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à¤Ã×èͧÁ×ͧ͢ËÑÇ¢éÍ | ¤é¹ËÒã¹ËÑÇ¢é͹Õé |
#1
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ᨡ੾ÒСԨ : Ãͺáá
â·É·Õ¤ÃѺ à¾Ô觡ÅѺ¶Ö§ºéÒ¹
¤Ó¶ÒÁ·Ñé§ 7 ¢éÍ ã¹Ãͺáá Áմѧ¹Õé¤ÃѺ 1. (3 ¤Ðá¹¹) AB à»ç¹àÊé¹¼èÒ¹ÈÙ¹Âì¡Åҧǧ¡ÅÁ ¤ÍÃì´ AC µÑ´¡Ñº¤ÍÃì´ BD ·Õè P «Öè§à»ç¹¨Ø´ã¹Ç§¡ÅÁ ¾ÔÊÙ¨¹ìÇèÒ $ AB^2= (AC)(AP)+ (BD)(BP) $ (Edit: à´ÔÁ⨷Âì¢Ñ´áÂ駡ѹàͧ) 2. (3 ¤Ðá¹¹) ËÒ¤èÒ $$ \sum_{n=1}^{\infty}\bigg( \frac{4n+5}{n^2+n}\Bigg)\Bigg(\frac{1}{5}\Bigg)^n $$ 3. (4 ¤Ðá¹¹ ) ¶éÒ $ x ,y $ à»ç¹àŢⴴ·ÕèäÁèà»ç¹ 0 ¾ÃéÍÁ¡Ñ¹ áÅÐÊÍ´¤Åéͧ¡ÑºÃкºÊÁ¡Òà $ xy^2=50y^2-5xy-406x $ $ x^2y=25y^2-2x^2-200x $ ËÒ¤èÒ $100x^2+y^2 $ ·Õèà»ç¹ä»ä´é·Ñé§ËÁ´ 4. (4 ¤Ðá¹¹) ¡Ó˹´ $ \large a_1 =1 $ áÅÐ $ \large a_{n+1}= \frac{\sqrt{1+a_{n}^2}-1}{a_n} $ ·Ø¡ $ n \geq 1 $ ËÒ $n$ (¶éÒÁÕ) ·Õè·ÓãËé $ |a_n-\tan\frac{\pi}{60}| $ ÁÕ¤èÒ¹éÍ·ÕèÊØ´ 5. (5 ¤Ðá¹¹) àÅ×Í¡·Ó 1 ¢éÍà·èÒ¹Ñé¹ (A) ãËé $n$ à»ç¹¨Ó¹Ç¹¹Ñº áÅÐ $a_1,a_2,\cdots,a_n > 0 $ ¾ÔÊÙ¨¹ìÇèÒ $$ \sum_{k=1}^n a_k \leq (n-1)+ \prod_{k=1}^n max(1,a_k) $$ ËÁÒÂà赯 : $ max (x,y) $ ¨Ð¤×¹¤èÒ·ÕèÁÒ¡¡ÇèÒàÁ×èÍà·ÕºÃÐËÇèÒ§ x ¡Ñº y àªè¹ $ max(1,5) =5 $ (B) ¾ÔÊÙ¨¹ìÇèÒ $$ \sum_{n=1}^{100} \frac{n^2-n+1}{n^4-n^3+n^2-n+1} < 1.99 $$ 6. à¹×èͧ㹠ÇâáÒÊ ·Õèã¹ËÅǧ·Ã§¤ÃͧÊÔÃÔÃÒªÊÁºÑµÔ¤Ãº 60 »Õ ã¹Çѹ·Õè 9 ÁÔ.Â. 2549 ·Õè¨ÐÁÒ¶Ö§ ¼ÁÍÂÒ¡ãËé ¹ÓàÅ¢ 9 , 6 áÅÐ 49 (ÍÂèÒ§ÅÐ 1 µÑÇà·èÒ¹Ñé¹) ÁÒ´Óà¹Ô¹¡Ò÷ҧ¤³ÔµÈÒʵÃìÍÂèÒ§äáçä´é ·ÕèäÁèãªè¡Òà àÍÒàÅ¢ÁÒµè͡ѹ (concatenation) â´Âµéͧ¡ÒüÅÅѾ¸ìÊØ´·éÒÂà»ç¹ 60 ËÁÒÂà赯 : (1)ãËéÇÔ¸ÕÅÐ 2 ¤Ðá¹¹ áÅеͺä´éäÁèà¡Ô¹¤¹ÅÐ 3 ÇÔ¸Õ ÊÓËÃѺÇÔ¸Õ·Õè«éӡѺ·ÕèÁÕ¤¹µÍºä»áÅéÇ ¨ÐäÁèä´é¤Ðá¹¹ (2) à¤Ã×èͧËÁÒ square root ÊÒÁÒöãªéä´é·Ñ¹·Õ¤ÃѺ áµè¶éҶʹÃÒ¡Í×è¹ ¨ÐµéͧÊÃéÒ§àÅ¢¢Í§ÃÒ¡¹Ñé¹¢Öé¹ÁÒàͧ àªè¹ ¨Ð¶Í´ÃÒ¡·Õè 3 ¡çµéͧ àÍÒàÅ¢·Õè¼ÁãËéÁÒ ÁÒÊÃéÒ§à»ç¹ 3 ¡è͹ (3) ËéÒÁÊÃéÒ§ operation ¢Öé¹ÁÒàͧ¹Ð¤ÃѺ Edit : à¾ÔèÁËÁÒÂà˵آéÍ 3 7. (6 ¤Ðá¹¹) ¡Ó˹´ $n$ à»ç¹¨Ó¹Ç¹¤Õè·ÕèÁÒ¡¡ÇèÒËÃ×Íà·èҡѺ 3 ¾ÔÊÙ¨¹ìÇèÒ ÁըӹǹàµçÁ $c_i $ «Öè§ $ |c_i| \leq 4 $ ·Ø¡ $ i=1,2,\cdots,\frac{n+1}{2} $ ·Õè·ÓãËé $$ c_1{n \choose 1} + c_2{n \choose 3}+\cdots + c_{(n+1)/2}{n \choose n} = 0 $$ (â´Â $ c_i$ äÁèà»ç¹ 0 ¾ÃéÍÁ¡Ñ¹) Ãͺ¹ÕéäÁè¹èÒ¨ÐÂÒ¡ÁÒ¡à¾ÃÒÐà»ç¹Ãͺáá áµèÃͺµèÍä» ÍÔ ÍÔ ÍÔ ¡µÔ¡Ò·Ñé§ËÁ´´Ùä´é ·Õè¹Õè ¤ÃѺ áÅÐÍÂèÒÅ×ÁÇèÒÃͺáá ËÁ´à¢µµÍº ÇѹÍѧ¤Ò÷Õè 2 ¾.¤. àÇÅÒ 23.30 ¹. ¹Ð¤ÃѺ ¢ÍãËé⪤´Õ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 01 ¾ÄÉÀÒ¤Á 2006 11:51 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#2
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¢éÍ 2. à¹×èͧ¨Ò¡ \[ \sum_{n=1}^{\infty} \bigl( \frac{4n+5}{n(n+1)} \bigl) \bigl( \frac{1}{5} \bigl)^n \; \; = \; \; \sum_{n=1}^{\infty} \big( \frac{5}{n} - \frac{1}{n+1} \big) \bigl( \frac{1}{5} \bigl)^n \; \; = \; \; \sum_{n=1}^{\infty}\frac{1}{n} \bigl( \frac{1}{5} \bigl) ^{n-1} \; - \; \sum_{n=1}^{\infty}\frac{1}{n+1} \bigl( \frac{1}{5} \bigl) ^n \]
¾Ô¨ÒóÒà·ÍÁ \[ \sum_{n=1}^{\infty}\frac{1}{n} \bigl( \frac{1}{5} \bigl) ^{n-1} = 1 + \sum_{n=2}^{\infty}\frac{1}{n} \bigl( \frac{1}{5} \bigl) ^{n-1} = 1 + \sum_{k=1}^{\infty}\frac{1}{k+1} \bigl( \frac{1}{5} \bigl) ^{k}\] «Ö觨Ðä´éÇèÒà·ÍÁ¢éÒ§ËÅѧµÑ´¡Ñ¹ä» ´Ñ§¹Ñ鹵ͺ \[ \sum_{n=1}^{\infty} \bigl( \frac{4n+5}{n^2+n} \bigl) \bigl( \frac{1}{5} \bigl)^n \; \; = \; \; 1 \]
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PaTa PatA pAtA Pon! |
#3
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6.$$\frac{\Gamma(6)}{9-\sqrt{49}}$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#4
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àË繤ӶÒÁáÅéÇÍÂÒ¡Ê¡Ñ´´ÒÇÃØ觪ÐÁÑ´ àÊÕ´ÒÂäÁèÁÕÊÔ·¸Ôìá¢è§ ¢éÍË¡§èÒÂÊØ´æÃÕºæÁÒà¡çº¹Ð¤ÃѺ
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. |
#5
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¢éÍ 6
(1) $60=\sqrt{49}\times9-\lceil\sqrt{6}\rceil$ Êèǹ¢éÍ 1 ¼ÁÇèÒ⨷ÂìÁÕ»ÑËҹФÃѺ
