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Difference between revisions of "1966 IMO Problems/Problem 2"

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Let <math>A</math>, <math>B</math>, and <math>C</math> be the lengths of the sides of a triangle, and <math> \alpha,\beta,\gamma </math> respectively, the angles opposite these sides.
+
==Problem==
  
−
<cmath> a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) </cmath>
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Let <math>a</math>, <math>b</math>, and <math>c</math> be the lengths of the sides of a triangle, and <math> \alpha,\beta,\gamma </math> respectively, the angles opposite these sides. Prove that if
  
−
Prove that if the triangle is isosceles.
+
<cmath> a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}), </cmath>
 +
 
 +
the triangle is isosceles.
 +
 
 +
 
 +
==Solution==
 +
 
 +
We'll prove that the triangle is isosceles with <math>a=b</math>.
 +
We'll prove that <math>a=b</math>. Assume by way of contradiction WLOG that <math>a>b</math>.
 +
First notice that as <math>\gamma = \pi -\alpha-\beta</math> then and the identity <math>\tan\left(\frac \pi 2 - x \right)=\cot x</math> our equation becomes:
 +
<cmath>a+b=\cot \frac{\alpha +\beta}{2}\left(a\tan \alpha + b\tan \beta \right)</cmath><cmath>\iff a\tan\frac{\alpha +\beta}{2}+b\tan \frac{\alpha +\beta}{2}=a\tan \alpha + b\tan \beta  </cmath>
 +
<cmath>\iff a\left(\tan \alpha -\tan \frac{\alpha +\beta}{2}\right)+b\left(\tan \beta -\tan \frac{\alpha +\beta}{2}  \right)=0 </cmath>
 +
Using the identity <math>\tan (A-B)=\frac {\tan A-\tan B}{1+\tan A\tan B}</math> <math> \iff \tan A-\tan B=\tan(A-B)(1+\tan A\tan B)</math>
 +
and inserting this into the above equation we get:
 +
<cmath>\iff a\tan \frac{\alpha -\beta}{2}\left(1+\tan \alpha \tan \frac{\alpha +\beta}{2}\right)+b\tan \frac{\beta -\alpha}{2}\left(1+\tan \beta \tan \frac{\alpha +\beta}{2} \right)=0 </cmath>
 +
<cmath>\underbrace{\iff}_{\tan -A=-\tan A}a\tan \frac{\alpha -\beta}{2}\left(1+\tan \alpha \tan \frac{\alpha +\beta}{2}\right)-b\tan \frac{\alpha -\beta}{2}\left(1+\tan \beta \tan \frac{\alpha +\beta}{2} \right)=0  </cmath>
 +
<cmath>\iff \tan \frac{\alpha -\beta}{2}\left(a-b+\tan \frac{\alpha +\beta}{2}(a\tan\alpha -b\tan \beta) \right)=0 </cmath>
 +
Now, since <math>a>b</math> and the definitions of <math>a,b,\alpha,\beta</math> being part of the definition of a triangle, <math>\alpha >\beta</math>.
 +
Now, <math>\pi >\alpha -\beta >0</math> (as <math>\alpha+\beta +\gamma = \pi</math> and the angles are positive), <math>\tan \frac{\alpha -\beta}{2}\neq 0</math>, and furthermore, <math>\tan \frac{\alpha+\beta}{2}>0</math>. By all the above, <cmath>\left(a-b+\tan \frac{\alpha +\beta}{2}(a\tan\alpha -b\tan \beta) \right)>0</cmath>
 +
Which contradicts our assumption, thus <math>a\leq b</math>. By the symmetry of the condition, using the same arguments, <math>a\geq b</math>. Hence <math>a=b</math>.
 +
 
 +
 
 +
==Solution 2==
 +
 
 +
First, we'll prove that both <math>\alpha</math> and <math>\beta</math> are acute.
 +
At least one of them has to be acute because these are angles
 +
of a triangle.  We can assume that <math>\alpha</math> is acute.  We want
 +
to show that <math>\beta</math> is acute as well.  For a proof by
 +
contradiction, assume <math>\beta \ge \frac{\pi}{2}</math>.
 +
 
