2011组合问题选讲

2011�Ip¥êÆémÀeùŒ |ܯKÀù o d liqian.jmtlf@gmail.com 2011.10 1. �mÚn´‰½�Œu1��ê, a1 < a2 < · · · < amÑ´�ê. y²: 3�ê8�˜‡f8T , Ù �ƒ‡ê |T | ≤ 1 + am − a1 2n+ 1 , …éz‡i ∈ {1, 2, · · · ,m}, þkt ∈ T9s ∈ [−n, n], ��ai = t+ s. )‰ -a1 = a, am = b, Š‘{Ø{b − a = (2n + 1)q + r, Ù¥q, r ∈ Z…0 ≤ r ≤ 2n. �T = {a+ n+ (2n+ 1)k|k = 0, 1, · · · , q}, K|T | = q + 1 ≤ 1 + b− a 2n+ 1 , …8Ü B = {t+ s|t ∈ T, s = −n,−n+ 1, · · · , n} = {a, a+ 1, · · · , a+ (2n+ 1)q + 2n}. 5¿�a+ (2n+ 1)q + 2n ≥ a+ (2n+ 1)q + r = b, Ïdz‡aiþ3B¥, l (ؤá. 2. ‰½�ên ≥ 3. y²: 8ÜX = {1, 2, 3, · · · , n2 − n}U�¤ü‡Øƒ��š˜f8�¿, ��z ˜‡f8þ؝¹n‡�ƒa1, a2, · · · , an, a1 < a2 < · · · < an, ÷vak ≤ ak−1 + ak+1 2 , k = 2, · · · , n− 1. )‰ ½ÂSk = {k 2 − k + 1, · · · , k2}, Tk = {k 2 + 1, · · · , k2 + k}, k = 1, 2, · · · , n− 1. -S = n−1⋃ k=1 Sk, T = n−1⋃ k=1 Tk. e¡y²SÚT=Ǒ÷vK8‡��ü‡f8. ÄkS ∩ T = Ø, …S ∪ T = X . Ùg, XJS¥3n‡�ƒa1, a2, · · · , an, a1 < a2 < · · · < an, ÷v ak ≤ ak−1 + ak+1 2 , k = 2, · · · , n− 1. K ak − ak−1 ≤ ak+1 − ak, k = 2, · · · , n− 1. (∗) ؔ�a1 ∈ Si, d|Sn−1| < n, Œi < n − 1. a1, a2, · · · , anùn‡ê–�kn − |Si| = n − i‡ 3Si+1 ∪ · · · ∪ Sn−1¥. ŠâÄT�K, 7k,‡Sj (i < j < n) ¥¹kÙ¥–�ü‡ê, �Ù¥ ��˜‡Ǒak, Kak, ak+1 ∈ Sj , ak−1 ∈ S1 ∪ · · · ∪ Sj−1. u´ak+1 − ak ≤ |Sj| − 1 = j − 1, ak − ak−1 ≥ |Tj−1| + 1 = j. l ak+1 − ak < ak − ak−1, †(∗)gñ. =S¥Ø3n‡�ƒ÷vK¥b �. Ón, T¥½Ø3ù��n‡�ƒ. ùL²SÚT=Ǒ÷vK¥‡��ü‡f8. 3. X㤫,�/�Y³�©Ǒ2n (n ≥ 5)‡“‚f” . ·‚rkú�…p(ú�>½ú�l) �“‚ f” ¡Ǒƒ��, l , z‡“‚f” Ñkn‡�‚. 1 Y³¥˜�a\ 4n + 1“�, “�JuS·�?. ‡,‡“‚f” ¥kØ�un“�, � o, ´�˜½¬kÙ¥n©OӞa n‡ØÓ�‚. y²: ‡²L˜ãžmƒ�, “�B¬3Y ³¥Œ—©Ùþ!. 5: ¤¢Œ—©Ùþ!, Ò´?�Ù¥˜‡“‚f” ,½ö§p¡k“�, ½ö§�n‡�‚pÑk “�. )‰·‚r˜‡‚f¥Ñy˜gn“�Ӟ©Oa•n‡�‚�¯‡¡ǑT‚fu)˜g“� u” ;r˜‡‚f½ö´§p¡k“�,½ö´§�n‡ƒ��‚fp¡Ñk“�,¡ǑT‚f?u“² ïG�” . ´wÑ,˜‡‚f‡˜�k“�a\, �o,§Ò˜†?u“²ïG�” . ¯¢þ, ‡Øu)“� u” , �o, T‚f¥�“�جÄ, §�,?u“²ïG�” ; XJu)“�u” , �o, §�n‡� ‚¥ÒÑk“�, ¿…‡n‡�‚ÑØu)“�u” , §Ò˜†?u“²ïG�” ; ØØ=‡�‚u )“�u” , Ѭk“�a�§p¡, §p¡Ò˜½k“�, ¤±, §˜†?u“²ïG�” . ù�˜5, Ǒy²K¥äó, ‡y²: ?ۘ‡‚fÑ´�¬k“�a\. ?�˜‡‚f, r§¡ǑA‚, r§¤3�÷/¡Ǒ1Ò÷/, rT÷/¥�,˜‡‚f¡ǑB‚. ·‚‡y², A‚´�¬k“�a\. Xã, U^ž�•gòÙ{÷/�X?Ǒ2–nÒ. Äky², 1Ò÷/´�¬k“�a\. b�1Ò÷/¥[�“�a\,�o,Òجk“��L1Ò ÷/†nÒ÷/ƒm�…p. ·‚5 “�¤3�÷/?Ò�²Ú. duvk“�a\1Ò÷/( Ùvk“��L1Ò÷/†nÒ÷/ƒm�…p) ,¤±, U´kn“�d,‡k (3 ≤ k ≤ n− 1)Ò ÷/©Oa\k − 1, kÚk + 1Ò÷/ˆ˜. Ïd, ²Ú�CzþǑ (k − 1)2 + k2 + (k + 1)2 − 3k2 = 2, =O\2. ˜¡, du“��aÄجʎ(ÏǑok˜‡‚fpkØ�un“�) , ¤±, ² Ú�O\ª³Ø¬ÊŽ; ,˜¡, “�¤3÷/?Ò�²Ú،U[�Ž¸/O\e�(جŒ u(4n+ 1)n2) , dd�)gñ. ¤±, ´�¬k“��L1Ò÷/†nÒ÷/ƒm�…p, a\1Ò÷/. ·‚25y², 1Ò÷/´�¬kn“�a\. XJ1Ò÷/¥–õkü“�a\, �o, §‚ Ñجar, ¿…g©–ªþ㲐ږõkügC�(U3ü“��L1Ò÷/†nÒ÷/ƒm �…pžC�) , ±�B˜†±YØä/þ,, l , q­�fâ�gñ. 2 ¤±, 1Ò÷/´�¬kn“�a\. XJùn“�¥k uA‚�, KA‚¥®²k“�a\; XJùn“��Ñ uB‚, KB‚ ´�¬u)“�u” . l , k“�a\A‚. 4. ,úiI‡¹^˜¶“Ö, �k10<�¶, úi²nû½Uì�…�¶�^SŇ¡Á, 3‡< ¡Á�˜½Ø¹^. g14‡ A2 > · · · > A8 = A9 = A10; (2) Túik‡L70%�ŒU5¹��Uår�3‡<ƒ˜, k؇L10%�ŒU5¹^� Uåf�3‡<ƒ˜. )‰˜ ò 3‡¡Áö¥Uår�ü¶¶gPǑa. w,, a ≤ 8. òdžUåü¶1k�<�À þ�ü��8ÜPǑAk(a), ƒA�ü�ê8PǑ|Ak(a)|. (1)´,�a = 1ž,7,˜L ¡9‡<,¹^�˜‡¡Á�<,džØUå11�<ƒ ,Ù{ ˆ<Ŭþ�, ØJŽ� |Ak(1)| = 3× 8! := r1, k = 2, 3, · · · , 10. Ù¥, “:=” L«“PǑ” . �2 ≤ a ≤ 8, éuUåü¶1k�<�¹^Ŭ. éu1 ≤ k < a, džÅ¬þ�. ¯¢þ, džUåü¶1a�<ü3 n‡, k3«ÀJ ˜�{. Uåü¶11–1a− 1�< Ñü3�7‡ ˜þ, ¿…X u�‚ƒÄÒ´X�¹^, kü{Ca−17 (a− 2)!