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[[:://R-Tummykung de Lamar\\::]] || (a,b,c > 0,a+b+c=3) $$\sqrt a+\sqrt b+\sqrt c\geq ab+ac+bc$$ |
#6
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¢éÍ 5 ¤ÃѺ
㹡óշÕè $n=1$ ¨Ðä´éÇèÒ \[a_1 \leq (1-1)+max(1,a_1)\] «Öè§à»ç¹¨ÃÔ§àÊÁÍ ÊÁÁµÔãËé \[\sum^m_{k=1} a_k \leq (m-1)+\prod^m_{k=1} max(1,a_k)\] â´Â·Õè $m$ à»ç¹¨Ó¹Ç¹àµçÁºÇ¡ ¡Ã³Õ·Õè 1 $a_{m+1} \leq 1$ ¨Ðä´éÇèÒ \[\begin{array}{rcl}\sum^{m+1}_{k=1} a_k = \sum^m_{k=1} a_k +a_{m+1} &\leq& (m-1)+a_{m+1}+\prod^m_{k=1} max(1,a_k) \\ &\leq& m+\prod^m_{k=1} max(1,a_k) \\ &=& m+\prod^{m+1}_{k=1} max(1,a_k) \end{array}\] ¡Ã³Õ·Õè 2 $a_{m+1} > 1$ àËç¹ä´éªÑ´ÇèÒ $\prod^{m}_{k=1} > 1$ ¨Ðä´éÇèÒ $a_{m+1}-1 < \prod^m_{k=1} max(1,a_k)[a_{m+1}-1]$ ´Ñ§¹Ñé¹ \[\begin{array}{rcl}\sum^{m+1}_{k=1} a_k = \sum^m_{k=1} a_k +a_{m+1} &\leq& (m-1)+a_{m+1}+\prod^m_{k=1} max(1,a_k) \\ &<& m+\prod^m_{k=1} max(1,a_k)[a_{m+1}-1]+\prod^m_{k=1} max(1,a_k) \\ &=&m+\prod^{m+1}_{k=1} max(1,a_k) \end{array}\] ´Ñ§¹Ñé¹â´ÂÍØ»¹Ñ·ҧ¤³ÔµÈÒʵÃì¨Ðä´éÇèÒ \[\sum^n_{k=1} a_k \leq (n-1)+\prod^n_{k=1} max(1,a_k)\] à»ç¹¨ÃÔ§·Ø¡¨Ó¹Ç¹àµçÁºÇ¡ $n$ 30 àÁÉÒ¹ 2006 21:25 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ gools |
#7
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¢éÍ 6 Ẻ·Õè 2 ¢Í§¼Á
$$60=\bigg\lceil\frac{9}{\cos \sqrt{49}^\circ}\bigg\rceil\times 6$$ Ẻ·Õè 3 $$60=\bigg\lfloor\frac{\sinh 6^\circ}{\sqrt{9}}-\sqrt{49}\bigg\rfloor$$ Edit : à»ÅÕè¹ $\lfloor$ à»ç¹ $\rfloor$ ·ÕèµÑÇÊØ´·éÒÂ
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[[:://R-Tummykung de Lamar\\::]] || (a,b,c > 0,a+b+c=3) $$\sqrt a+\sqrt b+\sqrt c\geq ab+ac+bc$$ 30 àÁÉÒ¹ 2006 20:59 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ R-Tummykung de Lamar |
#8
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¢éÍ 1 ¼Á¤§¨ÐÁÖ¹ä»Ë¹èÍ µÍ¹¹Õéä» á¡éä¢â¨·ÂìãËéáÅéǹФÃѺ
