% %ȉ̕RgƂ̂łC_ł͂܂ł܂B
% ̃^Cg
[Title]
104_WC

% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
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[usepackage]
\usepackage{color,amssymb,graphicx,ascmac}

% ꂼ̖𓚂$\displaystyle $tꍇ́C@ON ܂1
% ꂼ̖𓚂$\displaystyle $tȂꍇ́COFF܂0
% up̕ҏWv|u[U[ݒv̉ɂݒ
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% LqȂ΁Cup̕ҏWv|u[U[ݒv
% @@@@@@@@@@@@@@@@@@@ɂݒ肪D悳܂B
[displaystyle]
OFF


% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
W̗vfibLqj
̏WCvfׂĕ\ȂB\\
1̐̋̏W$A$

$A=\{ 2,\,4,\,6,\,8 \}$

̏WCvfׂĕ\ȂB\\
15̖̐񐔂̏W$B$

$B=\{ 1,\,3,\,5,\,15 \}$

̏WCvfׂĕ\ȂB\\
1ȏ20ȉ3̔{̏W$A$

$A=\{ 3,\,6,\,9,\,12,\,15,\,18 \}$

̏WCvfׂĕ\ȂB\\
16̖̐񐔂̏W$B$

$B=\{ 1,\,2,\,4,\,8,\,16 \}$

̏WCvfׂĕ\ȂB\\
24̖̐񐔂̏W$A$

$A=\{ 1,\,2,\,3,\,4,\,6,\,8,\,12,\,24 \}$

1ȏ50ȉ7̔{̏W$B$

$B=\{ 7,\,14,\,21,\,28,\,35,\,42,\,49\}$

̏WCvfׂ@ŕ\B\\
$\{ x\,|\,-3<x<5,\,x͐ \}$

$\{ -2,\,-1,\,0,\,1,\,2,\,3,\,4 \}$

̏WCvfׂ@ŕ\B\\
$\{ x\,|\,x36̖̐ \}$

$\{ 1,\,2,\,3,\,4,\,6,\,9,\,12,\,18,\,36 \}$

̏WCvfׂ@ŕ\B\\
$\{ x\,|\,2x-8<0,\,x͎R \}$

process
$\{ x\,|\,x<4,\,x͎R \}$

$\{ 1,\,2,\,3 \}$

̏WCvfׂ@ŕ\B\\
$\{ 3n-1\,|\,1<n<5,\,n͐ \}$

process
$n=2,\,3,\,4$
$3n-1=5,\,8,\,11$

$\{ 5,\,8,\,11 \}$

2̏WɂāC$ A \cap B$$ A\cup B$
vfׂ@ŕ\B\\
$A=\{ 2,\,3,\,5,\,7,\,11,\,13 \}$\\
$B=\{ 1,\,2,\,4,\,5,\,10 \}$

$ A \cap B=\{ 2,\,5 \}$C\\
$ A\cup B=\{ 1,\,2,\,3,\,4,\,5,\,7,\,10,\,11,\,13 \}$

2̏W\\
$A=\{ 1,\,3,\,5,\,7,\,9,\,11 \}$\\
$B=\{ 3,\,6,\,9,\,12 \}$
ɂāC̖ɓȂB\\
\textcircled{\small 1}\,$A\cap B$߂ȂB\\
\textcircled{\small 2}\,$A\cup B$߂ȂB

\textcircled{\small 1}\,$A\cap B=\{ 3,\,9 \}$\\
\textcircled{\small 2}\,$A\cup B=\{ 1,\,3,\,5,\,6,\,7,\,9,\,11,\,12 \}$









[Level2]
WCWibIj
̏ŴC$A=\{1,\,2,\,3,\,4,\,5,\,6,\,7\}$̕Włׂ̂ĂȂB\\
$B=\{ 2,\,3,\,4,\,5 \}$C$C=\{ 1,\,2,\,4,\,8 \}$C$D=\{ 1,\,3,\,7 \}$

$B$C$D$

̋󗓂̒$\in$܂$\notin$̂ꂩB
C$N$͎RŜ̏W\B\\
$A=\{ x\,|\,x^2+4x+3=0 \}$̂ƂC2\,\fbox{@A@}$A$C$-3$\,\fbox{@C@}$A$

process
$(x+1)(x+3)=0 \therefore x=-1,\,-3$

AF$\notin$CCF$\in$

̋󗓂̒$\in$܂$\notin$̂ꂩB
C$N$͎RŜ̏W\B\\
$B=\{ x\,|\,x<4,\,x\in N \}$̂ƂC2\,\fbox{@A@}$B$C$-3$\,\fbox{@C@}$B$

process
$\{1,\,2,\,3\}$

AF$\in$CCF$\notin$

̏ŴC
$A=\{ 1,\,2,\,3,\,4,\,6,\,9,\,12 \}$
̕Włׂ̂ĂȂB\\
$B=\{ 1,\,4,\,6 \}$\\
$C=\{ 2,\,4,\,6,\,8 \}$\\
$D=\{ 3,\,6,\,9,\,12 \}$

$B$C$D$

S̏W$U=\{ 5,\,6,\,7,\,8,\,9,\,10,\,11,\,12 \}$
ƂƂC
W$A=\{ 5,\,8,\,11 \}$
̕W$\overline{A}$
ׂĕ\ȂB

$\overline{A}=\{ 6,\,7,\,9,\,10,\,12 \}$








[Level3]
ʕƘaW̗vfibIj
2̏W${\rm A}=\{ x|\,1<x<5 \}$C
$B=\{ x|\,2<x<8 \}$ɂāC\\
$A\cup B=\{x| $\fbox{\, 11 \,}$<x<$\fbox{\, 12 \,}$\}$
łB\\
\textcircled{\small 1}\,$1<x<2$~~~
\textcircled{\small 2}\,$2<x<5$\\
\textcircled{\small 3}\,$5<x<8$~~~
\textcircled{\small 4}\,$1<x<8$

