数学竞赛中的组合问题

êÆ �šŽùŒ êÆ¿m¥�|ܯK o d liqian.jmtlf@gmail.com 2010.05 �N5†AÏ5 1. (1)†‚þ�z‡:Ñ�/Ǒù7�ڃ˜,¿…ü«�Ú�:Ñk. y²: Œ±3T†‚þé� n‡�må�ÓÚ:. (2) ²¡þ�z‡:Ñ�/Ǒù7�ڃ˜, ¿…ü«�Ú�:Ñk. y²: é?Ûa > 0, (a) Ñ Œ±3T²¡þé�ü‡ålǑa�ÓÚ:; (b) ь±3T²¡þé�ü‡ålǑa�ÉÚ:. (3) ²¡þ�z‡:Ñ�/Ǒù7�ڃ˜,¿…ü«�Ú�:Ñk. y²: Œ±3T²¡þé� ˜‡�nÆ/, §�n‡º:ÓÚ; §�>½ö�u1, ½ö�u √ 3. 2. ‹I²¡þ�z‡�:Ñ�/Ǒn«�ڃ˜, ¿…n«�Ú�:Ñk©y²: Œ±é�˜‡† ÆnÆ/, §�n‡º:´n«ØÓ�Ú�:© )‰ò²¡þ��:¡Ǒ(:. XJ3z˜^熆‚þ�¤k(:ѴӘ«�Ú�,K?À Ù¥˜‡(:(�ÙǑ1ÒÚ) .Šü^²L§�ƒpR†�†‚, �§‚†ç†•Ñ�¤45◦�Æ,¿ …3§‚þ¡©O�ј‡2ÒÚ�:ژ‡3ÒÚ�:(ù´Œ±‰��, ÏǑ3äkù �Ú� :�熆‚) . ¤�ƒ†ÆnÆ/=Ǒ¤�. XJz˜^Y²†‚þ�¤k(:ѴӘ«�Ú�, KŒaq/)‰. XJ3˜^熆‚v, Ùþ�(:©O�/Ǒü«ØÓ�Ú(�Ǒ1ÒÚÚ2ÒÚ) .·‚?�˜ ‡3ÒÚ�:C, 2�ц:CÓ3˜^Y²†‚þ��‡ u†‚vþ�:A (ؔ�ÙǑ1ÒÚ) , ± 9 u†‚vþ�äk2ÒÚ�:B, =�¤�. XJ3˜^熆‚v, Ùþ�(:©O�/Ǒn«ØÓ�Ú, �o·‚2�˜^Y²†‚h, ��Ùþ�:شӘ«�Ú. ؔ�§‚��:A´1ÒÚ�. ·‚3†‚hþ�˜:B�ƒ†AÉ Ú(�ÙǑ2ÒÚ) , 23†‚vþ�˜‡3ÒÚ�:C=Œ. 3. �N�ˆ‡º:þ©O�X�ê1–8, ˆ^ þ©O�XÙü‡à:þ�êƒ��ýéŠ. Á ¯, 12^ þ–õŒÑyõ�‡p؃Ó�ê? –�Q? 4. 5gŽSØÓ{°�33‡¬óÓ3˜[úi‹ó. ί�‚¥z‡<ü‡Ó��¯K:3Ù{32< ¥kA‡†\Ó{, kA‡†\Ó Ñ)? (Juy, 3¤���£‰¥¹ l0�10�¤k�ê. y²: 3�‚¥k�,ü<�£ž>ǑùÚ,Ø�£žǑ7Ú,�ë�1003‡n Æ/. enÆ/¥�,‡Æ�ü>ÓÚ, Ò¡Ǒ“ÓÚÆ” . z‡nÆ/–�k˜‡ÓÚÆ, ØÓnÆ/ �ÓÚÆØÓ, KÓÚÆØ�u10063‡. §‚©Ù3C23 × 1002 = 3× 1002‡:éþ, l ,7k˜‡: é¤Ü�ÓÚÆØ�u [ 100 3 ] + 1 = 34‡,Ù¥7k17‡ǑӘÚ. ù‡:é=Ǒ¤�. (2) òz‡óŠ< ÑéAǑ²¡þ�˜‡:(?Ûn:Ø�‚) , òn‡” ¬�óŠ< ¤é A�:8©OPǑA, B, C. 3?ÛØáuӘ:8�ü:ƒmë˜^‚ã: eéA�ü‡<ƒp�£, Òëù‚; eØ�£, Òë7‚. ù�¤��ã¡ǑnÜã. �nÜã�A −B − C�º:êǑ(a, b, c). ŠâK¿, A−B�z^ùÚ þTk10‡ùÚnÆ/, z^7Ú þTk10‡7ÚnÆ/. z ‡ÓÚnÆ/Tk˜>ǑA−B, K�ã�ÓÚnÆ/�‡êǑ10ab. Ón, §Ǒ�u10bc, 10ca. Ïd, a = b = c. ù�˜5, =ÓÚnÆ/�‡êǑ10a2. u´, ÉÚnÆ/�‡êǑa3 − 10a2. duz‡ÉÚn Æ/Tk˜‡ÓÚÆ, z‡ÓÚnÆ/kn‡ÓÚÆ, KÓÚÆ�o‡êǑ3× 10a2 + a3 − 10a2 = a3 + 20a2. ,˜¡, z^ Ü20‡ÓÚÆ, KÓÚÆ�‡êq�u20 × 3a2 = 60a2. -a3 + 20a2 = 60a2, )�a = 40. Ïd, o<êǑ3a = 120. 5 ˜„5`, ez^ ÑTÜn‡ùÚÆ9n‡7ÚÆ, Kº:oêǑ12n. 6. 30Ük¡þ©O�k?ÒǑ1, 2, · · · , 30, §‚�?¿/�må/{˜3˜‡�±þ. NN�†? Ûü܃�k¡� ˜.3²L˜X�ù«C†ƒ�,uyzÜk¡Ñ u�5 ˜�é»:þ(Ә †»�üàpǑé»:) . y²: 3þã�†L§�,˜Úþ, 7küÜk¡þ�ÒèƒÚ�u31. 7. (1) nm‡ÓÆ�¤n×m� �. lz1ÓÆ¥éсp�ÓÆ, 2l�‚¥éсL�˜‡, ¡ Ǒ`; lz�ÓÆ¥éсL�ÓÆ, 2l�‚¥éсp�˜‡, ¡Ǒ¯. Á¯, `¯�<¥X�p? (2) én× n�êL, ?1Xeü«öŠ: 2 öŠ1 lL¥ÀсŒ�ê, ¿ò§¤3�1†��Üí�, ,�3e�(n − 1) × (n − 1)�ê L¥2éсŒ�ê, ¿í�§¤3�1†�, Xd˜†e�. öŠ2 lL¥Àс��ê, ¿ò§¤3�1†��Üí�, ,�3e�(n − 1) × (n − 1)�ê L¥2éс��ê, ¿í�§¤3�1†�, Xd˜†e�. Á¯, ´Ä3ù��n×nêL, ��UöŠ2ÀÑ�n‡ê�oڇ 'UöŠ1ÀÑ�n‡ê� oڄŒ? )‰ (2) ·‚òy², Ø3÷v^‡�n× n�êL. òUöŠ1¤ÀÑ�n‡êPǑa1, a2, · · · , an, òUöŠ2¤ÀÑ�n‡êPǑb1, b2, · · · , bn, Kk a1 ≥ a2 ≥ · · · ≥ an, b1 ≤ b2 ≤ · · · ≤ bn. ·‚5y², éz‡k ∈ {1, 2, · · · , n}, Ñkak ≥ bn+1−k. 5¿�, 3UöŠ1ÀÑakƒ , êL¥ÿn + 1 − k1, n+ 1 − k�; 3UöŠ2ÀÑbn+1−kƒ , êL¥ÿk1, k�; du(n+ 1− k) + k = n+ 1 > n, ¤±7k˜1, Ǒ7k˜�, 3ü«œ¹e Ñ3uêLƒ¥. PT1ÚT�ƒ�?�êǑx, �oÒk ak ≥ x ≥ bn+1−k. 8. (1) 3�N�z‡º:þÑÊk˜ñG. É�¯h�, §‚�ќå. L ˜ãžm�, §‚ qÑá £�, E,´z‡º:þ˜, �ؘ½Ñ£�� . y²: Ù¥7knñG, §‚圃 ¤3�n‡º:¤¤�nÆ/†§‚á£�ƒ�¤3�n‡º:¤¤�nÆ/��. (2)3�n>/�z‡º:þˆÊk˜U&. óɯh, ¯U&ќ�. ˜ãžm�, §‚qÑ£ �ù º:þ, E´z‡º:þ˜,�™7Ñ£��5�º:. �¤k��ên, ��˜½33U &, ±§‚ �¤3�º:©O/¤�nÆ/½ÓǑbÆnÆ/, ½ÓǑ†ÆnÆ/, ½ÓǑðÆn Æ/. )‰ (2) �K�(J´, n�ŒUŠǑ¤kØ�u3…Ø�u5���ê. �n = 5ž, k‡~X㤫. Ù¥z‡�5�bÆnÆ/ÑCǑðÆnÆ/, z‡�5�ðÆ nÆ/ÑCǑbÆnÆ/, ¤±n 6= 5. ŠÑ�n>/� ��. éun = 2m (m ≥ 2, m ∈ N+) , Ä?¿˜é�5é»( uӘ†» üà) �U&AÚB. XJ§‚�5E,é», �o§‚†?ۘO�U&C �¤3�º:©O/ ¤�nÆ/Ñ´†ÆnÆ/; XJ§‚�5Ø2é», K�5†Aé»�´,O�U&, PǑD. ´ A,B,DnU& �¤3�º:©O/¤�nÆ/ǑÑ´†ÆnÆ/, ¤±nŒ±Ǒ˜ƒØ�u4� óê. n = 3�œ/w,. éun = 2m+1 (m ≥ 3, m ∈ N+) . ·‚84?¿˜U&A,LA¤3�º: Š ���†»AA1,duA1Ø´�2m+1>/�º:,¤±:A1ÑvkU&,džAA1�üýw,ˆ kmU&. Ä uAA1,˜ý�mU&,dum ≥ 3,ŠâÄT�K,Ù¥73üU&B,C, §‚á£�Ǒ uAA1�Әý. ù�, A,B,C �¤3�n‡º:þ�¤ðÆnÆ/. ¤±nŒ±Ǒ ˜ƒØ�u3…Ø�u5�Ûê. 3 nþ¤ã, n�ŒUŠǑ¤kØ�u3…Ø�u5���ê. ,) ¡�)‰Ó{˜, òn = 2m+ 1 (m ≥ 3, m ∈ N+) �œ/UyXe. Äky²XeÚn: Ún �n = 2m+ 1 (m ≥ 3, m ∈ N+) ž, ëѤkéƂ��n>/¥, ðÆnÆ/�‡êõu ¤knÆ/�‡ê�˜Œ. Ún�y²��n>/ǑA1A2 · · ·An. ùž,±éƂA1Aj+1Ǒ>�ðÆnÆ/�‡êǑj − 1. ¤±, ðÆnÆ/�oêǑ S = (2m+ 1)(1 + 2 + · · ·+ (m− 1)) = 1 2 (2m+ 1)m(m− 1). �n>/¥�nÆ/�oêǑ t = C3n = 1 3 (2m+ 1)m(2m− 1). dum > 1 3 (2m− 1)é¤km > 2þ¤á, ÏdS > 1 2 t, =Ún¤á. dÚnŒ, �n = 2m+1 (m ≥ 3, m ∈ N+)ž, o3nU&,§‚ �¤2º:©O/¤� nÆ/ÓǑðÆnÆ/. 9. ,‡ãäkn ≥ 8‡º:. Á¯,ù º:�ÝêŒÄ©OǑ4, 5, 6, · · · , n−4, n−3, n−2, n−2, n− 2, n− 1, n− 1, n− 1? )‰ �n = 8, 9ž, 3÷v^‡�ã(ãÑ) . b��n ≥ 10ž, 3÷v^‡�ã, Tã�n‡º:©OǑA1, A2, · · · , An, òù º:©Šn‡ 8Ü, ©OPǑ M1 = {A1, A2, A3}, M2 = {A4, A5, A6}, M3 = {A7, A8, · · · , An}. Ù¥M1¥n‡º:�ÝêþǑn − 1, M2¥n‡º:�ÝêþǑn − 2, M3¥�n − 6‡º:�Ýê gǑn− 3, n− 4, · · · , 6, 5, 4. w,, M1¥z‡º:ÑA†Ù{º:ƒë, M2¥z‡º:і�†M3¥ �n− 2− 3− 2 = n− 7‡º:ƒë, ù«dM2ë•M3�‚ãØ�u3(n− 7) = 3(n− 8) + 2 + 1^. d uAnÚAn−1�Ýê©O´4Ú5, §‚qÑ®†M1¥ëkn^‚ã, ¤±U©O2†M2¥�1‡: Ú2‡:ƒë, ù`²M3¥Ù{�º:Ñ7LÓM2¥�z‡º:ƒë, âU�ë‚�ê8ˆ�3(n − 7)^. ù�˜5, dAn, An−1, An−2¤ëÑ�‚ãê8Ò®²©Oˆ� 