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1961 IMO Problems/Problem 2

Revision as of 21:31, 3 December 2022 by Mathboy100 (talk | contribs) (→‎Solution 2 (duality principle))

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 know that \[4b^2c^2-(b^2 + c^2 - a^2)^2 = 4a^2b^2-(a^2 + b^2 - c^2)^2 = 4c^2a^2-(c^2 + a^2 - b^2)^2.\] This is because the area of the triangle stays the same if we switch around the values of $a$, $b$, and $c$.

Thus, \[4S\sqrt{3} = \sqrt{4a^2b^2 + 4b^2c^2 + 4c^2a^2 - (b^2 + c^2 - a^2)^2 - (c^2 + a^2 - b^2)^2 - (a^2 + b^2 - c^2)^2}.\] 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 is reduced to \[A + B + C \geq \sqrt{4AB + 4BC + 4CA - (B + C - A)^2 - (C + A - B)^2 - (A + B - C)^2}.\] We will now simplify the RHS.

For any real numbers $x$, $y$, and $z$, \[(x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2xz - 2yz,\] and thus \[(x + y - z)^2 + (y + z - x)^2 + (z + x - y^2) = 3x^2 + 3y^2 + 3z^2 - 2xy - 2xz - 2yz.\] Applying this to the equation, we obtain \[\sqrt{4AB + 4BC + 4CA - 3A^2 - 3B^2 - 3C^2 + 2AB + 2BC + 2CA}\] \[\sqrt{6AB + 6BC + 6CA - 3A^2 - 3B^2 - 3C^2}.\] 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.$\blacksquare$

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.

1961 IMO (Problems) • Resources
Preceded by
Problem 1 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 3
All IMO Problems and Solutions

~PEKKA

Video Solution

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

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