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ÊÁѤÃÊÁÒªÔ¡ | ¤ÙèÁ×Í¡ÒÃãªé | ÃÒª×èÍÊÁÒªÔ¡ | »¯Ô·Ô¹ | ¢éͤÇÒÁÇѹ¹Õé | ¤é¹ËÒ |
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à¤Ã×èͧÁ×ͧ͢ËÑÇ¢éÍ | ¤é¹ËÒã¹ËÑÇ¢é͹Õé |
#16
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¾Õè passer-by ÃÇÁ¤Ðá¹¹¢éÍ O.D.E. ãËé¼Á´éǤÃéÒº
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¤ Ç Ò Á ÃÑ º ¼Ô ´ ª Í º $$|I-U|\rightarrow \infty $$ |
#17
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¾Ô¨ÒóÒÊÒÁàËÅÕèÂÁã´æ 2 ÃÙ»·ÕèÁÕ°Ò¹ÃèÇÁ¡Ñ¹¨Ðä´éÇèÒ´éÒ¹·ÕèÃÇÁ¡Ñ¹ 2 ´éÒ¹¢Í§ÊÒÁàËÅÕèÂÁÃٻ㹨йéÍ¡ÇèÒ´éÒ¹·ÕèàËÅ×ÍÊͧ´éÒ¹¢Í§ÊÒÁàËÅÕèÂÁÃÙ»¹Í¡ àÊÁÍ
**** ¨Ò¡ÃÙ»¹Ð¤ÃѺ¼ÁÅÒ¡ DF ¼èÒ¹¨Ø´ B µÑ´ AC ·Õè F ¹Ð¤ÃѺ¨Ðä´éÇèÒ ÁØÁ BAF < ÁØÁ DAF â´Â·ÕèÁÕÁØÁ AFB à·èҡѹ ´Ñ§¹Ñ鹨Ðä´éÇèÒ ÁØÁ ABF > ÁØÁ ADF àÊÁÍ ¨Ðä´éÇèÒ ãËéÁØÁ ABF = A ÁØÁ ABF = B ÁØÁ AFB= C SinA/AF=SinC/AB áÅÐ SinB/AF=SinC/AD ¨Ðä´éÇèÒ AB= SinC•AF/SinA AD= SinC•AF/SinB ¨Ò¡ SinA>SinB ä´éÇèÒ 1/SinA< 1/SinB ¤Ù³´éÇ SinC µÅÍ´¨Ðä´éÇèÒ AB<AD µÒÁµéͧ¡ÒÃÊèǹ´éÒ¹ DC > BC ¡ç·Ó¹Í§à´ÕÂǡѹ¹ÔáËÅФÃѺ¡ç¨Ðä´éÇèÒ AD+DC>AB+BC....
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Rose_joker @Thailand Serendipity 25 ¾ÄȨԡÒ¹ 2007 18:49 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ RoSe-JoKer à˵ؼÅ: -*- ·ÓäÁ¼Áãªé Latex äÁèä´éÍФÃѺ ËÃ×ÍÇèÒ¼ÁãªéäÁèà»ç¹àͧ 55 |
#18
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à¢éÒÁÒ´Ù
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠|
#19
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ÍéÒ§ÍÔ§:
¢Ñé¹µé¹ $(2^1+1,2^3-1)=(5,7)$ ÁÕ 6 «Öè§ $6=3+3$ ´Ñ§¹Ñé¹ $P(1)$ à»ç¹¨ÃÔ§ ¢Ñé¹ÍØ»¹Ñ ÊÁÁµÔÇèÒ $P(k)$ à»ç¹¨ÃÔ§ $k\geq 2$ ¹Ñ蹤×ͨÐÁÕ $x \in \mathbb{N}$ «Öè§ $2^{k+1}-1>x>a^k+1$ áÅÐÁըӹǹ੾ÒÐ $p_1,p_2,p_3,...,p_k$ «Öè§ $x=p_1+p_2+p_3+...+p_k$ ¨Ò¡ $2^{k+1}-2^k=2^k>1$ ¨Ò¡·Äɮպ· Tschebychev ($n \in \mathbb{N},n\geq2$¨ÐÁըӹǹ੾ÒÐã¹ÅӴѺ $n,n+1,n+2,...,2n$) «Öè§ $2(2^{k+1}-2^k)>p_{k+1}>2^{k+1}-2^k$ $\therefore 2^{k+1}-1+2^{k+2}-2^{k+1}>x+p_{k+1}>2^k+2^k+1-2^k+1$ $2^{k+2}-1>x+p_{k+1}>2^{k+1}+1$ ¨ÐàËç¹ÇèÒÁÕ $y$ «Ö觷ÓãËé $y=p_1+p_2+p_3+...+p_k+p_{k+1}$ ¹Ñ蹤×Í $P_{k+1}$ à»ç¹¨ÃÔ§ ´Ñ§¹Ñé¹ ÊÓËÃѺ ¨Ó¹Ç¹¹Ñº $ n \geq 2 $ 㹪èǧ $ (2^n+1 ,2^{n+1}-1)$ ºÃèبӹǹ àµçÁ·ÕèÊÒÁÒöà¢Õ¹ä´éã¹ÃÙ»¼ÅºÇ¡¢Í§¨Ó¹Ç¹à©¾ÒÐ n µÑÇ (·Õè«éӡѹä´é) à»ç¹¨ÃÔ§â´ÂËÅÑ¡ÍØ»¹ÑÂàªÔ§¤³ÔµÈÒʵÃì »Å. à˹×èÍÂàËÁ×͹¡Ñ¹áËÐ
