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#1
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ᨡ੾ÒСԨ (â¤Ã§¡Òà 2) : ÈØ¡Ãì¹Õé !
¡ÅѺÁÒáÅéǤÃѺ ÊÓËÃѺᨡ੾ÒСԨ â¤Ã§¡Òà 2 µé͹ÃѺà´×͹áË觤ÇÒÁÃÑ¡
â´Â¡ÒÃᨡ¤ÃÒǹÕé ¨ÐËÒ¼Ùé·Õèä´é¤Ðá¹¹ÊÙ§ÊØ´à¾Õ§¤¹à´ÕÂÇ·Õè¨Ðä´é¢Í§ä» Êèǹ¡ÒÃá¢è§¢Ñ¹ ¨Ðá¢è§áººÃͺà´ÕÂǨº¤ÃѺ ¤Ó¶ÒÁ·Ñé§ËÁ´ 15 ¢éÍ ¨Ð¶Ù¡ post »ÃÐÁÒ³ 1 ·ØèÁ¤ÃÖ觢ͧÇѹÈØ¡Ãì·Õè 2 ¡.¾. 2550 áÅШлԴÃѺ·Ø¡¤ÓµÍºã¹àÇÅÒ 6 âÁ§àªéҢͧÇѹàÊÒÃì·Õè 3 ¡ØÁÀҾѹ¸ì à·èҡѺàÇÅÒÁÕàÇÅÒ»ÃÐÁÒ³ 10 ªÑèÇâÁ§ 㹡ÒäԴ⨷Âì ÃдѺ¤ÇÒÁÂÒ¡§èÒ ¡çã¡Åéà¤Õ§¡Ñ¹¤ÃѺ à¾ÃÒмÁãËé¤Ðá¹¹àµçÁ·Ø¡¢éÍà·èҡѹ à¹×éÍËÒ·Õèãªé ¡çÁÕµÑé§áµè á¡éÊÁ¡Òà , µÃÕ⡳ÁÔµÔ , àÁµÃÔ¡«ì , ÍÊÁ¡Òà , àâҤ³Ôµ , ¿Ñ§¡ìªÑ¹ , Number theory , ¤ÇÒÁ¹èÒ¨Ðà»ç¹ áÅФÇÒÁÃÙé·ÑèÇä»à¡ÕèÂǡѺ Maths ¤ÃѺ ¡µÔ¡Ò ¡ç¤ÅéÒÂæà´ÔÁ¤ÃѺ ¤×Í (i) ʧǹÊÔ·¸Ôì੾ÒмÙé·ÕèÂѧäÁè ¨º ».µÃÕ ËÃ×Í ¼Ùé·ÕèäÁèÍÂÙèã¹¢éÍËéÒÁ 3 ¢é͵èÍ仹Õé (A) à»ç¹ moderator ·Õè¹Õè (B) ÁÕÇØ²Ô â·-àÍ¡ ·Ò§¤³ÔµÈÒʵÃì (C) ¡ÓÅѧÈÖ¡ÉÒ â·-àÍ¡ ·Ò§ ¤³ÔµÈÒʵÃì ¶éÒÍÂÙè㹡ÅØèÁ (A) ËÃ×Í (B) ËÃ×Í (C) ¡çËÁ´ÊÔ·¸ÔìàÅè¹ ¤ÃѺ (ii) áÊ´§ÇÔ¸Õ·Ó¾ÍÊѧࢻ áÅÐÊÓËÃѺ¢éÍà´ÕÂǡѹ 2 ¤¹¨ÐµÍº«éӡѹä´é áµèµéͧãªéÇÔ¸Õ·ÕèµèÒ§¡Ñ¹(ÍÂèÒ§àËç¹ä´éªÑ´) (iii) á¡é䢤ӵͺ·ÕèµÍºä»áÅéÇ¡Õè¤ÃÑ駡çä´é áµèµéͧ·Ó¡è͹»Ô´ÃѺ¤ÓµÍº (iv) ÍÂèÒ¾ÂÒÂÒÁ⡧ äÁèÇèÒÇÔ¸Õã´¡çµÒÁ ¢ÍãËé«×èÍÊѵÂì¡ÑºµÑÇàͧ¤ÃѺ ã¤ÃÁÕ¢éÍʧÊÑÂÍÐäáçÊͺ¶ÒÁä´é¡è͹¡ÒÃᨡ¨ÐàÃÔèÁã¹ÈØ¡Ãì¹Õé¤ÃѺ Êèǹ¢Í§ÃÒ§ÇÑÅ·Õè¨ÐᨡÊÓËÃѺ¼Ùé·Õèä´é¤Ðá¹¹ÊÙ§ÊØ´ ¤×Í Ë¹Ñ§Ê×Í¢éÍÊͺ¾ÃéÍÁà©Å Singapore Maths Olympiad (2001-2002) ¤ÃѺ (àÍÒ˹éÒ»¡ä»´Ù¾ÅÒ§æ¡è͹áÅéǡѹ)
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#2
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âÍéÇ ¼ÁÃдѺã¡Å騺».µÃÕ ¤ÃѺ ÂѧàÅè¹ä´é áËÐææ
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PaTa PatA pAtA Pon! |
#3
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ã¡Åéà¡ÉÕ³à¡×ͺ60 àÅè¹´éÇÂä´éäËÁ
