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[Title]
s̏ؖ(ʏo)

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% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̕sؖB

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% Level1̖BȉLevel7܂œlB
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[Level1]
yx>yzt

$x>ŷƂ$\\
@s$2x+y>x+2y$ؖB

\hfill@\begin{flushleft}
$(2x+y)-(x+2y)=x-y$\\
$x>yAx-y>0ł邩$\\
$(2x+y)-(x+2y)>0$\\
$2x+y>x+2y$\\
\end{flushleft}


$x>ŷƂ$\\
@s$3x-4y>2x-3y$ؖB

\hfill@\begin{flushleft}
$(3x-4y)-(2x-3y)=x-y$\\
$x>yAx-y>0ł邩$\\
$(3x-4y)-(2x-3y)>0$\\
$3x-4y>2x-3y$\\
\end{flushleft}


$x>ŷƂ$\\
@s$3x+4y>2x+5y$ؖB

\hfill@\begin{flushleft}
$(3x+4y)-(2x+5y)=x-y$\\
$x>yAx-y>0ł邩$\\
$(3x+4y)-(2x+5y)>0$\\
$3x+4y>2x+5y$\\
\end{flushleft}


$x>ŷƂ$\\
@s$5x-3y>3x-y$ؖB

\hfill@\begin{flushleft}
$(5x-3y)-(3x-y)=2x-2y=2(x-y)$\\
$x>yA2(x-y)>0ł邩$\\
$(5x-3y)-(3x-y)>0$\\
$5x-3y>3x-y$\\
\end{flushleft}


[Level2]
yx>a,y>bzt

$x>1,y>1̂Ƃ$\\
@s$xy+1>x+y$ؖB

\hfill@\begin{flushleft}
$@(xy+1)-(x+y)$\\
$@=xy-x-y+1$\\
$@=x(y-1)-(y-1)$\\
$@=(x-1)(y-1)$\\
$x>1,y>1A$\\
$@x-1>0,y-1>0ł邩$\\
$(x-1)(y-1)>0$\\
$(xy+1)-(x+y)>0$\\
$xy+1>x+y$\\
\end{flushleft}


$x>2,y>3̂Ƃ$\\
@s$xy+6>3x+2y$ؖB

\hfill@\begin{flushleft}
$@(xy+6)-(3x+2y)$\\
$@=xy-3x-2y+6$\\
$@=x(y-3)-2(y-3)$\\
$@=(x-2)(y-3)$\\
$x>2,y>3A$\\
$@x-2>0,y-3>0ł邩$\\
$(x-2)(y-3)>0$\\
$(xy+6)-(3x+2y)>0$\\
$xy+6>3x+2y$\\
\end{flushleft}


$x>1,y>2̂Ƃ$\\
@s$xy+2>2x+y$ؖB

\hfill@\begin{flushleft}
$@(xy+2)-(2x+y)$\\
$@=xy-2x-y+2$\\
$@=x(y-2)-(y-2)$\\
$@=(x-1)(y-2)$\\
$x>1,y>2A$\\
$@x-1>0,y-2>0ł邩$\\
$(x-1)(y-2)>0$\\
$(xy+2)-(2x+y)>0$\\
$xy+2>2x+y$\\
\end{flushleft}


$x>4,y>5̂Ƃ$\\
@s$xy+20>5x+4y$ؖB

\hfill@\begin{flushleft}
$@(xy+20)-(5x+4y)$\\
$@=xy-5x-4y+20$\\
$@=x(y-5)-4(y-5)$\\
$@=(x-4)(y-5)$\\
$x>4,y>5A$\\
$@x-4>0,y-5>0ł邩$\\
$(x-4)(y-5)>0$\\
$(xy+20)-(5x+4y)>0$\\
$xy+20>5x+4y$\\
\end{flushleft}


$x>5,y>3̂Ƃ$\\
@s$xy+15>3x+5y$ؖB

\hfill@\begin{flushleft}
$@(xy+15)-(3x+5y)$\\
$@=xy-3x-5y+15$\\
$@=x(y-3)-5(y-3)$\\
$@=(x-5)(y-3)$\\
$x>5,y>3A$\\
$@x-5>0,y-3>0ł邩$\\
$(x-5)(y-3)>0$\\
$(xy+15)-(3x+5y)>0$\\
$xy+15>3x+5y$\\
\end{flushleft}