\textcircled{\small 4}

$A=\{ 1,\,2,\,4,\,8 \}$C$B=\{  1,\,2,\,3,\,4,\,6,\,12 \}$̂ƂC
$A\cap B$C$A \cup B$ꂼ\WƂĐg\fbox{\,\bf I\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$A\cap B=\{ 1,\,2,\,4 \}$C~~~ $A\cup B=\{ 3,\,6,\,8,\,12 \}$\\
\textcircled{\small 2}\,$A\cap B=\{ 1,\,2,\,4 \}$C \\ $A\cup B=\{ 1,\,2,\,3,\,4,\,6,\,8,\,12 \}$\\
\textcircled{\small 3}\,$A\cap B=\{ 3,\,6,\,12 \}$C~~~ $A\cup B=\{ 3,\,6,\,8,\,12 \}$\\
\textcircled{\small 4}\,$A\cap B=\{ 3,\,6,\,12 \}$C\\ $A\cup B=\{ 1,\,2,\,3,\,4,\,6,\,8,\,12 \}$

\textcircled{\small 2}

$A=\{ 1,\,2,\,3 \}$C$B=\{  2,\,3,\,4 \}$̂ƂC
$A\cap B$C$A \cup B$ꂼ\WƂĐg\fbox{\,\bf I\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$A\cap B=\{ 2,\,3 \}$C~~~ $A\cup B=\{ 1,\,2,\,3,\,4 \}$\\
\textcircled{\small 2}\,$A\cap B=\{ 2,\,3 \}$C~~~ $A\cup B=\{ 1,\,4 \}$\\
\textcircled{\small 3}\,$A\cap B=\{ 1,\,4 \}$C~~~ $A\cup B=\{ 1,\,2,\,3,\,4 \}$\\
\textcircled{\small 4}\,$A\cap B=\{ 1,\,2,\,3,\,4 \}$C~~~ $A\cup B=\{ 1,\,4 \}$

\textcircled{\small 1}

W$U=\{ 1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9,\,10 \}$̕W$A$C$B$ɂāC\\
$\overline{A}\cap \overline{B}=\{ 1,\,2,\,5,\,8 \}$C$A\cap B=\{ 3 \}$C\\
$\overline{A}\cap B=\{ 4,\,7,\,10 \}$\\
킩ĂƂC
$A\cap \overline{B}=$\fbox{@\textcircled{\small 1}@}C$B=\fbox{@\textcircled{\small 2}@}$łB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic41.tex}
\end{center}

$A\cap \overline{B}=\{ 6,\,9 \}$C$B=\{ 3,\,4,\,7,\,10 \}$

S̏W$U=\{ 1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9 \}$ƂC̕W
$P=\{ x\,|\,x͋ \}$C$Q=\{ x\,|\,x3̔{ \}$ƂƂC
W$P\cup Q$߂B

process
$P=\{ 2,\,4,\,6,\,8 \}$C$Q=\{ 3,\,6,\,9 \}$

$P\cup Q=\{ 2,\,3,\,4,\,6,\,8,\,9 \}$

S̏W$U=\{ 1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9 \}$ƂC
̕W$P$C$Q$ꂼ$P=\{ x\,|\,x͋ \}$C$Q=\{ x\,|\,x3̔{ \}$ƂB
ŏWP̕W$\overline{P}$ƂƂC$\overline{P} \cap Q$߂B

process
$\overline{P}=\{ 1,\,3,\,5,\,7,\,9 \}$C$Q=\{ 3,\,6,\,9 \}$

$\overline{P} \cap Q=\{ 3,\,9\}$

$U=\{ 1,2,3,4,5,6,7,8,9 \}$S̏WƂB
$U$̕W$A=\{ 1,3,5,7,9 \}$C$B=\{ 2,3,6,8 \}$ɂāC
$A \cap B$̗vfł̂\fbox{\,\bf J\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$1$~~~
\textcircled{\small 2}\,$2$~~~
\textcircled{\small 3}\,$3$~~~
\textcircled{\small 4}\,$4$

\textcircled{\small 3}

W$U$Ƃ̕W$A$C$B$̊֌WC}̂悤ɂȂĂB
̂ƂC$A\cap \overline{B}$\̂\fbox{\,\bf J\,}̐}̎ΐłB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/h2802101.tex}
\end{center}
\begin{figure}[!hbt]\begin{center}
 \includegraphics[width=76mm,clip]{D:/texlive/2018/bin/win32/mathtex/104_syuugou/h2802m102.eps}
 \end{center}\end{figure} 

\textcircled{\small 3}

S̏W$U=\{ 1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9 \}$ƂC
̕W\\
$A=\{ 1,\,3,\,5,\,7 \}$C
$B=\{ 1,\,2,\,3,\,7,\,8,\,9 \}$
ƂƂC
W${A} \cup \overline{ B}=\fbox{\hspace{8pt} 7 \hspace{8pt}}$
łB\\
\textcircled{\small 1}\,$\{ 4,\,5,\,6 \}$~~~
\textcircled{\small 2}\,$\{ 1,\,3,\,5,\,7,\,8 \}$\\
\textcircled{\small 3}\,$\{ 2,\,4,\,6,\,8,\,9 \}$~~~
\textcircled{\small 4}\,$\{ 1,\,3,\,4,\,5,\,6,\,7 \}$

process
$\overline{\rm B}=\{ 4,\,5,\,6 \}$ł邩C\\
${\rm A} \cup \overline{\rm B}=\{ 1,\,3,\,4,\,5,\,6,\,7 \} $