 +
From the hypothesis, it follows that
 +
<math>(a + b) \tan \frac{\alpha + \beta}{2} = a \tan \alpha + b \tan \beta</math>.
 +
 
 +
From <math>\alpha < \frac{\pi}{2} \le \beta</math> it follows that <math>a < b</math>.  So,
 +
 
 +
<math>b \tan \beta = (a + b) \tan \frac{\alpha + \beta}{2} - a \tan \alpha >
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2a \tan \frac{\alpha + \beta}{2} - a \tan \alpha \ge
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a (2 \tan \left( \frac{\alpha}{2} + \frac{\pi}{4} \right) - \tan \alpha) =</math>
 +
 
 +
<math>2a \left( \frac{\tan \frac{\alpha}{2} + 1}{1 - \tan \frac{\alpha}{2}} -
 +
\frac{\tan \frac{\alpha}{2}}{1 - \tan^2 \frac{\alpha}{2}} \right) =
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2a \cdot \frac{\tan^2 \frac{\alpha}{2} + \tan \frac{\alpha}{2} + 1}
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{1 - \tan^2 \frac{\alpha}{2}} > 0</math>
 +
 
 +
because the numerator is <math>> 0</math> (because <math>Y^2 + Y + 1 > 0</math> for any real
 +
<math>Y</math>), and the denominator is also <math>> 0</math> (because <math>\alpha < \frac{\pi}{2}</math>,
 +
so <math>\tan \frac{\alpha}{2} < \tan \frac{\pi}{4} = 1</math>).
 +
 
 +
It follows that <math>\tan \beta > 0</math>, so it can not be that
 +
<math>\beta \ge \frac{\pi}{2}</math>.
 +
 
 +
Now, we will prove that
 +
<math>(a + b) \tan \frac{\alpha + \beta}{2} = a \tan \alpha + b \tan \beta</math>
 +
implies <math>\alpha = \beta</math>.
 +
 
 +
Replace <math>a = \sin \alpha \cdot 2R</math> and <math>b = \sin \beta \cdot 2R</math>
 +
(in fact, we don't care that <math>R</math> is the radius of the circumscribed
 +
circle), and simplify by <math>2R</math>.  We get
 +
 
 +
<math>(\sin \alpha + \sin \beta) \tan \frac{\alpha + \beta}{2} =
 +
\sin \alpha \tan \alpha + \sin \beta \tan \beta</math>.
 +
 
 +
This becomes
 +
 
 +
<math> \left( \sin \frac{\alpha + \beta}{2} \tan \frac{\alpha + \beta}{2} \right)
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\cdot \cos \frac{\alpha - \beta}{2} =
 +
\frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta)</math>
 +
 
 +
We will show that the function <math>f(x) = \tan x \sin x</math> is convex on
 +
the interval <math> \left( 0, \frac{\pi}{2} \right)</math>.  Indeed, the first
 +
derivative is <math>f'(x) = \frac{\sin x}{\cos^2 x} + \sin x</math>, and the
 +
second derivative is <math>f''(x) = \frac{\cos^4 x - \cos ^2 x + 2}{\cos^3 x}</math>.
 +
 
 +
We have <math>f''(x) > 0</math> on <math> \left( 0, \frac{\pi}{2} \right)</math> since the
 +
numerator is <math>> 0</math> (because <math>Y^2 - Y + 1 > 0</math> for any real <math>Y</math>), and
 +
the denominator is <math>> 0</math> on the interval
 +
<math> \left( 0, \frac{\pi}{2} \right)</math>.  It follows that <math>f(x) = \tan x \sin x</math>
 +
is convex on the interval <math> \left( 0, \frac{\pi}{2} \right)</math>.
 +
 
 +
Using the convexity we have
 +
<math>f \left( \frac{x + y}{2} \right) \le \frac{1}{2} (f(x) + f(y))</math>.  In
 +
our case, we have
 +
 