[Level3]
()^20𗘗p

s$x^2+y^2 \geqq 2xy$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(x^2+y^2)-2xy=x^2 -2xy +y^2=(x-y)^2\geqq 0$\\
$x^2+y^2\geqq 2xy$\\
̕sœ̂́A\\
@$x-y=0$Ȃ킿$x=y$̎łB
\end{flushleft}


s$x^2+4y^2 \geqq 4xy$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(x^2+4y^2)-4xy=x^2 -4xy +4y^2=(x-2y)^2\geqq 0$\\
$x^2+4y^2\geqq 4xy$\\
̕sœ̂́A\\
@$x-2y=0$Ȃ킿$x=2y$̎łB
\end{flushleft}


s$(x+y)^2 \geqq 4xy$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(x+y)^2-4xy=x^2 -2xy +y^2=(x-y)^2\geqq 0$\\
$(x+y)^2\geqq 4xy$\\
̕sœ̂́A\\
@$x-y=0$Ȃ킿$x=y$̎łB
\end{flushleft}




[Level4]
()^2{()^20̗p

s$a^2+5b^2 \geqq 4ab$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(a^2+5b^2)-4ab$\\
$=a^2 -4ab +5b^2$\\
$=a^2 -4ab +4b^2 +b^2$\\
$=(a -2b)^2 +b^2$\\
$(a-2b)^2 \geqq 0Cb^2 \geqq 0$ł邩\\
$(a -2b)^2 +b^2 \geqq 0$\\
$a^2+5b^2 \geqq 4ab$\\
̕sœ̂́A\\
@$a-2b=0$$b=0$\\
@Ȃ킿$a=b=0$̎łB
\end{flushleft}


s$a^2+10b^2 \geqq 6ab$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(a^2+10b^2)-6ab$\\
$=a^2 -6ab +10b^2$\\
$=a^2 -6ab +9b^2 +b^2$\\
$=(a -3b)^2 +b^2$\\
$(a-3b)^2 \geqq 0Cb^2 \geqq 0$ł邩\\
$(a -3b)^2 +b^2 \geqq 0$\\
$a^2+10b^2 \geqq 6ab$\\
̕sœ̂́A\\
@$a-3b=0$$b=0$\\
@Ȃ킿$a=b=0$̎łB
\end{flushleft}


s$a^2+9b^2 \geqq 4ab$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(a^2+9b^2)-4ab$\\
$=a^2 -4ab +9b^2$\\
$=a^2 -4ab +4b^2 +5b^2$\\
$=(a -2b)^2 +5b^2$\\
$(a-2b)^2 \geqq 0C5b^2 \geqq 0$ł邩\\
$(a -2b)^2 +5b^2 \geqq 0$\\
$a^2+9b^2 \geqq 4ab$\\
̕sœ̂́A\\
@$a-2b=0$$2b=0$\\
@Ȃ킿$a=b=0$̎łB
\end{flushleft}


s$a^2+6b^2 \geqq 2ab$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
$(a^2+6b^2)-2ab$\\
$=a^2 -2ab +6b^2$\\
$=a^2 -2ab +b^2 +5b^2$\\
$=(a -b)^2 +5b^2$\\
$(a-b)^2 \geqq 0C5b^2 \geqq 0$ł邩\\
$(a -b)^2 +5b^2 \geqq 0$\\
$a^2+6b^2 \geqq 2ab$\\
̕sœ̂́A\\
@$a-b=0$$5b=0$\\
@Ȃ킿$a=b=0$̎łB
\end{flushleft}


[Level5]
(ニ[g܂ގ)^2̗p

$a>0Cb>0̂Ƃ$\\
@s$\sqrt{a}+\sqrt{b}>\sqrt{a+b}$ؖB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(\sqrt{a}+\sqrt{b})^2-(\sqrt{a+b})^2$\\
$=(a+2\sqrt{ab}+b)-(a+b)$\\
$=2\sqrt{ab}>0$\\
$(\sqrt{a}+\sqrt{b})^2>(\sqrt{a+b})^2$\\
$\sqrt{a}+\sqrt{b}>0C\sqrt{a+b}>0$ł邩\\
@$\sqrt{a}+\sqrt{b}>\sqrt{a+b}$
\end{flushleft}


$x>0̂Ƃ$\\
@s$1+x>\sqrt{1+2x}$ؖB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(1+x)^2-(\sqrt{1+2x})^2$\\
$=(1+2x+x^2)-(1+2x)=x^2>0$\\
$(1+x)^2>(\sqrt{1+2x})^2$\\
$1+x>0C\sqrt{1+2x}>0$ł邩\\
@$1+x>\sqrt{1+2x}$
\end{flushleft}


$a>0Cb>0̂Ƃ$\\
@s$\sqrt{a}+2\sqrt{b}>\sqrt{a+4b}$ؖB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(\sqrt{a}+2\sqrt{b})^2-(\sqrt{a+4b})^2$\\
$=(a+4\sqrt{ab}+4b)-(a+4b)$\\
$=4\sqrt{ab}>0$\\
$(\sqrt{a}+2\sqrt{b})^2>(\sqrt{a+4b})^2$\\
$\sqrt{a}+2\sqrt{b}>0C\sqrt{a+4b}>0$ł邩\\
@$\sqrt{a}+2\sqrt{b}>\sqrt{a+4b}$
\end{flushleft}


$a>0Cb>0̂Ƃ$\\
@s$2\sqrt{a}+3\sqrt{b}>\sqrt{4a+9b}$ؖB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(2\sqrt{a}+3\sqrt{b})^2-(\sqrt{4a+9b})^2$\\
$=(4a+12\sqrt{ab}+9b)-(4a+9b)$\\
$=12\sqrt{ab}>0$\\
$(2\sqrt{a}+3\sqrt{b})^2>(\sqrt{4a+9b})^2$\\
$2\sqrt{a}+3\sqrt{b}>0C\sqrt{4a+9b}>0$ł邩\\
@$2\sqrt{a}+3\sqrt{b}>\sqrt{4a+9b}$
\end{flushleft}


[Level6]
|Βl܂ގ|^2̗p

s$|a|+|b|\geqq|a+b|$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(|a|+|b|)^2-|a+b|^2$\\
$=|a|^2 +2|a||b|+|b|^2 -(a+b)^2$\\
$=a^2 +2|ab|+b^2 -(a^2 +2ab+b^2)$\\
$=2(|ab|-ab) \geqq 0$\\
$(|a|+|b|)^2 \geqq |a+b|^2$\\
$|a|+|b|\geqq 0C@ |a+b| \geqq 0$ł邩\\
@$|a|+|b| \geqq |a+b|$\\
̕sœ̂́A\\
@$|ab|=ab$Ȃ킿$ab\geqq0$̎łB
\end{flushleft}


s$|a|+2|b|\geqq|a+2b|$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(|a|+2|b|)^2-|a+2b|^2$\\
$=|a|^2 +4|a||b|+4|b|^2 -(a+2b)^2$\\
$=a^2 +4|ab|+4b^2 -(a^2 +4ab+4b^2)$\\
$=4(|ab|-ab) \geqq 0$\\
$(|a|+2|b|)^2 \geqq |a+2b|^2$\\
$|a|+2|b|\geqq 0C@ |a+2b| \geqq 0$ł邩\\
@$|a|+2|b| \geqq |a+2b|$\\
̕sœ̂́A\\
@$|ab|=ab$Ȃ킿$ab\geqq0$̎łB
\end{flushleft}


s$|a|+3|b|\geqq|a+3b|$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(|a|+3|b|)^2-|a+3b|^2$\\
$=|a|^2 +6|a||b|+9|b|^2 -(a+3b)^2$\\
$=a^2 +6|ab|+9b^2 -(a^2 +6ab+9b^2)$\\
$=9(|ab|-ab) \geqq 0$\\
$(|a|+3|b|)^2 \geqq |a+3b|^2$\\
$|a|+3|b|\geqq 0C@ |a+3b| \geqq 0$ł邩\\
@$|a|+3|b| \geqq |a+3b|$\\
̕sœ̂́A\\
@$|ab|=ab$Ȃ킿$ab\geqq0$̎łB
\end{flushleft}


s$4|a|+|b|\geqq|4a+b|$ؖB\\
@܂A藧𒲂ׂB

\hfill@\begin{flushleft}
ӂ̍̕l\\
$(4|a|+|b|)^2-|4a+b|^2$\\
$=16|a|^2 +8|a||b|+|b|^2 -(4a+b)^2$\\
$=16a^2 +8|ab|+b^2 -(16a^2 +8ab+b^2)$\\
$=8(|ab|-ab) \geqq 0$\\
$(4|a|+|b|)^2 \geqq |4a+b|^2$\\
$4|a|+|b|\geqq 0C@ |4a+b| \geqq 0$ł邩\\
@$4|a|+|b| \geqq |4a+b|$\\
̕sœ̂́A\\
@$|ab|=ab$Ȃ킿$ab\geqq0$̎łB
\end{flushleft}


