1997 JBMO Problems/Problem 4

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 $n.$

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

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