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#31
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¢éÍ 12 ¤ÃѺ
à¹×èͧ¨Ò¡ 2549 à»ç¹¨Ó¹Ç¹à©¾ÒÐ(µÃǨÊͺä´éâ´Â¡ÒÃËÒôéǨӹǹ੾ÒеÑé§áµè 3 ¨¹¶Ö§ 49 - -") ´Ñ§¹Ñé¹ $10^{2548}-1 \equiv 0 (\bmod \ 2549)$ ¾Ô¨ÒÃ³Ò $2549 | \frac{10^{2548}-1}{10-1}$ à¹×èͧ¨Ò¡ $9 \not| 2549$ áÅÐ $\frac{10^{2548}-1}{9}=10^{2547}+10^{2546}+\ldots +1$ ´Ñ§¹Ñé¹ $2549 | 10^{2547}+10^{2546}+\ldots +1$ áÊ´§ÇèÒ $2549 | (111...1)_{10}$ (ÁÕ 1 ÍÂÙè 2548 µÑÇ) áÅШÐä´éÍÕ¡ÇèÒ $2549 | 10^n[(111...1)_{10}]$ àÁ×èÍ $n$ à»ç¹¨Ó¹Ç¹àµçÁ·ÕèäÁèà»ç¹Åº áÅÐà¹×èͧ¨Ò¡ $10^n[(111...1)_{10}] \in A$ ´Ñ§¹Ñ鹤èÒ¤ÇÒÁ¨ÃÔ§¢Í§»Ãо¨¹ì¢éÒ§µé¹¤×Í "¨ÃÔ§" |
#32
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¢éÍ 11 ¤ÃѺ
¨Ðä´éÇèÒ $(x^2+4)(x-1)=-1$ à¹×èͧ¨Ò¡ $x^2+4 > 0$ ´Ñ§¹Ñé¹ $x-1<0$ áÅÐ $x^2+4 \geq 4$ áÅÐ $x-1<0$ ´Ñ§¹Ñé¹ $x-1 \geq -\frac{1}{4}$ ¨Ðä´éÇèÒ $\frac{3}{4} \leq x < 1$ ´Ñ§¹Ñé¹ $\lfloor r \rfloor =0$ |
#33
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Î×ÍÍÍÍ ¾ÅÒ´ä»áÅéǤÃѺ ¼Á¡çàÍÐã¨ÍÂÙèàÅç¡æÇèÒ·ÓäÁÁѹ§èÒ¼Դ»¡µÔà»èÒ ËØËØ ¹éͧæ ÍÂèÒ¾ÅÒ´àËÁ×͹¼Á¹Ð¤ÃéÒº ËÅØÁ¤Ø³ passer-by ÍѹµÃÒ ¨ÃÔ§æ
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#34
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¢éÍ 8 ¤ÃѺ
$|A|=\sin\theta + \frac{\sin^2 \theta}{\cos\theta}-( \frac{\sin^2 \theta}{\cos\theta}-\sin\theta)+(-sin^2\theta-\cos^2\theta)=2\sin\theta-1$ ´Ñ§¹Ñé¹ $\max\{\det(A^2)\}=\max\{(\det(A))^2\}=\max\{(2\sin\theta-1)^2\}=9$ â´Â·Õè $\sin\theta=-1$ áµè $|\sin\theta| \not= 1$ à¾ÃÒШзÓãËé $\tan\theta$ ËÒ¤èÒäÁèä´é ´Ñ§¹Ñé¹ $\max\{(2\sin\theta-1)^2\}=8$ à»ç¹¨Ó¹Ç¹àµçÁ·ÕèÁÕ¤èÒÁÒ¡·ÕèÊØ´«Öè§ $\tan\theta$ ËÒ¤èÒä´é |