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¤ Ç Ò Á ÃÑ º ¼Ô ´ ª Í º $$|I-U|\rightarrow \infty $$ |
#20
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¼ÁÇèÒ ¼ÁÃÇÁ¤Ðá¹¹ ¢éÍ ODE ãËé¤Ø³ kanakon áÅéǹФÃѺ
ÊÃØ» ¤×Í µÍ¹·Õè 1 ¤Ø³ kanakon (i) µÍº¢éÍ ODE ä´é 2 ¤Ðá¹¹ (ii) µÍº¢éÍ 4 ¶Ù¡ä» 4 ¢éÍÂèÍ ä´éÍÕ¡ ¢éÍÂèÍÂÅÐ 1 ¤Ðá¹¹ ÃÇÁà»ç¹ 4 ¤Ðá¹¹ ÃÇÁà»ç¹ 6 ¤Ðá¹¹ ÊèǹµÍ¹·Õè 2 ¡çµÍº¢éÍ 4 áÅÐ 2 ¶Ù¡ ¡çä´é¤Ðá¹¹ÃÇÁà»ç¹ 4 ¤Ðá¹¹ Êèǹ¤ÓµÍºà¾ÔèÁàµÔÁ¢Í§·Ñ駤س Rose-Joker áÅФس timestopper ¡çäÁèÁÕ»ÑËÒ¤ÃѺ (áµè¨ÃÔ§æ ¢éÍ 1 µÍ¹·Õè 2 ÊÒÁÒö͸ԺÒÂä´éÊÑ鹡ÇèÒ¹Õé¹Ð¤ÃѺ â´ÂäÁèµéͧÂØ觡Ѻ¡®¢Í§ sine ã¤ÃÍÂÒ¡¨ÐÅͧµÍº¡çàªÔµÒÁʺÒ¤ÃѺ) à·èҡѺµÍ¹¹Õé ¼Å¤Ðá¹¹áµèÅзèÒ¹à»ç¹´Ñ§¹Õé PART 1 Kanakon : 6 Timestopper:4 PART 2 Kanakon:4 Rose-Joker: 3 ÁÒ hint Íա˹èÍ´աÇèÒ ¢éÍ 3.2 µÍ¹·Õè 1 Åͧáºè§ $ b_n $ µÒÁ¨Ó¹Ç¹ËÅÑ¡´Ù¹Ð¤ÃѺ áÅéÇËÒµÑÇÁÒ bound ¢éÍ 5.1 ãËé¹Ö¡¶Ö§ binomial theorem Ẻ·ÕèàŢ¡¡ÓÅѧäÁèãªè¨Ó¹Ç¹¹Ñº¤ÃѺ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#21
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ÃÙéÊÖ¡µÍ¹¹Õé¨ÐÁÕ¤¹·ÓÍÂÙè 3 ¤¹áÅéǤس¾Õèà¢Ò¡ç¡Ô¹àÃÕº - -"
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Rose_joker @Thailand Serendipity |
#22
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¢éÍ 1 ÍÕ¡ÇÔ¸Õ¤ÃѺ... ¡Ñ¹¤¹áÂ觤Ðá¹¹ËÃ×;Õè passer-by ¨Ðà¾ÔèÁ¤Ðá¹¹ãËé¼Á¡çäÁèÇèҹР55+
ÅÒ¡ AF µÑ駩ҡ DF ¤ÃѺ áÅéÇ¡ç»Ô·Ò¡ÍÃÑʸÃÃÁ´Òæ¨Ò¡ DF>BF
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Rose_joker @Thailand Serendipity |
#23
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·Óä´éÅФÃѺ...¢éÍ 4 part 2
ãËé a>b ¨Ðä´éÇèÒ b µéͧËÒà a ŧµÑÇ ¾Ô¨ÒÃ³Ò b=13665001*13665001*13665001 = 2551696414821715995 "001" ä´éÇèÒãËé a=13665001*13665001*13665001*10....0 = "2551"696414821715995001....000... (ÁÕ 10 ¨Ó¹Ç¹ 3n µÑÇ) ¡ç¨Ðä´éÇèÒ b ËÒà a ŧµÑÇáÅÐà»ç¹ä»µÒÁà§×è͹䢢éÍ 4 ¤ÃѺ =>â´Â¨ÐàËç¹ä´éªÑ´ÇèÒÁÕ 0 à»ç¹¨Ó¹Ç¹Í¹Ñ¹µì´Ñ§¹Ñé¹ÁÕ¤ÙèÍѹ´Ñº a,b à»ç¹¨Ó¹Ç¹Í¹Ñ¹µìµÒÁ...
