Difference between revisions of "2014 AIME II Problems/Problem 12"
(→Solution 1: cleaned up wrong solution) |
(→Solution 1) |
||
Line 3: | Line 3: | ||
== Solution 1 == | == Solution 1 == | ||
− | Note that <math>\cos{3C}=-\cos{(3A+3B)}</math>. Thus, our expression is of the form <math>\cos{3A}+\cos{3B}-\cos{(3A+3B)}=1</math>. Let <math>\cos{3A}=x</math> and <math>\cos{3B}=y</math>. | + | Note that <math>\cos{3C}=-\cos{(3A+3B)}</math>. Thus, our expression is of the form <math>\cos{3A}+\cos{3B}-\cos{(3A+3B)}=1</math>. Let <math>\cos{3A}=x</math> and <math>\cos{3B}=y</math>. Using the fact that <math>\cos(3A+3B)=\cos 3A\cos 3B-\sin 3A\sin 3B=xy-\sqrt{1-x^2}\sqrt{1-y^2}</math>, we get <math>x+y-xy+\sqrt{1-x^2}\sqrt{1-y^2}=1</math>, or <math>\sqrt{1-x^2}\sqrt{1-y^2}=xy-x-y-1=(x-1)(y-1)</math>. |
Squaring both sides, we get <math>(1-x^2)(1-y^2) = [(x-1)(y-1)]^2</math>. Cancelling factors, <math>(1+x)(1+y) = (1-x)(1-y).</math> | Squaring both sides, we get <math>(1-x^2)(1-y^2) = [(x-1)(y-1)]^2</math>. Cancelling factors, <math>(1+x)(1+y) = (1-x)(1-y).</math> | ||
− | Expanding, <math> | + | Expanding, <math>1+x+y+xy= 1-x-y+xy\rightarrow x+y=-x-y\rightarrow 2x=-2y</math>. Simplification leads to <math>x=-y</math> and <math>x+y=0</math>. |
− | Therefore, cos(3C) | + | Therefore, <math>cos(3C)=1</math>. So <math>\angle C</math> could be <math>0^\circ</math> or <math>120^\circ</math>. We eliminate <math>0^\circ</math> and use law of cosines to get our answer: |
+ | <cmath>m=10^2+13^2-2\cdot 10\cdot 13\cos\angle C</cmath> | ||
+ | <cmath>\rightarrow m=269-260\cos 120^\circ=269-260\left(\frac{1}{2}\right)</cmath> | ||
+ | <cmath>\rightarrow m=269+130=399</cmath> | ||
+ | |||
+ | <math>\framebox{399}</math> | ||
==Solution 2== | ==Solution 2== |
Revision as of 16:02, 9 February 2016
Contents
Problem
Suppose that the angles of satisfy
Two sides of the triangle have lengths 10 and 13. There is a positive integer
so that the maximum possible length for the remaining side of
is
Find
Solution 1
Note that . Thus, our expression is of the form
. Let
and
. Using the fact that
, we get
, or
.
Squaring both sides, we get . Cancelling factors,
Expanding,
. Simplification leads to
and
.
Therefore, . So
could be
or
. We eliminate
and use law of cosines to get our answer:
Solution 2
As above, we can see that
Expanding, we get
Note that , or
Thus , or
.
Now we know that , so we can just use Law or Cosines to get
See also
2014 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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.