Difference between revisions of "2019 AIME I Problems/Problem 11"
Firebolt360 (talk | contribs) (→Solution 3) |
MRENTHUSIASM (talk | contribs) (Undo revision 147964 by Sugar rush (talk)--Guess-and-check is a valid way to obtain correct answers, although not recommended. This solution provides an alternate thought. So, it is okay.) (Tag: Undo) |
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− | ==Problem | + | ==Problem== |
− | In <math>\triangle ABC</math>, the sides have | + | In <math>\triangle ABC</math>, the sides have integer lengths and <math>AB=AC</math>. Circle <math>\omega</math> has its center at the incenter of <math>\triangle ABC</math>. An ''excircle'' of <math>\triangle ABC</math> is a circle in the exterior of <math>\triangle ABC</math> that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to <math>\overline{BC}</math> is internally tangent to <math>\omega</math>, and the other two excircles are both externally tangent to <math>\omega</math>. Find the minimum possible value of the perimeter of <math>\triangle ABC</math>. |
==Solution 1== | ==Solution 1== | ||
Line 9: | Line 9: | ||
Using this gives us the following equation: <cmath>\frac{2b*IB}{a}=r+\frac{2rs}{s-a}+\frac{rs}{s-b}</cmath> | Using this gives us the following equation: <cmath>\frac{2b*IB}{a}=r+\frac{2rs}{s-a}+\frac{rs}{s-b}</cmath> | ||
Motivated by the <math>s-a</math> and <math>s-b</math>, we make the following substitution: <math>x=s-a, y=s-b</math> | Motivated by the <math>s-a</math> and <math>s-b</math>, we make the following substitution: <math>x=s-a, y=s-b</math> | ||
− | This changes things quite a bit. Here's what we can get from it: <cmath>a=2y, b=x+y, s=x+2y</cmath> It is known (easily proved with Heron's and a=rs) that <cmath>r=\sqrt{\frac{(s-a)(s-b)(s-b)}{s}}=\sqrt{\frac{xy^2}{x+2y}}</cmath> Using this, we can also find <math>IB</math>: let the midpoint of <math>BC</math> be <math>N</math>. Using Pythagorean's Theorem on <math>\triangle INB</math>, <cmath>IB^2=r^2+(\frac{a}{2})^2=\frac{xy^2}{x+2y}+y^2=\frac{2xy^2+2y^3}{x+2y}=\frac{2y^2(x+y)}{x+2y} </cmath> We now look at the RHS of the main equation: <cmath>r+\frac{2rs}{s-a}+\frac{rs}{s-b}=r(1+\frac{2(x+2y)}{x}+\frac{x+2y}{y})=r(\frac{x^2+5xy+4y^2}{xy})=\frac{r(x+4y)(x+y)}{xy}=\frac{2(x+y)IB}{2y}</cmath> | + | This changes things quite a bit. Here's what we can get from it: <cmath>a=2y, b=x+y, s=x+2y</cmath> It is known (easily proved with Heron's and <math>a=rs</math>) that <cmath>r=\sqrt{\frac{(s-a)(s-b)(s-b)}{s}}=\sqrt{\frac{xy^2}{x+2y}}</cmath> Using this, we can also find <math>IB</math>: let the midpoint of <math>BC</math> be <math>N</math>. Using Pythagorean's Theorem on <math>\triangle INB</math>, <cmath>IB^2=r^2+(\frac{a}{2})^2=\frac{xy^2}{x+2y}+y^2=\frac{2xy^2+2y^3}{x+2y}=\frac{2y^2(x+y)}{x+2y} </cmath> We now look at the RHS of the main equation: <cmath>r+\frac{2rs}{s-a}+\frac{rs}{s-b}=r(1+\frac{2(x+2y)}{x}+\frac{x+2y}{y})=r(\frac{x^2+5xy+4y^2}{xy})=\frac{r(x+4y)(x+y)}{xy}=\frac{2(x+y)IB}{2y}</cmath> |