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Rose_joker @Thailand Serendipity 25 ¾ÄȨԡÒ¹ 2007 21:37 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ RoSe-JoKer |
#24
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ÍéÒ§ÍÔ§:
ÂѧÁÕ solution Í×è¹ÍÕ¡¹Ð¤ÃѺ ÊÓËÃѺ·èÒ¹Í×è¹ ÊÃØ»¤Ðá¹¹µÍ¹¹Õé PART 1 Kanakon : 6 (2+4) Timestopper:4 (1+3) PART 2 Kanakon:4 (2+2) Rose-Joker: 5 (1+4) »Õ¹Õé »ÃÔÁÒ³¤¹àÅè¹äÁèµèÒ§¨Ò¡à´ÔÁ áµè´Ù competitive ´Õ¹Ð¤ÃѺ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#25
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¤ÓµÍº ¢éÍ 3 Part 2 ¤×Íà»ç¹ä»äÁèä´é¤ÃѺ
ÇÔ¸Õ¤Ô´¡ç 1.ÊÃéÒ§ÀÒ¾©Ò¢ͧ·Ã§ÅÙ¡ºÒÈì¡ 2.·ÒÊÕ 2 ÊÕµÒÁ·ÕèàËç¹ã¹ÃÙ»¹Ð¤ÃѺ 3.¨ÐÊѧࡵàËç¹ä´éªÑ´ÇèÒ¶éÒÁ´à´Ô¹¼èÒ¹ÊÕÊéÁ ÊÕµèÍä»·Õè¨Ðµéͧà´Ô¹¼èÒ¹¤×ÍÊÕàËÅ×ͧà·èÒ¹Ñé¹ ¨Ðä´éÇèҨӹǹ¤ÃÑ駢ͧÊÕ·Õèà´Ô¹¼èÒ¹µéͧà·èҡѹËÃ×ÍÁÒ¡¡ÇèÒËÃ×͹éÍ¡Çèҡѹ "˹Öè§" à·èÒ¹Ñé¹ ¨Ðä´éÇèÒ ¶éÒãËé¼èÒ¹¨Ø´ÊÕá´§¨Ø´Ë¹Öè§ 25 ¤ÃÑé§ ¨Ðä´éÇèҨмèÒ¹¨Ø´ÊÕÊéÁ·Ñé§ËÁ´ 85 ¤ÃÑé§ áÅмèÒ¹¨Ø´ÊÕàËÅ×ͧ 80 ¤ÃÑé§ «Öè§à»ç¹ä»äÁèä´é¤ÃѺ ....
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Rose_joker @Thailand Serendipity 26 ¾ÄȨԡÒ¹ 2007 11:47 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ RoSe-JoKer |
#26
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¤Ø³ Rose-joker µÍº¶Ù¡àËÁ×͹µÒàËç¹à©Å àŹФÃѺ (á«ÇàÅè¹)
¢éÍ 3 µÍ¹·Õè 2 ´Ö§ÁÒ¨Ò¡ÍÍÊàµÃàÅÕ ·Õè¼Á¨Ðᨡ¹ÕèáËÅÐ ÊÃØ»¤Ðá¹¹ÍÕ¡·Õ PART 1 Kanakon : 6 (2+4) Timestopper:4 (1+3) PART 2 Kanakon:4 (2+2) Rose-Joker: 8 (1+4+3)
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#27
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ÍéÒ§ÍÔ§:
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¤ Ç Ò Á ÃÑ º ¼Ô ´ ª Í º $$|I-U|\rightarrow \infty $$ |
#28
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¢éÍ 2 Part 2 ¤ÃѺ
¤×ͨÐÇèÒÍÐäÃäËÁ¶éÒ¼ÁãªéàªÍúÔà«¿àËÁ×͹¡Ñ¹... áµè¤Ô´ÇèÒá¹Ç¤Ô´¤¹ÅÐá¹Ç¹Ð¤ÃѺ¨Ò¡¡ÒÃÊѧࡵ n=2 ¨Ðä´éÇèÒÁÕÊÁÒªÔ¡¤×Í 6 = 3+3 ¼ÁÊѧࡵàËç¹àÅ¢ 3 àŤԴÇèÒ¶éÒÁÕ 3 ÍÂÙè·Ñé§ËÁ´ n-1 µÑǨÐä´éÇèÒ¤èҢͧ¨Ó¹Ç¹·ÕèàËÅ×Í㹪èǧ¡ç¤×Í ($ 2^n $ - 3n + 4,$ 2^{n+1} $ -3n+2) ¾Ô¨ÒóҪèǧ ($ 2^n $+1,$ 2^{n+1} $ -1) Êѧࡵä´éÇèÒ ($ 2^n $ + 1,$ 2^{n+1} $ + 2) ¨ÐµéͧÁըӹǹ੾ÒÐÍÂÙè¨Ò¡ àªÍúÔિ «Öè§àËç¹ä´éªÑ´ÇèҨеéͧÁըӹǹÍÂÙè㹪èǧ ($ 2^n $ + 1,$ 2^{n+1} $ -1) ·ÕèÊÒÁÒöà¢Õ¹ä´éã¹ÃÙ» x=3(n-1)+ $ p_i $ µÑÇÍÂèÒ§àªè¹ $ (17,31) $ ÁÕ 22 ·Õèà¢Õ¹ä´éã¹ÃÙ» $ 22=3*3 + 13 $ à»ç¹µé¹ ....
