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Difference between revisions of "1995 IMO Problems/Problem 2"

(→‎Solution 7 from Brilliant Wiki (Muirheads) =)
(→‎Solution 6)
 
(13 intermediate revisions by 4 users not shown)
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Proceed as in Solution 1, to arrive at the equivalent inequality
 
Proceed as in Solution 1, to arrive at the equivalent inequality
 
<cmath> \frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} \ge \frac{3}{2} . </cmath>
 
<cmath> \frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} \ge \frac{3}{2} . </cmath>
−
But we know that <cmath>x + y + z \ge 3xyz \ge 3</cmath> by AM-GM. Furthermore,
+
But we know that <cmath>x + y + z \ge 3xyz = 3</cmath> by AM-GM. Furthermore,
 
<cmath> (x + y + y + z + x + z) (\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y}) \ge (x + y + z)^2 </cmath>
 
<cmath> (x + y + y + z + x + z) (\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y}) \ge (x + y + z)^2 </cmath>
 
by Cauchy-Schwarz, and so dividing by <math>2(x + y + z)</math> gives
 
by Cauchy-Schwarz, and so dividing by <math>2(x + y + z)</math> gives
−
<cmath> \begin{align*}\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} &\ge \frac{(x + y + z)}{2} \\ &\ge \frac{3}{2} \end{align*},</cmath>
+
<cmath> \begin{align*}\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} &\ge \frac{(x + y + z)}{2} \\ &\ge \frac{3}{2} \end{align*}</cmath>
 
as desired.
 
as desired.
  
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Without clever substitutions, and only AM-GM!
 
Without clever substitutions, and only AM-GM!
  
−
Note that <math>abc = 1 \implies a = \frac{1}{bc}</math>. The cyclic sum becomes <math>\sum_{cyc}\frac{(bc)^3}{b + c}</math>. Note that by AM-GM, the cyclic sum is greater than or equal to <math>3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}</math>. We now see that we have the three so we must be on the right path. We now only need to show that <math>\frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^\frac13</math>. Notice that by AM-GM, <math>a + b \geq 2\sqrt{ab}</math>, <math>b + c \geq 2\sqrt{bc}</math>, and <math>a + c \geq 2\sqrt{ac}</math>. Thus, we see that <math>(a+b)(b+c)(a+c) \geq 8</math>, concluding that <math>\sum_{cyc} \frac{(bc)^3}{b+c} \geq \frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}</math>
+
Note that <math>abc = 1 \implies a = \frac{1}{bc}</math>. The cyclic sum becomes <math>\sum_{cyc}\frac{(bc)^3}{b + c}</math>. Note that by AM-GM, the cyclic sum is greater than or equal to <math>3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}</math>. We now see that we have the three so we must be on the right path. We now only need to show that <math>\frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^\frac13</math>. Notice that by AM-GM, <math>a + b \geq 2\sqrt{ab}</math>, <math>b + c \geq 2\sqrt{bc}</math>, and <math>a + c \geq 2\sqrt{ac}</math>. Thus, we see that <math>(a+b)(b+c)(a+c) \geq 8</math>, concluding that <math>\sum_{cyc} \frac{(bc)^3}{b+c} \geq \frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}</math>.
  
 +
^ This solution is incorrect, as it does not prove inequalities in the right direction. Proving that <math>A \geq B</math>, and <math>C \geq B</math> does not show that <math>A \geq C \geq B</math>.
  
 
=== Solution 7 from Brilliant Wiki (Muirheads) ====
 
=== Solution 7 from Brilliant Wiki (Muirheads) ====
 
https://brilliant.org/wiki/muirhead-inequality/
 
https://brilliant.org/wiki/muirhead-inequality/
 +
 +
 +
=== Solution 8 (fast Titu's Lemma no substitutions) ===
 +
Rewrite <math>\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)}</math> as <math>\frac{(1/a)^2}{a(b+c)} + \frac{(1/b)^2}{b(a+c)} + \frac{(1/c)^2}{c(a+b)}</math>.
 +
 +
Now applying Titu's lemma yields <math>\frac{(1/a)^2}{a(b+c)} + \frac{(1/b)^2}{b(a+c)} + \frac{(1/c)^2}{c(a+b)} \geq \frac{(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})^2}{a(b+c) + b(a+c) + c(a+b)} = \frac{(ab + bc + ca)^2}{2(ab + bc + ca)} = \frac{ab + bc + ca}{2}</math>.
 +
 +
Now applying the AM-GM inequality on <math>ab + bc +ca \geq 3((abc)^2)^{\frac{1}{3}} = 3</math>. The result now follows.
 +
 +
Note: <math>ab + bc + ca = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}</math>, because <math>abc = 1</math>. (Why? Because <math>a = \frac{1}{bc}</math>, and hence <math>\frac{1}{a} = bc</math>).
 +
 +
~th1nq3r
 +
  
