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

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.

Resources

1995 IMO (Problems) • Resources
Preceded by Problem 1	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 3
All IMO Problems and Solutions

@@ Line 49: / Line 49: @@
 === Solution 4 ===
+After the setting <math>a=\frac{1}{x}, b=\frac{1}{y}, c=\frac{1}{z},</math> and as <math>abc=1</math> so <math>\left(\frac{1}{a}\cdot\frac{1}{b}\cdot\frac{1} {c}=1\right)</math> concluding <math>x y z=1 .</math>
+<math>\textsf{Claim}:</math>
+<cmath>
+\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{3}{2}
+</cmath>
+By Titu Lemma,
+<cmath>
+\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)}
+</cmath>
+<cmath>
+\implies\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{(x+y+z)}{2}
+</cmath>
+Now by AM-GM we know that <cmath> (x+y+z)\geq3\sqrt[3]{xyz}
+</cmath>and <math>xyz=1</math> which concludes to <math>\implies  (x+y+z)\geq3\sqrt[3]{1}</math>
+Therefore we get
+<cmath>
+\implies\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \geq \frac{3}{2}
+</cmath>Hence our claim is proved ~~ Aritra12
+=== Solution 5 ===
 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>
-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>
 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.
+=== Solution 6 ===
+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>.
+^ 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) ====
+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
 {{alternate solutions}}
@@ Line 66: / Line 113: @@
 [[Category:Olympiad Algebra Problems]]
+{{IMO box|year=1995|num-b=1|num-a=3}}

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