\textcircled{\small 4}

2̏WACBɂāC
$A=\{ 1,\,2,\,3,\,4,\,5,\,6 \}$C
$B=\{ 3,\,6,\,9 \}$
̂ƂC
$A \cap B=$\fbox{\bf\, I \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂IׁB\\
\textcircled{\small 1}\,$\{ 9 \}$~~~
\textcircled{\small 2}\,$\{ 3,\,6 \}$~~~
\textcircled{\small 3}\,$\{ 1,\,2,\,4,\,5 \}$\\
\textcircled{\small 4}\,$\{ 1,\,2,\,3,\,4,\,5,\,6,\,9 \}$

\textcircled{\small 2}

2̏WACBɂāC
$A=\{ 1,\,2,\,3,\,4,\,5,\,6 \}$C
$B=\{ 3,\,6,\,9 \}$
̂ƂC
$A \cup  B=$\fbox{\bf\, I \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂IׁB\\
\textcircled{\small 1}\,$\{ 9 \}$~~~
\textcircled{\small 2}\,$\{ 3,\,6 \}$~~~
\textcircled{\small 3}\,$\{ 1,\,2,\,4,\,5 \}$\\
\textcircled{\small 4}\,$\{ 1,\,2,\,3,\,4,\,5,\,6,\,9 \}$

\textcircled{\small 4}

$A=\{ 1,\,2,\,4,\,8 \}$C$B=\{ 1,\,2,\,5,\,10 \}$
̂ƂC$A \cup B$ƓW
\fbox{\bf\, J \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,$\{ 1,\,2 \}$~~~
\textcircled{\small 2}\,$\{ 4,\,5,\,8,\,10 \}$\\
\textcircled{\small 3}\,$\{ 2,\,4,\,9,\,18 \}$~~~
\textcircled{\small 4}\,$\{ 1,\,2,\,4,\,5,\,8,\,10 \}$

\textcircled{\small 4}

$A=\{ 1,\,2,\,4,\,8 \}$C$B=\{ 1,\,2,\,5,\,10 \}$
̂ƂC$A \cap B$ƓW
\fbox{\bf\, J \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,$\{ 1,\,2 \}$~~~
\textcircled{\small 2}\,$\{ 4,\,5,\,8,\,10 \}$\\
\textcircled{\small 3}\,$\{ 2,\,4,\,9,\,18 \}$~~~
\textcircled{\small 4}\,$\{ 1,\,2,\,4,\,5,\,8,\,10 \}$

\textcircled{\small 1}

W$A,\, B$$A=\{ 2,\,4,\,6,\,8 \}$C
$B=\{ 2,\,3,\,6,\,7 \}$ƂƂCW$A \cap B$
߂ȂB\\
\textcircled{\small 1}\,$\phi $~~~
\textcircled{\small 2}\,$\{ 2,\,6 \}$~~~
\textcircled{\small 3}\,$\{ 3,\,7 \}$~~~
\textcircled{\small 4}\,$\{ 4,\,8 \}$\\
\textcircled{\small 5}\,$\{ 2,\,3,\,4,\,6,\,7,\,8 \}$

\textcircled{\small 2}

S̏W${\rm Z}=\{ 1,\,2,\,3,\,4,\,5,\,6 \}$
̕W
$A=\{ 1,\,3,\,5 \}$C$B=\{ 2,\,3,\,6 \}$
ƂƂC̏WCCDAB̊֌Wŕ\̂C

\textcircled{\small 1}$\sim$\textcircled{\small 5}
̒1IсC̔ԍœȂB\\
\hspace{10pt} ${\rm C}=\{ 1,\,2,\,3,\,5,\,6 \}=$\fbox{\, 17 \,}\\
\textcircled{\small 1}\,${\rm A}\cup {\rm B}$~~~
\textcircled{\small 2}\,${\rm A}\cap {\rm B}$~~~
\textcircled{\small 3}\,${\rm \overline{A}}\cup {\rm B}$~~~
\textcircled{\small 4}\,${\rm A}\cap {\rm \overline{B}}$~~~
\textcircled{\small 5}\,${\rm \overline{A}}\cap {\rm \overline{B}}$

\textcircled{\small 1}

S̏W$U=\{ 1,\,2,\,3,\,4,\,5,\,6 \}$
̕W
$A=\{ 1,\,3,\,5 \}$C$B=\{ 2,\,3,\,6 \}$
ƂƂC̏WCCDAB̊֌Wŕ\̂C

\textcircled{\small 1}$\sim$\textcircled{\small 5}
̒1IсC̔ԍœȂB\\
\hspace{10pt} ${\rm D}=\{ 1,\,5 \}=$\fbox{\, 18 \,}\\
\textcircled{\small 1}\,${\rm A}\cup {\rm B}$~~~
\textcircled{\small 2}\,${\rm A}\cap {\rm B}$~~~
\textcircled{\small 3}\,${\rm \overline{A}}\cup {\rm B}$~~~
\textcircled{\small 4}\,${\rm A}\cap {\rm \overline{B}}$~~~
\textcircled{\small 5}\,${\rm \overline{A}}\cap {\rm \overline{B}}$