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#4
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ÍéÒ§ÍÔ§:
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àÁ×èͤԴ¨Ð·ÓÍÐäà ËÒ¡¤Ô´ÁÒ¡ä» àÁ×èÍäËÃè¨Ðä´éŧÁ×Í·Ó |
#5
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ÊÓËÃѺ ¹éͧ M@gpie äÁèÁÕ»ÑËÒ¤ÃѺ ¤ÃÒǹÕé confirm ÇèÒ ¹éͧ M@gpie ¨ÐäÁèâ´¹¡µÔ¡Ò¡´¤ÐṹẺ¤ÃÒÇ·ÕèáÅéÇá¹è¹Í¹
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#6
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PRACTICE PROBLEMS (ãËé«éÍÁÁ×Íà©Âæ áµèäÁè»ÃÒ¡¯ã¹¤Ó¶ÒÁ¢Í§Çѹ¨ÃÔ§¹Ð¤ÃéÒº)
1. Calculate $$ \prod_{i=1}^{89} \sin i^{\circ} $$ 2. ËҤӵͺ·Ñé§ËÁ´·Õèà»ç¹¨Ó¹Ç¹¨ÃÔ§¢Í§ÊÁ¡Òà $ x^2-x-1000\sqrt{1+8000x}=1000 $ 3. ¡Ó˹´ $ F_1=1 , F_2=1 $ áÅÐ $ F_n = F_{n-1}+F_{n-2} \,\, (n \geq 3) $ ËÒ¤èÒ $$ \sum_{n=2}^{\infty} \frac{F_n}{F_{n-1}F_{n+1}} $$ 4. ¡Ó˹´àÁµÃÔ¡«ì¢¹Ò´ n x n á·¹´éÇ $D_n$ áÅÐÁÕÊÁÒªÔ¡áµèÅеÑǴѧ¹Õé $ d_{ij}= \left\{\begin{array}{ll} 5 & ,i= j \\ 2 & ,\mid i-j \mid = 1 \\ 0 & ,\text{otherwise} \end{array} \right. $ ËÒàÈÉ·Ñé§ËÁ´·Õèä´é¨Ò¡¡ÒÃËÒà $ det(D_p) $ ´éÇ 5 àÁ×èÍ p à»ç¹¨Ó¹Ç¹à©¾Òкǡ 5. ¡Ó˹´ÊÕèàËÅÕèÂÁ¨ÑµØÃÑÊ ABCD áÅÐ O à»ç¹Ç§¡ÅÁ¨Ø´ÈÙ¹Âì¡ÅÒ§ A ÃÑÈÁÕ AB ¶éÒ P, M à»ç¹¨Ø´º¹ CD, BC µÒÁÅӴѺ «Öè§ PM ÊÑÁ¼ÑÊǧ¡ÅÁ O ãËé AP , AM µÑ´ BD ·Õè Q, N µÒÁÅӴѺ ¾ÔÊÙ¨¹ìÇèÒ ÊÒÁÒöÊÃéҧǧ¡ÅÁ·ÕèÁÕ P, Q, N, M , C ÍÂÙ躹àÊé¹Ãͺǧä´é
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#7
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1. ÍéÒ§ä´éàÅÂÃÖà»ÅèÒ¤ÃѺÇèÒ $$\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$$
Hence, $$\prod_{i=1}^{89} \sin i^{\circ}=\sqrt{\frac{180}{2^{179}}}$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#8
