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#16
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¹éͧ Tummykun ·Ó¢éÍˡẺ·ÕèÊÒÁ¼Ô´¹Ð¤ÃѺ à¾ÃÒÐ $\sinh 6\ne\sinh 6^{\circ}=\sinh{\frac{\pi}{30}}$
¼èÒ¹ä»äÁè¡ÕèªÑèÇâÁ§µÍº¡Ñ¹¢¹Ò´¹Õé ËÇѧÇèÒÊÁÒªÔ¡¤¹Í×蹤§¨ÐäÁè¢ÇÑ˹մսèÍäÁè¡ÅéÒ¾ÔÁ¾ì¤ÓµÍº áÅСçËÇѧÇèҤس Passer-by ¨ÐäÁèà¢éÁ§Ç´¡Ñº¡µÔ¡Ò(ËéÒÁá¡é)ÁÒ¡¹Ð¤ÃѺ
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. 01 ¾ÄÉÀÒ¤Á 2006 00:01 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nongtum |
#17
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¢Íº¤Ø³ ¤Ø³ nongtum ·ÕèªèǵÃǨ·Ò¹ãËé¤ÃѺ
à´ÕëÂǼÁ¤§µéͧµÃǨ§Ò¹ áÅÐÃÇÁ¤Ðá¹¹¢Ñé¹µé¹äÇé¡è͹´Õ¡ÇèÒ ¡è͹¨Ð·ÐÅÑ¡·èÇÁ·é¹ 仡ÇèÒ¹Õé ÍéÍ ! ÁÕ¢èÒÇ´ÕÁҺ͡ ¤×ͼÁ¨Ðá»Ðà¾ÔèÁãËéÍÕ¡ 5 ¢éÍ à¾ÃÒÐÃÙéÊÖ¡¨Ð¢Ò´¤Ó¶ÒÁÊäµÅì Á.»ÅÒÂä» â´Â¼Á¨Ðá»ÐäÇé·Õè¡ÃзÙé¹Õé µÍ¹àÇÅÒ 10.00 ¢Í§Çѹáç§Ò¹áË觪ҵԹФÃѺ Êèǹ deadline àËÁ×͹à´ÔÁ Êèǹ¢éÍ·Ñ¡·éǧ¢Í§¤Ø³ M@gpie à´ÕëÂǼÁä»àªç¤ÍÕ¡·Õ áÅШÐÁҺ͡ãËéàÃçÇ·ÕèÊØ´ (áµè¼ÁÇèÒäÁè¹èÒÁÕ»ÑËÒ¹Ð) áÅéǤس M@gpie ¡çÍÂèÒà¾Ô觶ʹã¨á¾éà´ç¡¹Ð¤ÃѺ p.s. ÊÓËÃѺàÃ×èͧËéÒÁá¡é ¤§µéͧËÂǹæáÅéÇÅèÐÁÑé§ áµèÃͺ˹éÒÍÂèҷӹФéÒº
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#18
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ÍéÒ§ÍÔ§:
Åͧàªç¤ $\sin 30$ Âѧà·èҡѺ 0.5 ÍÂÙèààáµèä˧ $\sinh 6$ Áѹ¡ÅÒÂà»ç¹ radian ÍФѺ ÃØè¹ CASIO fx-4500PA ËØËØ
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[[:://R-Tummykung de Lamar\\::]] || (a,b,c > 0,a+b+c=3) $$\sqrt a+\sqrt b+\sqrt c\geq ab+ac+bc$$ |
#19
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¡ÃзÙé¹ÕéÁÕ¤¹µÍºàÃçÇÁÒ¡... (⨷ÂìÃÙéÊÖ¡ÇèÒÂÒ¡ÍФÃѺ)
áµèäÁèà»ç¹äÃÃÍÍÕ¡ 10 ªÑèÇâÁ§´Õ¡ÇèÒ
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#20
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á·¹ $a_n=\tan\theta_n,0 \leq \theta_n < \frac{\pi}{2}$
¨Ò¡ \[\begin{array}{rcl} \frac{\sqrt{a^2+1}-1}{a} &=& \frac{\sqrt{\tan\theta^2+1}-1}{\tan\theta} \\ &=& \frac{|\sec \theta| -1}{\tan\theta} \\ &=& \frac{\sec\theta -1}{\tan\theta} \\ &=& \frac{1-\cos\theta}{\sin\theta} \\ &=& \tan \frac{\theta}{2} \\ \end{array} \] ¨Ðä´éÇèÒ $\tan\theta_{n+1}=\tan \frac{\theta_n}{2}$ à¹×èͧ¨Ò¡ $a_1=1=\tan \frac{\pi}{4} $ ¨Ðä´éÇèÒ $a_4=\tan \frac{\pi}{32},a_5=\tan \frac{\pi}{64}$ ´Ñ§¹Ñé¹ $n=5$ ·ÓãËé $|a_n-\tan\frac{\pi}{60}|$ ÁÕ¤èÒ¹éÍ·ÕèÊØ´ |