Cancelling some terms, we have <cmath>\frac{r(x+4y)}{x}=IB</cmath> | Cancelling some terms, we have <cmath>\frac{r(x+4y)}{x}=IB</cmath> | ||
Squaring, <cmath>\frac{2y^2(x+y)}{x+2y}=\frac{(x+4y)^2*xy^2}{x^2(x+2y)}\to \frac{(x+4y)^2}{x}=2(x+y)</cmath> Expanding and moving terms around gives <cmath>(x-8y)(x+2y)=0\to x=8y</cmath> Reverse substituting, <cmath>s-a=8s-8b\to b=\frac{9}{2}a</cmath> Clearly the smallest solution is <math>a=2</math> and <math>b=9</math>, so our answer is <math>2+9+9=\boxed{020}</math> | Squaring, <cmath>\frac{2y^2(x+y)}{x+2y}=\frac{(x+4y)^2*xy^2}{x^2(x+2y)}\to \frac{(x+4y)^2}{x}=2(x+y)</cmath> Expanding and moving terms around gives <cmath>(x-8y)(x+2y)=0\to x=8y</cmath> Reverse substituting, <cmath>s-a=8s-8b\to b=\frac{9}{2}a</cmath> Clearly the smallest solution is <math>a=2</math> and <math>b=9</math>, so our answer is <math>2+9+9=\boxed{020}</math> | ||
-franchester | -franchester | ||
− | ==Solution 2== | + | ==Solution 2 (Lots of Pythagorean Theorem)== |
− | Video | + | <asy> |
+ | |||
+ | unitsize(1cm); | ||
+ | |||
+ | |||
+ | var x = 9; | ||
+ | |||
+ | pair A = (0,sqrt(x^2-1)); | ||
+ | pair B = (-1,0); | ||
+ | pair C = (1,0); | ||
+ | |||
+ | dot(Label("$A$",A,NE),A); | ||
+ | dot(Label("$B$",B,SW),B); | ||
+ | dot(Label("$C$",C,SE),C); | ||
+ | |||
+ | draw(A--B--C--cycle); | ||
+ | |||
+ | |||
+ | var r = sqrt((x-1)/(x+1)); | ||
+ | |||
+ | pair I = (0,r); | ||
+ | dot(Label("$I$",I,SE),I); | ||
+ | draw(circle(I,r)); | ||
+ | draw(Label("$r$"),I--I+r*SSW,dashed); | ||
+ | |||
+ | |||
+ | pair M = intersectionpoint(A--B,circle(I,r)); | ||
+ | pair N = (0,0); | ||
+ | pair O = intersectionpoint(A--C,circle(I,r)); | ||
+ | |||
+ | dot(Label("$M$",M,W),M); | ||
+ | dot(Label("$N$",N,S),N); | ||
+ | dot(Label("$O$",O,E),O); | ||
+ | |||
+ | var rN = sqrt((x+1)/(x-1)); | ||
+ | |||
+ | pair EN = (0,-rN); | ||
+ | dot(Label("$E_N$",EN,SE),EN); | ||
+ | draw(circle(EN,rN)); | ||
+ | draw(Label("$r_N$"),EN--EN+rN*SSW,dashed); | ||
+ | |||
+ | |||
+ | pair AB = (-1-2/(x-1),-2rN); | ||
+ | pair AC = (1+2/(x-1),-2rN); | ||
+ | |||
+ | draw(B--AB,EndArrow); | ||
+ | draw(C--AC,EndArrow); | ||
+ | |||
+ | pair H = intersectionpoint(B--AB,circle(EN,rN)); | ||
+ | dot(Label("$H$",H,W),H); | ||
+ | |||
+ | |||
+ | var rM = sqrt(x^2-1); | ||
+ | |||
+ | pair EM = (-x,rM); | ||
+ | dot(Label("$E_M$",EM,SW),EM); | ||
+ | draw(Label("$r_M$"),EM--EM+rM*SSE,dashed); | ||
+ | |||
+ | |||
+ | pair CB = (-x-1,0); | ||
+ | pair CA = (-2/x,sqrt(x^2-1)+2(sqrt(x^2-1)/x)); | ||
+ | |||
+ | draw(B--CB,EndArrow); | ||
+ | draw(A--CA,EndArrow); | ||
+ | |||
+ | |||
+ | pair J = intersectionpoint(A--B,circle(EM,rM)); | ||
+ | pair K = intersectionpoint(B--CB,circle(EM,rM)); | ||
+ | |||
+ | dot(Label("$J$",J,W),J); | ||
+ | dot(Label("$K$",K,S),K); | ||
+ | |||
+ | |||
+ | draw(arc(EM,rM,-100,15),Arrows); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | First, assume <math>BC=2</math> and <math>AB=AC=x</math>. The triangle can be scaled later if necessary. Let <math>I</math> be the incenter and let <math>r</math> be the inradius. Let the points at which the incircle intersects <math>AB</math>, <math>BC</math>, and <math>CA</math> be denoted <math>M</math>, <math>N</math>, and <math>O</math>, respectively. | ||
