Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2000 USAMO Problems/Problem 2"

(solution)
 
m (→‎Solution 2)
 
(13 intermediate revisions by 5 users not shown)
Line 9: Line 9:
 
We let <math>x = AP = s - a, y = BQ = s-b, z = CR = s-c</math>, and [[without loss of generality]] let <math>x \le y \le z</math>. Then <math>x + y + z = 3s - (a+b+c) = s</math>, so <math>r = \frac {A}{s} = \frac{\sqrt{(x+y+z)xyz}}{x+y+z} = \sqrt{\frac{xyz}{x+y+z}}</math>. Thus,  
 
We let <math>x = AP = s - a, y = BQ = s-b, z = CR = s-c</math>, and [[without loss of generality]] let <math>x \le y \le z</math>. Then <math>x + y + z = 3s - (a+b+c) = s</math>, so <math>r = \frac {A}{s} = \frac{\sqrt{(x+y+z)xyz}}{x+y+z} = \sqrt{\frac{xyz}{x+y+z}}</math>. Thus,  
  
−
<cmath>6\sqrt{\frac{xyz}{x+y+z}} = \frac{2yz + 5xy + 5xz}{xyz}</cmath>  
+
<cmath>6\sqrt{\frac{x+y+z}{xyz}} = \frac{2yz + 5xy + 5xz}{xyz}</cmath>  
  
−
Squaring yields  
+
Squaring and simplifying yields (after much grueling work)
  
 
<cmath>0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz</cmath>
 
<cmath>0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz</cmath>
Line 19: Line 19:
 
<cmath>\begin{align}0 \le 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz \tag{*} \end{align}</cmath>
 
<cmath>\begin{align}0 \le 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz \tag{*} \end{align}</cmath>
  
−
holds true, with equality [[iff]] <math>4x = y = z</math>. Then <math>s = x+y+z = 9x</math>, and <math>s-a = x, s-b = 4x, s-c = 4x</math> yields <math>\boxed{a:b:c = 5:8:8}</math>.
+
holds true, with equality [[iff]] <math>4x = y = z</math>.
  
  
Line 30: Line 30:
 
<cmath>\Delta = z^2(14-16z)^2 - 4(4z^2 - 16z + 25)(25z^2) = -144(z-4)^2 \le 0</cmath>
 
<cmath>\Delta = z^2(14-16z)^2 - 4(4z^2 - 16z + 25)(25z^2) = -144(z-4)^2 \le 0</cmath>
  
−
as desired. Equality comes when <math>z = 4</math>; since <math>(*)</math> is symmetric in <math>y</math> and <math>z</math>, it follows that <math>y = 4</math> is also necessary for equality. Reversing our scaling, it follows that <math>x:y:z = 1:4:4</math>. <math>\blacksquare</math>
+
as desired. Equality comes when <math>z = 4</math>; since <math>(*)</math> is symmetric in <math>y</math> and <math>z</math>, it follows that <math>y = 4</math> is also necessary for equality. Reversing our scaling, it follows that <math>x:y:z = 1:4:4</math>.
  
−
== See also ==
+
Then <math>s = x+y+z = 9x</math>, and <math>s-a = x, s-b = 4x, s-c = 4x</math> yields <math>a:b:c = 8:5:5</math>. Thus, we have proved that all possible ABC are isosceles (as b = c), and that they are similar to a 5-5-8 triangle.
 +
<math>\blacksquare</math>
 +
== Solution 2 ==
 +
Let <math>A</math>, <math>B</math>, <math>C</math> be the three angles of the triangle, then <math>r/AP = \tan{(A/2)}</math>, <math>r/BQ=\tan{(B/2)}</math>, and <math>r/CR=\tan{(C/2)}</math>. Without loss of generality, assume <math>A \leq B \leq C</math>, then <math>CR=min{(AP,BQ,CR)}</math>. Denote <math>x=\tan{(A/2)}</math>, <math>x=\tan{(B/2)}</math>, <math>z=\tan{(C/2)}</math>, then we have,
 +
<cmath>5(x+y+z)- 3z=6 \qquad \qquad (1) </cmath>
 +
Meanwhile,
 +
<cmath>z = \tan{(C/2)} = \tan{(90- (A+B)/2))} = \frac{1-xy}{x+y} </cmath>
 +
Substitute <math>z</math> in <math>(1)</math>, we get
 +
<cmath>5x^{2}+8xy+5y^{2} - 6x - 6y+2=0 \qquad \qquad (2) </cmath>
 +
Treating (2) as a quadratic equation for <math>x</math>, the discriminant is:
 +
<cmath>\Delta = (8y-6)^2 - 4*5(5y^2-6y+2)=-36y^2+24y-4 = -(6y-2)^2 \leq 0</cmath>
 +
For <math>x</math>, <math>y</math> to be both real numbers, <math>\Delta</math> must not be negative, so <math>\Delta = 0</math>, which yields <math>y=1/3</math>. Solving for x, we get <math>x=\frac{-(8y-6)}{2*5}=1/3</math>. Using the formula <math>\tan(A)=\frac{2\tan(A/2)}{1-\tan^2(A/2)}</math>, we have <math>\tan(A)=\tan(B)=3/4</math>. Therefore <math>A=B=\arctan{(3/4)}</math>. This proves that all possible triangle <math>ABC</math> are isosceles with the same set of angles, i.e., they are similar to each other.
 +
 