#35
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#36
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µÍºÍÕ¡
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#37
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à¤ÅÕÂÃì·ÕÅлÃÐà´ç¹ àŹФÃѺ
(1) àÃ×èͧ¤Ø³ M@gpie ¼Á͹ØâÅÁãËé¢é͹֧áÅéǡѹ¹Ð¤ÃѺ â´Â¡ÒÃäÁèµÔ´Åº¤Ðá¹¹¢éÍàÁµÃÔ¡«ì (2) ¼ÁªÍº¤ÓµÍº¤Ø³ gools ¢éÍ 11,12 ÁÒ¡¤ÃѺ ´ÙäÁè«Ñº«é͹´Õ á¶ÁµÍ¹¹Õé ¤Ðá¹¹¤Ø³ gools ¡çÂѧ¹Óâ´è§àªè¹à´ÔÁ µÒÁÁÒ´éǤس Mastermander áÅйéͧ Tummykung µÍ¹¹Õéã¤Ã·ÕèµÒÁæ ÍÂÙè ¡çÃÕºà¾ÔèÁ¤Ðá¹¹ãËéµÑÇàͧ´èǹàŹФÃѺ ¡è͹¨Ðà˹×èÍÂã¹ÃͺÊͧ ·ÕèäÁèÁÕ¤Ó¶ÒÁẺ¤Ó¹Ç³áÁéáµè¢éÍà´ÕÂÇ áÅмÁµÑ´ÊÔ¹ã¨áÅéÇÇèҨТÂѺ deadline Ãͺ¹ÕéãËéàÃçÇ¢Öé¹ à»ç¹ Íѧ¤Òà 2 ¾.¤. àÇÅÒ 9.00 ¹. à¾ÃÒЪèǧ¹Õé ¼ÁÁÕÀÒÃеéͧ·ÓÁҡ˹èͤÃѺ ÊÃØ»ÇèÒ¤Ó¶ÒÁÃͺÊͧÁÒ¾ÃØ觹Õé á¹è¹Í¹ ËÅѧ 9.00 ¹. (3) ÊÓËÃѺ¢éÍ 7 ¼Á Hint ãËé¹Ô´¹Ö§ÇèÒ ãªé ËÅÑ¡Ãѧ¹¡¾ÔÃÒº¡çä´é¤ÃѺ áµè ËÒÃѧ ËÒ¹¡ ãËéà¨ÍáÅéǡѹ Êèǹ¢éÍ 5(B) ¼Áä´éáçºÑ¹´ÒÅ㨠¨Ò¡à·¤¹Ô¤àÅç¡æ áµèÁÕ»ÃÐ⪹ì·Õè¤Ø³ Punk à¤Âŧ㹠My Maths àÅèÁàÁ×èͻշÕèáÅéÇ «Ñ¡àÅèÁ¹ÕèáËÅФÃѺ áÅТéÍ 11 ¡çÂѧ·ÓẺÍ×è¹æ ä´éÍÕ¡¹Ð¤ÃѺ àªè¹à´ÕÂǡѺ¢éÍ 12 ¡çãªéÇÔ¸ÕẺÍ×è¹ ÍÂèÒ§ combinatorics ÁÒªèÇÂä´é¹Ð (4) ¤ÓµÍº¤Ø³ gnopy ¢éÍ 2 ú¡Ç¹ªèÇ¢ÂÒ¤ÇÒÁ ·ÕèÁҢͧ¼ÅºÇ¡ -0.5 ãËéªÑ´à¨¹´éǤÃѺ µÍ¹¹Õé¼Á¶×ÍÇèÒãËé¤ÃÖè§Ë¹Ö觢ͧ¤Ðá¹¹àµçÁ¢é͹Ñ鹡è͹áÅéǡѹ áÅжéҤس gnopy ÁÒàµÔÁàµçÁ·Õè¼Áµéͧ¡ÒÃä´é ¡ç¨Ðä´é¤Ðá¹¹àµçÁ¢é͹Ñé¹ä» Êèǹ¢éÍ 7¢Í§¤Ø³ gnopy ÂѧäÁè¶Ù¡¹Ð¤ÃѺ
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#38
