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Difference between revisions of "1997 JBMO Problems/Problem 4"

(WIP Solution)
(Solution to Problem 4)
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\sin(\theta) &= \frac{b+c}{2\sqrt{bc}}
 
\sin(\theta) &= \frac{b+c}{2\sqrt{bc}}
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
Since <math>\theta</math> is inside a triangle, <math>0 < \sin{\theta} \le 1</math>.
+
Since <math>\theta</math> is inside a triangle, <math>0 < \sin{\theta} \le 1</math>.  Substitution yields
 +
<cmath>0 < \frac{b+c}{2\sqrt{bc}} \le 1.</cmath>
 +
Note that <math>2\sqrt{bc}</math>, so multiplying both sides by that value would not change the inequality sign.  This means
 +
<cmath>0 < b+c \le 2\sqrt{bc}.</cmath>
 +
Since all values in the inequality are positive, squaring both sides would not change the inequality sign, so
 +
<cmath>0 < b^2 + 2bc + c^2 \le 4bc</cmath>
 +
<cmath>-4bc < b^2 - 2bc + c^2 \le 0</cmath>
 +
<cmath>-4bc < (b-c)^2 \le 0</cmath>
 +
By the [[Trivial Inequality]], <math>(b-c)^2 \ge 0</math> for all <math>b</math> and <math>c,</math> so the only values of <math>b</math> and <math>c</math> that satisfies is when <math>(b-c)^2 = 0</math>.  Thus, <math>b = c</math>.  Since <math>-4bc < 0</math> for positive <math>b</math> and <math>c</math>, the value <math>b=c</math> truly satisfies all conditions.
  
 +
<br>
 +
That means <math>\sin(\theta) = \frac{2b}{2\sqrt{b^2}} = 1,</math> so <math>\theta = 90^\circ.</math>  That means the only truangle that satisfies all the conditions is a 45-45-90 triangle where <math>a</math> is the longest side.  In other words, <math>(a,b,c) \rightarrow \boxed{(n\sqrt{2},n,n)}</math> for all positive <math>n.</math>
  
 
== See Also ==
 
== See Also ==

Revision as of 14:24, 4 August 2018

Problem

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$.

Solution

Solving for $R$ yields $R = \tfrac{a\sqrt{bc}}{b+c}$. We can substitute $R$ into the area formula $A = \tfrac{abc}{4R}$ to get \begin{align*} A &= \frac{abc}{4 \cdot \tfrac{a\sqrt{bc}}{b+c} } \\ &= \frac{abc}{4a\sqrt{bc}} \cdot (b+c) \\ &= \frac{(b+c)\sqrt{bc}}{4}. \end{align*} We also know that $A = \tfrac{1}{2}ab \sin(\theta)$, where $\theta$ is the angle between sides $b$ and $c.$ Substituting this yields \begin{align*} \tfrac{1}{2}ab \sin(\theta) &= \frac{(b+c)\sqrt{bc}}{4} \\ 2\sqrt{bc} \cdot \sin(\theta) &= b+c \\ \sin(\theta) &= \frac{b+c}{2\sqrt{bc}} \end{align*} Since $\theta$ is inside a triangle, $0 < \sin{\theta} \le 1$. Substitution yields \[0 < \frac{b+c}{2\sqrt{bc}} \le 1.\] Note that $2\sqrt{bc}$, so multiplying both sides by that value would not change the inequality sign. This means \[0 < b+c \le 2\sqrt{bc}.\] Since all values in the inequality are positive, squaring both sides would not change the inequality sign, so \[0 < b^2 + 2bc + c^2 \le 4bc\] \[-4bc < b^2 - 2bc + c^2 \le 0\] \[-4bc < (b-c)^2 \le 0\] By the Trivial Inequality, $(b-c)^2 \ge 0$ for all $b$ and $c,$ so the only values of $b$ and $c$ that satisfies is when $(b-c)^2 = 0$. Thus, $b = c$. Since $-4bc < 0$ for positive $b$ and $c$, the value $b=c$ truly satisfies all conditions.


That means $\sin(\theta) = \frac{2b}{2\sqrt{b^2}} = 1,$ so $\theta = 90^\circ.$ That means the only truangle that satisfies all the conditions is a 45-45-90 triangle where $a$ is the longest side. In other words, $(a,b,c) \rightarrow \boxed{(n\sqrt{2},n,n)}$ for all positive $n.$

See Also

1997 JBMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5
All JBMO Problems and Solutions
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