#21
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ËÇѧÇèҤӵͺ¢Í§¤Ø³ gools ã¹¢éÍ 4 ¨ÐµÍº¤Ó¶ÒÁ¤Ø³ M@gpie ä´éáÅéǹФÃѺ
µÍ¹¹Õé¡ÓÅѧªÑè§ã¨ÍÂÙèÇèÒ ¨ÐãËé¤ÓµÍº¢éÍ 3 ¢Í§¤Ø³ gools ¡Õè¤Ðá¹¹ ËÃ×ÍäÁèãËé¤Ðá¹¹´Õ à¾ÃÒÐâ¤Ã§ÊÃéÒ§ÇÔ¸Õ·Ó¤ÅéÒ¡Ѻ¢Í§¹éͧ tummykung áÁé¨ÐäÁèàËÁ×͹«Ð·Õà´ÕÂÇ »ÃСͺ¡Ñº¹éͧ tummykung ¡çµÍºàÊÃ稡è͹ áµè¶Ö§äÁèãËé¤Ðá¹¹¢é͹Ñé¹ µÍ¹¹Õé¤Ðá¹¹¤Ø³ gools ¡ç¾Ø觻ÃÕê´áÅéǹÐà¹Õè ¹éͧ tummykung ËÃ×ͤ¹Í×è¹æ àµÃÕÂÁÊ¡Ñ´´ÒÇÃØ觡ѹ´éǹФÃѺ àÍÒà»ç¹ÇèÒ à´ÕëÂÇ µÍ¹post ÍÕ¡ 5 ¢éÍ ¨Ð¡ÅѺÁÒá¨é§ãËé·ÃÒºÇèÒ ¨ÐãËéËÃ×ÍäÁèãËé¤Ðá¹¹¢éÍ 3 ¢Í§¤Ø³ gools ´Õ Êèǹ¢éÍ 5 (A) ẺäÁè induction ¡Ñº 5 (B) áÅТéÍ 7 ¡çÂѧÃͤӵͺÍÂÙè¹Ð¤ÃѺ
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#22
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ÍéÒ§ÍÔ§:
».Å. ⨷Âì¢Í§¤Ø³ passer-by ¹ÕèÂÒ¡¡ÇèÒ·Õè¼Á¤Ô´äÇéÁÒ¡æàŤÃѺ ÁÕ¢éÍ 6. à·èÒ¹Ñé¹·Õèᨡ¤Ðá¹¹ ¹Í¡¹Ñé¹äÁèÁÕ¢éÍä˹ obvious ÊÓËÃѺ¼ÁàŤÃѺ áµè¡çÁÕ¤¹·Óä´éÍÂèÒ§ÃÇ´àÃçÇÍÂÙè´Õ |
#23
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Íѹáá¹èÒªéÓã¨ÍصÊÒ¤Ô´àÁ×èͤ׹ ¾Í´ÕºéÒ¹äÁèµÔ´à¹çµµéͧÁÒàÅè¹·ÕèÃéÒ¹ÍÔ¹àµÍÃìà¹çµàŵͺªéÒ¡ÇèÒã¤Ãà¾×è͹´Ù¢éÍ 6 ¡è͹ÅС¹
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#24
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ÂÍ´àÂÕèÂÁÍÕ¡áÅéǤÃѺÊÓËÃѺ ¢éÍ 4 555
àËç¹´éÇ¡Ѻ¤Ø³ warut ·Õà´ÕÂǤÃѺ ÂѧäÁè·Ñ¹à¤é¹áµèÅТéÍÍÍ¡ÁÒä´é ¡ç¶Ù¡¹éÍ§æ «ÔÇä»ËÁ´áÅéÇ ÇÃÂØ·¸ìà˹×ͪÑ鹨ԧæ ÍÔÍÔ
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PaTa PatA pAtA Pon! |
#25
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Ãͤӵͺ¡è͹¹Ð¤ÃѺ¾Í´Õ¼ÁãªéÅÒà·ç¡«ìäÁèà»ç¹àŵéͧ¾ÔÁ¾ìà»ç¹ÃÙ»áÅéÇÁÒá»Ð ÇèÒ§æ¨ÐÈÖ¡ÉÒÎÐÃÍà»Ô´à·ÍÁ¡è͹ ¢Íà¡çºâ¨·Âì令Դ áÅéǨÒÁÒá»ÐäÁèÇèҨҫӢͧã¤ÃÍÐà»ÅèÒ áµè¼Á¨ÐµÍºãËéà¡×ͺËÁ´·Ø¡¢éÍàÅ ÍÂÒ¡ä´é˹ѧÊ×ÍÁÒ¡ÍÐ
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#26
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à¾Ôè§à¢éÒã¨ÇèÒ¡Ò÷ӻÑËÒªÔ§ÃÒ§ÇÑÅ Áѹà˹×èÍÂÍÂèÒ§¹Õé¹Õèàͧ¤ÃѺ¤Ø³ Warut àÅè¹àÍÒ¼Áµ×è¹ÊÒÂâ´è§»èÒ¹¹Õé
