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Difference between revisions of "1961 IMO Problems/Problem 2"

 
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Let ''a'',''b'', and ''c'' be the lengths of a triangle whose area is ''S''.  Prove that
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==Problem==
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Let <math>a</math>, <math>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''.  Prove that
  
 
<math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
 
<math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
  
 
In what case does equality hold?
 
In what case does equality hold?
 +
 +
==Solution==
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By Heron's formula, we have
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<cmath>S = \sqrt{s(s-a)(s-b)(s-c)}.</cmath>
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This can be simplified to
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<cmath>S = \sqrt{\left(\frac{a+b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)\left(\frac{a-b+c}{2}\right)\left(\frac{a+b-c}{2}\right)}.</cmath>
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Next, we can factor out all of the <math>2</math>s and use a clever difference of squares:
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<cmath>S = \frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}</cmath>
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<cmath>S = \frac{1}{4}\sqrt{((b+c)^2 - a^2)(a^2 - (b-c)^2)}</cmath>
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<cmath>S = \frac{1}{4}\sqrt{(2bc + b^2 + c^2 - a^2)(2bc - b^2 - c^2 + a^2)}.</cmath>
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We can now use difference of squares again:
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<cmath>S = \frac{1}{4}\sqrt{4b^2c^2-(b^2 + c^2 - a^2)^2}</cmath>
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<cmath>4S\sqrt{3} = \sqrt{3(4b^2c^2 - (b^2 + c^2 - a^2)^2)}</cmath>
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We must prove that the RHS of this equation is less than or equal to <math>a^2 + b^2 + c^2</math>.
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Let <math>a^2 = A</math>, <math>b^2 = B</math>, <math>c^2 = C</math>. Then, our inequality can be reduced to
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<cmath>A + B + C \geq \sqrt{6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2}.</cmath>
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We now have to prove
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<cmath>(A + B + C)^2 \geq 6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2.</cmath>
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We can simplify:
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<cmath>A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \geq 6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2</cmath>
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<cmath>4A^2 + 4B^2 + 4C^2 \geq 4AB + 4BC + 4CA</cmath>
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<cmath>A^2 + B^2 + C^2 \geq AB + BC + CA.</cmath>
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Finally, we can apply AM-GM:
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<cmath>\frac{A^2 + B^2}{2} \geq AB</cmath>
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<cmath>\frac{B^2 + C^2}{2} \geq BC</cmath>
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<cmath>\frac{C^2 + A^2}{2} \geq CA</cmath>
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Adding these all up, we have the desired inequality
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<cmath>A^2 + B^2 + C^2 \geq AB + BC + CA, </cmath>
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and so the proof is complete.<math>\square</math>
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To have <math>A + B + C = 4S\sqrt{3}</math>, we must satisfy
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<cmath>\frac{A^2 + B^2}{2} = AB,</cmath>
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<cmath>\frac{B^2 + C^2}{2} = BC,</cmath>
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<cmath>\frac{C^2 + A^2}{2} = CA.</cmath>
