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#31
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¢éÍË¡·ÓãËéÔ¶Ö¡¹éÍÂŧ¡ÇèÒ¹Õéä´é¤ÃѺ ÁÕá¹Ç¤Ô´´Ñ§¹Õé
à¾ÃÒÐ $A = 1^{2002}\cdot 2^{1001} \cdot 3^{1000} \cdot 4^{999}\cdots 1002^1$ $B = (1003!)^{1003} \cdot 1004^{1003} \cdot1005^{1002} \cdot 1006^{1001}\cdots2006^{1} $ ¡Ó¨Ñ´à·ÍÁ·ÕèÁÕàÅ¢¡ÓÅѧ¤Ùè·Ôé§ä»ãËéËÁ´ ÊèǹµÑÇ¡ÓÅѧà»ç¹¤Õè ´Ö§ÍÍ¡ÁÒ˹Ö觵ÑÇ áÅéǡӨѴ¡ÓÅѧ¤Ùè·Õè´Ö§ÍÍ¡·Ôé§ ¡ç¨Ðä´é $A_{\text{new}}=2\cdot4\cdots1002$ áÅÐ $B_{\text{new}}=1003!\cdot1004\cdot1006\cdots2006$ ´Ö§ÊͧÍÍ¡¨Ò¡àÅ¢¤ÙèáµèÅеÑÇÍÕ¡Ãͺ áÅéǨѺ¤Ù³¡Ñ¹ ¨ÐàËç¹ÇèÒä´éÊͧÁÒ 1003 µÑÇ áÅÐ 1003! àÍÒÊͧ¤Ù³à¢éÒä»ÍÕ¡µÑÇ àÅ¢ªÕé¡ÓÅѧ¢Í§ÊͧáÅÐ 1003! ¡ç¨Ðà»ç¹àÅ¢¤Ùè à»ç¹ÍѹàÊÃç¨¾Ô¸Õ 4. ÊÓËÃѺ $x,y>0$ à¹×èͧ¨Ò¡ $$x^3+y^3=(x+y)(x^2-xy+y^2)\ge (\frac{x^2+y^2}{2})(x+y)$$ ´Ñ§¹Ñé¹ $$\frac{x^3+y^3}{x^2+y^2}\ge\frac{x+y}2$$ àÍÒä» apply ¡Ñºà·ÍÁã¹â¨·Âì ¨Ðä´é $$\frac{a_1^3+a_2^3}{a_1^2 +a_2^2}+\frac{a_2^3+a_3^3}{a_2^2 +a_3^2}+ \cdots +\frac{a_n^3+a_1^3}{a_n^2 +a_1^2} \ge \sum a_i=1$$ ´Ñ§¹Ñé¹à¾Õ§¾Í·Õè¨ÐáÊ´§ÇèéÒ $$A:=\frac{a_1^3}{a_1^2 +a_2^2}+\frac{a_2^3}{a_2^2 +a_3^2}+ \cdots +\frac{a_n^3}{a_n^2 +a_1^2}= \frac{a_2^3}{a_1^2 +a_2^2}+\frac{a_3^3}{a_2^2 +a_3^2}+ \cdots +\frac{a_1^3}{a_n^2 +a_1^2}=:B$$ ¾Ô¨ÒÃ³Ò $$\frac{x^3-y^3}{x^2+y^2}=(x-y)(1+\frac{xy}{x^2+y^2})\le\frac32(x-y)$$ áÅÐ $$\frac{y^3-x^3}{x^2+y^2}=(y-x)(1+\frac{xy}{x^2+y^2})\le\frac32(y-x)$$ apply ¡Ñº $a_i$ ¨Ðä´é $A-B\le0$ áÅÐ $B-A\le0$ ¹Ñ蹤×Í $A=B$
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. 03 ¡ØÁÀҾѹ¸ì 2007 19:55 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nongtum |