\textcircled{\small 4}

$U=\{ x\,|\,x10$菬R$ \}$S̏WƂB
W$A$C$B$
$A=\{ 3,\,6,\,9 \}$C
$B=\{ 1,\,3,\,5,\,7,\,9 \}$ƂƂC
W$\overline{A} \cap B$
߂ȂBC
$\overline{A}$$A$̕WłB\\
\textcircled{\small 1}\,$\phi $~~~
\textcircled{\small 2}\,$\{ 6 \}$~~~
\textcircled{\small 3}\,$\{ 3,\,9 \}$\\
\textcircled{\small 4}\,$\{ 1,\,5,\,7 \}$~~~
\textcircled{\small 5}\,$\{ 1,\,3,\,5,\,6,\,7,\,9 \}$

\textcircled{\small 4}





[Level4]
ʕƘaW̗vfibIj
鍂Z̃NX49l̒ŁC킪l27lCƖl5lCƂȂl9lB
킪ȂŖl͉lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic01.tex}
\end{center}
$ 49-9=40 $lC$ 40-27=13 $l

13l

40l̐kΏۂɁCs̃AP[gsƂCkCɍsƂ̂҂18lC
BɍsƂ̂҂13lłB܂CkCƋB̗ɍsƂ̂҂5lłB
ƂsƂ̂Ȃk͉lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic02.tex}
\end{center}
$ 40-(18+13-5)=40-26=14 $l

14l

cƉۂ53l̏]ƈɃAP[gƂCejXDȐl38lC
ejXƃTbJ[̗DȐl17lCejXTbJ[DłȂl9lB
TbJ[DȐl͉l邩B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic04.tex}
\end{center}
$ 38+x-17=53-9 $B$x=23$

23l

40l̃NXŁCNu̒ƂC
^ɏĂl16lŁCɏĂl19lłB
܂CǂɂĂȂl7lłB
ǂɂĂl͉lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic05.tex}
\end{center}
$ 16+19-x=40-7 $B$x=2$

2l

1100܂ł̐ŁC2܂3̔{͂邩B

process
2̔{50C3̔{33B̂C6̔{16dĂ邩C\\
$50+33-16=67$

67

70ȉ̎R̂C2܂3̔{͂邩B

process
2̔{35C3̔{23B̂C6̔{11dĂ邩C\\
$35+23-11=47$

47

70ȉ̎R̂C3܂4̔{͂邩B

process
3̔{23C3̔{17B̂C12̔{5dĂ邩C\\
$23+17-5=35$

35

180܂ł̐ŁC2܂3̔{͂邩B

process
2̔{40C3̔{26B̂C6̔{13dĂ邩C\\
$40+26-13=53$

53

190܂ł̐ŁC2܂3̔{͂邩B

process
2̔{45C3̔{30B̂C6̔{15dĂ邩C\\
$45+30-15=60$

60

180܂ł̐ŁC3܂4̔{͂邩B

process
3̔{26C4̔{20B̂C12̔{6dĂ邩C\\
$26+20-6=40$

40

190܂ł̐ŁC3܂4̔{͂邩B

process
3̔{30C4̔{22B̂C12̔{7dĂ邩C\\
$30+22-7=45$

45

1100܂ł̐̂C3܂4̔{͂邩B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic23.tex}
\end{center}
3̔{33C4̔{25C12̔{8B
$33+25-8=50$

50

1100܂ł̐̂C3܂5̔{͂邩B

process
3̔{33C5̔{20B̂C15̔{6dĂ邩C\\
$33+20-6=47$

47

1100܂ł̐ŁC3܂11̔{͂邩B

process
3̔{33C11̔{9B̂C33̔{3dĂ邩C\\
$33+9-3=39$

39

42lw̋ŒƂCꂪDȎqǂ15lC
ZȎqǂ25lCǂDłȂqǂ10lB
Z͍DC͍DłȂqǂ͉l邩B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic21.tex}
\end{center}
ZDȎqǂ̐$x$lƂƁC\\
$15+25-x=42-10$C$x=8$C\\
ӂ̐́C
$25-8=17$

17l

wZ200l̊wɑ΂CACB\,2ނ̎{ƂC
A̍i҂110lCB̕si҂83lC
ACBƂɍi҂48lłB
ACBƂɕsi҂ł҂͉lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic22.tex}
\end{center}
$83-62=21$l

21l

wZ100lΏۂɒƂĈ悤ȌʂɂȂB\\
싅Ƃ܂H\\
F78lCȂF22l\\
닅Ƃ܂H\\
F45lCȂF55l\\
싅닅ƂȂl͂̊wZɂ7lƂC
싅닅Ƃl͉l邩H

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic09.tex}
\end{center}
$78+45-x=100-7$lB$x=30l$

30l

k45l̃NXŁCẐ鐶k28lC
ô鐶k23lCǂȂk8lłƂC
Zo鐶k͉lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic25.tex}
\end{center}
$ 45-8=28+23-x$C$x=14$l

14l

NCỶ𓚎15l̂CPڂ̐҂͂WlCPڂƂQڂ̗̐҂͂UlCƂłȂ̂͂RlłBQڂ̐҂͉lB\\

process
$8+x-6=15-3$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic31.tex}
\end{center}

10l

Xp30lɁC[Ɣ~̍DɂĕƂC[DȐl20lC~DȐl15lłBƂDȐl10lłƂCƂDł͂Ȃl͉lB

process
$20+15-10=30-x$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic32.tex}
\end{center}

Tl

Ђ̎͂ႢЈ50lɁCΖ̎͋ɉgĂ邩ׂƂCዾ30lCR^Ng35lCǂgȂl4lłBዾƃR^Ng𗼕gl͉lB

process
$30+35-x=50-4$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic33.tex}
\end{center}

19l

C^[lbg̗pɂāC120lɃAP[gsB
gт݂̂ł̗p҂30lCgтƃp\R̗ł̗p҂60lCgтp\RpȂl20lłBp\R݂̂ł̗p҂͉lB

process
$30+60+x=120-20$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic34.tex}
\end{center}

