Difference between revisions of "2005 AIME I Problems/Problem 15"
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Now, by [[Stewart's Theorem]] in triangle <math>\triangle ABC</math> with [[cevian]] <math>\overline{AD}</math>, we have | Now, by [[Stewart's Theorem]] in triangle <math>\triangle ABC</math> with [[cevian]] <math>\overline{AD}</math>, we have | ||
− | <cmath>(3m)^2\cdot 20 + 20\cdot10\cdot10 = | + | <cmath>(3m)^2\cdot 20 + 20\cdot10\cdot10 = 1^2\cdot0 + (30 - 2c)^2\cdot 10.</cmath> |
Our earlier result from Power of a Point was that <math>2m^2 = (10 - c)^2</math>, so we combine these two results to solve for <math>c</math> and we get | Our earlier result from Power of a Point was that <math>2m^2 = (10 - c)^2</math>, so we combine these two results to solve for <math>c</math> and we get | ||
− | <cmath> | + | <cmath>(10 - c)^2 + 200 = 100 + (30 - 2c)^2 \quad \Longrightarrow \quad c^2 - 12c + 20 = 0.</cmath>Oh yeah we are awesome come on did you know |
− | Thus <math>c = 2</math> or <math> | + | Thus <math>c = 2</math> or <math> = 10</math>. We discard the value <math>c = 10</math> as extraneous (it gives us an [[equilateral triangle]]) and are left with <math>c = 2</math>, so our triangle has sides cdot 18 \cdot 8 \cdot 2} = 24\sqrt{14}<math> and so the answer is </math>24 + 14 = \boxed{038}$. |
== See also == | == See also == |
Revision as of 20:49, 17 October 2013
Problem
Triangle has
The incircle of the triangle evenly trisects the median
If the area of the triangle is
where
and
are integers and
is not divisible by the square of a prime, find
Solution
![[asy] size(300); pointpen=black;pathpen=black+linewidth(0.65); pen s = fontsize(10); pair A=(0,0),B=(26,0),C=IP(circle(A,10),circle(B,20)),D=(B+C)/2,I=incenter(A,B,C); path cir = incircle(A,B,C); pair E1=IP(cir,B--C),F=IP(cir,A--C),G=IP(cir,A--B),P=IP(A--D,cir),Q=OP(A--D,cir); D(MP("A",A,s)--MP("B",B,s)--MP("C",C,N,s)--cycle); D(cir); D(A--MP("D",D,NE,s)); D(MP("E",E1,NE,s)); D(MP("F",F,NW,s)); D(MP("G",G,s)); D(MP("P",P,SW,s)); D(MP("Q",Q,SE,s)); MP("10",(B+D)/2,NE); MP("10",(C+D)/2,NE); [/asy]](http://latex.artofproblemsolving.com/e/b/b/ebb5c6eb90edb59c2c2fc2cf3dfef0dcf818f4f9.png)
Let ,
and
be the points of tangency of the incircle with
,
and
, respectively. Without loss of generality, let
, so that
is between
and
. Let the length of the median be
. Then by two applications of the Power of a Point Theorem,
, so
. Now,
and
are two tangents to a circle from the same point, so
and thus
. Then
so
and thus
.
Now, by Stewart's Theorem in triangle with cevian
, we have
Our earlier result from Power of a Point was that , so we combine these two results to solve for
and we get
Oh yeah we are awesome come on did you know
Thus or
. We discard the value
as extraneous (it gives us an equilateral triangle) and are left with
, so our triangle has sides cdot 18 \cdot 8 \cdot 2} = 24\sqrt{14}
24 + 14 = \boxed{038}$.
See also
2005 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.