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¼Á¤§µéͧ¶Í¹µÑÇÃͺáá仡è͹¹Ð¤ÃѺ¾Í´ÕäÁè¤ÍèÂÊдǡºéÒ¹äÁèµÔ´à¹çµ¤ÓµÍºàŪéÒ áÅмÁ¡ç¾ÔÁ¾ìÅÒà·ç¡«×äÁèà»ç¹à´ÕëÂÇÃͼÁÈÖ¡ÉÒä´éáÅéǤèÍÂÁÒ»ÃÐÅͧãËÁè ÍÕ¡ÍÂèÒ§ÇÃÂØ·¸ì¢Í§¼Á»ÅÒÂá¶ÇÁҡ˹ӫéÓÂѧÍÂÙèÁ.»ÅÒ´éÇ ¼ÁÇèÒ¼Á仢Âѹ¡è͹´Õ¡ÇèÒ¤èÍÂÁÒÊÙéãËÁè ÊèǹÃͺÊͧ¼Á¨Òŧá¢è§¢Ñ¹à¾ÃÒÐäÁèä´é¤Ó¹Ç³ÎèÒææ
».Å ¤Ø³ gool ·ÓäÁäÁèÁյç·ÕèãËé¤Ðá¹¹ ¼Á´Ù»ÃÐÇѵÔäÁèä´éàÅÂÍФÃѺ ¶ÒÁàÅÂÅСѹ àÃÕ¹ÃдѺä˹ áÅÐÍÂÙèªÑé¹ÍÐäà ËÃ×ÍÇèÒÁËÒÅÑ à¡è§ÁÒ¡àŤÃѺ ¤ÇÒÁ¤Ô´á¹Ç´Õ ¼ÁÇèÒ¼ÁÍÂÙè㹺ÍÃì´¹Õ餧¨Ðà¡è§¢Öé¹ÊÑ¡Çѹá¹è¹Í¹ |
#39
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¢Öé¹ Á.5 ¤ÃѺ
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#40
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¢éÍ 7 ¤ÃѺ
ÊÁÁµÔãËé $|b_i|\leq 2$ ¨Ò¡ $ {n \choose 1} + {n \choose 3}+\cdots + {n \choose n}= {n \choose 0} + {n \choose 2}+\cdots + {n \choose n-1}$ ¨Ðä´éÇèÒ $ {n \choose 1} + {n \choose 3}+\cdots + {n \choose n}=\frac{\sum^n_{k=0} {n \choose k}}{2}=2^{n-1}$ ´Ñ§¹Ñé¹ $ b_1{n \choose 1} + b_2{n \choose 3}+\cdots + b_{(n+1)/2}{n \choose n}=S \in [-2^n,2^n]$ ¶éÒ $S$ ÁÕ¤èÒà·èҡѺÈÙ¹Âì¨Ð·ÓãËé»ÑËÒ¶Ù¡á¡é·Ñ¹·Õ ´Ñ§¹Ñé¹ÊÁÁµÔãËé $S \not= 0$ ´Ñ§¹Ñé¹ $S$ ÁÕ¤èÒä´é·Ñé§ËÁ´ $2^{n+1}$ ¤èÒ à¹×èͧ¨Ò¡ $b_i$ ã´æÁÕ¤èÒ·Õèà»ç¹ä»ä´é·Ñé§ËÁ´ 5 ¤èÒ¤×Í -2,-1,0,1,2 áÅÐãËé $b_i$ äÁèà»ç¹ÈÙ¹Âì¾ÃéÍÁ¡Ñ¹ ¾Ô¨ÒóҤèÒ·Ñé§ËÁ´·Õèà»ç¹ä»ä´é¢Í§ $S$ ¤×Í $5^{\frac{n+1}{2}}-1$ ¤èÒ ÊÁÁµÔãËéà»ç¹ $S_1,S_2,\ldots,S_{5^{\frac{n+1}{2}}-1}$ à¹×èͧ¨Ò¡ \[\begin{array}{rcl} 5^{\frac{n+1}{2}}-1 &=& (5-1)(5^{\frac{n+1}{2}-1}+\ldots+1) > 2^2(4^{\frac{n+1}{2}-1}+\ldots+1) \\ &=& 2^2(2^{n-1}+\ldots+1) > 2^{n+1} \end{array}\] ´Ñ§¹Ñé¹µÒÁËÅÑ¡Ãѧ¹¡¾ÔÃÒº¨ÐÁըӹǹ¹Ñº $i$ áÅÐ $j$ ·Õè $1\leq i < j \leq 5^{\frac{n+1}{2}}-1$ áÅÐ $S_i=S_j$ ´Ñ§¹Ñé¹ $S_i-S_j=c_1{n \choose 1} + c_2{n \choose 3}+\cdots + c_{(n+1)/2}{n \choose n}=0$ â´Â·Õè $|c_i| \leq 4$ |