10l

鏬wɏKɂĕĂ݂ƁCjKĂq25lCsAmKĂq20lCKĂq15lCǂKĂȂq10lłBSŉl̏wɕB

process
$(25+20-15)+10=40$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic35.tex}
\end{center}

40l

Ђ200lɑ΂ĕیɉĂl𒲂ׂƂCԕیɉĂl50lCیɉĂl130lC
ԕیƐی̗ɉĂl35lBԕیƐی̂ǂɂ̐l͉lB

process
$200-(50+130-35)=55$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic36.tex}
\end{center}

55l





[Level5]
W_ʕƘaWibLqj
鍂Z3NŁCIT@̏L󋵂ɂ100lɃAP[gB
Wvʂ͎̂悤ɂȂB
fWJƃv^[̗ƂĂȂƓl36lB
fWJƃv^[̗Ăl͉l邩B
\begin{table}[!h]
\begin{center}
\begin{tabular}{ccc}
   & Ă & ĂȂ \\
 \hline
 p\R & 74 & 26 \\
 \hline
 gѓdb & 86 & 14\\
 \hline
 fWJ & 48 & 52\\
 \hline
 v^[ & 37 & 63\\
 \hline
\end{tabular}
\end{center}
\end{table}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic11.tex}
\end{center}
$48+37-x=100-36$B$x=21l$

21l

鍂Z3NŁCIT@̏L󋵂ɂ100lɃAP[gB
Wvʂ͎̂悤ɂȂB
gт͎Ă邪p\R͎ĂȂl6lꍇC
ĂȂl͉l邩B
\begin{table}[!h]
\begin{center}
\begin{tabular}{ccc}
   & Ă & ĂȂ \\
 \hline
 p\R & 74 & 26 \\
 \hline
 gѓdb & 86 & 14\\
 \hline
 fWJ & 48 & 52\\
 \hline
 v^[ & 37 & 63\\
 \hline
\end{tabular}
\end{center}
\end{table}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic12.tex}
\end{center}
$74+6=80$B$100-80=20l$

20l

鍂Zł߂˂ɂĂ̒B
߂˂Ăl͑Ŝ28\%B
܂jq̊48\%ŁĈ$\displaystyle \frac{1}{4}$͂߂˂ĂB
߂˂q40l̂ƂC̊w͉̑lB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic03.tex}
\end{center}
߂˂Ăjq$ 48\% \times \displaystyle \frac{1}{4}=12\% $B
āC߂˂Ă鏗q͑Ŝ\\
$28 \%-12 \%=16 \%$B
$\displaystyle \frac{x}{100}=\displaystyle \frac{40}{16}$B$x=250$

250l

wZŒʊŵƂɎg蕨̒ƂC
dԂpl28\%CoXpl12\%C]Ԃ𗘗pl30\%C
3ƂpȂl16\%łB
dԂƃoX𗘗p邪C]Ԃ͗pȂl͉\%邩B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic06.tex}
\end{center}
$30+28+12+x+16=100$B$x=14$

14$\%$

42l̃NXŎƂC1ڂ̐҂35lC
2ڂ̐҂20lB
2Ƃ̐l͍łȂꍇlB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic24.tex}
\end{center}
}C\\
$35+20-x \leqq 42$C$x \geqq 13$

13l

2̎RCȂ킿1099܂ł̎R̂C3܂5̔{͂邩B

process
$3̔{=(1`99܂ł3̔{)-(1`9܂ł3̔{)=33-3=30$C\\
$5̔{=(1`99܂ł5̔{)-(1`9܂ł5̔{)=19-1=18$B\\
$15̔{=(1`99܂ł15̔{)-(1`9܂ł15̔{)=6-0=6$B\\
$30+18-6=42$

42

Ђ200lɑ΂ĕیɉĂl𒲂ׂƂCԕیɉĂl50lC΍ЕیɉĂl90lłB
܂Cԕی̐l4΍ЕیɉĂBԕیƉ΍Еی̗ɉĂl͉lB

process
}̎ΐ̐ĺC$150 \times 0.4=60$lB߂l$90-60=30$l
\begin{center}
 \input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic37p.tex}
\end{center}

30l

wCɎQl̓𕪐͂Ă݂BjƏ̔$5:3$łC
lƎqǂ̐l̔$4:1$łB
̒ŁClƎqǂ̔$11:4$łC̎qǂ̐l24lB
j̒̑l͉lB

process
\begin{center}
 \input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic10.tex}
\end{center}
ɏq̒̎qǂ̐4ƂƁC\̂悤ɂȂB
ċ߂鐔́C$21 \times \displaystyle \frac{24}{4}=126$