 
Scroll all the way down
 
Scroll all the way down
 
{{alternate solutions}}
 
{{alternate solutions}}
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=== Solution 8 (AM-GM only)===
 
−
 
−
We are given that <math>a,b,</math> and <math>c</math> are positive real numbers such that <math>abc=1.</math> This means that by AM-GM, we have <cmath>\frac{a+b+c}{3} \geq \sqrt[3]{abc}=\sqrt[3]{1},</cmath> so <math>a+b+c>3.</math> Let <math>a+b+c=3+k</math> for some nonnegative real number <math>k.</math> By AM-GM, we also have <cmath>\frac{a+(b+c)}{2} \geq \sqrt{a(b+c)}</cmath> <cmath>\frac{3+k}{2} \geq \frac{3}{2} \geq \sqrt{a(b+c)}.</cmath> Squaring the last two parts of this inequality chain, we have <cmath>a(b+c) \leq \frac{3}{2} \leq \frac{9}{4}.</cmath> Manipulating the first two parts of this inequality chain gives <cmath>\frac{2}{3} \leq \frac{1}{a(b+c)}.</cmath> <cmath>\frac{1}{a^3(b+c)} \geq \frac{2}{3a^2}.</cmath> Analogously, we obtain the two inequalities <math>\frac{1}{b^3(a+c)} \geq \frac{2}{3b^2}</math> and <math>\frac{1}{c^3(a+b)} \geq \frac{2}{3c^2}.</math> Now by AM-GM, we have <cmath>\frac{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}{3} \geq \sqrt[3]{\frac{1}{a^2 \cdot b^2 \cdot c^2}}=1,</cmath> so <math>\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \geq 3.</math> Using all of our information, we can conclude that <cmath>\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)} \geq \frac{2}{3a^2}+\frac{2}{3b^2} + \frac{2}{3c^2} = \frac{2}{3} \left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \right) \geq \frac{2}{3} \cdot 3 \geq \frac{3}{2},</cmath> and the result follows. <math>\mathbb{Q.E.D.}</math>
 
  
 
== Resources ==
 
== Resources ==
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[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]
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{{IMO box|year=1995|num-b=1|num-a=3}}

Latest revision as of 00:01, 26 July 2024

Problem

(Nazar Agakhanov, Russia) Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that \[\frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}.\]

Solution

Solution 1

We make the substitution $x= 1/a$, $y=1/b$, $z=1/c$. Then \begin{align*} \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} &= \frac{x^3}{xyz(1/y+1/z)} + \frac{y^3}{xyz(1/z+1/x)} + \frac{z^3}{xyz(1/x+1/z)} \\ &= \frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} . \end{align*} Since $(x^2,y^2,z^2)$ and $\bigl( 1/(y+z), 1/(z+x), 1/(x+y) \bigr)$ are similarly sorted sequences, it follows from the Rearrangement Inequality that \[\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} \ge \frac{1}{2} \left( \frac{y^2+z^2}{y+z} + \frac{z^2+x^2}{z+x} + \frac{x^2+y^2}{x+y} \right) .\] By the Power Mean Inequality, \[\frac{y^2+z^2}{y+z} \ge \frac{(y+z)^2}{2(x+y)} = \frac{x+y}{2} .\] Symmetric application of this argument yields \[\frac{1}{2}\left( \frac{y^2+z^2}{y+z} + \frac{z^2+x^2}{z+x} + \frac{x^2+y^2}{x+y} \right) \ge \frac{1}{2}(x+y+z) .\] Finally, AM-GM gives us \[\frac{1}{2}(x+y+z) \ge \frac{3}{2}xyz = \frac{3}{2},\] as desired. $\blacksquare$

Solution 2

We make the same substitution as in the first solution. We note that in general, \[\frac{p}{q+r} = \frac{(p+q+r)}{(p+q+r)-p} - 1 .\] It follows that $(x,y,z)$ and $\bigl(x/(y+z), y/(z+x), z/(x+y)\bigr)$ are similarly sorted sequences. Then by Chebyshev's Inequality, \[\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} \ge \frac{1}{3}(x+y+z) \left(\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \right) .\] By AM-GM, $\frac{x+y+z}{3} \ge \sqrt[3]{xyz}=1$, and by Nesbitt's Inequality, \[\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \ge \frac{3}{2}.\] The desired conclusion follows. $\blacksquare$

Solution 3

Without clever substitutions: By Cauchy-Schwarz, \[\left(\sum_{cyc}\dfrac{1}{a^3 (b+c)}\right)\left(\sum_{cyc}a(b+c)\right)\geq \left( \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right) ^2=(ab+ac+bc)^2\] Dividing by $2(ab+bc+ac)$ gives \[\dfrac{1}{a^3 (b+c)}+\dfrac{1}{b^3 (a+c)}+\dfrac{1}{c^3 (a+b)}\geq \dfrac{1}{2}(ab+bc+ac)\geq \dfrac{3}{2}\] by AM-GM.