+ | |||
+ | |||
+ | Next, we calculate <math>r</math> in terms of <math>x</math>. Note the right triangle formed by <math>A</math>, <math>I</math>, and <math>M</math>. The length <math>IM</math> is equal to <math>r</math>. Using the Pythagorean Theorem, the length <math>AN</math> is <math>\sqrt{x^2-1}</math>, so the length <math>AI</math> is <math>\sqrt{x^2-1}-r</math>. Note that <math>BN</math> is half of <math>BC=2</math>, and by symmetry caused by the incircle, <math>BN=BM</math> and <math>BM=1</math>, so <math>MA=x-1</math>. Applying the Pythagorean Theorem to <math>AIM</math>, we get | ||
+ | <cmath>r^2+(x-1)^2=\left(\sqrt{x^2-1}-r\right)^2.</cmath> | ||
+ | Expanding yields | ||
+ | <cmath>r^2+x^2-2x+1=x^2-1-2r\sqrt{x^2-1}+r^2,</cmath> | ||
+ | which can be simplified to | ||
+ | <cmath>2r\sqrt{x^2-1}=2x-2.</cmath> | ||
+ | Dividing by <math>2</math> and then squaring results in | ||
+ | <cmath>r^2(x^2-1)=(x-1)^2,</cmath> | ||
+ | and isolating <math>r^2</math> gets us | ||
+ | <cmath>r^2=\frac{(x-1)^2}{x^2-1}=\frac{(x-1)^2}{(x+1)(x-1)}=\frac{x-1}{x+1},</cmath> | ||
+ | so <math>r=\sqrt{\frac{x-1}{x+1}}</math>. | ||
+ | |||
+ | |||
+ | We then calculate the radius of the excircle tangent to <math>BC</math>. We denote the center of the excircle <math>E_N</math> and the radius <math>r_N</math>. | ||
+ | |||
+ | Consider the quadrilateral formed by <math>M</math>, <math>I</math>, <math>E_N</math>, and the point at which the excircle intersects the extension of <math>AB</math>, which we denote <math>H</math>. By symmetry caused by the excircle, <math>BN=BH</math>, so <math>BH=1</math>. | ||
+ | |||
+ | Note that triangles <math>MBI</math> and <math>NBI</math> are congruent, and <math>HBE</math> and <math>NBE</math> are also congruent. Denoting the measure of angles <math>MBI</math> and <math>NBI</math> measure <math>\alpha</math> and the measure of angles <math>HBE</math> and <math>NBE</math> measure <math>\beta</math>, straight angle <math>MBH=2\alpha+2\beta</math>, so <math>\alpha + \beta=90^\circ</math>. This means that angle <math>IBE</math> is a right angle, so it forms a right triangle. | ||
+ | |||
+ | Setting the base of the right triangle to <math>IE</math>, the height is <math>BN=1</math> and the base consists of <math>IN=r</math> and <math>EN=r_N</math>. Triangles <math>INB</math> and <math>BNE</math> are similar to <math>IBE</math>, so <math>\frac{IN}{BN}=\frac{BN}{EN}</math>, or <math>\frac{r}{1}=\frac{1}{r_N}</math>. This makes <math>r_N</math> the reciprocal of <math>r</math>, so <math>r_N=\sqrt{\frac{x+1}{x-1}}</math>. | ||
+ | |||
+ | |||
+ | Circle <math>\omega</math>'s radius can be expressed by the distance from the incenter <math>I</math> to the bottom of the excircle with center <math>E_N</math>. This length is equal to <math>r+2r_N</math>, or <math>\sqrt{\frac{x-1}{x+1}}+2\sqrt{\frac{x+1}{x-1}}</math>. Denote this value <math>r_\omega</math>. | ||
