Difference between revisions of "AoPS Wiki talk:Problem of the Day/June 21, 2011"

Latest revision as of 16:14, 22 June 2011

Problem

AoPSWiki:Problem of the Day/June 21, 2011

Solutions

First Solution

$(2x+3y)^2 + (3x-2y)^2 = 13(x^2+y^2)=26$ . Hence $(2x+3y)^2 \le 26$ or $2x+3y = \sqrt{26}$ . If $3x-2y=0$ and $2x+3y=\boxed{\sqrt{26}}$ , then $2x+3y$ attains this maximum value on the circle $x^2+y^2=2$ .

Second Solution

Let $x$ and $y$ be real numbers such that $x^2+y^2=2$ . Note that $|x|^2+|y|^2=2 \text{ and } 2|x|+3|y|\ge 2x+3y$ thus, we may assume that $x$ and $y$ are positive. Furthermore, by the Cauchy-Schwarz Inequality, we have $(4+9)(x^2+y^2)\ge (2x+3y)^2$ but since $x^2+y^2=2$ , the inequality is equivalent with $26\ge (2x+3y)^2$ or $\sqrt{26}\ge 2x+3y$ so the maximum is $\boxed{\sqrt{26}}$ and it is reached when $\frac{4}{x^2}=\frac{9}{y^2}\implies 2y=3x$ .

Third Solution

Imagine the equations graphed in the coordinate plane. $x^2+y^2=2$ is a circle centered at the origin with

radius $\sqrt{2}$ . $2x+3y=k$ is a line. We want to find the largest value of $k$ such that the line

intersects the circle, giving real number solutions for $x$ and $y$ . This occurs when $2x+3y=k$ is tangent

to the circle, and thus when the distance from the line to the origin is $\sqrt{2}$ . The distance from a point $(x_0, y_0)$ to a line $Ax+By+C=0$ is

$\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$ .

Plugging in $A=2, B=3, C=-k, x_0=0$ , and $y_0=0$ and setting the expression equal to $\sqrt{2}$ yields

$\pm\frac{k}{\sqrt{13}}=\sqrt{2}$ , or $k=\pm\sqrt{26}$ . We want the largest value of $k$ , so

$k=\boxed{\sqrt{26}}$ is the highest possible value.

Fourth Solution

By AM-QM, $\frac{2x + 3y}{13} = \frac{4 \cdot \frac{x}{2} + 9 \cdot \frac{y}{3}}{13} \le \sqrt{\frac{4 \cdot \left(\frac{x}{2}\right)^2 + 9 \left(\frac{y}{3}\right)^2}{13}} = \sqrt{\frac{2}{13}}$ , so $2x + 3y \le \sqrt{26}$ , equality when $\frac x2 = \frac y3$ .

@@ Line 1: / Line 1: @@
 ==Problem==
 {{:AoPSWiki:Problem of the Day/June 21, 2011}}
-==Solution==
+==Solutions==
+=== First Solution ===
 <math>(2x+3y)^2 + (3x-2y)^2 = 13(x^2+y^2)=26</math>.  Hence <math>(2x+3y)^2 \le 26</math> or <math>2x+3y = \sqrt{26}</math>.
-If <math>3x-2y=0</math> and <math>2x+3y=\sqrt{26}</math>, then <math>2x+3y</math> attains this maximum value on the circle <math>x^2+y^2=2</math>.
+If <math>3x-2y=0</math> and <math>2x+3y=\boxed{\sqrt{26}}</math>, then <math>2x+3y</math> attains this maximum value on the circle <math>x^2+y^2=2</math>.
+=== Second Solution ===
+Let <math>x</math> and <math>y</math> be real numbers such that <math>x^2+y^2=2</math>. Note that
+<cmath>|x|^2+|y|^2=2 \text{ and } 2|x|+3|y|\ge 2x+3y</cmath>
+thus, we may assume that <math>x</math> and <math>y</math> are positive. Furthermore, by the [[Cauchy-Schwarz Inequality]], we have
+<cmath>(4+9)(x^2+y^2)\ge (2x+3y)^2</cmath>
+but since <math>x^2+y^2=2</math>, the inequality is equivalent with
+<cmath>26\ge (2x+3y)^2</cmath>
+or
+<cmath>\sqrt{26}\ge 2x+3y</cmath>
+so the maximum is <math>\boxed{\sqrt{26}}</math> and it is reached when <math>\frac{4}{x^2}=\frac{9}{y^2}\implies 2y=3x</math>.
+===Third Solution===
+Imagine the equations graphed in the coordinate plane. <math> x^2+y^2=2 </math> is a circle centered at the origin with
+radius <math> \sqrt{2} </math>. <math> 2x+3y=k </math> is a line. We want to find the largest value of <math> k </math> such that the line
+intersects the circle, giving real number solutions for <math> x </math> and <math> y </math>. This occurs when <math> 2x+3y=k </math> is tangent
+to the circle, and thus when the distance from the line to the origin is <math> \sqrt{2} </math>. The distance from a point <math>
+(x_0, y_0) </math> to a line <math> Ax+By+C=0 </math> is
+<math> \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}} </math>.
+Plugging in <math> A=2, B=3, C=-k, x_0=0 </math>, and <math> y_0=0 </math> and setting the expression equal to <math> \sqrt{2} </math> yields
+<math> \pm\frac{k}{\sqrt{13}}=\sqrt{2} </math>, or <math> k=\pm\sqrt{26} </math>. We want the largest value of <math> k </math>, so
+<math> k=\boxed{\sqrt{26}} </math> is the highest possible value.
+=== Fourth Solution ===
+By [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality|AM-QM]], <math>\frac{2x + 3y}{13} = \frac{4 \cdot \frac{x}{2} + 9 \cdot \frac{y}{3}}{13} \le \sqrt{\frac{4 \cdot \left(\frac{x}{2}\right)^2 + 9 \left(\frac{y}{3}\right)^2}{13}} = \sqrt{\frac{2}{13}}</math>, so <math>2x + 3y \le \sqrt{26}</math>, equality when <math>\frac x2 = \frac y3</math>.

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