#32
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Í×ÁÁÁÁ ËÅѧ¨Ò¡·Õè¾Õèæ »Å´»Åè;ÅѧáÅéÇáµèÅТéÍ¡ç §èÒ´ѧ¾ÅÔ¡½èÒÁ×ͨÃÔ§æ¤ÃѺ 555
Í×ÁÁÁÁ ÃÙéÊÖ¡ÇèÒ ¹éͧæ㹺ÍÃì´àÃÒ¨ÐŴŧæ ʧÊѨÐäÁè¤èÍÂä´éáÇÐÁÒàÂÕèÂÁàÂÕ¹
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PaTa PatA pAtA Pon! |
#33
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ÁÒà¡çºµ¡¤ÃÑ駷Õè 1 ¡Ñ¹¡è͹¤ÃѺ à´ÕëÂÇ´Ô¹¾Í¡ËÒ§ËÁÙ
¢éÍ 3(A),12 : ´Ùä´é¨Ò¡¤ÓµÍº¢Í§¤Ø³ prachya (ªÍº diagram ¢éÍ 12 ÁÒ¡¤ÃѺ) ¢éÍ 4,5,7,9 : ´Ùä´é¨Ò¡¤ÓµÍº¢Í§¤Ø³ nooonuii (¢Íº¤Ø³ÊÓËÃѺ additional analysis ã¹¢éÍ 9 ¤ÃѺ) ¢éÍ 6 : ¼Áà©ÅÂä»áÅéÇ ¢éÍ 14 : ´Ùä´é¨Ò¡¤ÓµÍº¢Í§¤Ø³ M@gpie ¢éÍ 15 : ´Ùä´é¨Ò¡¤ÓµÍº¤Ø³ Warut ·ÕèàËÅ×ͼÁ ¢Íáºè§à»ç¹ 2 ÊèǹáÅéǡѹ ¤×Í solutions ¡Ñº comment à¾ÔèÁàµÔÁ㹺ҧ¢éÍ (I) Comment ¢éÍ 1 Âѧ·Óä´éÍÕ¡ÇÔ¸Õ¤ÃѺ ãËé $ A=2x+\frac{1}{x+y} $ áÅÐ $ B=2y+\frac{1}{x+y} $ ÃкºÊÁ¡Òâé͹Õé¨Ð¡ÅÒÂà»ç¹ $ A^2+AB+B^2 =\frac{85}{3}+6 $ $ A= \frac{13}{3} $ áÅçÇ¡ç solve ÍÍ¡ÁÒµÒÁ»¡µÔ¤ÃѺ ¢éÍ 2 à¹×èͧ¨Ò¡ $ a^{\log_7 b}= b^{\log_7 a} $ ÊÁ¡Òè֧à¢Õ¹ãËÁèä´éà»ç¹ $ (x^{\log_7 3}+4)^{\log_7 3 }= x - 4 \cdots(*)$ ãËé $ y= x^{\log_7 3}+4 \Rightarrow x^{\log_7 3}= y-4 \cdots(1) $ ¢³Ðà´ÕÂǡѹ ¨Ò¡ (*) ¡ç¨Ðä´é $ y^{\log_7 3}=x-4 \cdots(2) $ $ (1)-(2) ; \,\, x^{\log_7 3}-y^{\log_7 3}= y-x $ ¶éÒ 0<x<y ËÃ×Í 0<y<x ¨Ð·ÓãËé sign ·Ñé§ 2 ¢éÒ§¢Í§ÊÁ¡Òà äÁèàËÁ×͹¡Ñ¹ ´Ñ§¹Ñé¹ x=y ¹Ñ蹤×Í $ x= x^{\log_7 3}+4 \rightarrow \log_3 (x-4)= \log_7 x $ ¡Ó˹´ $ a= \log_3 (x-4)= \log_7 x \rightarrow 7^a -3^a = 4 $ à¾ÃÒÐ $ f(a) = 7^a-3^a $ à»ç¹ 1-1 & strictly increasing function º¹ positive real number ´Ñ§¹Ñé¹ $ a= 1 \rightarrow x=7 $ à»ç¹ unique real solution ¢Í§ÊÁ¡ÒäÃѺ ¢éÍ 10 ¾Ô¨ÒÃ³Ò $$ N = 1!2!3!