+ | |||
+ | |||
+ | Finally, we calculate the distance from the incenter <math>I</math> to the closest point on the excircle tangent to <math>AB</math>, which forms another radius of circle <math>\omega</math> and is equal to <math>r_\omega</math>. We denote the center of the excircle <math>E_M</math> and the radius <math>r_M</math>. We also denote the points where the excircle intersects <math>AB</math> and the extension of <math>BC</math> using <math>J</math> and <math>K</math>, respectively. In order to calculate the distance, we must find the distance between <math>I</math> and <math>E_M</math> and subtract off the radius <math>r_M</math>. | ||
+ | |||
+ | We first must calculate the radius of the excircle. Because the excircle is tangent to both <math>AB</math> and the extension of <math>AC</math>, its center must lie on the angle bisector formed by the two lines, which is parallel to <math>BC</math>. This means that the distance from <math>E_M</math> to <math>K</math> is equal to the length of <math>AN</math>, so the radius is also <math>\sqrt{x^2-1}</math>. | ||
+ | |||
+ | Next, we find the length of <math>IE_M</math>. We can do this by forming the right triangle <math>IAE_M</math>. The length of leg <math>AI</math> is equal to <math>AN</math> minus <math>r</math>, or <math>\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}</math>. In order to calculate the length of leg <math>AE_M</math>, note that right triangles <math>AJE_M</math> and <math>BNA</math> are congruent, as <math>JE_M</math> and <math>NA</math> share a length of <math>\sqrt{x^2-1}</math>, and angles <math>E_MAJ</math> and <math>NAB</math> add up to the right angle <math>NAE_M</math>. This means that <math>AE_M=BA=x</math>. | ||
+ | |||
+ | Using Pythagorean Theorem, we get | ||
+ | <cmath>IE_M=\sqrt{\left(\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}\right)^2+x^2}.</cmath> | ||
+ | Bringing back | ||
+ | <cmath>r_\omega=IE_M-r_M</cmath> | ||
+ | and substituting in some values, the equation becomes | ||
+ | <cmath>r_\omega=\sqrt{\left(\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}\right)^2+x^2}-\sqrt{x^2-1}.</cmath> | ||
+ | Rearranging and squaring both sides gets | ||
+ | <cmath>\left(r_\omega+\sqrt{x^2-1}\right)^2=\left(\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}\right)^2+x^2.</cmath> | ||
+ | Distributing both sides yields | ||
+ | <cmath>r_\omega^2+2r_\omega\sqrt{x^2-1}+x^2-1=x^2-1-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}+\frac{x-1}{x+1}+x^2.</cmath> | ||
+ | Canceling terms results in | ||
+ | <cmath>r_\omega^2+2r_\omega\sqrt{x^2-1}=-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}+\frac{x-1}{x+1}+x^2.</cmath> | ||
+ | Since | ||
+ | <cmath>-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}=-2\sqrt{(x+1)(x-1)\frac{x-1}{x+1}}=-2(x-1),</cmath> | ||
+ | We can further simplify to | ||
+ | <cmath>r_\omega^2+2r_\omega\sqrt{x^2-1}=-2(x-1)+\frac{x-1}{x+1}+x^2.</cmath> | ||
+ | Substituting out <math>r_\omega</math> gets | ||