126l








[Level6]
Ɛ^UCے
̖̐^U𒲂ׂȂB\\
$\sqrt{(-3)^2}=-3$

U

̖̐^U𒲂ׂȂB\\
9̖̐񐔂1C3C9łB

^

̖̐^U𒲂ׂȂB\\
$5^2+12^2=13^2$

^

̖̐^U𒲂ׂȂB\\
8͊łB

U

̖̐^U𒲂ׁCUłꍇɂ͔ȂB\\
$3x=6\Longrightarrow  x=2$

^

̖̐^U𒲂ׁCUłꍇɂ͔ȂB\\
$x=1 \Longrightarrow x^2=1$

^

̖̐^U𒲂ׁCUłꍇɂ͔ȂB\\
$x^2=9 \Longrightarrow x=3$

UB$x=-3$

̏̔ےqׂȂB\\
R$n$͊łB

R$n$͋łB

̏̔ےqׂȂB\\
$x \leqq 3$

$x>3$

WlāC̖̐^U𒲂ׂȂB\\
$-2<x<1 \Longrightarrow -3\leqq x\leqq 2 $

^

WlāC̖̐^U𒲂ׂȂB\\
$-1<x<2 \Longrightarrow 0 < x < 3 $

U

WlāC̖̐^U𒲂ׂȂB\\
$1<x<3$Ȃ$1<x<4$

^

WlāC̖̐^U𒲂ׂȂB\\
$0<x<1$Ȃ$-1<x<1$

^

̖̐^U𒲂ׂȂB\\
$x+2=0$Ȃ$x=2$łB

U

̖̐^U𒲂ׂȂB\\
͊̕łB

process
$2n-1$i$n=1,\,2,\,3,\,\cdots$jƕ\Ƃł邩C
$(2n-1)^2=4n^2-4n+1$͊B

^

̖̐^U𒲂ׂȂB\\
$A=90^\circ$Ȃ$\triangle {\rm ABC}$͒pOp`łB

^

̖̐^U𒲂ׂȂB\\
$x^2=4$Ȃ$x=2$łB

UiF$x=-2$j

̏̔ےqׂB\\
$x\geqq 3$

$x<3$

̏̔ےqׂB\\
$1\leqq x<4$

$x<1܂4\leqq x$

̏̔ےqׂB\\
$x=0$$y=0$

$x\not= 0$܂$y\not= 0$

̏̔ےqׂB\\
$x<2 $܂$y\geqq -3$

$x\geqq 2$$y<-3$

2̖iPjCiQjB\\
ij-----------------------------------------------------\\
iPjOp`ABC񓙕ӎOp`ȂΎOp`ABC͐Op`łB\\
iQj$n$12̔{Ȃ$n$4̔{łB\\
-----------------------------------------------------------------\\
iPjCiQj̐^ȖgƂĐ̂\fbox{\bf\,L\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB\\
\textcircled{\small 1}\,iPj^CiQj^\\
\textcircled{\small 2}\,iPj^CiQjU\\
\textcircled{\small 3}\,iPjUCiQj^\\
\textcircled{\small 4}\,iPjUCiQjU

\textcircled{\small 3}

̖̐^U𒲂ׁCUłꍇɂ͔ȂB\\
$a^2>0$Ȃ$a>1$łB

UB$a=-2$Bi$a^2>0$͐藧$a>1$Ȃj

$x$͎ƂB
\textcircled{\small 1}$\sim$\textcircled{\small 4} ̖̂C^ł̂͂ǂꂩB
̂IׁB\\
\textcircled{\small 1}\,$x^2=3 \Longrightarrow x=\sqrt{3}$\\
\textcircled{\small 2}\,$x^2=1 \Longrightarrow x=-1$\\
\textcircled{\small 3}\,$3x=9 \Longrightarrow x=3$\\
\textcircled{\small 4}\,$x^2>0 \Longrightarrow x>0$

\textcircled{\small 3}

S̏W$U$Ŝ̏WƂC̕
$A=\{ x | -6<x<0 \}$C
$B=\{ x | -3\leqq x \leqq 7 \}$ƂƂC
W$\overline{A \cap B} =$\fbox{\hspace{8pt} 7 \hspace{8pt}}łB\\
\textcircled{\small 1}\,$\{ x | x \leqq -6, 7 < x \}$~~~
\textcircled{\small 2}\,$\{ x | -6 \leqq < 7 \}$\\
\textcircled{\small 3}\,$\{ x | x < -3, 0 \leqq x \}$~~~
\textcircled{\small 4}\,$\{ x | -3 \leqq x < 0 \}$~~~

process
$A\cap B=\{ x | -3 \leqq x <0 \}$
łB

\textcircled{\small 3}


u$x>2$܂$x<-3$v
\fbox{\hspace{8pt} 19 \hspace{8pt}}
́u
$-3\leqq x \leqq 2$
v
łB\\
\textcircled{\small 1}\,~~~
\textcircled{\small 2}\,t~~~
\textcircled{\small 3}\,ҋ~~~
\textcircled{\small 4}\,ے

\textcircled{\small 4}






[Level7]
\ƕKv
$x$̂ƂC$|\, x \,|\leqq 0$$x=0$ł邽߂\fbox{\, 6 \,}B\\
\textcircled{\small 1}\,KvłB\\
\textcircled{\small 2}\,\łB\\
\textcircled{\small 3}\,Kv\łB\\
\textcircled{\small 4}\,Kvł\łȂB

process
u$|\,x\,|\geqq 0$vł邩C\\
u$|\, x \,|\leqq 0$v$\Longleftrightarrow $u$x=0$v

\textcircled{\small 1}

$a$ł邱Ƃ́C$a$ł邽߂\fbox{\, 5 \,}B\\
\textcircled{\small 1}\,KvłB\\
\textcircled{\small 2}\,\łB\\
\textcircled{\small 3}\,Kv\łB\\
\textcircled{\small 4}\,Kvł\łȂB

process
u$a$łv$\Longleftarrow $u$a$łv

\textcircled{\small 1}

$x$̂ƂCu$1\leqq x<2$vł邽߂̕KvƂȂ蓾̂́C
̏̒Ŕԍ\fbox{\, 15 \,}Ԃ̏łB\\
\textcircled{\small 1}\,$x>1$~~~
\textcircled{\small 2}\,$1<x\leqq 2$~~~
\textcircled{\small 3}\,$x\leqq 2$