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¤ÓµÍº¡ç¶Ù¡áËÅФÃѺ áµè¼ÁÇèÒ ÇÔ¸ÕÁѹÅÑ´à¡Ô¹ä»ÁÑ駤ÃѺ ¹éͧ mastermander
ÊÓËÃѺÇѹ¨ÃÔ§ 㹡óշÕ辺ÇèÒ á·¹¤èÒÊÙµÃáÅéÇÍÍ¡ áµèÊÙµÃäÁèãªè standard well-known formula ¡ç¹èҨк͡·Õèä»·ÕèÁҢͧÊٵùԴ¹Ö§¡ç´Õ¤ÃѺ p.s. ¼Á guide ãËé¹Ô´¹Ö§ÇèÒ ã¹Çѹ¨ÃÔ§ ãËéÅͧ scan ¤Ó¶ÒÁ·Ñé§ËÁ´¡è͹¤ÃѺ à¾ÃÒÐÁÕºÒ§¢éÍ·ÕèÁѹ§èÒÂÍÂèÒ§àËç¹ä´éªÑ´ ὧÍÂÙè´éÇÂ
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#9
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àÅè¹äÁèä´é §Ñé¹áÇÐÁÒ»ÅèͤÓãºé¤Ó¶ÒÁ«éÍÁÁ×ͺҧ¢éÍÅСѹ¤ÃѺ
2. à·¤¹Ô¤¤ÅéÒÂæ¢éÍ 4. ¡ÃзÙé sequence and series marathon 3. ÊÁ¡ÒÃ⨷ÂìãËé¢éÍÁÙÅà¡ÕèÂǡѺà·ÍÁ·ÕèµÔ´ÃÒ¡ÍÂèÒ§äà ¢é͹ÕéµÍº 2001 4. ¾Ô¨ÒóҤÇÒÁÊÑÁ¾Ñ¹¸ìÃÐËÇèÒ§ $\det(D_{2k+1})$ áÅÐ $\det(D_{2k})$ àÁ×èÍ $k=0,1,2,\dots$ 5. ÅÒ¡¨Ò¡¨Ø´ A ä»ËÒàÊé¹ÊÑÁ¼ÑÊ ¨Ø´ P áÅÐ M
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. |
#10
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¾ÃéÍÁÍÂÙè¡Ñ¹¢éÒÁ¤×¹ËÃ×ÍÂѧ¤ÃѺ §Ñé¹ÁÒàÃÔèÁ¡Ñ¹àÅÂ
¹Õè¤×Í ¢éÍ 1-14 (¢éÍÅÐ 2 ¤Ðá¹¹) 1. ËÒ (x,y) ·Ñé§ËÁ´·Õèà»ç¹¨Ó¹Ç¹¨ÃÔ§ áÅÐÊÍ´¤Åéͧ¡Ñº $$ 4xy+4(x^2+y^2)+ \frac{3}{(x+y)^2}= \frac{85}{3} $$ $$ 2x+\frac{1}{x+y}= \frac{13}{3} $$ 2. ËÒÃÒ¡¨ÃÔ§·Ñé§ËÁ´¢Í§ÊÁ¡Òà $$ (3^{\log_7 x}+4)^{\log_7 3 }= x - 4 $$ 3. àÅ×Í¡·Ó 1 ¢éÍà·èÒ¹Ñé¹ (A) Calculate $$ \prod_{i=1}^{29} (\sqrt{3}+\tan i^{\circ}) $$ (B) Simplify $$ \frac{1}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{1}{\cos 1^{\circ} \cos 2^{\circ}}+\cdots +\frac{1}{\cos 88^{\circ} \cos 89^{\circ}}$$ 4. ¡Ó˹´ $ a_i>0 $ áÅÐ $ \sum_{i=1}^n a_i = 1 $ ¾ÔÊÙ¨¹ìÇèÒ $$ \frac{a_1^3}{a_1^2 +a_2^2}+\frac{a_2^3}{a_2^2 +a_3^2}+ \cdots +\frac{a_n^3}{a_n^2 +a_1^2} \geq \frac{1}{2} $$ 5. ãËé $ p_1 , p_2 , \cdots , p_{2006} $ à»ç¹¨Ó¹Ç¹¹Ñº·ÕèµèÒ§¡Ñ¹ áÅÐÁÒ¡¡ÇèÒ 1 ¾ÔÊÙ¨¹ìÇèÒ $$ \prod_{i=1}^{2006} \big( 1- \frac{1}{p_i^2} \big ) > \frac{1}{2} $$ 6. ãËé Q ᷹૵¢Í§¨Ó¹Ç¹µÃáÂÐ ¡Ó˹´ $ f:N \rightarrow Q $ «Öè§ $ f(1)= \frac{3}{2} $ áÅÐ $ f(x+y)= (1+ \frac{y}{x+1})f(x)+ (1+ \frac{x}{y+1})f(y)+ x^2y+xy+xy^2 $ ·Ø¡¨Ó¹Ç¹¹Ñº $ x,y $ ËÒ¤èÒ f(20) 7. ãËé $ a \geq 0$ ¾ÔÊÙ¨¹ìÇèÒ $ \sqrt{a}+\sqrt[3]{a}+ \sqrt[6]{a} \leq a+2 $ 8. ¡Ó˹´Ç§¡ÅÁ¨Ø´ÈÙ¹Âì¡ÅÒ§ I ÃÑÈÁÕ r Ṻã¹ÊÒÁàËÅÕèÂÁ ABC ¶éÒàÊé¹·Õèàª×èÍÁ I áÅШش¡Ö觡ÅÒ§ BC µÑ´ÊèǹÊÙ§¨Ò¡ A ·Õè X ¾ÔÊÙ¨¹ìÇèÒ ¢¹Ò´¢Í§ AX à·èҡѺ r 9. Ëҿѧ¡ìªÑ¹ 1-1 ·ÑèǶ֧ $ f :[0,1) \rightarrow (0,1) $ â´Â $ f(x) \neq x $ for infinitely many x 10. $ A= 1!2!3! \cdots 1002! $ áÅÐ $ B = 1004!1005!1006! \cdots 2006! $ ¾ÔÊÙ¨¹ìÇèÒ 2AB à¢Õ¹ã¹ÃÙ»¡ÓÅѧÊͧ¢Í§¨Ó¹Ç¹¹Ñºä´é 11. Êèǹ¢Í§àÊ鹵çÂÒÇ 5 ˹èÇ ¶éÒÊØèÁàÅ×Í¡¨Ø´ 2 ¨Ø´ÍÂèÒ§ÊØèÁº¹Êèǹ¢Í§àÊ鹵ç à¾×èÍáºè§Êèǹ¢Í§àÊ鹵çà»ç¹ 3 Êèǹ ËÒ¤ÇÒÁ¹èÒ¨Ðà»ç¹·Õè·Ñé§ 3 Êèǹ ÂÒÇäÁèà¡Ô¹ 3 ˹èÇ 12. $ a_1 , a_2, \cdots a_{13} \in R $ â´Â $ \mid a_{n+1} ?\, a_n \mid =1 $ àÁ×èÍ $ n =1,2,\cdots 12 $ áÅÐ $a_1=1 , a_{13}=5 $ ËÒÇèÒÁÕ $ a_1 , a_2, \cdots a_{13} $ ¡ÕèªØ´ 13. $k$ à»ç¹ÃÒ¡¨Ó¹Ç¹¨ÃÔ§ºÇ¡¢Í§ÊÁ¡Òà $ x^2-2550x-1=0 $ ¶éÒ¡Ó˹´ $x_0=1 , x_{n+1}= \lfloor kx_n \rfloor \,\, (n \geq 0)$ ËÒàÈɨҡ¡ÒÃËÒà $ x_{2550}$ ´éÇ 2550 14. Simplify det(A) àÁ×èÍ A = $\bmatrix{a-b & b-c &c-a \\ b-c & c-a & a-b \\c-a &a-b &b-c} $
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#11
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áÅТéÍ·Õè 15 ãËé 3 ¤Ðá¹¹¤ÃѺ