+ | <cmath>\left(\sqrt{\frac{x-1}{x+1}}+2\sqrt{\frac{x+1}{x-1}}\right)^2+2\left(\sqrt{\frac{x-1}{x+1}}+2\sqrt{\frac{x+1}{x-1}}\right)\sqrt{x^2-1}=-2(x-1)+\frac{x-1}{x+1}+x^2</cmath> | ||
+ | which when distributed yields | ||
+ | <cmath>\frac{x-1}{x+1}+4+4\left(\frac{x+1}{x-1}\right)+2(x-1+2(x+1))=-2(x-1)+\frac{x-1}{x+1}+x^2.</cmath> | ||
+ | After some canceling, distributing, and rearranging, we obtain | ||
+ | <cmath>4\left(\frac{x+1}{x-1}\right)=x^2-8x-4.</cmath> | ||
+ | Multiplying both sides by <math>x-1</math> results in | ||
+ | <cmath>4x+4=x^3-x^2-8x^2+8x-4x+4,</cmath> | ||
+ | which can be rearranged into | ||
+ | <cmath>x^3-9x^2=0</cmath> | ||
+ | and factored into | ||
+ | <cmath>x^2(x-9)=0.</cmath> | ||
+ | This means that <math>x</math> equals <math>0</math> or <math>9</math>, and since a side length of <math>0</math> cannot exist, <math>x=9</math>. | ||
+ | |||
+ | |||
+ | As a result, the triangle must have sides in the ratio of <math>9:2:9</math>. Since the triangle must have integer side lengths, and these values share no common factors greater than <math>1</math>, the triangle with the smallest possible perimeter under these restrictions has a perimeter of <math>9+2+9=\boxed{020}</math>. ~[[User:emerald_block|emerald_block]] | ||
+ | |||
+ | ==Solution 3 (Not that hard construction)== | ||
+ | Notice that the <math>A</math>-excircle would have to be very small to fit the property that it is internally tangent to <math>\omega</math> and the other two excircles are both externally tangent, given that circle <math>\omega</math>'s centre is at the incenter of <math>\triangle ABC</math>. If <math>BC=2</math>, we see that <math>AB=AC</math> must be somewhere in the <math>6</math> to <math>13</math> range. If we test <math>6</math> by construction, we notice the <math>A</math>-excircle is too big for it to be internally tangent to <math>\omega</math> while the other two are externally tangent. This means we should test <math>8</math> or <math>9</math> next. I actually did this and found that <math>9</math> worked, so the answer is <math>2+9+9=\boxed{20}</math>. Note that <math>BC</math> cannot be <math>1</math> because then <math>AB=AC</math> would have to be <math>4.5</math> which is not an integer. | ||
+ | |||
+ | ~[[User:Icematrix2|icematrix2]] | ||
+ | |||
+ | ==Video Solution (On the Spot STEM)== | ||
+ | |||
+ | https://www.youtube.com/watch?v=zKHwTJBhKdM | ||
+ | |||
+ | ==Video Solution 2 (I would recommend this one because it's more concise)== | ||
+ | |||
+ | https://www.youtube.com/watch?v=ldr4yi3t6hQ | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=I|num-b=10|num-a=12}} | {{AIME box|year=2019|n=I|num-b=10|num-a=12}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 11:07, 26 February 2021
Contents
Problem
In , the sides have integer lengths and . Circle has its center at the incenter of . An excircle of is a circle in the exterior of that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to is internally tangent to , and the other two excircles are both externally tangent to . Find the minimum possible value of the perimeter of .
