answer78 - 7801 m n n(1 m n c ª ‚ m Þ n...

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Unformatted text preview: 7801 ( m, n ) n (1) m, n c * ª · ‚ m, Þ n pë·ª*ÐYE (2) (3) m 2 + n 2 hÞë· ‚·ª* (2), (3‚ )·ª m = n+ 2p m>n pn m = n + 2 p , m 2 + n 2 = 2k 2 •k c *ª 2 2 2 2 2 2 ( n + 2 p ) + n = 2k ª · * m + n = 2k 2 p* [k + (n + p)][k − (n + p )] = p [k + (n + p)] = p 2 ,[k − (n + p)] = 1 ♠ • ( p − 1)2 − 2 pn nn 99· ‚♠ ∗ 15· ‚♠ ∗ 2 p = 3,5, 7,11,13‚ * ª· (m, n, p) = (7,1,3), (17, 7,5), (31,17, 7), (71, 49,11), (97, 71,13) · ‚ª (m, n, p) = (71, 49,11)ª‚ · (m, n) = (71, 49), (49, 71) n= 1 99 Z [ @ E ë * ª·‚ , E E ë ë [ Y * …`YëE ÿ p ð ¹ ` B þ ƒ * ª· ‚ 7802 ¸ 2 ›Ê ˆ @ E 5[ E @ ë Y @ * [ ¹ p ` B þ ƒ ë , * 1¶ & * 214 8 ª 2 è ª · ¬? 824 , ( m, n ) n n 5m − 4 n 2n + 3 ª· ‚ m (i), 5m − 4 2n + 3 ≥ 2n ≥ 5n n m 5m − 4 2n + 3 < 2n <5n n m 5m − 4 ≥ 2 n 2n + 3 E≥Y *5ªÐ m: Π ë ´Ô (ii), 5m − 4 <2 n ´: Ô H 5m − 4 = 1n n m = 1,5 n n n = 5m −ª 4‚ · 2n + 3 10m − 5 5 = = 10 − m m m (m, n) = (1,1), (5,21) n (iii), 2n + 3 <5n m 2n + 3 2n + 3 = 1,2,3,4 n = 2,4 n m m (ii) ( ) 2n + 3 = 3n m n = 1,3 n 2n + 3 = 1,3 n m n 3m = 2n + 3 n n ª· xÐê ª · ‚ c ª* · ‚ c Ü6 @ ZëE@[ 7803 , 61 ‚· ª* , ë 6· Û ª*ÐYëE , 12 ê Ð x· ª* ˜Ûë· Ü 6 , , @[ëE•ƒþB`¹*p 3 Ýë n 2n + 3 = 1n m 3× (m, n) = (5,1), (25,11) n 5m − 4 10n + 3 3 = = 10 + ξ· ª* Ðê n n n (m, n) = (3,3) n n (m, n) = (1,1), (5,21), (5,1), (25,11), (3,3) 6 Ü 824 ¨· ë Ý ª*ÐYëE 41 ]¹* áE] , 214 1 ‚· ª* , , 720 1 ‚· ª* 6 8· ¬ ? ª*è&2 ¶1 1,2, ,12 M ê 6 x· Ð ª* C 78 = 39 · ‚ ª* 2 26 Ü 12 6 1 + 2 + + 12 = 78 M = 924 n a + b + c + d + e = 39 PÝë· 6 + 7 + 8 + 9 + 10 = 40 ≥ 39 (a, b, c, d , e) 6 a ≤ 5n b ≤ 8n Ü 1 5 10 11 12 3 1 6 9 11 12 3 1 7 8 11 12 3 1 7 9 10 12 3 1 8 9 10 11 3 2 4 10 11 12 3 2 5 9 11 12 3 2 6 8 11 12 3 2 6 9 10 12 4 2 7 8 10 12 4 2 7 9 10 11 4 4 5 5 , 4 5 5 6 6 6 7 7 5 6 6 7 6 9 8 9 7 8 9 8 8 7 7 8 8 7 11 11 10 11 10 10 9 10 11 10 9 9 9 12 12 12 12 12 11 12 11 12 12 12 11 12 5 