µéͧ¢Íâ·É·Ø¡¤¹ÁÒ¡æ·Õè late µÍ¹¹Õé ¹éͧ Tummy kung µéͧä»à¢éÒ¤èÒÂÊÍǹ. áÅÐÊͺãËè ã¹Çѹ·Õè 8 ¾.¤. ¡çàŶ͹µÑÇä» 1 ¤¹ àÊÕ´Ò¨ѧ áµè¡çÍÒ¨¨Ðà»ç¹âÍ¡ÒÊ·Õè´ÕÊÓËÃѺ¤¹Í×è¹ ·Õè¨Ð·Ó¤Ðá¹¹´éÇ Êèǹ¡Ã³Õ¢Í§¹éͧ gools ã¹¢éÍ 3 ¼ÁµÑ´ÊÔ¹ã¨áÅéÇÇèÒ ¨ÐäÁèãËé¤Ðá¹¹ ´éÇÂà˵ؼŷÕè¡ÅèÒÇä»áÅéÇ »ÃСͺ¡ÑºµÍ¹¹Õé ¤Ðá¹¹¹éͧ¹ÓÅÔèÇæ 件֧ 18 ¤Ðá¹¹áÅéÇ ¤¹Í×è¹ÍÂèÒà¾Ô觶ʹ㨹ФÃѺ à¾ÃÒÐÂѧÁÕÍÕ¡ 5 ¢éÍã¹Çѹ¹Õé áÅÐ 50 ¤Ðá¹¹ã¹ÃͺµÑ´àª×Í¡ ÊÓËÃѺ¤ÓµÍº¢Í§¤Ø³ gnopy ¤ÓµÍºááä»àËÁ×͹¡Ñº¢Í§¤Ø³ gools ¼ÁàÅÂäÁèãËé¤Ðá¹¹¹Ð¤ÃѺ Êèǹ¤ÓµÍº·ÕèÊͧ àÅè¹áçÁÒ¡æ ¡Ó˹´ operation àͧàÊÃç¨ÊÃþ àÍÒà»ç¹ÇèÒ ¼ÁãËéáÅéǡѹ(áÅСçÁÒá¡éºÃ÷Ѵ·Õèà»ç¹ 50 ´éǤÃѺ) áÅéÇà´ÕëÂǼÁ¨Ðä»à¾ÔèÁàµÔÁ㹤ӶÒÁ à»ç¹ËÁÒÂà˵آéÍ·Õè 3 ÇèÒËÅѧ¨Ò¡¹Õé ËéÒÁã¤ÃÊÃéÒ§ operation àͧ¹Ð¤ÃѺ áÅйÕè¤×Í 5 ¢éÍ·ÕèàËÅ×ͤÃѺ ¶éҷӡѹÃÇ´àÃçÇẺ·Õè¼èÒ¹ÁÒ ÍÒ¨¨ÐàÅ×è͹ deadline à¢éÒÁÒ à¾ÃÒмÁ¡çÍÂҡᨡàÃçÇæ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 01 ¾ÄÉÀÒ¤Á 2006 11:33 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#27
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9. Let $\sqrt{2^x+1} =a$
$$a+1=\frac{4a^2-4}{a^2}$$ $$a^3+a^2=4a^2-4$$ $$a^3-3a^2+4=0$$ $$(a+1)(a^2-4a+4)=0$$ $$a=-1,2$$ $$a \geq 0$$ $$\sqrt{2^x+1} =2$$ $$2^x+1=4$$ $$2^x=3$$ $$x=\log_2 3$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠03 ¾ÄÉÀÒ¤Á 2006 20:31 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#28
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¢éÍ 8. ¹Ð¤ÃéÒº
\[det(A) = \vmatrix{1 & \sin \theta & \cos \theta \\ 1 & \cos \theta & -\sin \theta \\ 1 & \tan \theta & \tan \theta} = 2 \sin \theta -1 \] à¹×èͧ¨Ò¡ \( det(A^2) =(det(A))^2 = (2\sin \theta -1 )^2 \) «Öè§ÁÕ¤èÒÁÒ¡·ÕèÊØ´¤×Í \( max (det(A))^2 = 9 \)
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PaTa PatA pAtA Pon! |
#29
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10. $\because 2\sin^2 a = 1-\cos 2a$
$$2(1-\cos(6x+\frac{\pi}{2})) = 1+4\sin 4x\cos 2x$$ $$2+2\sin 6x = 1+2(\sin 6x +\sin 2x)$$ $$1+2\sin 6x =2\sin 6x + 2\sin 2x$$ $$\sin 2x = \frac12$$ $$\cos 4x = 1-2\sin^2 2x=1-2(\frac12)^2=\frac12 $$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠01 ¾ÄÉÀÒ¤Á 2006 11:50 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#30
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ã¹·ÕèÊØ´¤Ø³ M@gpie ¡ç¾ÅÒ´¨¹ä´é ¢éÍ 8 ÍÒ¨¨Ð´ÙàËÁ×͹µÍº 9 áµèäÁèãªè¹Ð¤ÃѺ
¤¹Í×è¹ÂѧÊÒÁÒöµÍº¢éÍ 8 ä´é¤ÃѺ áÅдٷÕèàÁµÃÔ¡«ì´Õæ´éÇÂ
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