Solution 1
Let the tangent circle be . Some notation first: let , , be the semiperimeter, , and be the inradius. Intuition tells us that the radius of is (using the exradius formula). However, the sum of the radius of and is equivalent to the distance between the incenter and the the excenter. Denote the B excenter as and the incenter as . Lemma: We draw the circumcircle of . Let the angle bisector of hit the circumcircle at a second point . By the incenter-excenter lemma, . Let this distance be . Ptolemy's theorem on gives us Again, by the incenter-excenter lemma, so as desired. Using this gives us the following equation: Motivated by the and , we make the following substitution: This changes things quite a bit. Here's what we can get from it: It is known (easily proved with Heron's and ) that Using this, we can also find : let the midpoint of be . Using Pythagorean's Theorem on , We now look at the RHS of the main equation: Cancelling some terms, we have Squaring, Expanding and moving terms around gives Reverse substituting, Clearly the smallest solution is and , so our answer is -franchester
Solution 2 (Lots of Pythagorean Theorem)
First, assume and . The triangle can be scaled later if necessary. Let be the incenter and let be the inradius. Let the points at which the incircle intersects , , and be denoted , , and , respectively.
Next, we calculate in terms of . Note the right triangle formed by , , and . The length is equal to . Using the Pythagorean Theorem, the length is , so the length is . Note that is half of , and by symmetry caused by the incircle, and , so . Applying the Pythagorean Theorem to , we get
Expanding yields
which can be simplified to
Dividing by and then squaring results in
and isolating gets us
so .
We then calculate the radius of the excircle tangent to . We denote the center of the excircle and the radius .
Consider the quadrilateral formed by , , , and the point at which the excircle intersects the extension of , which we denote . By symmetry caused by the excircle, , so .
Note that triangles and are congruent, and and are also congruent. Denoting the measure of angles and measure and the measure of angles and measure , straight angle , so . This means that angle is a right angle, so it forms a right triangle.
Setting the base of the right triangle to , the height is and the base consists of and . Triangles and are similar to , so , or . This makes the reciprocal of , so .
Circle 's radius can be expressed by the distance from the incenter to the bottom of the excircle with center . This length is equal to , or . Denote this value .
Finally, we calculate the distance from the incenter to the closest point on the excircle tangent to , which forms another radius of circle and is equal to . We denote the center of the excircle and the radius . We also denote the points where the excircle intersects and the extension of using and , respectively. In order to calculate the distance, we must find the distance between and and subtract off the radius .
We first must calculate the radius of the excircle. Because the excircle is tangent to both and the extension of , its center must lie on the angle bisector formed by the two lines, which is parallel to . This means that the distance from to is equal to the length of , so the radius is also .
Next, we find the length of . We can do this by forming the right triangle . The length of leg is equal to minus , or . In order to calculate the length of leg , note that right triangles and are congruent, as and share a length of , and angles and add up to the right angle . This means that .
Using Pythagorean Theorem, we get Bringing back and substituting in some values, the equation becomes Rearranging and squaring both sides gets Distributing both sides yields Canceling terms results in Since We can further simplify to Substituting out gets which when distributed yields After some canceling, distributing, and rearranging, we obtain Multiplying both sides by results in which can be rearranged into and factored into This means that equals or , and since a side length of cannot exist, .
As a result, the triangle must have sides in the ratio of . Since the triangle must have integer side lengths, and these values share no common factors greater than , the triangle with the smallest possible perimeter under these restrictions has a perimeter of . ~emerald_block
Solution 3 (Not that hard construction)
Notice that the -excircle would have to be very small to fit the property that it is internally tangent to and the other two excircles are both externally tangent, given that circle 's centre is at the incenter of . If , we see that must be somewhere in the to range. If we test by construction, we notice the -excircle is too big for it to be internally tangent to while the other two are externally tangent. This means we should test or next. I actually did this and found that worked, so the answer is . Note that cannot be because then would have to be which is not an integer.
Video Solution (On the Spot STEM)
https://www.youtube.com/watch?v=zKHwTJBhKdM
Video Solution 2 (I would recommend this one because it's more concise)
https://www.youtube.com/watch?v=ldr4yi3t6hQ
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.