 +
It's inconsequential to the proof, but it can be easily found that, since <math>tan(A) = 3/4</math>, the altitude to the base of the isosceles triangle divides the triangle into two 3-4-5 right triangles with <math>4</math> on the base; therefore, the entire triangle is 5-5-8. <math>\square</math>.
 +
 
 +
Solution by <math>Mathdummy</math>.
 +
 
 +
== See Also ==
 
{{USAMO newbox|year=2000|num-b=1|num-a=3}}
 
{{USAMO newbox|year=2000|num-b=1|num-a=3}}
  
 
[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 17:40, 3 January 2019

Problem

Let $S$ be the set of all triangles $ABC$ for which

\[5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},\]

where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.

Solution

We let $x = AP = s - a, y = BQ = s-b, z = CR = s-c$, and without loss of generality let $x \le y \le z$. Then $x + y + z = 3s - (a+b+c) = s$, so $r = \frac {A}{s} = \frac{\sqrt{(x+y+z)xyz}}{x+y+z} = \sqrt{\frac{xyz}{x+y+z}}$. Thus,

\[6\sqrt{\frac{x+y+z}{xyz}} = \frac{2yz + 5xy + 5xz}{xyz}\]

Squaring and simplifying yields (after much grueling work)

\[0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz\]

We claim that the inequality

\begin{align}0 \le 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz \tag{*} \end{align}

holds true, with equality iff $4x = y = z$.


Note that $(*)$ is homogeneous in $x,y,z$, so without loss of generality, scale so that $x=1$. Then

\[0 \le y^2 (4z^2 - 16z + 25) + y(14z - 16z^2) + 25z^2\]

which is a quadratic in $y$. As $4z^2 - 16z + 25 = 4(z-4)^2 + 9 \ge 0$, it suffices to show that the quadratic cannot have more than one root, or the discriminant $\Delta \le 0$. Then,

\[\Delta = z^2(14-16z)^2 - 4(4z^2 - 16z + 25)(25z^2) = -144(z-4)^2 \le 0\]

as desired. Equality comes when $z = 4$; since $(*)$ is symmetric in $y$ and $z$, it follows that $y = 4$ is also necessary for equality. Reversing our scaling, it follows that $x:y:z = 1:4:4$.

Then $s = x+y+z = 9x$, and $s-a = x, s-b = 4x, s-c = 4x$ yields $a:b:c = 8:5:5$. Thus, we have proved that all possible ABC are isosceles (as b = c), and that they are similar to a 5-5-8 triangle. $\blacksquare$

Solution 2

Let $A$, $B$, $C$ be the three angles of the triangle, then $r/AP = \tan{(A/2)}$, $r/BQ=\tan{(B/2)}$, and $r/CR=\tan{(C/2)}$. Without loss of generality, assume $A \leq B \leq C$, then $CR=min{(AP,BQ,CR)}$. Denote $x=\tan{(A/2)}$, $x=\tan{(B/2)}$, $z=\tan{(C/2)}$, then we have, \[5(x+y+z)- 3z=6 \qquad \qquad (1)\] Meanwhile, \[z = \tan{(C/2)} = \tan{(90- (A+B)/2))} = \frac{1-xy}{x+y}\] Substitute $z$ in $(1)$, we get \[5x^{2}+8xy+5y^{2} - 6x - 6y+2=0 \qquad \qquad (2)\] Treating (2) as a quadratic equation for $x$, the discriminant is: \[\Delta = (8y-6)^2 - 4*5(5y^2-6y+2)=-36y^2+24y-4 = -(6y-2)^2 \leq 0\] For $x$, $y$ to be both real numbers, $\Delta$ must not be negative, so $\Delta = 0$, which yields $y=1/3$. Solving for x, we get $x=\frac{-(8y-6)}{2*5}=1/3$. Using the formula $\tan(A)=\frac{2\tan(A/2)}{1-\tan^2(A/2)}$, we have $\tan(A)=\tan(B)=3/4$. Therefore $A=B=\arctan{(3/4)}$. This proves that all possible triangle $ABC$ are isosceles with the same set of angles, i.e., they are similar to each other.

It's inconsequential to the proof, but it can be easily found that, since $tan(A) = 3/4$, the altitude to the base of the isosceles triangle divides the triangle into two 3-4-5 right triangles with $4$ on the base; therefore, the entire triangle is 5-5-8. $\square$.

Solution by $Mathdummy$.

See Also

2000 USAMO (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2000_USAMO_Problems/Problem_2&oldid=100024"