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This is only true when <math>A = B = C</math>, and thus <math>a = b = c</math>. Therefore, equality happens when the triangle is equilateral.
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~mathboy100
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==Solution 2 (duality principle)==
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We firstly use the duality principle.
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<math>a=x+y~~b=x+z~~c=y+z</math>
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The LHS becomes <math>(x+y)^2+(x+z)^2+(y+z)^2</math>
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and the RHS becomes <math>4\sqrt{3}\sqrt{(x+y+z)xyz}</math> If we use Heron's formula.
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By AM-GM <math>\frac{(x+y+z)^3}{27} \ge xyz</math>
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Making this substitution <math>[ABC]</math> becomes
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<math>\sqrt{(x+y+z)^4\frac{1}{27}}</math>
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and once we take the square root of the area then our RHS becomes
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<math>\frac{4\sqrt{3}}{3\sqrt{3}}(x+y+z)^2=\frac{4}{3}(x+y+z)^2</math>
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Multiplying the RHS and the LHS by 3 we get the LHS to be
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<math>3((x+y)^2+(x+z)^2+(y+z)^2)=6(x^2+y^2+z^2+xy+yz+xz).</math>
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Our RHS becomes
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<math>4(x^2+y^2+z^2)+8(xy+yz+xz).</math>
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Subtracting <math>4(x^2+y^2+z^2)+6(xy+yz+xz)</math> we have the LHS equal to <math>(2(x^2+y^2+z^2))</math> and the RHS being <math>2(xy+xz+yz)</math>
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If LHS <math>\ge</math> RHS then LHS-RHS<math>\ge 0</math>
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LHS-RHS=<math>2(x^2+y^2+z^2)-2(xy+xz+yz)=x^2-2xy+y^2+x^2-2x+z^2+y^2-2yz+z^2=(x-y)^2+(x-z)^2+(y-z)^2.</math>
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<math>(x-y)^2+(x-z)^2+(y-z)^2 \ge 0</math> by the trivial inequality so therefore, <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math> and we're done.
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~PEKKA
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==Solution 3 (Trigonometry)==
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Let <math>\theta</math> be the angle between vertices <math>a</math> and <math>b</math>. We use two formulae:
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<cmath>\begin{align}c^2=a^2+b^2-2\cos\theta\\T=\frac12ab\sin\theta\end{align}</cmath>
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 +
Going backwards, 
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 +
<math>\begin{rcases}(a-b)^2=a^2-2ab+b^2\leq0\\-2\leq2\sin x\geq2\end{rcases}\forall a,b,x\in\Bbb R^+:a^2-2ab\sin x+b^2\geq0</math>
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<cmath>\begin{align*}a^2-2ab\sin(\theta+30^\circ)+b^2&\geq0\\
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a^2-2ab(\sin\theta\cos30^\circ+\cos\theta\sin30^\circ)+b^2&\geq0\\a^2-ab\sin\theta\sqrt3-ab\cos\theta+b^2&\geq0\\a^2-ab\cos\theta+b^2&\geq ab\sin\theta\sqrt3\\a^2-ab\cos\theta+b^2&\geq2\left(\frac12ab\sin\theta\right)\sqrt3\\a^2-ab\cos\theta+b^2&\geq2T\sqrt3\\2a^2-2ab\cos\theta+2b^2&\geq4T\sqrt3\\a^2+b^2+(a^2-2ab\cos\theta+b^2)&\geq4\sqrt3\ T\\\boxed{a^2+b^2+c^2\geq4\sqrt3\ T}\ \blacksquare\end{align*}</cmath>
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From <math>a^2-2ab\sin x+b^2\geq0</math>,
 +
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Equality occurs when <math>a=b</math> and <math>\sin(\theta+30^\circ)=1\implies\theta=60^\circ</math>, which is when <math>a=b=c</math> so that they form an equilateral triangle.
 +
 +
~ztilB
 +
 +
==Video Solution==
 +
 +
https://www.youtube.com/watch?v=ZYOB-KSEF3k&list=PLa8j0YHOYQQJGzkvK2Sm00zrh0aIQnof8&index=4
 +
- AMBRIGGS
 +
{{IMO box|year=1961|num-b=1|num-a=3}}

