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#16
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âË ! ¼ÁËź仫×éͧ͢á»êºà´ÕÂÇ ¤Ø³ M@gpie â«éÂä»ËÅÒ¢éÍ«ÐáÅéÇ
áµè ÁÕ¼Ô´ÍÂÙè¢é͹֧ ¹Ð¤ÃѺ ÃÕºæá¡é à´ÕëÂÇÁÕã¤ÃµÑ´Ë¹éÒ仡è͹ ÊÓËÃѺ¢éÍ 7 ÂѧÁÕÇÔ¸ÕẺäÁèãªé calculus ¤ÃѺ ¶éÒÁÕ·èÒ¹Í×蹵ͺÇԸմѧ¡ÅèÒÇÁÒ áÅéǶ١ ¼Á¡çÂѧãËé¤Ðá¹¹ÍÂÙè¹Ð ¢éÍ 3(A) ¢Í§¹éͧ prachya à¡×ͺ¶Ù¡áÅéÇÍèÐ ÃÕºæÁÒá¡é´éǹФÃѺ ¢éÍ 6 ¡Ñº 12 ÊÓËÃѺ¹éͧÁѸÂÁ ¡çäÁèÂÒ¡¹Ð¤ÃѺ ¨Ð clarify ¢éÍ 15 ¹Ô´¹Ö§ ¤ÃѺ ÃÙ»·ÕèãËéÁÒ à»ç¹á¤è¡Òà repeat 2 ¤ÃÑé§áá à·èÒ¹Ñ鹹ФÃѺ áµè¼ÁÍÂÒ¡·ÃÒº ÇèÒ¶éÒ·Óä»àÃ×èÍÂæ ºÃ÷ѴÊØ´·éÒ ¨ÐÁÕÁÔµÔà»ç¹à·èÒäà ÊÓËÃѺ¼Ùé·ÕèäÁèà¤Â¼èÒ¹µÒ cantor set ¡è͹ Åͧ¤é¹¤ÇéÒ¢éÍÁÙÅà¡ÕèÂǡѺ cantor set Ẻ standard ã¹ google áÅéÇÅͧ refer à¢éÒËÒÃÙ»¹Õé´Ù ¡ç¤§äÁèÂÒ¡à¡Ô¹ä»·Õè¨ÐµÍº¢é͹Õé¤ÃѺ ÇèÒáµè µÍ¹¹Õé ÁÕàÅè¹á¤è 3 ¤¹ãªèäËÁ¤ÃѺà¹ÕèÂ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#17
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âÍठ¤ÃѺ ¹éͧ prachya
Êèǹ àÃ×èͧÁԵԢͧ¹éͧ M@gpie ¤×Í ¶éÒà»ç¹ ÊÕèàËÅÕèÂÁ ¡ç 2 ÁÔµÔ ãªèäËÁ¤ÃѺ ÅÙ¡ºÒÈ¡ì 3 ÁÔµÔ àÊé¹â¤é§¤ÇÒÁÂÒǨӡѴ ¡ç 1 ÁÔµÔ áµè Cantor set ¢é͹Õé´ÙàËÁ×͹ÁÔµÔ¨Ðà»ç¹ 1 ãªèäËÁ¤ÃѺà¾ÃÒÐà»ç¹àÊé¹(·ÕèÊÑé¹áÅжÕèÁÒ¡æ) áµè¡ÅѺäÁèãªè á¶ÁÁÔµÔ¡çäÁèà»ç¹¨Ó¹Ç¹àµçÁ«Ð§Ñé¹ ¼Á¡çãºéãËéä´éá¤è¹ÕéáËÅÐ ¨Ø´»ÃÐʧ¤ì¢Í§¼Á ¡ç¤×Í ÍÂÒ¡ãËéÅͧ search ¢éÍÁÙÅ´éǵÑÇàͧ áÅéǹéͧ¡ç¨Ð·ÃÒºàͧÇèÒ à¢Ò¤Ô´ÁÔµÔÃÙ»¹Õé¡Ñ¹ÍÂèÒ§äà (Êٵà simple ÁÒ¡¤ÃѺ)
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 02 ¡ØÁÀҾѹ¸ì 2007 22:55 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#18
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á¶Á hint ãËéàÎ×Í¡ÊØ´·éÒ ÊÓËÃѺºÒ§¢éÍ à¼×èÍÁÕã¤Ã post ÍÕ¡
