AC,AB$ at $E,F,$ respectively. Given that $AE=1,EF=\sqrt{3}, AB=21,$ the sum of all possible values of $BC$ can be expressed as $\sqrt{a}+\sqrt{b},$ where $a,b$ are positive integers. What is $a+b$?

Solution

Problem 14

The point $(10,26)$ is a focus of a non-degenerate ellipse tangent to the positive $x$ and $y$ axes. The locus of the center of the ellipse lies along graph of, $ax-by+c=0,$ where $a,b,c$ are positive integers with no common factor other than $1$. Find $a+b+c.$

Solution

Problem 15

Two circles $\omega_1,\omega_2$ radius $28,112$ respectively intersect at $P,Q$. $A$ is on $\omega_1$ and $B$ on $\omega_2$ such that $A,P,B$ are collinear. Tangents to $\omega_1,\omega_2$ at $A,B$ respectively meet at $T$. Suppose $\angle AQT=\angle BQT=60^{\circ}.$ The length of $TQ$ can be expressed in the form $a\sqrt{b}$ where $a,b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

Solution

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