Latest revision as of 13:12, 13 February 2024

Problem

Let $a$, $b$, and $c$ be the lengths of a triangle whose area is S. Prove that

$a^2 + b^2 + c^2 \ge 4S\sqrt{3}$

In what case does equality hold?

Solution

By Heron's formula, we have \[S = \sqrt{s(s-a)(s-b)(s-c)}.\] This can be simplified to \[S = \sqrt{\left(\frac{a+b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)\left(\frac{a-b+c}{2}\right)\left(\frac{a+b-c}{2}\right)}.\] Next, we can factor out all of the $2$s and use a clever difference of squares: \[S = \frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\] \[S = \frac{1}{4}\sqrt{((b+c)^2 - a^2)(a^2 - (b-c)^2)}\] \[S = \frac{1}{4}\sqrt{(2bc + b^2 + c^2 - a^2)(2bc - b^2 - c^2 + a^2)}.\] We can now use difference of squares again: \[S = \frac{1}{4}\sqrt{4b^2c^2-(b^2 + c^2 - a^2)^2}\] \[4S\sqrt{3} = \sqrt{3(4b^2c^2 - (b^2 + c^2 - a^2)^2)}\] We must prove that the RHS of this equation is less than or equal to $a^2 + b^2 + c^2$.

Let $a^2 = A$, $b^2 = B$, $c^2 = C$. Then, our inequality can be reduced to \[A + B + C \geq \sqrt{6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2}.\] We now have to prove \[(A + B + C)^2 \geq 6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2.\] We can simplify: \[A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \geq 6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2\] \[4A^2 + 4B^2 + 4C^2 \geq 4AB + 4BC + 4CA\] \[A^2 + B^2 + C^2 \geq AB + BC + CA.\] Finally, we can apply AM-GM: \[\frac{A^2 + B^2}{2} \geq AB\] \[\frac{B^2 + C^2}{2} \geq BC\] \[\frac{C^2 + A^2}{2} \geq CA\] Adding these all up, we have the desired inequality \[A^2 + B^2 + C^2 \geq AB + BC + CA,\] and so the proof is complete.$\square$

To have $A + B + C = 4S\sqrt{3}$, we must satisfy \[\frac{A^2 + B^2}{2} = AB,\] \[\frac{B^2 + C^2}{2} = BC,\] \[\frac{C^2 + A^2}{2} = CA.\] This is only true when $A = B = C$, and thus $a = b = c$. Therefore, equality happens when the triangle is equilateral.

~mathboy100

Solution 2 (duality principle)

We firstly use the duality principle. $a=x+y~~b=x+z~~c=y+z$ The LHS becomes $(x+y)^2+(x+z)^2+(y+z)^2$ and the RHS becomes $4\sqrt{3}\sqrt{(x+y+z)xyz}$ If we use Heron's formula. By AM-GM $\frac{(x+y+z)^3}{27} \ge xyz$ Making this substitution $[ABC]$ becomes $\sqrt{(x+y+z)^4\frac{1}{27}}$ and once we take the square root of the area then our RHS becomes $\frac{4\sqrt{3}}{3\sqrt{3}}(x+y+z)^2=\frac{4}{3}(x+y+z)^2$ Multiplying the RHS and the LHS by 3 we get the LHS to be $3((x+y)^2+(x+z)^2+(y+z)^2)=6(x^2+y^2+z^2+xy+yz+xz).$ Our RHS becomes $4(x^2+y^2+z^2)+8(xy+yz+xz).$ Subtracting $4(x^2+y^2+z^2)+6(xy+yz+xz)$ we have the LHS equal to $(2(x^2+y^2+z^2))$ and the RHS being $2(xy+xz+yz)$ If LHS $\ge$ RHS then LHS-RHS$\ge 0$ LHS-RHS=$2(x^2+y^2+z^2)-2(xy+xz+yz)=x^2-2xy+y^2+x^2-2x+z^2+y^2-2yz+z^2=(x-y)^2+(x-z)^2+(y-z)^2.$ $(x-y)^2+(x-z)^2+(y-z)^2 \ge 0$ by the trivial inequality so therefore, $a^2 + b^2 + c^2 \ge 4S\sqrt{3}$ and we're done.

~PEKKA

Solution 3 (Trigonometry)

Let $\theta$ be the angle between vertices $a$ and $b$. We use two formulae:

\begin{align}c^2=a^2+b^2-2\cos\theta\\T=\frac12ab\sin\theta\end{align}

Going backwards,

$\begin{rcases}(a-b)^2=a^2-2ab+b^2\leq0\\-2\leq2\sin x\geq2\end{rcases}\forall a,b,x\in\Bbb R^+:a^2-2ab\sin x+b^2\geq0$

\begin{align*}a^2-2ab\sin(\theta+30^\circ)+b^2&\geq0\\ a^2-2ab(\sin\theta\cos30^\circ+\cos\theta\sin30^\circ)+b^2&\geq0\\a^2-ab\sin\theta\sqrt3-ab\cos\theta+b^2&\geq0\\a^2-ab\cos\theta+b^2&\geq ab\sin\theta\sqrt3\\a^2-ab\cos\theta+b^2&\geq2\left(\frac12ab\sin\theta\right)\sqrt3\\a^2-ab\cos\theta+b^2&\geq2T\sqrt3\\2a^2-2ab\cos\theta+2b^2&\geq4T\sqrt3\\a^2+b^2+(a^2-2ab\cos\theta+b^2)&\geq4\sqrt3\ T\\\boxed{a^2+b^2+c^2\geq4\sqrt3\ T}\ \blacksquare\end{align*}

From $a^2-2ab\sin x+b^2\geq0$,

Equality occurs when $a=b$ and $\sin(\theta+30^\circ)=1\implies\theta=60^\circ$, which is when $a=b=c$ so that they form an equilateral triangle.

~ztilB

Video Solution

https://www.youtube.com/watch?v=ZYOB-KSEF3k&list=PLa8j0YHOYQQJGzkvK2Sm00zrh0aIQnof8&index=4 - AMBRIGGS

1961 IMO (Problems) • Resources
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