ÊÓËÃѺ¹éͧ tummy kung ¼Á⺹ÑÊãËé 1 ¤Ðá¹¹äÇé¡è͹áÅéǡѹ·Õ辺·Õè¼Ô´ Êèǹ¤¹Í×è¹æ ÃÇÁ·Ñ駹éͧ Tummy àͧ ¡çÂѧµÍº¢éÍ 1 ·Õèá¡éä¢áÅéÇ ä´é¹Ð¤ÃѺ â´Â¤Ðá¹¹àµçÁ 3 ¤Ðá¹¹àËÁ×͹à´ÔÁ ÊÓËÃѺ¢éÍ 6 âªÇì¾Åѧ¡Ñ¹ÊØ´ÂÍ´¨ÃÔ§æ ¢ÍªÁ¨Ò¡ã¨ ÍéÍ! ¹éͧ Mastermander Áҵͺ¢éÍ 6 à¾ÔèÁàµÔÁä´é¹Ð¤ÃѺ áµè¹éͧ tummy àµçÁâ¤ÇµÒ¢é͹ÕéáÅéǤÃѺ áÅТéÍ 5 (A) áÁéµÍ¹¹Õé ¼ÁÂѧäÁèä´éàªç¤ÇԸշӤس gool áµèºÍ¡¼ÙéÃèÇÁʹء·èÒ¹Í×è¹æä´éÇèÒÂѧÁÕÇÔ¸Õ·ÕèäÁèãªé induction ´éǹФÃѺ ¶éҵͺÁÒ¡çÂѧÁÕÊÔ·¸Ôìä´é¤Ðá¹¹àµçÁ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 30 àÁÉÒ¹ 2006 20:51 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#9
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àÅÕ¹Ẻ¤¹Í×蹨Ðä´éÁÑé¤ÃѺ(¨Ñº¼Ô´)
6. $$60=\bigg\lfloor\frac{\sinh 6}{\sqrt{9}}-\sqrt{49}\bigg\rfloor$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#10
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6.ÍÕ¡¤ÃÑ駤ÃѺ ä´éá¹Ç¤Ô´¨Ò¡¢Íºº¹-ÅèÒ§ àËÍÐæ
$$60=\bigg\lfloor\frac{49}{\cos 6}+9\bigg\rfloor$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#11
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¢éÍ 1 ¤ÃѺ
\[ (AC)(AP)+(BD)(BP)=AP(AP+PC)+BP(BP+PD)=AP^2+(AP)(PC)+BP^2+(BP)(PD)=AD^2+DP^2+(AP)(PC)+BP^2+(BP)(PD)\] ¨Ò¡ Power of a point Theorem ¨Ðä´éÇèÒ \[AD^2+DP^2+(AP)(PC)+BP^2+(BP)(PD)=AD^2+DP^2+BP^2+2(BP)(PD)=AD^2+(DP+BP)^2=AD^2+DB^2=AB^2\] 30 àÁÉÒ¹ 2006 21:23 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ gools |
#12
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¢éÍ 3
¶éÒ x à»ç¹ 0 ËÃ×Í y à»ç¹ 0 àÍÒä»á·¹ã¹ÊÁ¡Òèкѧ¤ÑºãËéÍÕ¡µÑÇà»ç¹ 0 «Öè§â¨·ÂìäÁèµéͧ¡Òà µèÍ仹Õé x,y à»ç¹¨Ó¹Ç¹àµçÁ·Õè $\large 1\leq x,y \leq 9$ ¡Ã³Õ $5|x$ ÁÕà¾Õ§µÑÇà´ÕÂǤ×Í $x=5$ 2(ÊÁ¡Ò÷Õè 2)-(ÊÁ¡Ò÷Õè 1) ; $$2x^2y-xy^2=5xy-4x^2+6x$$ $$2xy+4x=y^2+5y+6$$ $$2x(y+2)=(y+2)(y+3)$$ $y\not= -2$ ´Ñ§¹Ñé¹ $$2x=y+3$$ ¶éÒ $x=5$ ä´é $y=7$ àÍÒä»á·¹ã¹·Ñé§ÊͧÊÁ¡ÒÃáÅéÇà»ç¹¨ÃÔ§ µèÍ仹Õé ¨ÐÊÁÁµÔÇèÒ $5\not|x$ ÊÁ¡Ò÷Õè 2 ÂéÒ¢éÒ§ä»ä´é $$x(x(y+2)+200)=25y^2$$ ¨Ò¡ $5\not|x$ ¹Ñ蹤×Í $25|(y+2)$ äÁèÁÕ ¤ÓµÍºÍÕ¡áÅéÇ ¹Ñ蹤×Í $$100x^2+y^2=100(25)+(49)=2549$$ ËØËØ ÍÍ¡ÁÒà»ç¹»Õ¹Õé¾Í´ÕàÅÂ
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[[:://R-Tummykung de Lamar\\::]] || (a,b,c > 0,a+b+c=3) $$\sqrt a+\sqrt b+\sqrt c\geq ab+ac+bc$$ 30 àÁÉÒ¹ 2006 22:30 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ R-Tummykung de Lamar |
#13
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¢éÍ 3 ¤ÃѺ
\[\begin{array}{rcl} xy^2 &=& 50x^2-5xy-406x \ldots (1) \\ x^2y &=& 25y^2-2x^2-200x \ldots (2) \end{array}\] $(1)-2(2)$ ¨Ðä´éÇèÒ \[\begin{array}{rcl} xy^2-2x^2y &=& 4x^2-5xy-6x \\ y^2-2xy &=& 4x-5y-6 \\ y^2+2y &=& 4x-3y+2xy-6 \\ y(y+2) &=& (2x-3)(y+2) \\ (2x-y-3)(y+2) &=& 0 \\ \end{array} \] à¹×èͧ¨Ò¡ $y>0$ ´Ñ§¹Ñé¹ $2x-y=3 \ldots (3)$ áÅШҡ $(2)$ ¨Ðä´éÇèÒ \[x^2(y+2)=25(y^2-8x)\] à¹×èͧ¨Ò¡ $0<y+2<25$ ¨Ðä´éÇèÒ $5|x^2$ ´Ñ§¹Ñé¹ $5|x$ ´Ñ§¹Ñé¹ $x=5$ áÅШҡ $(3)$ ¨Ðä´éÇèÒ $y=7$ áÅÐàÁ×èÍá·¹¤èÒ $x,y$ ã¹ÊͧÊÁ¡ÒâéÒ§µé¹¨Ðä´éÇèÒà»ç¹¨ÃÔ§ ´Ñ§¹Ñé¹ $100x^2+y=2500+49=2549$ Edit: à»ÅÕ蹵çËÑÇ¢éͨҡ¢éÍ 4 à»ç¹¢éÍ 3 ¤ÃѺ 30 àÁÉÒ¹ 2006 22:44 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ gools |
#14
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¢éÍ 4 à¡ÕèÂǡѺÅӴѺ ÁÕ»ÑËÒÃÖà»ÅèÒ¤ÃѺ ú¡Ç¹¤Ø³passer-by àªç¤Ë¹èÍÂ
ËÃ×ÍÇèÒ ¼Á¤Ô´äÁèÍÍ¡àͧ á˧Р(ÂѧäÁèµÍº¹Ð¤ÃѺà´ÕëÂÇàÊÕÂÊÔ·¸Ôì ÍÔÍÔ ) »Å. ÊÙé¹éͧæäÁèä´éàÅ 555 ÂÍ´àÂÕèÂÁ¡Ñ¹´Õ¤ÃѺ
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PaTa PatA pAtA Pon! |
#15
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¢éÍ 6 ¤ÃѺ
$60=6(\sqrt{49}+\sqrt{9})$ $60=49+9+\lfloor\sqrt{6}\rfloor$ $60=\lfloor 49+9+\sqrt{6}\rfloor$ 30 àÁÉÒ¹ 2006 22:46 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ gools |
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