15. ¨Ò¡ÃÙ»´éÒ¹ÅèÒ§ à»ç¹Êèǹ˹Ö觢ͧ¡ÒÃÊÃéÒ§ Cantor set (¶Ö§áÁéäÁèãªè standard version) «Öè§à¡Ô´¨Ò¡¡Òà ÅÒ¡àÊ鹵çÂÒÇ 1 ˹èÇ , áºè§à»ç¹ 5 Êèǹà·èÒæ¡Ñ¹ áÅÐ remove Êèǹ·Õè 2 ¡Ñº 4 ·Ôé§ä» (´Ñ§ºÃ÷Ѵ·Õè 2) ¨Ò¡¹Ñ鹡ç·Ó«éÓàªè¹¹Õéä»àÃ×èÍÂæ ¶Ö§Í¹Ñ¹µì ÊÔ觷Õè¹èÒʹ㨤×Í àÁ×èÍ·Óàªè¹¹Õé件֧͹ѹµì ºÃ÷ѴÊØ´·éÒ·Õèä´é ÁÕÁÔµÔà·èÒäËÃè ¨ÐàËÁ×͹¡Ñº ¡Ã³Õ ÁԵԢͧàÊ鹵ç ÃÙ»àËÅÕèÂÁ ËÃ×ÍÅÙ¡ºÒÈ¡ìËÃ×ÍäÁè ´Ñ§¹Ñé¹ ¤Ó¶ÒÁ¢é͹Õé ¡ç¤×ͼÁ ÍÂÒ¡ÃÙéÇèÒ ÁÔµÔ (dimension) ¢Í§Ãٻ㹺Ã÷ѴÊØ´·éÒ àÁ×èÍ·Ó件֧͹ѹµì à»ç¹à·èÒã´ (µÍºà»ç¹·È¹ÔÂÁ 4 µÓá˹è§) (Hint: Íѹ´Ñºáá ¤ÇèÐËÒãËéä´é¡è͹ÇèÒ ¨ÐËÒÁԵԢͧÃÙ»ä´é¨Ò¡ÊÙµÃÍÐäÃ) ¢ÍãËé⪤´Õ¤ÃѺ
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#12
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¢Í͹ØÒµ«ÔÇ¢éͧèÒ¡è͹¹Ð¤ÃѺ ¾ÃØ觹ÕéÁÕÊͺàÊÕ´éÇ ÍÔÍÔ
¢éÍ 1. ¾Ô¨ÒÃ³Ò \[ 4xy +4(x^2+y^2) +\frac{3}{(x+y)^2} = \frac{85}{3} \Rightarrow 4(x+y)^2-4xy + \frac{3}{(x+y)^2} = \frac{85}{3} .........(*) \] ãËé $u=x+y, \; v=x-y$ ¨Ðä´éÇèÒ $u=\frac{x+y}{2}, \; v=\frac{x-y}{2}$ á·¹¤èÒ ã¹ÊÁ¡Ò÷Ñé§Êͧ ¨Ðä´éÇèÒ \[ 3(u^2+\frac{1}{u^2}) + v^2 = \frac{85}{3}\] \[ 3(u+\frac{1}{u})^2 - 6 +v^2 = \frac{85}{3} \] áÅÐã¹ÊÁ¡Ò÷ÕèÊͧã¹â¨·Âì¨Ðä´éà»ç¹ \[ u+\frac{1}{u} = \frac{13}{3} - v\] á¡éÊÁ¡Ò÷Ñé§Êͧ ¨Ðä´é $v=\frac{11}{2}, \;\; 1$ áÅйÓä»á·¹¤èÒËÒ u àÁ×èÍ $v=1$ ¨Ðä´é¤ÓµÍº·Õèà»ç¹¨Ó¹Ç¹¨ÃÔ§ ¤×Í $ \{ (\frac{1}{3},1),(3,1) \}$ àÁ×èÍ $v=\frac{11}{2}$ ¨Ðä´é $u$ äÁèà»ç¹¨Ó¹Ç¹¨ÃÔ§ ¨Ðä´éÇèÒ \[ (u,v) \in \{ (\frac{1}{3},1),(3,1) \}\] ¤ÓµÍº·Õèá·é¨ÃÔ§¤×Í \[ (x,y) \in \{ (\frac{2}{3},-\frac{1}{3}),(2,1) \}\] ¢éÍ 3(A) ¨Ò¡â¨·Âì¨Ðä´éÇèÒ \[ \Pi_{i=1}^{29}(\sqrt{3}+\tan i^{\circ}) =\Pi_{i=1}^{29}\frac{\sin 60^{\circ} \cos i^{\circ} +\cos 60^{\circ}\sin i^{\circ}}{\cos 60^{\circ} \cos i^{\circ}} = \frac{1}{\cos^{29} 60^{\circ}}\Pi_{i=1}^{29}\frac{\cos (30-i)^{\circ}}{\cos i^{\circ}} = 2^{29}\] edit áÅéǤÃѺ ÁÖ¹æàÅ硹éͤÃѺ ÍÔÍÔ 7. ãËé $x=\sqrt[6]{a}$ ÍÊÁ¡Ò÷Õèµéͧ¡ÒþÔÊÙ¨¹ì¡ÅÒÂà»ç¹ $0 \leq x^6 -x^3-x^2-x+2,\; \; \forall x\geq 0$ ãËé $f(x)=x^6 -x^3-x^2-x+2$ àÃÒ¨ÐáÊ´§ÇèÒ $f(x) \geq 0$ ÊÓËÃѺ·Ø¡¤èÒ $x\geq 0$ ¡çà¾Õ§¾ÍáÅéÇ à¹×èͧ¨Ò¡ $f'(x)= (x-1)(6x^4+6x^3+6x^2+3x+1) = 0 \Rightarrow x=1$ áÅÐ $f''(1) >0$ ´Ñ§¹Ñé¹ $x=0$ à»ç¹¨Ø´µèÓÊØ´º¹ªèǧ $[0,\infty)$ ¨Ö§ä´éÇèÒ $f(x) \geq 0 $ ÊÓËÃѺ·Ø¡¤èÒ $x \geq 0$ ¨Ö§ä´éÍÊÁ¡ÒõÒÁ·Õè⨷Âìµéͧ¡Òà ¢éÍ 14. ãËé $u=a-b, \; v=b-c, \; w= c-a$ ¨Ðä´éÇèÒ $u+v+w =0 $ \[ \begin{array}{ccl} \det (A) &= & \left\vert \begin{array}{ccc} u &v &w \\ v&w&u \\ w &u &v \end{array} \right\vert \\ &=& \left\vert \begin{array}{ccc} u &v &w \\ u+v &v+w &w+u \\ w &u &v \end{array}\right\vert \; \; (R_1+R_2\rightarrow R_2)\\ &=& \left\vert \begin{array}{ccc} u &v &w \\ -w &-u &-v \\ w &u &v \end{array} \right\vert \\ &=& 0 \end{array} \] ´Ñ§¹Ñ鹵ͺ $\det (A)=0$
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PaTa PatA pAtA Pon! 02 ¡ØÁÀҾѹ¸ì 2007 23:04 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 7 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ M@gpie |
#13
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5. ¨ÐáÊ´§ÇèÒ $$\prod_{n=2}^\infty(1-\frac1{n^2})=\frac12 $$
$$\prod_{n=2}^N(\frac{n-1}{n})\prod_{n=2}^N(\frac{n+1}{n})$$ \[ \left( {\frac{3}{2} \cdot \frac{4}{3} \cdot ... \cdot \frac{{N + 1}}{N}} \right)\left( {\frac{1}{2} \cdot \frac{2}{3} \cdot ... \cdot \frac{{N - 1}}{N}} \right) = \frac{{N + 1}}{{2N}} \] \[ \prod\limits_{n = 2}^\infty {\left( {1 - \frac{1}{{n^2 }}} \right)} = \mathop {\lim }\limits_{N \to \infty } \frac{{N + 1}}{{2N}} = \frac{1}{2} \] à¾ÃÒÐÇèÒ·Ø¡µÑǤٳ·ÕèÁÕ¤èÒ¹éÍ¡ÇèÒ˹Öè§ àÁ×èÍàÃÒ¤Ù³à¢éÒ仡Ѻ¨Ó¹Ç¹¨ÃÔ§ºÇ¡ã´æáÅéÇ·ÓãËé¨Ó¹Ç¹¹Ñé¹ÁÕ¤èÒ¹éÍ¡ÇèÒà´ÔÁ (´Ñ§¹Ñé¹àÃÒËÒÃÍÍ¡¨Ö§ÁÕ¤èÒÁÒ¡¡ÇèÒà´ÔÁ) ´Ñ§¹Ñ鹶éÒàÃÒàÅ×Í¡ÁÒà¾Õ§ 2006 µÑÇ ¤èÒ·Õèä´é¨Ö§ÁÒ¡¡ÇèÒ $\frac12$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠03 ¡ØÁÀҾѹ¸ì 2007 14:43 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#14