4,5,6^, ØU2ÓÙ�º:ƒ ë . , A7´˜‡n− 3Ý�º:,Ø ®†M1ÚM2ëk�6^‚ã , –�„A†M3¥n− 3− 6 = n − 9‡º:ƒë. �´, =�A7†A8, A9, · · · , An−3уë, Ǒ–õ=kn− 10‡º:Œ±ƒë, gñ. ùL², �n ≥ 10ž, Ø3÷v^‡�ã. |ÜOê¯K 10. �ؽ§x1 + x2 + x3 + 3x4 + 3x5 + 5x6 = 21���ê)�|ê. )‰ -x1 + x2 + x3 = x, x4 + x5 = y, x6 = z, Kx ≥ 3, y ≥ 2, z ≥ 1. kÄؽ§x + 3y + 5z = 21÷vx ≥ 3, y ≥ 2, z ≥ 1���ê), dž5z = 21− x− 3y ≤ 12, K1 ≤ z ≤ 2. �z = 1ž, kx+ 3y = 16, d§÷vx ≥ 3, y ≥ 2���ê)Ǒ(x, y) = (10, 2), (7, 3), (4, 4). �z = 2ž, kx+ 3y = 11, d§÷vx ≥ 3, y ≥ 2���ê)Ǒ(x, y) = (5, 2). ¤±Ø½§x + 3y + 5z = 21÷vx ≥ 3, y ≥ 2, z ≥ 1���ê)Ǒ(x, y, z) = (10, 2, 1), (7, 3, 1), (4, 4, 1), (5, 2, 2). 4 q§x1 + x2 + x3 = x (x ∈ N+, x ≥ 3) ���ê)�|êǑC2x−1, §x4 + x5 = y (y ∈ N+, y ≥ 2) ���ê)�|êǑC1y−1 , Ïd, d©ÚOê�n, �ؽ§���ê)�|êǑC29C11 + C26C 1 2 +C 2 3C 1 3 +C 2 4C 1 1 = 36 + 30 + 9 + 6 = 81. 11. �8>/ABCDEF´>Ǒ1��8>/, O´8>/�¥%, Ø 8>/�z˜^>, ·‚„ l:O�z‡º:ë˜^‚ã, ���12^ÝǑ1�‚ã. ˜^´»´l:OÑu, ÷X‚ã� q£�:O. Á�ÝǑ2003�´»�^ê. )‰˜ �anL«lO�O�ÝǑn�´»^ê, bnL«lA�O�ÝǑn�´»^ê, Kk{ an = 6bn−1, bn = an−1 + 2bn−1, é?Û��ên¤á. ÏL“†Úz{� an+1 − 2an+1 − 6an = 0. ÙAƐ§Ǒλ2 − 2λ− 6 = 0, )�λ = 1±√7. Ïd, an = A(1 + √ 7)n +B(1−√7)n, Ù¥AÚB´~ ê. |^�©^‡a0 = 1, a1 = 0, Œ�{ 1 = A+B, 0 = (1 + √ 7)A+ (1−√7)B. )�A = 7−√7 14 , B = 7 + √ 7 14 . Ïd, an = 1 14 [ (7− √ 7)(1 + √ 7)n + (7 + √ 7)(1 − √ 7)n ] . AO/, ¤��‰YǑ a2003 = 1 14 [ (7 − √ 7)(1 + √ 7)2003 + (7 + √ 7)(1− √ 7)2003 ] . )‰� �lO�O�ÝǑn− 2�´»�kan−2^, �ù ´»¥�?ۘ^,ŠǑ(å:O, · ‚2lO�6‡º:¥�?ۘ‡, 2ˆ£�:O, u´Ò�� ˜^lO�O�ÝǑn�´», ù� �ÀJk6«. 2�lO�O�ÝǑn − 1�´»�kan−1^, éù ´»¥�?ۘ^, ˜½Œ±é�˜‡ º:(~Xº:A) �O�˜^>Ǒ�˜^>, lAk�B½F (2«ÀJ) , 2£�O, Ò�E ü^ lO�O�ÝǑn�´». w,, þ¡ü«�EÑ´ØÓ�ÝǑn�´», …§‚|¤ ÝǑn� �Ü´», Ïd an = 2an−1 + 6an−2. ±eÓ)‰˜. 12. ‰½˜‡2008 × 2008�Ú�, Ú�þz‡�‚��Úþ؃Ó. 3Ú��z˜‡�‚¥W \C, G, M , Où4‡i1¥�˜‡, eÚ�¥z˜‡2 × 2��Ú�¥ÑkC, G, M , Où4‡i1, K¡ ù‡Ú�Ǒ“Ú�Ú�” . ¯kõ�«ØÓ�“Ú�Ú�” ? )‰ k12× 22008 − 24«ØÓ�“Ú�Ú�” . ·‚Äky²e¡ù‡(Ø: 3z‡“Ú�Ú�” ¥, –�Ñy±eœ¹¥�,˜«: (1) z˜1Ñ´,ü‡i1�OÑy; (2) z˜�Ñ´,ü‡i1�OÑy. Ù¢, b�,˜1Ø´�O�, Kù˜17½¹n‡ƒ���‚WkØÓ�i1. ؔ˜ „5, b�ùn‡i1ǑC, G, M , Xã1¤«. ùéN´��X2 = X5 = O, ¿…X1 = X4 = M , X3 = X6 = C, Xã2¤«. 5 ã1 X1 X2 X3 C G M X4 X5 X6 ã2 M O C C G M M O C Ón,·‚ÒŒ±��ùn�Ñ´ü‡i1�OÑy. l N´��z˜�Ñ´,ü‡i1�OÑy. y3·‚5OŽ“Ú�Ú�” �‡ê. XJ†>˜�´,ü‡i1('XCÚM) �OÑy, êþ Œ±��SÒǑÛê��Ñ´ùü‡i1�OÑy,…SÒǑóê��Ñ´, ü‡i1('XGÚO)� OÑy. z˜��1˜‡i1Œ±´ù˜�¤¹�ü‡i1�?