 +
<math>\frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta) =
 +
\left( \sin \frac{\alpha + \beta}{2} \tan \frac{\alpha + \beta}{2} \right)
 +
\cdot \cos \frac{\alpha - \beta}{2} \le
 +
\frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta) \cdot
 +
\cos \frac{\alpha - \beta}{2}</math>.
 +
 
 +
We can simplify by <math>\sin \alpha \tan \alpha + \sin \beta \tan \beta</math>
 +
because it is positive (because both <math>\alpha, \beta</math> are acute!),
 +
and we get
 +
 
 +
<math>1 \le \cos \frac{\alpha - \beta}{2}</math>.  This is possible only when
 +
<math>\cos \frac{\alpha - \beta}{2} = 1</math>, i.e. <math>\alpha = \beta</math>.
 +
 
 +
[Solution by pf02, September 2024]
 +
 
 +
 
 +
==See Also==
 +
{{IMO box|year=1966|num-b=1|num-a=3}}

Latest revision as of 19:17, 10 November 2024

Problem

Let $a$, $b$, and $c$ be the lengths of the sides of a triangle, and $\alpha,\beta,\gamma$ respectively, the angles opposite these sides. Prove that if

\[a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}),\]

the triangle is isosceles.


Solution

We'll prove that the triangle is isosceles with $a=b$. We'll prove that $a=b$. Assume by way of contradiction WLOG that $a>b$. First notice that as $\gamma = \pi -\alpha-\beta$ then and the identity $\tan\left(\frac \pi 2 - x \right)=\cot x$ our equation becomes: \[a+b=\cot \frac{\alpha +\beta}{2}\left(a\tan \alpha + b\tan \beta \right)\]\[\iff a\tan\frac{\alpha +\beta}{2}+b\tan \frac{\alpha +\beta}{2}=a\tan \alpha + b\tan \beta\] \[\iff a\left(\tan \alpha -\tan \frac{\alpha +\beta}{2}\right)+b\left(\tan \beta -\tan \frac{\alpha +\beta}{2} \right)=0\] Using the identity $\tan (A-B)=\frac {\tan A-\tan B}{1+\tan A\tan B}$ $\iff \tan A-\tan B=\tan(A-B)(1+\tan A\tan B)$ and inserting this into the above equation we get: \[\iff a\tan \frac{\alpha -\beta}{2}\left(1+\tan \alpha \tan \frac{\alpha +\beta}{2}\right)+b\tan \frac{\beta -\alpha}{2}\left(1+\tan \beta \tan \frac{\alpha +\beta}{2} \right)=0\] \[\underbrace{\iff}_{\tan -A=-\tan A}a\tan \frac{\alpha -\beta}{2}\left(1+\tan \alpha \tan \frac{\alpha +\beta}{2}\right)-b\tan \frac{\alpha -\beta}{2}\left(1+\tan \beta \tan \frac{\alpha +\beta}{2} \right)=0\] \[\iff \tan \frac{\alpha -\beta}{2}\left(a-b+\tan \frac{\alpha +\beta}{2}(a\tan\alpha -b\tan \beta) \right)=0\] Now, since $a>b$ and the definitions of $a,b,\alpha,\beta$ being part of the definition of a triangle, $\alpha >\beta$. Now, $\pi >\alpha -\beta >0$ (as $\alpha+\beta +\gamma = \pi$ and the angles are positive), $\tan \frac{\alpha -\beta}{2}\neq 0$, and furthermore, $\tan \frac{\alpha+\beta}{2}>0$. By all the above, \[\left(a-b+\tan \frac{\alpha +\beta}{2}(a\tan\alpha -b\tan \beta) \right)>0\] Which contradicts our assumption, thus $a\leq b$. By the symmetry of the condition, using the same arguments, $a\geq b$. Hence $a=b$.


Solution 2

First, we'll prove that both $\alpha$ and $\beta$ are acute. At least one of them has to be acute because these are angles of a triangle. We can assume that $\alpha$ is acute. We want to show that $\beta$ is acute as well. For a proof by contradiction, assume $\beta \ge \frac{\pi}{2}$.

