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\section[ª‰‘÷‰Ã¢û‰Ş‰ƒ‰ß  ¢ø¤ù )�õ‰µ‰½‰‘ö  �‰ú‰‘¤ô(, ª‰ú‰Âş‰�¤ 7731]
{õ‰Æ‰\hamze ‘ó‰‚û‰‘ı  ª‰‘÷‰Ã¢û‰Ş‰ƒ‰ß  ¢ø¤\hamze ù �󉪉³‰ƒ‰‘¢ ¤ş‰‘®‰ü
 ¢�÷‰Ç�õ‰�¥�ö î‰È‰�¤\\*
)�õ‰µ‰½‰‘ö  �‰ú‰‘¤ô(, \InE{}~\EnE{}ª‰ú‰Âş‰�¤ õ‰‘ù\InE{}~\EnE{}7731}
\begin{enumerate}
\item î‰Ü‰ƒ‰\hamze ‚ �ä‰À�¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$n$\EnE{} ¤� •‰ƒ‰À� ‰ƒ‰À  ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ \InE{}$m$\EnE{} õ‰�›‰�¢
 “‰‘ª‰À \InE{}$2^n-1$\EnE{} õ‰Ö‰Æ‰�ô ä‰Ü‰ƒ‰‚ \InE{}$m^2+9$\EnE{} “‰‘ª‰À.
\item ¢¤ ª‰Ç ®‰Ü‰ã‰ü õ‰½‰À’ \InE{}$ABCDEF$\EnE{} ¢�¤ş‰İ \InE{}$AB=BC$\EnE{} , \InE{}$CD=DE$\EnE{} ø \InE{}$EF=FA$\EnE{}
 ™‰‘“‰´ ‰ƒ‰À:
$$\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\geq \frac{3}{2}$$
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\item õ‰·‰Ü‰¶ \InE{}$ABC$\EnE{} ¢¤ ¬‰Ô‰½‰‚ õ‰Ô‰Âø­ �¨‰´. ÷‰È‰‘ö ¢û‰ƒ‰À �ğ‰Â ¬‰Ô‰½‰‚ ¤� “‰‘ ¢ø ¤÷‰Ù ì‰Âõ‰Ã ø ¨‰±‰Ã
¤÷‰Ù�õ‰ƒ‰Ãı ‰ƒ‰İ ş‰‘ ¢ø ÷‰Ö‰Î‰\hamze ‚ ì‰Âõ‰Ã “‰‚ ꉑ¬‰Ü‰\hamze ‚ ø�Ÿ‰À ø›‰�¢ ¢�¤¢ ş‰‘ ¨‰‚ ÷‰Ö‰Î‰\hamze ‚ ¨‰±‰Ã 
õ‰·‰Ü‰·‰ü õ‰Æ‰‘øı “‰‘ õ‰·‰Ü‰¶ \InE{}$ABC$\EnE{} õ‰ü ¨‰‘¥÷‰À.
\item �ä‰À�¢ Ÿ‰Ö‰ƒ‰Ö‰ü \InE{}$r_1$\EnE{}, \InE{}$r_2$\EnE{}, \InE{}$\dots$\EnE{}, \InE{}$r_n$\EnE{} õ‰Ô‰Âø­�÷‰À.
 ™‰‘“‰´ ‰ƒ‰À ¥ş‰Â õ‰¹‰Ş‰�ä‰\hamze ‚ \InE{}$A$\EnE{} �¥ \InE{}$\left\{1,2,\ldots ,n \right\}$\EnE{}
  õ‰�›‰�¢ �¨‰´ “‰‚ω�¤ı ª‰‘õ‰Û û‰ƒ‰º ¨‰‚ ä‰À¢
õ‰µ‰��ó‰ü ÷‰±‰‘ª‰À øó‰ü �¥ û‰Â ¨‰‚ ä‰À¢ õ‰µ‰��ó‰ü „�ì‰Û õ‰È‰µ‰Ş‰Û “‰Â ş‰Ø‰ü �¥ �÷‰ú‰‘ “‰‘ª‰À ø
 ¢�ª‰µ‰‚ “‰‘ª‰ƒ‰İ:
$$ \left|\sum _{j \in A} r_j\right| \geq  \frac{1}{6} 
\sum _{j=1}^{n}|r_j|$$
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\item ¢¤ õ‰·‰Ü‰¶ \InE{}$ABC$\EnE{} ê‰Â­ ‰ƒ‰À \InE{}$D$\EnE{} õ‰½‰Û “‰Â¡‰�¤¢ ÷‰ƒ‰Ş‰Æ‰‘¥ ¥�øş‰\hamze ‚ \InE{}$A$\EnE{} “‰‘ ®‰Ü‰â \InE{}$BC$\EnE{}
“‰‘ª‰À. ¢�ş‰Âù�ı õ‰Ş‰‘§ “‰Â \InE{}$BC$\EnE{} ¢¤ ÷‰Ö‰Î‰\hamze ‚ \InE{}$D$\EnE{} ¤�   �¥ \InE{}$A$\EnE{} û‰İ õ‰üğ‰Á¤¢  ¤¨‰İ õ‰ü‰ƒ‰İ.
÷‰Ö‰Î‰\hamze ‚ ¢øô —‰Ö‰‘ω⠢�ş‰Âù “‰‘ \InE{}$AC$\EnE{} ¤� \InE{}$M$\EnE{} õ‰üğ‰ƒ‰Âş‰İ ø ê‰Â­ ‰ƒ‰À \InE{}$BM$\EnE{} ¢�ş‰Âù ¤� ¢¤ \InE{}$P$\EnE{}
ì‰Î‰â ‰À. ™‰‘“‰´ ‰ƒ‰À \InE{}$AP$\EnE{} õ‰ƒ‰‘÷‰\hamze ‚ õ‰·‰Ü‰¶ \InE{}$ABD$\EnE{} �¨‰´.
\item ê‰Â­ ‰ƒ‰À \InE{}$n \geq k \geq 0 $\EnE{} �ä‰À�¢ı  ¬‰½‰ƒ‰¼ “‰‘ª‰€‰À. �ä‰À�¢ \InE{}$C(n,k)$\EnE{}
 “‰‚¬‰�¤– ¥ş‰Â —‰ã‰Âş‰Ó õ‰üª‰�÷‰À:
\InE{}$C(n,0)=C(n,n)=1$\EnE{} ø\\
$C(n+1,k)=2^k C(n,k)+C(n,k-1) \hspace{2cm} n \geq k \geq 1 $

™‰‘“‰´ ‰ƒ‰À:
\[C(n,k) = C(n,n-k),   \hspace{2.65cm} n \geq k \geq 0 \]
\end{enumerate}
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