3(B) multiplied by something and use telescopic technique 4 ãªéá¤è $ x^2+y^2 \geq 2xy $ ¡çÍÍ¡¤ÃѺ áµèãªéãËé¶Ù¡µÓá˹è§áÅéǡѹ 12 apply ÁҨҡ⨷Âìà¡ÕèÂǡѺ ¤¹àÁÒà´Ô¹à«ä»ÁÒ ã¹¤ÙèÁ×Í Á.»ÅÒ àÃ×èͧ ¨Ñ´ËÁÙè àÃÕ§ÊѺà»ÅÕè¹ ¹ÕèáËÅФÃѺ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#19
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ÍéÒ§ÍÔ§:
¨Ñ´ÃÙ»ÍѹáæãËé§èÒ¢Öé¹Ë¹èÍÂä´é $ f(x+y)= (x+y+1)( \frac{y}{x+1})f(x)+ \frac{x}{y+1} + xy) $ ...* ¡Ã³Õ x = y $ f(2x)= (2x+1)( \frac{2f(x)}{x+1}) + x^2) $ ...** $f(2) = (3)[( \frac{2f(1)}{2}) + 1] = \frac{15}{2}$ edit : à¨Í·Õè¼Ô´ÅФѺ Å×ÁãÊèǧàÅ纫ЧÑé¹ = = $f(3) = (4)[ \frac{f(2)}{3}+ \frac{f(1)}{2} + 2] = 21$ $f(5) = (6)[ \frac{f(2)}{3}+ \frac{f(3)}{4} + 6] = \frac{165}{2}$ $f(10) = (11)[( \frac{2f(5)}{6}) + 25] = \frac{(11)(105)}{2}$ $f(20) = (21)[( \frac{2f(10)}{11}) + 100] = 4305 $ »Å. äÁèä´é«ÕàÃÕÂÊÎÐ ÍÔÍÔ àËç¹¾Õè passer-by µÑé§â¨·ÂìàÍÒã¨Á.»ÅÒ ¡çàÅÂÁÒàÅè¹´éÇ«Ð˹èÍ à´ÕëÂǾÃØ觹Õé¼Áä»Ã´.áÅéÇ T_T (àÁ×èÍÇÒ¹ËÅѺ¤Ò¤ÍÁàžÕè 55) 03 ¡ØÁÀҾѹ¸ì 2007 19:13 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ prachya |
#20
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áÇÐÁÒàªÕÂÃ줹ÍÖ´àµçÁ¾Ô¡Ñ´¤ÃѺ
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#21
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¡ÅÒÂà»ç¹à¨éҢͧ§Ò¹áͺËÅѺä»äÁèÃÙéµÑǨ¹àÅ 6 âÁ§àªéÒ ÍÔ ÍÔ
ÁÒÊÃØ»¤Ðá¹¹¡Ñ¹´Õ¡ÇèÒ ¤Ø³ M@gpie ä´é 8 ¤Ðá¹¹ (¢éÍ 6 ¼ÁäÁè¤Ô´¤Ðá¹¹¹Ð) ¤Ø³ Prachya ä´é 2 ¤Ðá¹¹ (¢éÍ 6 äÁè¶Ù¡ÍèФÃѺ) ¤Ø³ Mastermander ä´é 2 ¤Ðá¹¹ (áÁé¨Ðà¢Õ¹ 2 ºÃ÷ѴÊØ´·éÒ äÁè rigorous à·èÒäËÃè) §Ñé¹ÊͺàÊÃç¨áÅéÇ¡ç Êè§ pm ÁÒÂ×¹ÂѹàÃ×èͧ·ÕèÍÂÙèÊ觢ͧ´éǹФÃѺ ¤Ø³ M@gpie áÅСçàËç¹ã¹¤ÇÒÁ¾ÂÒÂÒÁ¢Í§¹éͧ prachya ·Õè·Ó¢éÍ 6 §Ñé¹¼ÁÁբͧá¶ÁãËéáÅéǡѹ Ê觷ÕèÍÂÙèÁÒËÅѧäÁ¤ì¹Ð¤ÃѺ áÅéǨк͡ÇèÒ ¤×ÍÍÐäà ¼Á¢Íà©Å¢éÍ 6 ¡è͹áÅéǡѹ¤ÃѺ (¢é͹ÕéÁÒ¨Ò¡ Singapore MO 2006 (Senior section: First round) ) ¨Ñ´ÃÙ» functional equation ·ÕèãËéÁÒà»ç¹ $ f(x+y)= (x+y+1)(\frac{f(x)}{x+1}+\frac{f(y)}{y+1}+xy) $ $ \frac{f(x+y)}{x+y+1} = \frac{f(x)}{x+1}+\frac{f(y)}{y+1}+xy $ á·¹ x= n ,y= 1 ¨ÐàËç¹ÀÒ¾ªÑ´¢Ö鹤ÃѺ ´Ñ§¹Ñé¹ÊÁ¡ÒáÅÒÂà»ç¹ $ \frac{f(n+1)}{n+2} = \frac{f(n)}{n+1}+\frac{3}{4}+ n $ á·¹¤èÒ n= 1,2,...,19 áÅéǨѺ·Ñé§ 19 ÊÁ¡Òúǡ¡Ñ¹ ÊØ´·éÒ¨еѴ¡Ñ¹¨¹àËÅ×Í f(20)= 4305 ¤ÃѺ àÍÒà»ç¹ÇèÒ µÍ¹¹Õé ã¤Ã¨Ðà©ÅÂÍ×è¹æ¢éÍä˹ ¡çàµçÁ·ÕèàŤÃѺ áÅéǼÁ¨ÐÁÒà¡çºµ¡ÃÒÂÅÐàÍÕ´ãËéÍÕ¡¤ÃÑé§
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 03 ¡ØÁÀҾѹ¸ì 2007 07:54 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#22
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¢éÍ 15. àÃÒÊÒÁÒö¹ÔÂÒÁ "ÁÔµÔ" ẺºéÒ¹æä´éâ´ÂÊѧࡵÇèÒ
¢Í§·ÕèÁÕ 1 ÁÔµÔ ÍÂèÒ§àªè¹ Êèǹ¢Í§àÊ鹵ç ¶éÒàÃÒ¢ÂÒÂÁѹÍÍ¡à»ç¹ 2 à·èÒ àÃÒ¨Ðä´éÊèǹ¢Í§àÊ鹵çẺà´ÔÁà»ç¹¨Ó¹Ç¹ 2 ªØ´ ¢Í§·ÕèÁÕ 2 ÁÔµÔ ÍÂèÒ§àªè¹ ÊÕèàËÅÕèÂÁ¨ÑµØÃÑÊ ¶éÒàÃÒ¢ÂÒÂÁѹÍÍ¡à»ç¹ 2 à·èÒ àÃÒ¨Ðä´éÊÕèàËÅÕèÂÁ¨ÑµØÃÑÊẺà´ÔÁà»ç¹¨Ó¹Ç¹ 4 ªØ´ ¢Í§·ÕèÁÕ 3 ÁÔµÔ ÍÂèÒ§àªè¹ ÅÙ¡ºÒÈ¡ì ¶éÒàÃÒ¢ÂÒÂÁѹÍÍ¡à»ç¹ 2 à·èÒ àÃÒ¨Ðä´éÅÙ¡ºÒÈ¡ìẺà´ÔÁà»ç¹¨Ó¹Ç¹ 8 ªØ´ ¶éÒàÃÒ¢ÂÒÂÁѹÍÍ¡à»ç¹ 3 à·èÒ àÃÒ¨Ðä´éÅÙ¡ºÒÈ¡ìẺà´ÔÁà»ç¹¨Ó¹Ç¹ 27 ªØ´ ´Ñ§¹Ñ鹶éÒàÃÒ¢ÂÒ¢ͧªÔé¹Ë¹Ö觢Öé¹à»ç¹ $n$ à·èÒ áÅéÇ»ÃÒ¡®ÇèÒä´é¢Í§áººà´ÔÁÍÍ¡ÁÒ $x$ ªØ´ àÃÒ¡ç¨ÐÁÑèÇä´éÇèҢͧªÔé¹¹Ñé¹ÁÕ "ÁÔµÔ (ẺºéÒ¹æ)" à·èҡѺ $\log_nx$ ¤ÃѺ 㹡óբͧÃÙ»·Õè¶ÒÁ ¨ÐàËç¹ÇèÒ¶éÒàÃÒ¢ÂÒÂÁѹ¢Öé¹à»ç¹ 5 à·èÒ ¨Ðà¡Ô´ÃٻẺà´ÔÁ·Ñé§ËÁ´ 3 ªØ´ ´Ñ§¹Ñé¹ "ÁÔµÔ" ¢Í§Áѹ¡ç¤ÇèÐà»ç¹ $\log_53\approx0.6826$ ¤ÃѺ |
#23
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2.$$(3^{\log_7 x}+4)^{\log_7 3 }= x - 4$$