process
u$1\leqq x<2$v$\Longrightarrow $u$x\leqq 2$v

\textcircled{\small 2}

̋󗓂ɓĂ͂܂̂C
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̒1IєԍœȂB
C$x$͎ƂB\\
\hspace{10pt}
$1\leqq x\leqq 2$$x>1$ł邽߂
\fbox{\,18\,}B\\
\textcircled{\small 1}\,Kvł邪C\ł͂ȂB\\
\textcircled{\small 2}\,\ł邪CKvł͂ȂB\\
\textcircled{\small 3}\,Kv\łB\\
\textcircled{\small 4}\,Kvł\łȂB

process
$x>1$$1\leqq x\leqq 2$܂܂ȂB

\textcircled{\small 4}

̋󗓂ɓĂ͂܂̂C
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̒1IєԍœȂB\\
\hspace{10pt}
ł邱Ƃ͓ł邽߂\fbox{\,17\,}B\\
\textcircled{\small 1}\,Kvł邪C\ł͂ȂB\\
\textcircled{\small 2}\,\ł邪CKvł͂ȂB\\
\textcircled{\small 3}\,Kv\łB\\
\textcircled{\small 4}\,Kvł\łȂB

process
uv$\Longrightarrow $uv

\textcircled{\small 2}

$a>0$ł邱Ƃ́C$a^2>0$ł邽߂
\fbox{\hspace{8pt} 5 \hspace{8pt}}B
C͎B\\
\textcircled{\small 1}\,Kv\ł~~~
\textcircled{\small 2}\,Kvł\\
\textcircled{\small 3}\,\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
u$a>0 \Longrightarrow  a^2>0$v\\
u$a^2>0 \Longrightarrow a>0$v͐ȂBF$a=-3$

\textcircled{\small 3}

̋󗓂ɂĂ͂܂̂́C{\bf A}.u\v{\bf C}.uKvvLœȂB\\
$x=3$$x^2=9$ł邽߂\fbox{@@}łB

process
$x=3 \Longrightarrow x^2=9$

A

̋󗓂ɂĂ͂܂̂́C{\bf A}.u\v{\bf C}.uKvvLœȂB\\
$x^2=1$$x=-1$ł邽߂\fbox{@@}łB

process
$x^2=1 \Longleftarrow  x=-1$

C

$p$C$q$ɂāC$p$$q$ł邽߂̕Kv\ɂȂĂ̂͂ǂꂩB
LœȂB\\
\textcircled{\small 1}\,$p:~~x=6$C~~~$q:~~5x=30$\\
\textcircled{\small 2}\,$p:~~x=1$C~~~$q:~~x^2=1$

process
\textcircled{\small 2}\,$p:~~x=1 \Longrightarrow q:~~x^2=1$ł邪C\\
$p:~~x=1,\,-1~~~~\Longleftarrow q:~~x^2=1$

\textcircled{\small 1}

̏$p$C$q$ɂāC$p$$q$ł邽߂\\
{\bf A}\,u\vC{\bf C}\,uKvvC{\bf E}\,uKv\v{\bf G}\,uKvł\łȂv̂ł邩CłK̂LœB\\
$p\,:\,x^2-x-6=0~~~q\,:\,x=3$

process
$p\Leftrightarrow (x+2)(x-3)=0\Leftrightarrow x=-2,\,3$ł邩$p \Leftarrow q$

C

̏$p$C$q$ɂāC$p$$q$ł邽߂\\
{\bf A}\,u\vC{\bf C}\,uKvvC{\bf E}\,uKv\v{\bf G}\,uKvł\łȂv̂ł邩CłK̂LœB\\
$p\,:\,x+1<2x+1~~~q\,:\,x+1>0$

process
$p\Leftrightarrow 0<x,\,q\Rightarrow  ,-1<x$ł邩$p \Rightarrow  q$

A

̏$p$C$q$ɂāC$p$$q$ł邽߂\\
{\bf A}\,u\vC{\bf C}\,uKvvC{\bf E}\,uKv\v{\bf G}\,uKvł\łȂv̂ł邩CłK̂LœB\\
$p\,:\,x+y=0xy=0~~~q\,:\,x=y=0$

process
$p\Leftarrow q$͖炩B\\
$p \Rightarrow  q$𒲂ׂB$xy=0$̂Ƃ$x=0$܂$y=0$B\\
iij$x=0$̂Ƃ$x+y=0$$y=0$\\
iiij$y=0$̂Ƃ$x+y=0$$x=0$\\
āC$p \Rightarrow  q$藧B\\
ȏɂKv\B\\
iʁj
$p$$x$C$y$\\
$t^2-0\cdot t+0=0$Ȃ킿C$t^2=0$̉B$t=0$ƂȂ邩$x=y=0$\\
$p\Rightarrow q$

E

̋󗓂ɓĂ͂܂̂́Cu{\bf A}:\vu{\bf C}: Kvvu{\bf E}:Kv\vLœȂB\\
$p:\,x=y$$q:\,x^2=y^2$ł邽߂\fbox{\,@@\,}łB

process
$q:\,x^2-y^2=0\Leftrightarrow (x-y)(x+y)=0 \therefore x=\pm y$\\
āC$p\Rightarrow q$B$p$͏\B

A

̋󗓂ɓĂ͂܂̂́Cu{\bf A}:\vu{\bf C}: Kvvu{\bf E}:Kv\vLœȂB\\
$p:$uOp`ŁC3̓p̑傫vƂ́C
$q:$uOp`łv߂\fbox{\,@@\,}łB

process
Kv\

E

̋󗓂ɓĂ͂܂̂́Cu{\bf A}:\vu{\bf C}: Kvvu{\bf E}:Kv\vLœȂB\\
$p:$uR$n$łvƂ́C
$q:$u$n=4$łv߂\fbox{\,@@\,}łB

process
$p \Leftarrow q$

C

$n$RƂB$n$3̔{ł邱Ƃ́C$n$12̔{ł邽߂\fbox{\bf\,J\,}B
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,\ł邪CKvł͂Ȃ\\
\textcircled{\small 2}\,Kvł邪C\ł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
u$n$3̔{łv$\cdots \{3,\,6,\,9,\,12\cdots\}$\\
$\Leftarrow  $u$n$12̔{łv$\cdots \{12,\,24,\,36\cdots\}$

