Difference between revisions of "2014 AIME I Problems/Problem 15"
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Finally, using the Pythagorean Theorem on <math>\triangle BDE</math>, | Finally, using the Pythagorean Theorem on <math>\triangle BDE</math>, | ||
<cmath> \left(3-\frac{5x}{\sqrt{2}}\right)^2 + \left(4-\frac{5x}{\sqrt{2}}\right)^2 = (5x)^2</cmath> | <cmath> \left(3-\frac{5x}{\sqrt{2}}\right)^2 + \left(4-\frac{5x}{\sqrt{2}}\right)^2 = (5x)^2</cmath> | ||
− | Solving for <math>x</math>, we get that <math>x=\frac{5\sqrt{2}}{14}</math>, so <math>DE=5x=\frac{25\sqrt{2}}{14}</math>. Thus, the answer is <math>25+2+14=\boxed{041}</math>. | + | Solving for <math>x</math>, we get that <math>x=\frac{5\sqrt{2}}{14}</math>, so <math>DE= 5x =\frac{25 \sqrt{2}}{14}</math>. Thus, the answer is <math>25+2+14=\boxed{041}</math>. |
== Solution 2 == | == Solution 2 == | ||
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See inside the <math>\triangle DEF</math>, we can find that <math>AG>AF</math> since if <math>AG<AF</math>, we can see that Ptolemy Theorem inside cyclic quadrilateral <math>EFGD</math> doesn't work. Now let's see when <math>AG>AF</math>, since <math>\frac{DG}{EG} = \frac{3}{4}</math>, we can assume that <math>EG=4x;GD=3x;ED=5x</math>, since we know <math>EF=FD</math> so <math>\triangle EFD </math> is isosceles right triangle. We can denote <math>DF=EF=\frac{5x\sqrt{2}}{2}</math>.Applying Ptolemy Theorem inside the cyclic quadrilateral <math>EFGD</math> we can get the length of <math>FG</math> can be represented as <math>\frac{x\sqrt{2}}{2}</math>. After observing, we can see <math>\angle AFE=\angle EDG</math>, whereas <math>\angle A=\angle EDG</math> so we can see <math>\triangle AEF</math> is isosceles triangle. Since <math>\triangle ABC</math> is a <math>3-4-5</math> triangle so we can directly know that the length of AF can be written in the form of <math>3x\sqrt{2}</math>. Denoting a point <math>J</math> on side <math>AC</math> with that <math>DJ</math> is perpendicular to side <math>AC</math>. Now with the same reason, we can see that <math>\triangle DJG</math> is a isosceles right triangle, so we can get <math>GJ=\frac{3x\sqrt{2}}{2}</math> while the segment <math>CJ</math> is <math>2x\sqrt{2}</math> since its 3-4-5 again. Now adding all those segments together we can find that <math>AC=5=7x\sqrt{2}</math> and <math>x=\frac{5\sqrt{2}}{14}</math> and the desired <math>ED=5x=\frac{25\sqrt{2}}{14}</math> | See inside the <math>\triangle DEF</math>, we can find that <math>AG>AF</math> since if <math>AG<AF</math>, we can see that Ptolemy Theorem inside cyclic quadrilateral <math>EFGD</math> doesn't work. Now let's see when <math>AG>AF</math>, since <math>\frac{DG}{EG} = \frac{3}{4}</math>, we can assume that <math>EG=4x;GD=3x;ED=5x</math>, since we know <math>EF=FD</math> so <math>\triangle EFD </math> is isosceles right triangle. We can denote <math>DF=EF=\frac{5x\sqrt{2}}{2}</math>.Applying Ptolemy Theorem inside the cyclic quadrilateral <math>EFGD</math> we can get the length of <math>FG</math> can be represented as <math>\frac{x\sqrt{2}}{2}</math>. After observing, we can see <math>\angle AFE=\angle EDG</math>, whereas <math>\angle A=\angle EDG</math> so we can see <math>\triangle AEF</math> is isosceles triangle. Since <math>\triangle ABC</math> is a <math>3-4-5</math> triangle so we can directly know that the length of AF can be written in the form of <math>3x\sqrt{2}</math>. Denoting a point <math>J</math> on side <math>AC</math> with that <math>DJ</math> is perpendicular to side <math>AC</math>. Now with the same reason, we can see that <math>\triangle DJG</math> is a isosceles right triangle, so we can get <math>GJ=\frac{3x\sqrt{2}}{2}</math> while the segment <math>CJ</math> is <math>2x\sqrt{2}</math> since its 3-4-5 again. Now adding all those segments together we can find that <math>AC=5=7x\sqrt{2}</math> and <math>x=\frac{5\sqrt{2}}{14}</math> and the desired <math>ED=5x=\frac{25\sqrt{2}}{14}</math> | ||