¿˜‡; N´�y?¿ù��W� ь±��“Ú�Ú�” . l ·‚kC24 = 6«ØÓ�ªÀJ1˜��ü‡i1,…k2 2008 «ªû ½z˜��1˜‡i1,¤±,·‚k6×22008«W{��z˜�Ñ´�O�. Ó�,·‚Ǒk6×22008« W{��z˜1Ñ´�O�. y3¤‡‰�Ò´l¥~�OŽ üg�W{—1Ú�Ñ´�O�. w,, o‡ØÓi13†þ Æ�2× 2‚¥�?Ûü�ь±*¿��‡Ú���˜‡“Ú�Ú�” , ¿…1Ú�Ñ´�O�,Ù ¢, ‡kW� ü���§‚´�O�, 2W¤k�1��§‚´�O�=Œ; ‡L5, ù«V�O �W{d†þÆ�2 × 2‚˜û½. k4! = 24«ª3†þÆ�2 × 2‚¥ü�o‡ØÓ�i1. ¤±·‚��24«ØÓ�W{��1Ú�Ñ´�O�, ddŒ±��þ¡(J. 13. �÷v x− y x+ y + y − z y + z + z − u z + u + u− x u+ x > 0,…1 ≤ x, y, z, w ≤ 10�¤ko�kS�ê|(x, y, z, u)� ‡ê. )‰�f(a, b, c, d) = a− b a+ b + b− c b+ c + c− d c+ d + d− a d+ a . PA = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f(x, y, z, u) > 0}, B = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f(x, y, z, u) < 0}, C = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f(x, y, z, u) = 0}. w,Card(A) + Card(B) + Card(C) = 104. ·‚y²Card(A) = Card(B). éz˜‡(x, y, z, u) ∈ A, Ä(x, u, z, y). (x, y, z, u) ∈ A ⇔ f(x, y, z, u) > 0 ⇔ x− y x+ y + y − z y + z + z − u z + u + u− x u+ x > 0 ⇔ x− u x+ u + u− z u+ z + z − y z + y + y − x y + x < 0 ⇔ f(x, u, z, y) < 0⇔ (x, u, z, y) ∈ B. �XOŽCard(C). (x, y, z, u) ∈ C ⇔ xz − yu (x+ y)(z + u) = xz − yu (y + z)(u+ x) ⇔ (z−x)(u−y)(xz−yu) = 0. �C1 = {(x, y, z, u)|x = z, 1 ≤ x, y, z, u ≤ 10}, C2 = {(x, y, z, u)|x 6= z, y = u, 1 ≤ x, y, z, u ≤ 10}, C3 = {(x, y, z, u)|x 6= z, y 6= u, xz = yu, 1 ≤ x, y, z, u ≤ 10}. ÏǑ÷vab = cd, 1 ≤ a, b, c, d ≤ 10�üüØÓ��So�|k1 × 6 = 2 × 3, 1 × 8 = 2 × 4, 1× 10 = 2× 5, 2× 6 = 3× 4, 2× 9 = 3× 6, 2× 10 = 4× 5, 3× 8 = 4× 6, 3× 10 = 5× 6, 4× 10 = 5× 8. ÷vx 6= z, y 6= u, xz = yu�o�|�90‡, ÷vx = u, y = z, x 6= z�o�|�90‡, ÷vx = u, y = z, x 6= z�o�|�90‡, u´Card(C3) = 4 × 2 × 9 + 90 + 90 = 252. qCard(C1) = 1000, Card(C2) = 900,KCard(C) = 2152. ÏdCard(A) = 3924. 14. òê8A = {a1, a2, · · · , an}¥¤k�ƒ�Žâ²þŠPǑP (A) (P (A) = a1 + a2 + · · ·+ an n ). eB´A�š˜f8, …P (B) = P (A), K¡B´A�˜‡“þïf8” . Á�ê8M = {1, 2, 3, 4, 5, 6, 7, 8, 9}�¤k“þïf8”�‡ê. )‰ duP (M) = 5, -M ′ = {x − 5|x ∈ M} = {−4,−3,−2,−1, 0, 1, 2, 3, 4},KP (M ′) = 0, ì d²£'X, MÚM ′�þïf8Œ˜˜éA. ^f(k)L«M ′�k�þïf8�‡ê, w,kf(9) = 1, f(1) = 1 (M ′�9�þïf8kM ′, ˜�þïf8k{0}) . M ′���þïf8�o‡, ǑBi = {−i, i}, i = 1, 2, 3, 4,Ïdf(2) = 4. 