5 19 (ê·ª* Ò 924 − 26 = 449 2 6 6 7 8 10 11 9 11 @≅ @[ëE'ƒþB`¹*p 7804 76 @[ëE) 2010“í· · í “ ¨ª*ÐYëE n n 1“í· 2È“í· 2010 èª* ·ëx · ë x èª a1 “í · aê 2 (Òª* · aë 3 èxª* ak (Ò · êª* a2010 €5 a1 n a1 n · aí 1 ˆ•ª*ÐYëE ? , a1 n a1 n a2 n a2 n a3 n ak −1 ak n 1, M M k €•í· ] áE]¹*c · (êª*Ò · ê (Òª* , (1)€ =| a1 − a2 | + | a2 − a3 | + + | a2009 − a2010 | + | a2010 − a1 | = ( Max(a1 , a2 ) − Min(a1 , a2 ) ) + ( Max(a2 , a3 ) − Min(a2 , a3 ) ) + + ( Max (a2009 , a2010 ) − Min(a2009 , a2010 ) ) + ( Max (a2010 , a1 ) − Min(a2010 , a1 ) ) = ( Max(a1 , a2 ) + + Max(a2010 , a1 ) ) − ( Min(a1 , a2 ) + + Min( a2010 , a1 ) ) ≤ 2 ( 1006 + 1007 + + 2010 ) − 2 ( 1 + 2 + + 1005 ) = 2(1005 + 1005 + + 1005) = 2 ×1005 ×1005 = 2020050 (2)(· *ê ªÒ 2 a1 = 1, a2 = 1006, a3 = 2, a4 = 1007, , a2009 = 1005, a2010 = 2010 ♠ • ∗ èξ =| 1 − 1006 | + |1006 − 2 | + | 2 − 1007 | + |1007 − 3 | + + |1005 − 2010 | + | 2010 − 1| = 2 ( 1006 + 1007 + + 2010 ) − 2 ( 1 + 2 + + 1005 ) = 2(1005 + 1005 + + 1005) = 2 ×1005 ×1005 = 2020050 (3), (1),(2) ¨ 2020050 (2) ª È * ·ë s * 1 2& * è ª ¬ ª¶ · 8 E ë Y * Ð ª ˆë · Ú * 1*2& 8 ª ª¬ è ¶· ëÚ€ EëZ * ¹ ]]E á ëÚ€ ªß · ê 7, í¨ 4, E ë )[ @ @E ) 67 ë [@ " @ @ 67 4 @ 2020050 2020050 2020050 2020050 3, * 1*2& 8 è ª ª¬ ¶· * ¹ ]]E á 1ë Z E [@ A 7805 n AM n n ABC n BC 8·ª* ëy N AB n AC n n AM n Pn Qn Nn AB AC 2 AM + = n AP AQ AN Q P N A B M P N C Q B n BD // PQ n AC n Dn n Mn D E B M C ME // PQ , AC n En∵ BM = MC ∴ CE = ED ∴ AB AD = AP AQ AM AE = AN AQ AB AC AD AC ( AE − DE ) + ( AE + EC ) 2 AE 2 AM + = + = = = AP AQ AQ AQ AQ AQ AN ë8 y·ª @[ëE 7N @[ëE! 7N ì≅ à·ª*ÐYëE ë8 y·ª* , 3, , 6 Héì· 4 8y·ª ë 1, 7, 7 8àì· 1, 9 , 11 8ë·ª* y 6, 1, 2 14 8ë·ª* y 4, 1, *ª·ëÞx 0] ³ ...
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This note was uploaded on 03/25/2011 for the course MATH 1232 taught by Professor Dr.jiangzhengchien during the Spring '11 term at Nanjing University.

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