which our answer is <math>\boxed{041}</math> ~bluesoul | which our answer is <math>\boxed{041}</math> ~bluesoul | ||
+ | |||
+ | ==Solution 5== | ||
+ | [[File:2014 AIME II 15.png|450px|right]] | ||
+ | The main element of the solution is the proof that <math>BF</math> is bisector of <math>\angle B.</math> | ||
+ | |||
+ | Let <math>O</math> be the midpoint of <math>DE.</math> <math>\angle EBF = 90^\circ \implies</math> | ||
+ | |||
+ | <math>O</math> is the center of the circle <math>BDGFE.</math> | ||
+ | <math>\angle EOF = 90^\circ \implies \overset{\Large\frown} {EF} = 90^\circ \implies \angle EBF = 45^\circ \implies</math> | ||
+ | BF is bisector of <math>\angle ABC\implies BF = \frac {2AB \cdot BC}{AB+BC} \cos 45^\circ =\frac {12 \cdot \sqrt{2}}{7}.</math> | ||
+ | <cmath>\angle EGD = 90^\circ, \frac {EG}{GD}=\frac{4}{3} \implies</cmath> | ||
+ | <cmath>\angle GED = \angle GCD =\gamma \implies \overset{\Large\frown} {DG} = 2\gamma.</cmath> | ||
+ | <cmath>2\angle ACB = \overset{\Large\frown} {BEF} - \overset{\Large\frown} {DG} \implies \overset{\Large\frown} {BEF} = 4 \gamma \implies</cmath> | ||
+ | <cmath>\angle BOF = 4 \gamma \implies \angle OBF = \angle OFB = 90^\circ – 2 \gamma.</cmath> | ||
+ | Let <math>BO = EO = DO = r \implies BF = 2 r \cos(90^\circ – 2\gamma) =</math> | ||
+ | <cmath>=2 r \sin 2\gamma = 4r \sin \gamma \cdot \cos \gamma = 4 r\cdot \frac {3}{5} \cdot \frac {4}{5} = \frac {48}{25} = \frac {12 \cdot \sqrt{2}}{7}\implies</cmath> | ||
+ | <cmath>r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = 2r = \frac {25 \cdot \sqrt{2}}{14}\implies \boxed{\textbf{041}}.</cmath> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Solution 6 == | ||
+ | [[File:2014 AIME II 15a.png|450px|right]] | ||
+ | The main element of the solution is the proof that <math>G</math> is midpoint of <math>AC.</math> | ||
+ | |||
+ | As in Solution 5 we get <math>\angle GED = \angle DBG =\gamma \implies</math> | ||
+ | |||
+ | <math>\triangle BCG </math> is isosceles triangle with <math>BG=CG.</math> | ||
+ | |||
+ | Similarly <math>BG = AG \implies AG = CG = BG = \frac {AC}{2} =\frac {5}{2}.</math> | ||
+ | |||
+ | <cmath>\overset{\Large\frown} {FG} = 90^\circ – \overset{\Large\frown} {GD} = 90^\circ – 2\gamma \implies</cmath> | ||
+ | <cmath>\overset{\Large\frown} {BFG} = 4\gamma + 90^\circ – 2\gamma = 90^\circ + 2\gamma \implies</cmath> | ||
+ | <cmath>\angle BOG = 90^\circ + 2\gamma \implies \angle BGO = \angle GBO = 45^\circ - \gamma.</cmath> | ||
+ | Let <math>\hspace{10mm} BO = EO = DO = r \implies</math> | ||
+ | <cmath>BG = 2 r \cos(45^\circ – \gamma) = 2 r (\sin \gamma + \cos \gamma)\frac {\sqrt {2}}{2} =</cmath> | ||
+ | <cmath>r \biggl(\frac {3}{5} + \frac {4}{5}\biggr) \sqrt {2} = r \frac {7 \sqrt{2}}{5} = \frac {5}{2}\implies</cmath> | ||
+ | <cmath>r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = \frac {25 \cdot \sqrt{2}}{14}\implies \boxed{\textbf{041}}.</cmath> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Solution 7== | ||
+ | Let <math>(BEFGD) = \omega</math>. | ||
+ | By Incenter-Excenter(Fact <math>5</math>), <math>F</math> is the angle bisector of <math>\angle B</math>. | ||
+ | Then by Ratio Lemma we have | ||
+ | <cmath>\frac{AG}{CG} = \frac{\sin(ABG)}{\sin(CBG)} \cdot \frac{AB}{BC} = \frac{\sin(GDE)}{\sin(DEG)} \cdot \frac{3}{4} = 1</cmath> | ||
+ | Thus, <math>G</math> is the midpoint of <math>AC</math>. | ||
+ | |||
+ | We can calculate <math>AF</math> and <math>CF</math> to be <math>\frac{15}{7}</math> and <math>\frac{20}{7}</math> respectively. | ||
+ | And then by Power of a Point, we have | ||
+ | <math>\newline</math> | ||
+ | <cmath>\operatorname{Pow}_{\omega}(A) = AE \cdot AB = AF \cdot AG \implies AE = \frac{25}{14}</cmath> | ||