[Level7]
敽ς̗p

$a>0̂ƂAa+\nfrac{1}{a}\geqq 2$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$a>0,@\nfrac{1}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$a+\nfrac{1}{a} \geqq 2\sqrt{a\cdot \nfrac{1}{a}}=2$\\
@$a+\nfrac{1}{a} \geqq 2$\\
̕sœ̂́A\\
@$a>0a=\nfrac{1}{a}Ȃ킿a=1$̎łB
\end{flushleft}


$a>0̂ƂAa+\nfrac{4}{a}\geqq 4$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$a>0,@\nfrac{4}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$a+\nfrac{4}{a} \geqq 2\sqrt{a\cdot \nfrac{4}{a}}=4$\\
@$a+\nfrac{4}{a} \geqq 4$\\
̕sœ̂́A\\
@$a>0a=\nfrac{4}{a}Ȃ킿a=2$̎łB
\end{flushleft}


$a>0̂ƂAa+\nfrac{16}{a}\geqq 8$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$a>0,@\nfrac{16}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$a+\nfrac{16}{a} \geqq 2\sqrt{a\cdot \nfrac{16}{a}}=8$\\
@$a+\nfrac{16}{a} \geqq 8$\\
̕sœ̂́A\\
@$a>0a=\nfrac{16}{a}Ȃ킿a=4$̎łB
\end{flushleft}


$a>0̂ƂAa+\nfrac{25}{a}\geqq 10$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$a>0,@\nfrac{25}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$a+\nfrac{25}{a} \geqq 2\sqrt{a\cdot \nfrac{25}{a}}=10$\\
@$a+\nfrac{25}{a} \geqq 10$\\
̕sœ̂́A\\
@$a>0a=\nfrac{25}{a}Ȃ킿a=5$̎łB
\end{flushleft}


$a>0̂ƂA2a+\nfrac{1}{a}\geqq 2\sqrt{2}$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$2a>0,@\nfrac{1}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$2a+\nfrac{1}{a} \geqq 2\sqrt{2a\cdot \nfrac{1}{a}}=2\sqrt{2}$\\
@$2a+\nfrac{1}{a} \geqq 2\sqrt{2}$\\
̕sœ̂́A\\
@$a>02a=\nfrac{1}{a}Ȃ킿a=\nfrac{\sqrt{2}}{2}$̎łB
\end{flushleft}


$a>0,@b>0̂ƂA\nfrac{a}{b}+\nfrac{b}{a}\geqq 2$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$\nfrac{a}{b}>0,@\nfrac{b}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$\nfrac{a}{b}+\nfrac{b}{a} \geqq 2\sqrt{\nfrac{a}{b}\cdot \nfrac{b}{a}}=2$\\
@$\nfrac{a}{b}+\nfrac{b}{a} \geqq 2$\\
̕sœ̂́A\\
$a>0,b>0\nfrac{a}{b}=\nfrac{b}{a}Ȃ킿a=b$̎łB
\end{flushleft}


$a>0,@b>0̂ƂA\nfrac{a}{2b}+\nfrac{2b}{a}\geqq 2$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$\nfrac{a}{2b}>0,@\nfrac{2b}{a}>0$ł邩A\\
敽ς̊֌WɂA\\
$\nfrac{a}{2b}+\nfrac{2b}{a} \geqq 2\sqrt{\nfrac{a}{2b}\cdot \nfrac{2b}{a}}=2$\\
@$\nfrac{a}{2b}+\nfrac{2b}{a} \geqq 2$\\
̕sœ̂́A\\
$a>0,b>0\nfrac{a}{2b}=\nfrac{2b}{a}Ȃ킿a=2b$̎łB
\end{flushleft}


$a>0,@b>0̂ƂA\nfrac{b}{a}+\nfrac{4a}{b}\geqq 4$ؖ\\
܂藧𒲂ׂB

\hfill@\begin{flushleft}
$\nfrac{b}{a}>0,@\nfrac{4a}{b}>0$ł邩A\\
敽ς̊֌WɂA\\
$\nfrac{b}{a}+\nfrac{4a}{b} \geqq 2\sqrt{\nfrac{b}{a}\cdot \nfrac{4a}{b}}=4$\\
@$\nfrac{b}{a}+\nfrac{4a}{b} \geqq 4$\\
̕sœ̂́A\\
$a>0,b>0\nfrac{b}{a}=\nfrac{4a}{b}Ȃ킿2a=b$̎łB
\end{flushleft}


[EOF]