ãËé $x=7^u$ ÊÁ¡ÒáÅÒÂà»ç¹ $$(3^{u}+4)^{\log_7 3 }= 7^u - 4$$ ãËé $3^u+4=7^w$ ...(1) àËç¹ä´éÇèÒ $u=w=1$ à»ç¹¤ÓµÍºË¹Ö觢ͧÊÁ¡ÒùÕé ¹Óä»á·¹ÊÁ¡ÒÃà´ÔÁ¨Ðä´é $3^w=7^u-4$ ...(2) «Ö觷Ñé§ÊͧÊÁ¡ÒùÑé¹ $u=w=1$ à»ç¹¤ÓµÍº à¹×èͧ¨Ò¡ $3^x,7^y-4$ à»ç¹¿Ñ§¡ìªÑ¹ 1-1 áÅÐà»ç¹¿Ñ§¡ìªÑ¹à¾ÔèÁ¡ÃÒ¿¨Ö§µÑ´¡Ñ¹·Õè¨Ø´à´ÕÂÇ ´Ñ§¹Ñé¹ $u=w=1$ à»ç¹¤ÓµÍºà´ÕÂǢͧÊÁ¡Òà $\therefore\quad x=7$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#24
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§Ò¹¹Õé¡ÅÒÂà»ç¹ÃÒ¡ÒÃÍѨ©ÃÔÂТéÒÁ¤×¹àÇÍÃìªÑ¹ MATHCENTER ä»áÅéǤÃѺ
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site:mathcenter.net ¤Ó¤é¹ |
#25
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ÍéÒ§ÍÔ§:
¨Ó¹Ç¹ªØ´ ¡ç¤×ͨӹǹàÊé¹·Ò§·Õèà´Ô¹¨Ò¡ a1 ä» a13 $$ \frac{12!}{8!4!} = 495 $$ |
#26
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ÍéÒ§ÍÔ§:
$\displaystyle{ \frac{a_1^3}{a_1^2+a_2^2} = a_1 - a_2\Big(\frac{a_1a_2}{a_1^2+a_2^2}\Big) \geq a_1 - \frac{a_2}{2}}$ $\displaystyle{\frac{a_2^3}{a_2^2+a_3^2} = a_2 - a_3\Big(\frac{a_2a_3}{a_2^2+a_3^2}\Big) \geq a_2 - \frac{a_3}{2}}$ . . . $\displaystyle{\frac{a_n^3}{a_n^2+a_1^2} = a_n - a_1\Big(\frac{a_n a_1}{a_n^2+a_1^2}\Big) \geq a_n - \frac{a_1}{2}}$ ºÇ¡ÍÊÁ¡Ò÷Ñé§ËÁ´à¢éÒ´éÇ¡ѹ¨Ðä´é¤ÓµÍº¤ÃѺ 7. ãªé AM - GM $\begin{array}{rcl} \sqrt{a}+\sqrt[3]{a}+ \sqrt[6]{a} & = & \sqrt{a\cdot 1}+\sqrt[3]{a\cdot 1 \cdot 1}+ \sqrt[6]{a\cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1} \\ & \leq & \displaystyle{ \frac{a+1}{2} + \frac{a+2}{3} + \frac{a+5}{6} } \\ & = & a+2 \end{array} $ ÊÁ¡ÒÃà»ç¹¨ÃÔ§¡çµèÍàÁ×èÍ $a = 1$
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site:mathcenter.net ¤Ó¤é¹ |
#27
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ÍéÒ§ÍÔ§:
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠03 ¡ØÁÀҾѹ¸ì 2007 15:10 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#28
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ÍéÒ§ÍÔ§:
´Ñ§¹Ñé¹àÃÒ¨Ðä´éÇèÒ $p_i\geq i+1$ ·Ø¡¤èÒ $i=1,2,...,2006$ «Ö觨ҡÍÊÁ¡ÒùÕéàÃÒÊÒÁÒö¾ÔÊÙ¨¹ìµèÍä´éäÁèÂÒ¡ÇèÒ $$\frac{p_i^2-1}{p_i^2} \geq \frac{(i+1)^2-1}{(i+1)^2}$$ ·Ø¡¤èÒ $i=1,2,...,2006$ ´Ñ§¹Ñé¹ $$\begin{array}{rcl} \displaystyle{ \prod_{i=1}^{2006} (1-\frac{1}{p_i^2}) } & = & \displaystyle{ \prod_{i=1}^{2006} (\frac{p_i^2-1}{p_i^2}) } \\ & \geq & \displaystyle{ \frac{(2^2-1)(3^2-1)\cdots (2007^2-1)}{2^2\cdot 3^2 \cdots 2007^2} } \\ & = & \displaystyle{ \frac{(1\cdot 3)(2\cdot 4)\cdots (2006\cdot 2008)}{(2\cdot 2)(3\cdot 3)\cdots (2007\cdot 2007)} } \\ & = & \displaystyle{ \frac{2006!\cdot 2008!}{2\cdot 2007!\cdot 2007!} } \\ & = & \displaystyle{ \frac{2008}{2\cdot 2007} } \\ & > & \displaystyle{ \frac{1}{2} } \end{array}$$
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site:mathcenter.net ¤Ó¤é¹ |
#29
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ÍéÒ§ÍÔ§:
¨ÃÔ§æáÅéÇàÃÒÊÒÁÒö¾ÔÊÙ¨¹ìä´é¤ÃѺÇèÒ ·Ø¡¿Ñ§¡ìªÑ¹ $f:[0,1)\to (0,1)$ ·Õèà»ç¹¿Ñ§¡ìªÑ¹Ë¹Ö觵èÍ˹Öè§áÅзÑèǶ֧ ¨ÐÊÍ´¤Åéͧ¤Ø³ÊÁºÑµÔ·ÕèÇèÒ $f(x)\neq x$ à»ç¹¨Ó¹Ç¹Í¹Ñ¹µìàÊÁÍ Proof : ÊÁÁµÔÇèÒ $A=\{x\in [0,1) : f(x)\neq x\}$ à»ç¹à«µ¨Ó¡Ñ´ ÊѧࡵÇèÒ $0\in A$ ´Ñ§¹Ñé¹ $A$ äÁèà»ç¹à«µÇèÒ§ à¹×èͧ¨Ò¡ $f$ à»ç¹¿Ñ§¡ìªÑ¹Ë¹Ö觵èÍ˹Öè§ àÃÒ¨Ðä´éÇèÒ $f(A)\subseteq A$ áÅÐ $f_{|A}$ à»ç¹¿Ñ§¡ìªÑ¹Ë¹Ö觵èÍ˹Öè§ áµè $A$ à»ç¹à«µ¨Ó¡Ñ´ àÃÒ¨Ðä´éÇèÒ $f_{|A}$ µéͧà»ç¹¿Ñ§¡ìªÑ¹·ÑèǶ֧´éÇ ´Ñ§¹Ñ鹨еéͧÁըش㹠$A$ ·ÕèÊè§ä»ËÒ 0 «Ö觢ѴáÂé§