+ | And then similarly, we have <math>CD = AE = \frac{25}{14}</math>. | ||
+ | <math>\newline</math> | ||
+ | |||
+ | Then <math>EB = \frac{17}{14}</math> and <math>DB = \frac{31}{14}</math> and by Pythagorean we have <math>DE = \frac{25\sqrt{2}}{14}</math>, so our answer is <math>\boxed{\textbf{041}}.</math> | ||
+ | |||
+ | ~dolphinday | ||
+ | |||
+ | ==Video Solution by mop 2024== | ||
+ | https://youtu.be/GxxZYZrQl2A | ||
+ | |||
+ | ~r00tsOfUnity | ||
== See also == | == See also == | ||
{{AIME box|year=2014|n=I|num-b=14|after=Last Question}} | {{AIME box|year=2014|n=I|num-b=14|after=Last Question}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:30, 28 January 2024
Contents
Problem 15
In , , , and . Circle intersects at and , at and , and at and . Given that and , length , where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .
Solution 1
Since , is the diameter of . Then . But , so is a 45-45-90 triangle. Letting , we have that , , and .
Note that by SAS similarity, so and . Since is a cyclic quadrilateral, and , implying that and are isosceles. As a result, , so and .
Finally, using the Pythagorean Theorem on , Solving for , we get that , so . Thus, the answer is .
Solution 2
First we note that is an isosceles right triangle with hypotenuse the same as the diameter of . We also note that since is a right angle and the ratios of the sides are .
From congruent arc intersections, we know that , and that from similar triangles is also congruent to . Thus, is an isosceles triangle with , so is the midpoint of and . Similarly, we can find from angle chasing that . Therefore, is the angle bisector of . From the angle bisector theorem, we have , so and .
Lastly, we apply power of a point from points and with respect to and have and , so we can compute that and . From the Pythagorean Theorem, we result in , so
Also: . We can also use Ptolemy's Theorem on quadrilateral to figure what is in terms of :
Thus .
Solution 3
Call and as a result . Since is cyclic we just need to get and using LoS(for more detail see the nd paragraph of Solution ) we get and using a similar argument(use LoS again) and subtracting you get so you can use Ptolemy to get . ~First
Solution 4
See inside the , we can find that since if , we can see that Ptolemy Theorem inside cyclic quadrilateral doesn't work. Now let's see when , since , we can assume that , since we know so is isosceles right triangle. We can denote .Applying Ptolemy Theorem inside the cyclic quadrilateral we can get the length of can be represented as . After observing, we can see , whereas so we can see is isosceles triangle. Since is a triangle so we can directly know that the length of AF can be written in the form of . Denoting a point on side with that is perpendicular to side . Now with the same reason, we can see that is a isosceles right triangle, so we can get while the segment is since its 3-4-5 again. Now adding all those segments together we can find that and and the desired which our answer is ~bluesoul
Solution 5
The main element of the solution is the proof that is bisector of
Let be the midpoint of
is the center of the circle BF is bisector of Let vladimir.shelomovskii@gmail.com, vvsss
Solution 6
The main element of the solution is the proof that is midpoint of
As in Solution 5 we get
is isosceles triangle with
Similarly
Let vladimir.shelomovskii@gmail.com, vvsss
Solution 7
Let . By Incenter-Excenter(Fact ), is the angle bisector of . Then by Ratio Lemma we have Thus, is the midpoint of .
We can calculate and to be and respectively. And then by Power of a Point, we have And then similarly, we have .
Then and and by Pythagorean we have , so our answer is
~dolphinday
Video Solution by mop 2024
~r00tsOfUnity
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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