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ÍéÒ§ÍÔ§:
$$(\sqrt{3}+\tan 1^{\circ})(\sqrt{3}+\tan 2^{\circ})(\sqrt{3}+\tan 3^{\circ})....(\sqrt{3}+\tan 29^{\circ}) $$ $$(\sqrt{3}+\tan 1^{\circ})(\sqrt{3}+\tan 29^{\circ}) = 3 + \sqrt{3}(\tan 1^{\circ}+ \tan 29^{\circ})+\tan 1^{\circ}\tan 29^{\circ} $$ $$ \tan (1+29)^{\circ} = \frac{\tan 1^{\circ}+\tan 29^{\circ}}{1-\tan 1^{\circ}\tan 29^{\circ}} = \frac{1}{\sqrt{ 3}} --> \sqrt{3}(\tan 1^{\circ}+ \tan 29^{\circ}) = 1-\tan 1^{\circ}\tan 29^{\circ}$$ ´Ñ§¹Ñé¹ $$(\sqrt{3}+\tan 1^{\circ})(\sqrt{3}+\tan 29^{\circ}) = 4$$ àÁ×èͨѺ¤Ùè 1~29 , 2~28 , .... , 14~16 «Ö觨Ðä´é·Ñé§ËÁ´ 14 ¤Ùè áÅéÇàËÅ×Í 15 ·ÕèäÁèÁÕ¤Ùè áµè $$ \sqrt{3} +\tan 15^{\circ} = \sqrt{3} + (2 - \sqrt{3}) = 2 $$ ´Ñ§¹Ñé¹ $$ \prod_{i=1}^{29} (\sqrt{3}+\tan i^{\circ}) = 4^{14}*2 = 2^{29} $$ 02 ¡ØÁÀҾѹ¸ì 2007 22:31 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ prachya |
#15
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¢ÍàµÔÁá¹Ç¤Ô´ÅСѹ¹Ð¤ÃѺ µéͧ仹͹áÅéÇ ¾ÃØ觹ÕéÊͺ (Âѧ¨ÐÁÒàÅè¹ÍÕ¡á¹èÐ ) ¢éÍ 6. ·ÓµÒÁ¢Ñ鹵͹´Ñ§¹Õé 1. á·¹ $x=1, y=1$ ¨ÐÊÒÁÒöËÒ $f(2)$ ä´é 2. á·¹ $x=2, y=2$ ¨ÐÊÒÁÒöËÒ $f(4)$ ä´é 3. á·¹ $x=4, y=4$ ¨ÐÊÒÁÒöËÒ $f(8)$ ä´é 4. á·¹ $x=8, y=8$ ¨ÐÊÒÁÒöËÒ $f(16)$ ä´é 5. á·¹ $x=4, y=16$ ¨ÐÊÒÁÒöËÒ $f(20)$ ä´é ¤×ͤӵͺ ¤Ô´ÇèÒÁÕÇÔ¸Õ·Õè¶Ö¡¹éÍ¡ÇèÒ¹Õé¤ÃѺ äÇé¨ÐÁÒ¤Ô´´Ù
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PaTa PatA pAtA Pon! 03 ¡ØÁÀҾѹ¸ì 2007 02:08 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ M@gpie |
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