\cdots 2006! = \prod_{n=1}^{1003} (2n-1)!(2n)! $$ à¾ÃÒÐ $$ \begin{array}{lcr} \prod_{n=1}^{1003} (2n-1)!(2n)! &=& \prod_{n=1}^{1003}(2n)((2n-1)!)^2 \\ & = & 2^{1003}\cdot 1003! \cdot \text{Square term} \end{array} $$ ÊѧࡵÇèÒ $ \frac{N}{1003!}= AB $ «Öè§àÁ×èͤٳ´éÇ 2 ¨Ðä´é 2AB à»ç¹ square ·Ñ¹·Õ ¢éÍ 12 á¹Ç¤Ô´¢é͹Õé à·Õºà¤Õ§ä´é¡Ñº¤Ó¶ÒÁ㹤ÙèÁ×Í Á.»ÅÒ »ÃÐÁÒ³ÇèÒ ¶éÒÁÕ¤¹àÁÒà´Ô¹¡éÒÇ˹éÒ ¶ÍÂËÅѧ ÃÇÁ 12 ¡éÒÇ áÅéǾºÇèÒ à¢ÒÍÂÙè¢éҧ˹éҢͧ¨Ø´àÃÔèÁµé¹ 4 ¡éÒÇ à¢Ò¨ÐÁÕÇÔ¸Õà´Ô¹ä´é¡ÕèÇÔ¸Õ à§×èÍ¹ä¢ $ \mid a_{n+1} ?\, a_n \mid =1 $ ¡çàËÁ×͹¡Ñº ¡ÒáéÒÇ˹éÒ ¶ÍÂËÅѧ¹Ñè¹àͧ¤ÃѺ ÊÁÁµÔµéͧ¡éÒÇ仢éҧ˹éÒ x ¡éÒÇ ´Ñ§¹Ñé¹ x - (12-x) = 4 ËÃ×Í x= 8 ´Ñ§¹Ñ鹨ӹǹÇÔ¸Õ¡ÒÃà´Ô¹ãËéÍÂÙè˹éҨشàÃÔèÁµé¹ 4 ¡éÒÇ $( a_{13} -a_1 = 4)$ ¡ç¤×Í¡ÒÃàÃÕ§ÊѺà»ÅÕè¹ +1 (à´Ô¹Ë¹éÒ) 8 ¤ÃÑé§ áÅÐ -1 (¶ÍÂËÅѧ) 4 ¤ÃÑé§ «Ö觡ç¨Ðä´é¤ÓµÍº ¤×Í $ \frac{12!}{8!4!}$ ¢éÍ 15 ¢é͹Õé ¼Á¤§äÁèµéͧ͸ԺÒÂÍÐäÃÁÒ¡ à¾ÃÒФس Warut ¾Ù´ä»ËÁ´áÅéÇ ÊÒà˵طÕè¼ÁàÍÒ¢é͹ÕéÁÒ¶ÒÁ à¾ÃÒÐÍÂÒ¡ãËéÃÙé¨Ñ¡ fractal dimension ¤ÃѺ µÑÇÍÂèÒ§ fractal ·Õè¤Øé¹æ¡Ñ¹´Õ ¡çÍÂèÒ§ ÃÙ» Koch snowflake 㹡ÃзÙé Sequence series marathon «Öè§ÁÕÁÔµÔ»ÃÐÁÒ³ 1.26186 (Åͧ Verify àͧ¹Ð¤ÃѺ) à·èҡѺÇèÒ ÍÂÙèÃÐËÇèÒ§ÁԵԢͧàÊé¹â¤é§ ¡ÑºÁԵԢͧÃÙ»àËÅÕèÂÁ«Ð´éÇ µÑÇÍÂèÒ§¡ÒÃ¹Ó fractal ä»ãªé ·Õèà´è¹ªÑ´·ÕèÊش㹻Ѩ¨ØºÑ¹ ¡ç¤×Í ¡ÒþѲ¹ÒàÊÒàÍÒ¡ÒÈà¾×èÍãËéÃͧÃѺ ÂèÒ¹¤ÇÒÁ¶Õè¢Í§¤Å×è¹ÊÑÒ³·Õè¡ÇéÒ§¢Ö鹤ÃѺ ÊèǹÃÒÂÅÐàÍÕ´ ¤§µéͧÊͺ¶ÒÁ¨Ò¡¾Ç¡ÇÔÈÇ¡Ãâ·Ã¤Á¹Ò¤ÁáÅéÇÅèÐ
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#34
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ÍéÒ§ÍÔ§:
ãËé $\displaystyle{ x=\frac{1}{2},y=\frac{1}{3},z=\frac{1}{6} }$ ¨Ðä´éÇèÒ $\displaystyle{A = \frac{x^3}{x^2+y^2} + \frac{y^3}{y^2+z^2} + \frac{z^3}{z^2+x^2} = \frac{491}{780} }$ $\displaystyle{ B = \frac{y^3}{x^2+y^2} + \frac{z^3}{y^2+z^2} + \frac{x^3}{z^2+x^2} = \frac{457}{780} }$ ¼ÁÇèÒÍÊÁ¡ÒõèÍ仹ÕéäÁè¨ÃÔ§¤ÃѺ àÅÂà¡Ô´»ÑËÒ $\displaystyle{ (x-y)(1+\frac{xy}{x^2+y^2})\le\frac32(x-y) }$ $\displaystyle{ (y-x)(1+\frac{xy}{x^2+y^2})\le\frac32(y-x) }$ à¾ÃÒÐàÃÒäÁèÃÙéÇèÒ $x-y\geq 0$ ËÃ×ÍäÁè «Ö觶éÒàÃÒá¡éâ´Â¡ÒÃÊÁÁµÔãËé $x\geq y$ ¡çÂѧäÁèä´éÍÂÙè´Õà¾ÃÒÐàÃÒ¨Ðä´éÇèÒ $y-x\leq 0$ «Ö觨зÓãËéÍÕ¡ÍÊÁ¡ÒÃ˹Öè§äÁè¨ÃÔ§
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site:mathcenter.net ¤Ó¤é¹ |
#35
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¼Ô´ÍÕ¡áÅéÇ˹ÍàÃÒ
¶éÒ§Ñé¹ËÒ¡¨Ð·Óµèͨҡ·Õè¼Á·ÓÁÒ ãªé rearrangement ªèÇÂä´éäËÁ¤ÃѺ ËÒ¡äÁèä´é¨Ð·Óä´éÍÂèèÒ§äà ËÃ×ÍÇèÒ·Óä´éẺ·Õè¤Ø³ nooonuii Ẻà´ÕÂǤÃѺ
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. |
#36
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ÁÒà¡çºµ¡ ÍÕ¡¤ÃÑ駤ÃѺ
(II) Solutions 3(B) ãËé S ¤×ͼÅÅѾ¸ì·Õèµéͧ¡Òà $$ \begin{array}{rcl} \sin 1^{\circ}\cdot S &=& \frac{\sin 1^{\circ}}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{\sin 1^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\cdots +\frac{\sin 1^{\circ}}{\cos 88^{\circ} \cos 89^{\circ}} \\ &=& \tan 1^{\circ}+ \frac{\sin 2^{\circ}\cos 1^{\circ}-\sin 1^{\circ}\cos 2^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\cdots +\frac{\sin 89^{\circ}\cos 88^{\circ}-\sin 88^{\circ} \cos 89^{\circ}}{\cos 