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site:mathcenter.net ¤Ó¤é¹ 03 ¡ØÁÀҾѹ¸ì 2007 17:23 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nooonuii |
#30
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ÍéÒ§ÍÔ§:
$A = 2^{1001} . 3^{1000} . 4^{999} ..... 1002$ $B = 2^{1003} . 3^{1003} ..........1002^{1003} . 1003^{1003} . 1004^{1003} .1005^{1002} . 1006^{1001}.....2006^{1} $ $2AB = 2^{2005} . 3^{2003} . 4^{2002}.... 1002^{1004} . 1003^{1003} . 1004^{1003} . 1005^{1002} . 1006^{1001} .....2006$ ¡ÒÃà¢Õ¹ã¹ÃÙ» ¡ÓÅѧÊͧ¢Í§¨Ó¹Ç¹¹Ñºä´é áÊ´§ÇèÒ 2AB µéͧ¶Í´ÃÒ¡·Õè 2 ŧµÑÇ ´Ñ§¹Ñ鹡óշÕèàÅ¢ªÕé¡ÓÅѧà»ç¹¤Ùè¨Ð¶Í´ÃҡŧµÑÇàÊÁÍ ¼Á¨ÐµÑ´·Ôé§â´ÂÅÐäÁè¡ÅèÒǶ֧àŹФÃѺ $(2)(3.5.7.9.....1003)(1004 . 1006 . 1008 .1010 ...2006)$ $(2^{502+1})(3.5.7.9.....1003)(502 .503. 504 .......1003) --> (2^{503})(3.5.7.9...501)(502.504.506....1002) $ $(2^{503+251})(3.5.7.9...501)(251.252...501) --> (2^{754})(3.5.7...249)(252.254...500)$ $(2^{754+125})(3.5.7.9...249)(126.127.....250) --> (2^{879})(3.5.7..125)(126.128...250)$ $(2^{879+63})(3.5.7.9...125)(63.64.65...125) --> (2^{942})(3.5.7...61)(64.66.68...124)$ $(2^{942+31})(3.5.7.9...61)(32.33.34....62) --> (2^{973})(3.5.7..31)(32.34.36...62)$ $(2^{973+16})(3.5.7.9..29)(16.17.18...31) --> (2^{989})(3.5.7..15)(16.18.20....30)$ $(2^{989+8})(3.5.7..15)(8.9.10....15) --> (2^{997})(3.5.7)(8.10.12.14) $ $(2^{997+4})(3.5.7)(4.5.6.7) --> (2^{1001})(3)(4.6))$ $(2^{1001+3})(3)(3) $ ÊÃØ»ÇèÒ ¶Í´ÃÒ¡·Õè 2 ŧµÑǤѺ = =" »Å. ÊÁͧäÁèÊÙé áµèã¨ÊÙéÎÐ ÍÔÍÔ 03 ¡ØÁÀҾѹ¸ì 2007 18:41 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ prachya |
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