From the hypothesis, it follows that $(a + b) \tan \frac{\alpha + \beta}{2} = a \tan \alpha + b \tan \beta$.

From $\alpha < \frac{\pi}{2} \le \beta$ it follows that $a < b$. So,

$b \tan \beta = (a + b) \tan \frac{\alpha + \beta}{2} - a \tan \alpha > 2a \tan \frac{\alpha + \beta}{2} - a \tan \alpha \ge a (2 \tan \left( \frac{\alpha}{2} + \frac{\pi}{4} \right) - \tan \alpha) =$

$2a \left( \frac{\tan \frac{\alpha}{2} + 1}{1 - \tan \frac{\alpha}{2}} - \frac{\tan \frac{\alpha}{2}}{1 - \tan^2 \frac{\alpha}{2}} \right) = 2a \cdot \frac{\tan^2 \frac{\alpha}{2} + \tan \frac{\alpha}{2} + 1} {1 - \tan^2 \frac{\alpha}{2}} > 0$

because the numerator is $> 0$ (because $Y^2 + Y + 1 > 0$ for any real $Y$), and the denominator is also $> 0$ (because $\alpha < \frac{\pi}{2}$, so $\tan \frac{\alpha}{2} < \tan \frac{\pi}{4} = 1$).

It follows that $\tan \beta > 0$, so it can not be that $\beta \ge \frac{\pi}{2}$.

Now, we will prove that $(a + b) \tan \frac{\alpha + \beta}{2} = a \tan \alpha + b \tan \beta$ implies $\alpha = \beta$.

Replace $a = \sin \alpha \cdot 2R$ and $b = \sin \beta \cdot 2R$ (in fact, we don't care that $R$ is the radius of the circumscribed circle), and simplify by $2R$. We get

$(\sin \alpha + \sin \beta) \tan \frac{\alpha + \beta}{2} = \sin \alpha \tan \alpha + \sin \beta \tan \beta$.

This becomes

$\left( \sin \frac{\alpha + \beta}{2} \tan \frac{\alpha + \beta}{2} \right) \cdot \cos \frac{\alpha - \beta}{2} = \frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta)$

We will show that the function $f(x) = \tan x \sin x$ is convex on the interval $\left( 0, \frac{\pi}{2} \right)$. Indeed, the first derivative is $f'(x) = \frac{\sin x}{\cos^2 x} + \sin x$, and the second derivative is $f''(x) = \frac{\cos^4 x - \cos ^2 x + 2}{\cos^3 x}$.

We have $f''(x) > 0$ on $\left( 0, \frac{\pi}{2} \right)$ since the numerator is $> 0$ (because $Y^2 - Y + 1 > 0$ for any real $Y$), and the denominator is $> 0$ on the interval $\left( 0, \frac{\pi}{2} \right)$. It follows that $f(x) = \tan x \sin x$ is convex on the interval $\left( 0, \frac{\pi}{2} \right)$.

Using the convexity we have $f \left( \frac{x + y}{2} \right) \le \frac{1}{2} (f(x) + f(y))$. In our case, we have

$\frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta) = \left( \sin \frac{\alpha + \beta}{2} \tan \frac{\alpha + \beta}{2} \right) \cdot \cos \frac{\alpha - \beta}{2} \le \frac{1}{2}(\sin \alpha \tan \alpha + \sin \beta \tan \beta) \cdot \cos \frac{\alpha - \beta}{2}$.

We can simplify by $\sin \alpha \tan \alpha + \sin \beta \tan \beta$ because it is positive (because both $\alpha, \beta$ are acute!), and we get

$1 \le \cos \frac{\alpha - \beta}{2}$. This is possible only when $\cos \frac{\alpha - \beta}{2} = 1$, i.e. $\alpha = \beta$.

[Solution by pf02, September 2024]


See Also

1966 IMO (Problems) • Resources
Preceded by
Problem 1 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 3
All IMO Problems and Solutions
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