6 M ′�n�þïf8kü«œ¹: (1) ¹k�ƒ0�ǑBi ∪ {0} = {−i, 0, i}, i = 1, 2, 3, 4,�4‡; (2) ع�ƒ0�, du�ª3 = 1 + 2, 4 = 1 + 3Œ±L«Ǒ−3 + 1 + 2 = 0, 3 − 1 − 2 = 0± 9−4 + 1 + 3 = 0, 4 − 1 − 3 = 0��4‡þïf8{−3, 1, 2}, {3,−1,−2}, {−4, 1, 3}, {4,−1,−3}, Ï df(3) = 4 + 4 = 8. M ′�o�þïf8kn«œ¹: (1) zü‡��þïf8ƒ¿: Bi ∪Bj , 1 ≤ i < j ≤ 4, �6‡8; (2) ع�ƒ0�n�þïf8†{0}�¿8, �4‡8; (3)±þü«œ¹ƒ ö, du�ª1+4 = 2+3ŒLǑ−1−4+2+3 = 0±91+4−2−3 = 0�2‡ þïf8{−1,−4, 2, 3}, {1, 4,−2,−3},Ïdf(4) = 6 + 4 + 2 = 12. q5¿�, ØM ′�� , eB′´M ′�þïf8, �…=�ÙÖ8∁M ′B ′ Ǒ´M ′�þïf8, �ö ˜˜éA. Ïd, f(9− k) = f(k), k = 1, 2, 3, 4. l M ′�þïf8‡êǑ 9∑ k=1 f(k) = f(9) + 2 4∑ k=1 f(k) = 1 + 2(1 + 4 + 8 + 12) = 51. =M�þï f8k51‡. 15. ò±Ǒ24��±�©¤24ã, l24‡©:¥À�8‡:, ��Ù¥?Ûü:ƒm¤Y�lÑ Ø�u3Ú8. ¯÷v‡��8:|�ØÓ�{�kõ�«? )‰ ò24‡ê�¤3× 8�êL, u´, À�8‡êžz�T�˜‡ê. ùž, l11��˜‡ê, � k3«ØÓ��{. 11��½�, 12�¤��êØU†11�¤��êÓ1, Ïdkü«ØÓ�{. ±�z�Ñkü«ØÓ�{, �k3 × 27«ØÓ��{. �Ù¥1˜�¤��ê†18�¤��êÓ1 �¤k�{ÑØ÷v‡�. ePl3 × nêL¥z�T�˜‡ê…?ۃ�ü�()1n�†11�) ¤��êÑØÓ1�Ø Ó�{«êǑxn , Kþã(ØTǑx8 + x7 = 3 × 27. aq/Œ±��xn + xn−1 = 3 × 2n−1. dd4í =� x8 = 3× 27 − x7 = 3× 27 − (3 × 26 − x6) = 3× (27 − 26) + x6 = · · · · · · = 3× (27 − 26 + 25 − 24 + 23 − 22 + 2) = 3× 86 = 258. =÷vK¥‡���ØÓ�{oêǑ258. 16. ²¡Sk18‡:, Ù¥?¿3:Ø�‚, zü:똂ã, ù ‚ã^ù7üÚ/Ú, z^‚ãT /˜Ú, Ù¥¥,:AÑu�ùڂãkÛê^, lÙ{17‡º:Ñu�ùڂãêp؃Ó. �± ®:Ǒº:, ˆ>ÓǑùÚ�nÆ/�‡ê9kü>ǑùÚ,˜>Ǒ7Ú�nÆ/�‡ê. )‰˜ �¤k�nÆ/¥n>ǑùÚ, ü>ùژ>7Ú, ü>7ژ>ùÚ, n>Ǒ7Ú�n Æ/�‡ê©OǑm, n, p, q. ÏØA , Ù{ˆ:ÚÑ�ùڂãêp؃Ó,�Ǒ0,õǑ17,…0†17ØUӞÑy,u´ ke�ü«œ/: 0, 1, 2, · · · , 16½1, 2, · · · , 17. eǑ ö,�lAÑu�‚ãêǑ2k−1,Kã¥ùڂã�k1 2 (0+1+2+· · ·+16+2k−1) = 17×4+ k− 1 2 ^, gñ. l UǑ1�«œ/. �ØA , Ù{17‡:ǑB1, B2, · · · , B17…lBiÑuTki^ù ڂã(i = 1, 2, · · · , 17) ,u´B17†Ù{17:Ñëkù‚,l B1=†B17ëkù‚, B16†ØB1 �Ù {16‡:ëkù‚,l B2=†B17, B16ëkù‚.˜„`5, B17−i†ØB1, B2, · · · , Bi (i = 1, 2, · · · , 8) �Ù{17−i‡:ëkù‚, Bi=†B17, B16, · · · , B18−iëkù‚,Œ„A=†B17, B16, · · · , B9ù9: ëkù‚. Ødƒ , ¤ë�Ù�‚ãþǑ7‚. 7 ·‚¡l˜:Ñu�ü^ùڂãǑ˜‡ùÚÆ, l˜:Ñu�ü^7ڂãǑ˜‡7ÚÆ, u ´z˜‡n>ǑùÚ�nÆ/¥k3‡ùÚÆ, z˜‡ü>ǑùÚ,˜>Ǒ7Ú�nÆ/¥k˜‡ù ÚÆ, Ù�nÆ/¥�ùÚÆ, l ùÚÆ�oêǑ3m + n. ,˜¡, ±BiǑº:�ùÚÆkC 2 i (i = 1, 2, · · · , 17) ‡…�½C21 = 0. lAÑu�ùÚÆkC29‡, l ùÚÆ�oêǑ 17∑ i=2 C2i +C 2 9 = C 3 3 + 17∑ i=3 (C3i+3 − C3i ) + C29 = C318 + C29 = 852. ¤±, 3m+ n = 852. (1) aq/, ¤k�7ÚÆ�oêǑ p+ 3q = 17∑ i=2 C217−i +C 2 8 = C 3 17 +C 2 8 = 708. (2) qÏz˜^‚ã´C116 = 16‡nÆ/�ú�>, Kã¥ùڂãêǑ 1 16 (3m+ 2n+ p). ,˜¡ lBiÑuki^ùڂã(i = 1, 2, · · · , 17) ,…lAÑuk9^ùڂã,z^‚ãOŽ üg,Kùڂ ã�oêqǑ 1 2 (1 + 2 + · · ·+ 17 + 9) = 81, ¤± 3m+ 2n+ p = 16× 81 = 1296. (3) aqŒ�ã¥7ڂãêǑ 1 16 (n+ 2p+ 3q) = 1 2 (16 + 15 + · · ·+ 2 + 1 + 8) = 72, = n+ 2p+ 3q = 16× 72 = 1152. (4) éá(1) (2) (3) (4) , )�m = 204, n = 240, p = 204, q = 168. =n>ǑùÚ�nÆ/k204‡, ü>Ǒùژ>Ǒ7Ú�nÆ/k240‡. )‰� Ó)‰˜�©ÛÚPÒ�, -M = {B1, B2, · · · , B8}, N = {A,B9, B10, · · · , B17}, KM¥ ?¿ü:�ùڂãƒë, N¥?¿ü:ëkùڂã, …M¥?¿˜:Bi (1 ≤ i ≤ 8) , T†N¥ �i‡:ëkùڂã. u´, ±BiǑº:…n>ǑùÚ�nÆ/Ø3, ±Bi (2 ≤ i ≤ 8)Ǒº:…n >ǑùÚ�nÆ/kC2i‡, ±N¥?¿3:Ǒº:…n>ǑùÚ�nÆ/kC 3 10‡, l n>þǑùÚ �nÆ/‡êǑ C22 +C 2 3 + · · ·+C28 = C39 +C310 = 84 + 120 = 204. qü>ǑùÚ,˜>Ǒ7Ú�nÆ/©Ǒüa: 1˜a´nÆ/�˜‡º:ǑBi ∈M (1 ≤ i ≤ 8), ,ü‡º:áuN , …lBi•N¥ü:ÚÑ�ü^‚ã´˜ù˜7(N¥�ü:ë‚ǑùÚ) , ùan Æ/�‡êǑ 8∑ i=1 C1iC 1 10−i = 1× 9 + 2× 8 + 3× 7 + 4× 6 + 5× 5 + 6× 4 + 7× 3 + 8× 2 = 156. 1�anÆ/�˜‡º:ǑP ∈ N (PǑA½Bj (9 ≤ j ≤ 17) ) , ,ü‡áuº:M ,…lP•M¥ü: ¤Ú‚ãþǑùÚ(M¥ü:ë‚Ǒ7Ú) , ùanÆ/�‡êǑ 17∑ i=9 C2i−9 = C 2 2 +C 2 3 + · · ·+C28 = C29 = 84, ùp, C20 = C 2 1 = 0. Ïd, ü>ǑùÚ,˜>Ǒ7Ú�nÆ/�k156 + 84 = 240‡. 8 17. e˜‡àn>/�?¿n^éƂØ3Tõ>/SÜ�:, ÁOŽ¤kéƂrTõ>/©¤ �«�ê8. )‰˜ �nkL«n>/S£©¤�«¥k>/�‡ê(3 ≤ k ≤ m) , ¤kù õ>/�º:ê 8()­E3S) Ǒ3n3 +4n4 + · · ·+mnm. ,˜¡, z˜‡Sܺ:Ñ´ü^éƂ��:,¤± ´n>/SÜo‡«�ú�º:, Ï 3þãÚª¥, z‡Sܺ:Ñ­EOê og. qn>/± .þz‡º:w,TǑn− 2‡nÆ/�º:,KOê n− 2g,d ¤kSܺ:†±.º:¥¤k ŒU�o‡º:|ܤ˜˜éA, =Ǒn>/±.º:o�|�‡ê, l 3n3 + 4n4 + · · ·+mnm = 4C4n + (n− 2)n. (1) y3·‚r¤k�õ>/�Æ\å5, k>/ˆÆƒÚ(k− 2) · 180◦, ¤±�Üõ>/(w,þ´à �) Æ݃ÚǑn3 · 180◦ + n4 · 360◦ + · · ·+ nm · (m− 2) · 180◦. þãÆ݃ڄk˜Ž{, 3Sܺ: ?o‡ÆƒÚ´360◦, ¤±¤kSܺ:?ˆÆƒÚAǑC4n · 360◦. 3n>/�±.þˆº:?ˆÆƒ ÚǑ(n− 2) · 180◦, u´n3 · 180◦ + n4 · 360◦ + · · ·+ nm · (m− 2) · 180◦ = C4n · 360◦ + (n− 2) · 180◦, = n3 + 2n4 + 3n5 + · · ·+ (m− 2)nm = 2C4n + (n− 2). (2) (1)− (2), � 2n3 + 2n4 + · · ·+ 2nm = 2C4n + (n− 1)(n− 2), l ¤�«ên3 + n4 + · · ·+ nm = C4n +C2n−1. )‰� z�K˜^éƂ, K«�‡ê~�ai + 1‡, ùpai´TéƂ†„vk�K�Ù� éƂ�3/S��:ê, ÅÚòC2n − n^éƂ�K, �, e˜‡«, l ¤�«êǑS = 1 + C 2 n−n∑ i=1 (ai + 1), … C 2 n−n∑ i=1 ai = C 4 nT´éƂ3/S��:ê, ¤± S = 1 + C4n +C 2 n − n = C4n +C2n−1. 18. ‰½d n(n+ 1) 2 ‡:|¤��nÆ/: (X㤫) , P±: ¥n‡:Ǒº:�¤k�nÆ /�‡êǑf(n). �f(n)�Lˆª. )‰ P i1¤k:�¤��nÆ/‡êǑf(i) (i = 1, 2, · · · , n) . OŽf(1) = 0, f(2) = 1, f(3) = 5. e¡�f(n)†f(n− 1)�'X. w,, f(n)�uf(n− 1)\þ“kº:(1‡½2‡) 31n1þ�¤k�nÆ/�‡ê” , Ù¥, k2‡ º:31n1þ�¤k�nÆ/¡Š1Ia, §‚kC2n‡. 