#41
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ÊÓËÃѺ¤Ø³ gnopy ¡ç¢Âѹæ áÅÐà¾ÔèÁ¾Ù¹ÇÃÂØ·¸ìµÒÁ·ÕèµÑé§ã¨äÇéãËéä´é¹Ð¤ÃѺ ¢Íà»ç¹¡ÓÅѧã¨ãËé
áÅÐã¹·ÕèÊØ´ ¢éÍ 7 ¡çàÃÕºÃéÍÂâçàÃÕ¹¤Ø³ gools ÍÂèÒ§§´§ÒÁ ´Ù·èÒ·Ò§¤Ø³ gools ¹Õè¨Ðà»ç¹ "problem-solving man" µÑǨÃÔ§àÊÕ§¨ÃÔ§àŹФÃѺ ÊÓËÃѺ¤Ðá¹¹¶Ö§ ³ àÇÅÒ¹Õé à»ç¹´Ñ§¹Õé¤ÃѺ 1. gools 34 ¤Ðá¹¹ 2. Mastermander 12 ¤Ðá¹¹ 3. tummykung 9 ¤Ðá¹¹ 4. gnopy 3.5 ¤Ðá¹¹ 5. M@gpie 1.5 ¤Ðá¹¹ ¶éÒÁÕ¢éͼԴ¾ÅÒ´àÃ×èͧ¤Ðá¹¹¡çªÕéᨧä´é¹Ð¤ÃѺ (äÁèÃÙéÇèÒ¡è͹¶Ö§ 9.00 ¨ÐÁÕã¤ÃÁҵͺà¾ÔèÁÁÑéÂà¹ÕèÂ) Êèǹ deadline ¢Í§ÃͺÊͧ«Öè§à»ç¹ÃͺÊØ´·éÒ ¼Á¢Íà»ç¹ 00.30 ¢Í§Çѹ¾Ø¸·Õè 3 ¾.¤. ¹Ð¤ÃѺ àËç¹·Ó§Ò¹¡Ñ¹ÃÇ´àÃçÇ ¡çàÅÂÃè¹àÇÅÒãËéàÃçÇ¢Öé¹ ÊÓËÃѺÃͺ˹éÒ ¤Ó¶ÒÁÍÒ¨¨Ðµéͧãªé¡Òä鹤ÇéÒÁÒªèÇ »ÃСͺ¡Ñº¡ÒäҴ¤Ðà¹ã¹ºÒ§¢éͤÃѺ áÅÐÍÂèÒÅ×ÁÇèÒ µÍº·Ò§ pm ¹Ð¤ÃѺ ·ÕèÊÓ¤Ñ µÍºä´é¢éÍÂèÍÂÅÐ 1 ¤ÃÑé§ ¼Ô´áÅéǼԴàŹФÃѺ äÁèÁÕËÂǹáÅéǹФÃѺ Êèǹà§×è͹䢢ͧ¤Ø³ M@gpie ·Õè¼ÁºÍ¡ä»ã¹µÍ¹ááÇèÒ·Ó੾ÒТéÍ 4,5 ¼ÁźÍÍ¡ä»áÅéǹФÃѺ à¾×èͨÐä´éÃèÇÁʹء¡Ñ¹àµçÁ·Õè
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#42
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5(B) $$\sum_{n=1}^{100} \frac{n^2-n+1} {n^4-n^3+n^2-n+1} < \sum_{n=1}^\infty \frac{1}{n^2+ \displaystyle{ \frac{1-n}{n^2-n+1} }} < 1+ \sum_{n=2}^\infty \frac{1}{n^2-1} = 1.75 < 1.99 $$
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#43
