Cambodian Mathematics Y1

NOU CHANNARITH

Silpakorn University
channarithnoukh@gmail.com

May 25, 2020

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NOU CHANNARITH (SU) Inequality May 25, 2020 1 / 13
លំហាត់

េគឱ្យ 0 ≤ x, y, z ≤ 1 ។ ្រសាយបញ្ជក់ថា
( ) 81
(2x + 2y + 2z ) 2−x + 2y + 2−z ≤ ។
8
ដំេណាះ្រសាយ
Theorem (Cauchy Inequality)
√
ចំេពាះចំនួនពត
ិ មន
ិ អវ ិជ្ជមាន a នង
ិ b: a + b ≥ 2 ab ។
សមភាពេកីតមានកាលណា a = b ។

តាង a = 2x , b = 2y , c = 2z ⇒ 1 ≤ a, b, c ≤ 2
⇒ (a − 1)(a − 2) ≤ 0
2
⇒ a2 − 3a + 2 ≤ 0 ⇒ a + ≤3
a . . . . . . . . . . . . . . . . . . . .
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វិសមភាពកូស ី ុ

2 2
្រសាយដូចគា្នែដរ b + ≤ 3, c + ≤ 3
b ( c )
1 1 1
េយងី បាន (a + b + c) + 2 + + ≤9
√ a b c
( )
1 1 1
⇒ 9 ≥ 2 (a + b + c) × 2 + +
a b c
( )
1 1 1 81
⇒ (a + b + c) + + ≤
a b c 8
( ) 81
(2x + 2y + 2z ) 2−x + 2y + 2−z ≤ ពិត
8
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Bullet Points

Lorem ipsum dolor sit amet, consectetur adipiscing elit

Aliquam blandit faucibus nisi, sit amet dapibus enim tempus eu
Nulla commodo, erat quis gravida posuere, elit lacus lobortis est, quis
porttitor odio mauris at libero
Nam cursus est eget velit posuere pellentesque
Vestibulum faucibus velit a augue condimentum quis convallis nulla
gravida

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Blocks of Highlighted Text

Block 1
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Integer lectus nisl,
ultricies in feugiat rutrum, porttitor sit amet augue. Aliquam ut tortor
mauris. Sed volutpat ante purus, quis accumsan dolor.

Block 2
Pellentesque sed tellus purus. Class aptent taciti sociosqu ad litora
torquent per conubia nostra, per inceptos himenaeos. Vestibulum quis
magna at risus dictum tempor eu vitae velit.

Block 3
Suspendisse tincidunt sagittis gravida. Curabitur condimentum, enim sed
venenatis rutrum, ipsum neque consectetur orci, sed blandit justo nisi ac
lacus.
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Multiple Columns

Lorem ipsum dolor sit amet,

Heading consectetur adipiscing elit. Integer
1 Statement lectus nisl, ultricies in feugiat rutrum,
2 Explanation porttitor sit amet augue. Aliquam ut
3 Example tortor mauris. Sed volutpat ante
purus, quis accumsan dolor.

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Table

Treatments Response 1 Response 2

Treatment 1 0.0003262 0.562
Treatment 2 0.0015681 0.910
Treatment 3 0.0009271 0.296
Table: Table caption

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Theorem

Theorem (Mass–energy equivalence)

E = mc2

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Verbatim

Example (Theorem Slide Code)

\begin{frame}
\frametitle{Theorem}
\begin{theorem}[Mass--energy equivalence]
$E = mc^2$
\end{theorem}
\end{frame}

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Figure

Uncomment the code on this slide to include your own image from the
same directory as the template .TeX file.

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Citation

An example of the \cite command to cite within the presentation:

This statement requires citation [Smith, 2012].

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References

John Smith (2012)

Title of the publication
Journal Name 12(3), 45 – 678.

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 The End

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