9 �e5r=k1‡:31n1�¤k�nÆ/¡Š1IIa, ïÄهê. ��△ABC´1IIanÆ/, º:A31n1S, º:BÚC3 ¡�1S. L:BÚC©OŠnÆ /: >.‚�²1‚, †1n1�uPÚQü:. l , �△ABCkŽ˜�n:S(P,A,Q)†ƒéA. ‡L5, é1n1þ?¿n:�¤�:S(P,A,Q), ±ü‡>:PÚQǑº:3nÆ/: SŠÑ ˜‡�nÆ/,2±¥m:AǑº:ŠT�nÆ/�S��△ABC, w,,ù��nÆ/´Ž˜3�. u´, 31IIa�nÆ/†n:Sƒmïá ˜˜éA, �1IIanÆ/kC3n‡. Ïd, f(n) = f(n− 1) + C2n +C3n. 34íª¥nH{2, 3, · · · , n, ¿…®kf(1) = 0. �Ú=� f(n) = n∑ i=2 (C2n +C 3 n) = 1 6 n∑ i=2 (i3 − i) = 1 6 (13 + 23 + · · ·+ n3)− 1 6 (1 + 2 + · · ·+ n) = 1 6 × 1 4 n2(n+ 1)2 − 1 6 × 1 2 n(n+ 1) = (n− 1)n(n+ 1)(n+ 2) 24 . |܁Š¯K 19. �M´kê8,e®M�?Ûn‡�ƒ¥o3ü‡ê,§‚�ÚáuM ,Á¯M¥õk õ�‡ê? )‰ ¤�M¥�ƒ‡ê�ŒŠǑ7. e¡y²Tê8¥–õk7‡�ƒ. Äky²Tê8¥õk3‡�ê. b�ŒUkØ�u4‡� �ƒ, Ù¥Œ�4‡ê©OǑa1 , a2, a3, a4, …a1 < a2 < a3 < a4. ¯¢þ, ·‚ka3 + a4 > a2 + a4 > a1+a4 > a4,¤±Úêai+a4 6∈M(i = 1, 2, 3). Œua3��ƒka4˜‡,%ka2+a3 > a1+a3 > a3, u´38Ü{a1, a2, a4}½{a2, a3, a4}¥,–�k˜‡8Ü�?¿ü‡�ƒƒÚØ3M¥, ù†®gñ, �Tê8¥õk3‡�ê. Ón,Tê8¥õk3‡Kê, \þ˜‡0, l ê8M¥–õk7‡�ƒ. ¯¢þ, 8ÜM = {0,±1,±2,±3}÷v^‡,Ù¥Tk7‡�ƒ. 20. ��êa1, a2, · · · , a2006 (Œ±kƒÓ�) ,��a1 a2 , a2 a3 , · · · , a2005 a2006 üü؃�. ¯: a1, a2, · · · , a2006¥ �kõ�‡ØÓ�ê? )‰ du45‡p؃Ó���êüü�'Š–õk45 × 44 + 1 = 1981‡,Ïda1, a2, · · · , a2006¥ p؃Ó�êŒu45. e¡�E˜‡~f, `²46´Œ±���. �p1, p2, · · · , p46Ǒ46‡p؃Ó�Ÿê, �Ea1, a2, · · · , a2006Xe: p1, p1; p2, p1; p3, p2, p3, p1; p4, p3, p4, p2, p4, p1; · · · · · · pk, pk−1, pk, pk−2, · · · , pk, p2, pk, p1; · · · · · · p45, p44, p45, p43, · · · , p45, p2, p45, p1; p46, p45, p46, p44, · · · , p46, p34. ù2006‡��ê÷v‡�. ¤±, a1, a2, · · · , a2006¥�k46‡ØÓ�ê. 10 21. F1 {¥�“õÞÆ” d˜ ÞÚ¶f|¤,z˜^¶fë�ü‡Þ.zve˜ê,Œ±Íä d,˜‡ÞA¤ëÑ�¤k�Þ, �´dÞAá=ј #�¶f镤k�5؆§ƒë�Þ(z ‡Þë˜^¶f) . kr“õÞÆ” ÍǑü‡pØëÏ�Ü©, âŽÔ‘ §. Áéс��g, êN , ��é?Ûk100‡¶f�“õÞÆ” , –õ‡vØõuNê, Ҍ±Ô‘§. J« ©n«œ¹: (1) 3,‡ÞA, §–õ†Ù�10‡Þƒë; (2) 3,‡ÞB, §–õ†Ù�9‡ÞØë; (3)éu¤k�Þ,þ–�†Ù�11‡Þƒë, ¿–�†Ù�10‡ÞØë. dž–�k 1 2 ×22×11 = 121 > 100^¶, gñ. 22. 3˜�[ÌK8¥k10Üì¡, zÜì¡þÑk3‡I<, Õ3†>�<´¥m<�Æf, m >�<´¥m<�Š73. ®10Üì¡þ¥m�10‡�10‡ ÆfpØÓ|, =©Oáu10|. d , ù15‡<¥�©p�<�,áu,˜|. Ïd, 15‡<–� ‡©¤11|. ,˜¡, ©O Xeü«a.�|: (1) |¥z<Ñ3,Üì¡�¥m ˜ÑyL; (2) |¥–�k1‡�<´¥m<�Š73, ¤±ì¡¥m�<¤3�|–�k2‡<. q¥m  ˜k10‡<, �(1) a|–õ5|. ÏǑ10Üì¡þ�15‡<, Ù¥l™3ì¡¥m ˜Ñy�<k5‡, ¤±(2) a|Ǒ–õ5|, l 15‡<–õŒ©¤10|, gñ. nþ¤ã, 10Üì¡þ�k16‡ØÓ�<. 