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ÂÍ´àÂÕèÂÁ¡ÃÐà·ÕÂÁà¨ÕÂÇ ¤ÃѺ ¤Ø³ warut ¤ÒÃÇÐ 3 ¨Í¡
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#44
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ÃÐËÇèÒ§¹Õé ã¤Ã¨Ð post Alternative solution ¢éÍä˹¡çµÒÁʺÒÂàŹФÃѺ
¢éÍ 3 ãªéá¡éÊÁ¡ÒÃÍÂèÒ§à´ÕÂÇ¡çà¾Õ§¾Í¤ÃѺ áÅШÃÔ§æ ÃкºÊÁ¡ÒùÕéÁÕ 4 ¤ÓµÍº¤ÃѺ ¶éÒ x = 0 ËÃ×ÍãËé y=0 ¨Ðä´é ÍÕ¡µÑÇà»ç¹ 0 ´éÇ ´Ñ§¹Ñé¹ ä´é (0,0) à»ç¹¤ÓµÍºáá àÁ×èͨѴÃÙ»ÊÁ¡Ò÷Ñé§ÊͧãËÁè ¨Ðä´é $ x(y+3)(y+2) = 50(y^{2}-8x) $ $ x^{2}(y+2) = 25(y^{2}-8x) $ ¶éÒ y = -2 ·ÓãËé x= 0.5 ´Ñ§¹Ñé¹ (0.5,-2) à»ç¹¤ÓµÍº·ÕèÊͧ áÅÐàÁ×èÍ¹Ó 2 ÊÁ¡ÒÃÁÒËÒáѹ ¨Ðä´é y+3 = 2x ¨Ò¡¹Ñ鹡çá·¹¤èÒ x ã¹à·ÍÁ¢Í§ y Å§ä» ¡ç¨Ðä´é ¤ÓµÍºÁÒÍÕ¡ 2 ¤ÓµÍº ¤×Í (5,7) áÅÐ (45,87) ¤ÃѺ «Ö觤ӵͺ·ÕèÊÍ´¤Åéͧ¡Ñºâ¨·Âì¡ç¤×Í (5,7) à·èÒ¹Ñé¹ Êèǹ¢éÍ 5 (B) ´ÙàËÁ×͹¤Ø³ Warut ¨Ðä´é Upper bound ´Õ¡ÇèҢͧ¼ÁÍÕ¡¹Ðà¹Õè §Ñé¹¼Á¤§äÁèµéͧà©ÅÂÇÔ¸Õ¼ÁáÅéÇÁÑ駤ÃѺ ÊÓËÃѺ¢éÍ 12 ÍÒ¨·Óä´é´Ñ§¹Õé¤ÃѺ ¾Ô¨ÒÃ³Ò ¨Ó¹Ç¹ã¹ÃٻẺ $ 1+10^{2}+10^{4}+\cdots+10^{2k} \in A $ ÊÓËÃѺ¨Ó¹Ç¹¹Ñº k á¹è¹Í¹ÇèÒ 2549 ËÒèӹǹ㹡ÅØèÁ¹Õé ÂèÍÁµéͧÁÕ·ÕèàËÅ×ÍàÈÉà·èҡѹÍÂÙèà»ç¹¨Ó¹Ç¹Í¹Ñ¹µì (Åͧ¤Ô´´Ù¹Ð¤ÃѺ ÇèÒ·ÓäÁ) àÅ×Í¡ ÍÍ¡ÁÒ 2 ¨Ó¹Ç¹·ÕèàÈÉà·èҡѹ ÊÁÁµÔà»ç¹ $ x=1+10^{2}+10^{4}+\cdots+10^{2j} $ áÅÐ $ y=1+10^{2}+10^{4}+...+10^{2p} $ â´Âà»ç¹ä»ä´é·Õè¨ÐàÅ×Í¡ãËé j > p+1 ´Ñ§¹Ñé¹ $ 2549 | (x-y) = 10^{2(p+1)}+10^{2(p+2)}+\cdots+10^{2j} =10^{2(p+1)} (1+10^{2}+\cdots +10^{2(j-p-1)})$ áµè 2549 ËÒà $ 10^{2(p+1)}$ äÁèŧµÑÇ áÅÐ 2549 ¡çà»ç¹ ¨Ó¹Ç¹à©¾ÒÐ ´Ñ§¹Ñé¹ $ 2549| 1+10^{2}+\cdots +10^{2(j-p-1)} \in A $ ´Ñ§¹Ñé¹»Ãо¨¹ìÁÕ¤èÒ¤ÇÒÁ¨ÃÔ§à»ç¹¨ÃÔ§
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