\textcircled{\small 2}

̋󗓂ɓĂ͂܂̂IIсCLœȂB\\
$x \leqq 1$$y \leqq 3$ł邱Ƃ́C$x+y\leqq 4$ł邽߂\fbox{@@}B\\
\textcircled{\small 1}\,\ł邪CKvł͂Ȃ\\
\textcircled{\small 2}\,Kvł邪C\ł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
u$x \leqq 1$$y \leqq 3$v$\Rightarrow $u$x+y\leqq 4$v
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/104_syuugou/pic42.tex}
\end{center}

\textcircled{\small 1}

̋󗓂ɓĂ͂܂̂́Cu{\bf A}:\vu{\bf C}: Kvvu{\bf E}:Kv\vLœȂB\\
$p:\,x=1,\,y=1$$q:\,xy=1$ł邽߂\fbox{\,@@\,}łB

process
$p \Rightarrow q$͐B\\
$xy=1\Leftrightarrow (x,\,y)=(1,\,1),\,(-1,\,-1)$
$p \Leftarrow  q$͐藧ȂB
āC$p\Rightarrow q$B$p$͏\B

A

$x$C$y$𐮐ƂƂC
$x+y=3$$x=2$$y=1$
ł邽߂\fbox{\bf\, J \,}B
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,\ł邪CKvł͂Ȃ\\
\textcircled{\small 2}\,Kvł邪C\ł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
$x+y=3\Longleftarrow $u$x=2$$y=1$v

\textcircled{\small 2}

$x\geqq 3$$x\geqq 5$ł邽߂\fbox{\hspace{8pt} 6 \hspace{8pt}}B\\
\textcircled{\small 1}\,\ł邪CKvł͂Ȃ\\
\textcircled{\small 2}\,Kvł邪C\ł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
u$x\geqq 3$v$\Longleftarrow $u$x\geqq 5$v

\textcircled{\small 2}

Op`ł邱Ƃ́C񓙕ӎOp`ł邽߂\fbox{\hspace{8pt} 18 \hspace{8pt}} łB\\
\textcircled{\small 1}\,\ł邪CKvł͂Ȃ\\
\textcircled{\small 2}\,Kvł邪C\ł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

\textcircled{\small 2}

lp`${\rm ABCD}$`ł邱Ƃ́C
lp`${\rm ABCD}$`ł邽߂
\fbox{\bf\, J \,}B
\textcircled{\small 1}
$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,
Kvł邪\łȂ\\\textcircled{\small 2}\,
\ł邪KvłȂ\\\textcircled{\small 3}\,
Kv\ł\\\textcircled{\small 4}\,
Kvł\łȂ

\textcircled{\small 1}

R$m$C$n$ɂāC$m$C$n$ł邱Ƃ́C
$m+n$ł邽߂\fbox{\hspace{8pt} 7 \hspace{8pt}}B\\
\textcircled{\small 1}\,Kvł邪\ł͂Ȃ\\
\textcircled{\small 2}\,\ł邪Kvł͂Ȃ\\
\textcircled{\small 3}\,Kv\ł\\
\textcircled{\small 4}\,Kvł\łȂ

process
Fu$m=2$C$n=3$v@$\Longrightarrow $u$m+n=5$v\\
u$m=3$C$n=2$v@$\Longrightarrow $u$m+n=5$v\\

\textcircled{\small 2}

$x=1$$x^2-3x+2=0$ł邽߂\\
\textcircled{\small 1}\,Kv~~~
\textcircled{\small 2}\,\\\
\textcircled{\small 3}\,Kv\~~~
\textcircled{\small 4}\,łȂ

process
u$x^2-3x+2=0$v$\Longleftrightarrow$ u$x=1,2$vł邩C\\
u$x=1$v$\Longrightarrow $u$x^2-3x+2=0$v

\textcircled{\small 2}










[Level8]
tƑ΋

u$x=2 \Longrightarrow x^2=4$v̑΋ȂB

$x^2 \not= 4 \Longrightarrow x \not= 2$


$n$R̂ƂC\\
u$n$6̔{$\Longrightarrow n$3̔{v̑΋ȂB

$n$3̔{łȂ$\Longrightarrow n$6̔{łȂ

$n$͐ƂB\\
u$n^2$Ȃ΁C$n$͊łv\\
Ƃ΋𗘗pďؖB

΋\\
u$n$łȂ$n^2$͋v\\
$n$̂ƂC$n=2k$i$k$͐jƕ\ƂłB
̂ƂC$n^2=(2k)^2=4k^2$͋B\\
ė^炽́u^v

$\sqrt{3}$ł邱ƂpāC$5-2\sqrt{3}$ł邱Ƃw@ŏؖB

$5-2\sqrt{3}$LłƉ肷ƁC\\
$5-2\sqrt{3}=p$i$p$͗LjƂB
ό`$\displaystyle\frac{5-p}{2}=\sqrt{3}$\\
ӂ͗LCEӂ͖ƂȂ薵B$5-2\sqrt{3}$B

$n$R̂ƂC\\
u$n$͋$\Rightarrow n+1͊$v\\
̑΋ȂB

u$n+1$͋$\Rightarrow n$͊v






[EOF]