88^{\circ} \cos 89^{\circ}} \\ &=& \tan 1^{\circ}+ (\tan 2^{\circ}- \tan 1^{\circ} )+\cdots (\tan 89 ^{\circ}- \tan 88^{\circ}) \end{array} $$ ´Ñ§¹Ñé¹ $ S= \frac{\tan 89^{\circ}}{\sin 1^{\circ}} =\frac{\cos 1^{\circ}}{\sin^2 1^{\circ}} $ ¢éÍ 8 ,11 à´ÕëÂÇ¡ÅÒ§ÍÒ·ÔµÂì˹éҨРscan ÃÙ»¢éÍ 8 ¡Ñº¡ÃÒ¿¢éÍ 11 ÁÒãËé´Ù¤ÃѺ ËÃ×Íã¤ÃÍÂÒ¡à©Å 2 ¢é͹Õé仡è͹ ¡çµÒÁʺÒÂàŹФÃѺ ¢éÍ 13 àËç¹ä´éªÑ´ÇèÒ k à»ç¹¨Ó¹Ç¹ÍµÃáÂÐ ¹Í¡¨Ò¡¹Õé ÊÓËÃѺ ¨Ó¹Ç¹¹Ñº n ã´æ $ kx_{n-1} -1 < x_n < kx_{n-1} \Rightarrow \frac{x_n}{k}< x_{n-1}< \frac{x_n}{k}+\frac{1}{k}$ ´Ñ§¹Ñé¹ $ \lfloor \frac{x_n}{k} \rfloor = x_{n-1}-1 $ ¨Ò¡ÊÁ¡Ò÷Õè⨷Âì¡Ó˹´ ·ÓãËé $ kx_n = (2550 + \frac{1}{k})(x_n) = 2550x_n + \frac{x_n}{k} $ ´Ñ§¹Ñé¹ $ x_{n+1}= \lfloor kx_n \rfloor = 2550x_n + x_{n-1}-1 \equiv x_{n-1}-1 \pmod {2550} $ ¹Ñ蹤×Í $ x_{2550} \equiv x_0-1275 \equiv -1274 \pmod {2550} \equiv 1276 \pmod {2550} $ ´Ñ§¹Ñé¹ ¢é͹ÕéµÍº 1276 ¤ÃѺ p.s. ¡çµéͧ¢Íº¤Ø³¹éͧæ·Ñé§ 3 ¤¹ ·ÕèÊÅÐàÇÅÒÍѹÁÕ¤èÒÁÒá¨Áã¹ â¤Ã§¡Òà 2 ¹Õé¤ÃѺ áÁéÇèÒ ºÒ§¤¹¨ÐµÔ´Êͺ ºÒ§¤¹¡çã¡Åé¨Ðä»à¢Òª¹ä¡è ÍÂèÒ§¹éÍ ¡ç¨Ø´»ÃСÒ ·ÓãËé¼ÁÍÂÒ¡ËÒÍÐäÃÁÒᨡÍÕ¡ã¹»ÕµèÍæä» äÁèÇèÒ¨ÐÁÕ¤¹àÅ蹡Õ褹¡çµÒÁ
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#37
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ÍéÒ§ÍÔ§:
¼ÁÇèÒà˵ؼÅ˹Ö觷Õè⨷ÂìÍÊÁ¡ÒÃà»ç¹Ë¹Öè§ã¹Ë¡¢éÍÊͺ¤³ÔµÈÒʵÃìâÍÅÔÁ»Ô¡á·º·Ø¡»Õ¡çà¾ÃÒÐÇèÒ»ÑËÒÍÊÁ¡ÒùÑé¹Áդӵͺä´éËÅÒ¡ËÅÒ¤ÃѺ ¼Á¨Ö§¤Ô´ÇèÒ¢é͹ÕéµéͧÁÕÇÔ¸Õ¤Ô´·ÕèᵡµèÒ§ÍÍ¡ä» ¼Á¡ç¾ÂÒÂÒÁ¤Ô´â´ÂäÁèãªé Hint ·Õè¤Ø³ passer-by ãËéÁÒàËÁ×͹¡Ñ¹áµèÂѧäÁèÍÍ¡¤ÃѺ ÊÓËÃѺÇÔ¸Õ¤Ô´¢Í§¼Á ¼Á¡çÂѧÂ×¹ÂѹÇÔ¸Õ¡ÒÃà´ÔÁ¤×Íàªç¤ÊÁ¡Òáè͹¤ÃѺÇèÒà¡Ô´¢Öé¹àÁ×èÍäËÃè «Ö觨Ðä´éÇèÒÊÁ¡ÒÃà¡Ô´¢Öé¹àÁ×è͵ÑÇá»Ã·Ø¡µÑÇÁÕ¤èÒà·èҡѹ ¨Ò¡¹Ñ鹡çÁÒ´Ù·Õèà§×è͹ä¢â¨·Âì ¨ÐàËç¹ÇèÒ¶éÒàÃҨѺáµèÅо¨¹ìÁÒ·Ó¡ÒÃËÒÃÂÒÇ àÃÒ¨Ðä´é¼ÅÅѾ¸ìà»ç¹ $a_i$ â¼ÅèÁÒã¹áµèÅÐà·ÍÁ ÊèǹàÈÉ·Õèä´éÍÒ¨¨ÐÂѧäÁèµéͧʹ㨡çä´é «Ö觶éÒã¤Ã¤Ô´ÁÒä´é¶Ö§¨Ø´¹Õé¡ç make sense ·Õè¨Ð¤Ô´µèͤÃѺ à¾ÃÒÐà§×è͹ä¢â¨·ÂìÁѹªÕé¹ÓäÇéÍÂèÒ§¹Ñé¹ ¼Á¾ÂÒÂÒÁãªéÍÊÁ¡ÒÃ⤪մéÇÂàËÁ×͹¡Ñ¹¤ÃѺ áµè¨ÐµÔ´»ÑËÒ·Õèà§×è͹䢡ÒÃà»ç¹ÊÁ¡Òà à¾ÃÒÐÍÊÁ¡ÒÃ⤪ըÐÁÕà§×è͹䢡ÒÃà»ç¹ÊÁ¡Ò÷ÕèµèÒ§¨Ò¡ AM-GM ·ÓãËé¡ÒÃÊÃéÒ§àǤàµÍÃì·ÕèÊÁ¡ÒÃà»ç¹¨ÃÔ§àÁ×è͵ÑÇá»Ã·Ø¡µÑÇÁÕ¤èÒà·èҡѹ·Óä´éÂÒ¡¡ÇèÒÁÒ¡àŤÃѺ
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site:mathcenter.net ¤Ó¤é¹ |
#38
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µÍ¹áá¢éÍ 4. ¼Á¡ç¤×ͨÐàÃÕ§¤èÒµÑÇá»ÃàËÁ×͹¡Ñ¹¤ÃѺ áµè¾ºÇèÒÁѹǹ¡ÅѺÁÒà»ç¹§Ù¡Ô¹ËÒ§ àÅÂà¡Ô´»ÑËÒ¢Öé¹ àÅ §§ µèÍä» ÍÔÍÔ áµèÇÔ¸Õ¾Õè ¡çÂÒ¡ãªèÂè͹ТÍÃѺ¹Õè (ÊѧࡵÇèÒ¼Á·Óä´éáµèẺ§èÒÂæ) ÍÔÍÔ
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PaTa PatA pAtA Pon! |
#39
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à¡çºµ¡ 2 ¢éÍÊØ´·éÒ¤ÃѺ
¢éÍ 11 à»ç¹àÃ×èͧ¢Í§¤ÇÒÁ¹èÒ¨Ðà»ç¹ ¡Ã³Õ·Õè sample space à»ç¹à«µÍ¹Ñ¹µì ¤ÃѺ ãËé x,y á·¹ÃÐÂÐËèÒ§¨Ò¡«éÒÂÊØ´¢Í§àÊé¹ ´Ñ§¹Ñé¹ ÍÊÁ¡Ò÷Õè¼Áà¢Õ¹äÇéã¹ÃÙ»¢éÒ§ÅèÒ§ ·Ñé§ 2 ÊÁ¡Òà ¡ç¤×Í event ·Õèµéͧ¤Ô´ «Öè§àÁ×èÍÇÒ´ÃÙ»áÅéÇ ¡ç¨Ðä´é¾×é¹·Õè·ÕèáÃà§Ò Êèǹ sample space ¡ç¤×;×é¹·Õè¢Í§ÊÕèàËÅÕèÂÁ¨ÑµØÃÑÊ 5x5 ¹Ñè¹àͧ¤ÃѺ ´Ñ§¹Ñé¹ $ P(E)= \frac{\text{Shaded Area}}{\text{Square area}}= \frac{13}{25} $ ¢éÍ 8 ´ÙµÒÁ¤Ó͸ԺÒÂã¹ÃÙ»àŤÃѺ ¨Ò¡¹Ñé¹ à¾ÃÒÐ $ ME= IE\sin(\frac{A}{2}) $ áÅÐ $ AI= \frac{r}{\sin(\frac{A}{2})} $ (Åͧ Verify ´Ù¹Ð¤ÃѺ) ÊØ´·éÒ ¡ç¨Ðä´é ¢¹Ò´¢Í§ AX ¡ç¤×Í r ¹Ñè¹àͧ
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