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Rose_joker @Thailand Serendipity 26 ¾ÄȨԡÒ¹ 2007 20:05 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 7 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ RoSe-JoKer à˵ؼÅ: Add Latex... - -" |
#29
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5.1.(Part1)
Let $\displaystyle{f(x)=(1+x)^r,\forall r\in \mathbb{C},|x|<1}$ We can see that $\displaystyle{f^{(n)}(x)=r\cdot(r-1)\cdot ...\cdot(r-n+1)(1+x)^{r-n}}$ Expand our function with Maclaurin series we gonna get $\displaystyle{f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}=1+\sum_{n=1}^{\infty}\frac{r\cdot(r-1)\cdot ...\cdot(r-n+1)}{n!}x^n=1+\sum_{n=1}^{\infty}\binom{r}{n}x^n}$ Consider $\displaystyle{\frac{1}{\sqrt{1-x}}=(1-x)^{-0.5}=1+\sum_{n=1}^{\infty}\binom{-0.5}{n}(-x)^n=1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdot ...\cdot(2n-1)}{2\cdot 4\cdot 6\cdot ...\cdot(2n)}x^n}$ $\displaystyle{\therefore\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdot ...\cdot(2n-1)}{2\cdot 4\cdot 6\cdot ...\cdot(2n)}\left(\frac{1}{2}\right)^n=\frac{1}{\sqrt{1-0.5}}-1=\sqrt{2}-1}$ ãªèẺ¹ÕéËÃ×Íà»ÅèÒ¤ÃѺäÁèä´é·ÓTaylor¹Ò¹¾Í´Ù
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#30
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ÍéÒ§ÍÔ§:
Êèǹ¤ÓµÍº¤Ø³ Rose-Joker ¼ÁÂѧäÁèà¤ÅÕÂÃìµÃ§·ÕèàÍÒ $p_i$ 仺ǡãËéä´é x áÅéÇ x ÍÂÙè㹪èǧ·ÕèÇèÒ ÁÒ͸ԺÒ´éǤÃѺ PART 1 Kanakon : 6 (2+4) Timestopper:5.5 (1+3+1.5) PART 2 Kanakon:4 (2+2) Rose-Joker: 8 (1+4+3)
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
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¢éÍÊͺ¤Ñ´µÑÇ Olympiad ÊÊÇ·. 2545 | ToT | ¢éÍÊͺâÍÅÔÁ»Ô¡ | 33 | 28 ¾ÄȨԡÒ¹ 2007 18:04 |
Vietnam Mathematical Olympiad 2005 problem 4 | gools | ¢éÍÊͺâÍÅÔÁ»Ô¡ | 8 | 18 ÁԶعÒ¹ 2005 21:09 |
The First POSN-Mathematical Olympiad | Rovers | ¢èÒǤÃÒÇáǴǧ Á.»ÅÒ | 4 | 06 ¾ÄÉÀÒ¤Á 2005 09:55 |
The First POSN-Mathematical Olympiad | Rovers | »ÑËÒ¤³ÔµÈÒʵÃì·ÑèÇä» | 1 | 24 àÁÉÒ¹ 2005 02:12 |
British Mathematical Olympiad | Tony | »ÑËÒ¤³ÔµÈÒʵÃì·ÑèÇä» | 14 | 15 àÁÉÒ¹ 2005 08:59 |
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