Solution 3b

Without clever notation: By Cauchy-Schwarz, \[\left(a(b+c) + b(c+a) + c(a+b)\right) \cdot \left(\frac{1}{a^3 (b+c)} + \frac{1}{b^3 (c+a)} + \frac{1}{c^3 (a+b)}\right)\] \[\ge \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)^2\] \[= (ab + bc + ac)^2\]

Dividing by $2(ab + bc + ac)$ and noting that $ab + bc + ac \ge 3(a^2b^2c^2)^{\frac{1}{3}} = 3$ by AM-GM gives \[\frac{1}{a^3 (b+c)} + \frac{1}{b^3 (c+a)} + \frac{1}{c^3 (a+b)} \ge \frac{ab + bc + ac}{2} \ge \frac{3}{2},\] as desired.

Solution 4

After the setting $a=\frac{1}{x}, b=\frac{1}{y}, c=\frac{1}{z},$ and as $abc=1$ so $\left(\frac{1}{a}\cdot\frac{1}{b}\cdot\frac{1} {c}=1\right)$ concluding $x y z=1 .$

$\textsf{Claim}:$ \[\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{3}{2}\] By Titu Lemma, \[\implies\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{(x+y+z)^{2}}{2(x+y+z)}\] \[\implies\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{(x+y+z)}{2}\] Now by AM-GM we know that \[(x+y+z)\geq3\sqrt[3]{xyz}\]and $xyz=1$ which concludes to $\implies (x+y+z)\geq3\sqrt[3]{1}$

Therefore we get

\[\implies\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{3}{2}\]Hence our claim is proved ~~ Aritra12

Solution 5

Proceed as in Solution 1, to arrive at the equivalent inequality \[\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} \ge \frac{3}{2} .\] But we know that \[x + y + z \ge 3xyz = 3\] by AM-GM. Furthermore, \[(x + y + y + z + x + z) (\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y}) \ge (x + y + z)^2\] by Cauchy-Schwarz, and so dividing by $2(x + y + z)$ gives \begin{align*}\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y} &\ge \frac{(x + y + z)}{2} \\ &\ge \frac{3}{2} \end{align*} as desired.

Solution 6

Without clever substitutions, and only AM-GM!

Note that $abc = 1 \implies a = \frac{1}{bc}$. The cyclic sum becomes $\sum_{cyc}\frac{(bc)^3}{b + c}$. Note that by AM-GM, the cyclic sum is greater than or equal to $3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}$. We now see that we have the three so we must be on the right path. We now only need to show that $\frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^\frac13$. Notice that by AM-GM, $a + b \geq 2\sqrt{ab}$, $b + c \geq 2\sqrt{bc}$, and $a + c \geq 2\sqrt{ac}$. Thus, we see that $(a+b)(b+c)(a+c) \geq 8$, concluding that $\sum_{cyc} \frac{(bc)^3}{b+c} \geq \frac32 \geq 3\left(\frac{1}{(a+b)(b+c)(a+c)}\right)^{\frac13}$.

^ This solution is incorrect, as it does not prove inequalities in the right direction. Proving that $A \geq B$, and $C \geq B$ does not show that $A \geq C \geq B$.

Solution 7 from Brilliant Wiki (Muirheads) =

https://brilliant.org/wiki/muirhead-inequality/


Solution 8 (fast Titu's Lemma no substitutions)

Rewrite $\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)}$ as $\frac{(1/a)^2}{a(b+c)} + \frac{(1/b)^2}{b(a+c)} + \frac{(1/c)^2}{c(a+b)}$.

Now applying Titu's lemma yields $\frac{(1/a)^2}{a(b+c)} + \frac{(1/b)^2}{b(a+c)} + \frac{(1/c)^2}{c(a+b)} \geq \frac{(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})^2}{a(b+c) + b(a+c) + c(a+b)} = \frac{(ab + bc + ca)^2}{2(ab + bc + ca)} = \frac{ab + bc + ca}{2}$.

Now applying the AM-GM inequality on $ab + bc +ca \geq 3((abc)^2)^{\frac{1}{3}} = 3$. The result now follows.

Note: $ab + bc + ca = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$, because $abc = 1$. (Why? Because $a = \frac{1}{bc}$, and hence $\frac{1}{a} = bc$).

~th1nq3r


Scroll all the way down Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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1995 IMO (Problems) • Resources
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