23. ò>Ǒ��êm, n�Ý/y©¤eZ>þǑ��ê��/, z‡�/�>þ²1uÝ /�ƒA>, Á�ù �/>ƒÚ��Š. )‰ P¤��ŠǑf(m,n), Œ±y² f(m,n) = m+ n− (m,n). (∗) 11 Ù¥, (m,n)L«mÚn�Œú�ê. ¯¢þ, ؔ�m ≥ n. (1) ém8B. Œy²3˜«Ü�K¿�©{, �¤��/>ƒÚTǑm + n− (m,n). �m = 1ž, ·Kw,¤á. b��m ≤ kž, (ؤá(k ≥ 1) . �m = k + 1ž, en = k + 1, K·K¤á. en < k + 1, l Ý/ABCD¥ƒ��/AA1D1D, d8Bb�, Ý/A1BCD1k˜«©{��¤��/>ƒÚ TǑm − n + n − (m − n, n) = m − (m,n). u´�Ý/ABCDk˜«©{��¤��/>ƒÚ Ǒm+ n− (m,n). (2) ém8BŒy(∗)¤á. �m = 1ž, dun = 1, w,f(m,n) = 1 = m + n − (m,n). b��m ≤ kž, é?¿1 ≤ n ≤ m, kf(m,n) = m + n − (m,n). em = k + 1, �n = k + 1ž, w,f(m,n) = k + 1 = m + n − (m,n). �1 ≤ n ≤ kž,�Ý/U‡�©¤ p‡�/,Ù>©OǑa1, a2, · · · , ap,ؔ�a1 ≥ a2 ≥ · · · ≥ ap. w,a1 = n½a1 < n. ea1 < n,K3AD†BCƒm�†AD²1�?˜†‚–�BLü‡©¤��/(½Ù>.) ,u ´a1 + a2 + · · ·+ apØ�uAB†CDƒÚ. Ïd, a1 + a2 + · · ·+ ap ≥ 2m > m+ n− (m,n). ea1 = n,K˜‡>©OǑn−mÚn�Ý/ŒUK8‡�©¤>©OǑa2 , a3, · · · , ap��/. d8Bb�, a2+a3+· · ·+ap ≥ m−n+n−(m−n, n) = m−(m,n),l a1+a2+· · ·+ap ≥ m+n−(m,n). u´, �m = k + 1ž, f(m,n) ≥ m+ n− (m,n). 2d(1) Œ, f(m,n) = m+ n− (m,n). 24. 3100×25�/L‚¥z˜‚¥W\˜‡šK¢ê,1i11j�¥W\�êǑxi,j (i = 1, 2, · · · , 100; j = 1, 2, · · · , 25) (XL1) . ,�òL1z�¥�êUdŒ���gSlþ�e­#ü�Ǒ x′1,j ≥ x′2,j ≥ · · · ≥ x′100,j (j = 1, 2, · · · , 25). (XL2) �����êk, ��‡L1¥W\�ê÷v 25∑ j=1 xi,j ≤ 1 (i = 1, 2, · · · , 100), K�i ≥ kž, 3L2¥ÒU�y 25∑ j=1 x′i,j ≤ 1¤á. L1 x1,1 x1,2 · · · x1,25 x2,1 x2,2 · · · x2,25 ... ... ... x100,1 x100,2 · · · x100,25 L2 x′1,1 x ′ 1,2 · · · x′1,25 x′2,1 x ′ 2,2 · · · x′2,25 ... ... ... x′100,1 x ′ 100,2 · · · x′100,25 )‰ k��ŠǑ97. (1) � xi,j =  0, 4(j − 1) + 1 ≤ i ≤ 4j,1 24 , Ù{�i, (j = 1, 2, · · · , 25). ùž, 25∑ i=1 xi,j = 0 + 24× 1 24 (i = 1, 2, · · · , 100) , ÷vK�^‡, ­ü�k x′i,j =  1 24 , 1 ≤ i ≤ 96, 0, 97 ≤ i ≤ 100, (j = 1, 2, · · · , 25). 12 ùž, 25∑ i=1 x′ij = 25× 1 24 > 1 (1 ≤ i ≤ 96) . Ïdk��ŠØ�u97. (2)Äky²: L1¥7k˜1(�Ǒ1r1)�¤kêxr,1, xr,2, · · · , xr,2573­ü�¤�L2� 971 Ñy. ¯¢þ,eþã(Øؤá,KL1�z˜1¥–�k˜‡êØ3L2� 971Ñy, =L2� 971 ¥–õkL1¥�100× 24 = 2400‡ê. ù†L2� 971�k25× 97 = 2425‡êgñ. Ùg, d­ü‡�L2¥�z��êlþ�e´dlŒ��ü��, K�i ≥ 97ž, x′i,j ≤ x′97,j ≤ xr,j , j = 1, 2, · · · , 25. Ïd, �i ≥ 97ž, 25∑ j=1 x′i,j ≤ 25∑ j=1 xr,j ≤ 1. nþŒk��ŠǑ97. 25. (1) 3˜gk2n+ 1‡èë\�'m¥, z‡èцÙ��è?1 ˜|'m, …z|'m7k ˜‡è‘Ñ. XJA‘B, B‘C, Cq‘ A, K¡3‡è|¤�8Ü{A,B,C}Ǒ“̂�” . �̂�8Ü�‡ê�ŒŠ†�Š. (2) 14