«; Ù¥10 − a‡<Œ± 3e� ˜þ?¿ü�, k(10− a)!«ü{. �k |Ak(a)| = { 3Ca−17 (a− 2)!(10− a)! := ra, k = 1, 2, · · · , a− 1; 0, k = a, · · · , 10. þã(JL² A8 = A9 = A10 = r1 = 3× 8! > 0; (1) Ak = r1 + 8∑ a=k+1 ra, k = 2, 3, · · · , 7; (2) A1 = 8∑ a=2 ra. (3) dª(1) Ú(2)  A2 > A3 > · · · > A8 = A9 = A10 > 0; dª(2) Ú(3)  A1 −A2 = r2 − r1 = 3× 7× 8!− 3× 8! > 0. nþ¤ã, ¯K(1) ¼y. (2) dª(1)  A8 +A9 +A10 10! = 3r1 10! = 3× 3× 8! 10! = 10%, 3 =¹^�Uåf�n<ƒ˜�ŒU5�u10%. dª(2) Ú(3) Œ A1 = 8∑ a=2 ra = 8∑ a=2 3Ca−17 (a− 2)!(10− a)! = 3× 7! 8∑ a=2 (9− a)(10− a) a− 1 = 3× 7! 7∑ s=1 (8 − s)(9− s) s = 3× 7!× ( 56 + 21 + 10 + 5 + 12 5 + 1 + 2 7 ) = 3× 7!× 95 24 35 > 3× 7!× 95 2 3 = 287× 7!; A2 = r1 + 8∑ a=3 ra = 3× 8! + 3× 7!× ( 21 + 10 + 5 + 12 5 + 1 + 2 7 ) = 3× 7!× 47 24 35 > 3× 7!× 47 2 3 = 143× 7!; A3 = r1 + 8∑ a=4 ra = 3× 8! + 3× 7!× ( 10 + 5 + 12 5 + 1 + 2 7 ) = 3× 7!× 26 24 35 > 3× 7!× 26 2 3 = 80× 7!. ¤± A1 +A2 +A3 10! > 287 + 143 + 80 720 = 510 720 = 17 24 > 70%. =¹^�Uår�n<ƒ˜�ŒU5Œu70%. )‰� ±SkL«UåǑ1k�<(=ak) �¹^�¤kØÓ�¶^S�8Ü. KAk = |Sk|. N´w Ñ, éuk = 8, 9, 10,3áuSk�?ۘ«�¶^S¥, a1˜½‡3 3‡<¥, ak˜½´�˜‡� ¶ö. ¤±, Ak = |Sk| = 3× 8!, k = 8, 9, 10. �k = 2, 3, · · · , 7ž, 3?ÛáuSk�ü�α¥, akŒU´�˜‡�¶ö(dža173 3‡<¥) , akǑŒUØ´�˜‡�¶ö(džak−173Ù�) . ��ØÛ«œ¹, ‡�†α¥ak†ak−1�  ˜,¤��ü�α˜ÑáuSk−1,…éuØÓ�α ∈ Sk,¤���ü�α˜Ǒ؃Ó.ù��í�éuk = 8Ǒ ¤á. u´By� Ak−1 = |Sk−1| ≥ |Sk| = Ak, k = 2, 3, · · · , 8. ,˜¡, 5¿�þãN�Ø´dSk�Sk−1�÷�, ~Xα˜ := (a8, a9, ak, ak−1, · · · ) ∈ Sk−1, � ´α := (a8, a9, ak−1, ak, · · · ) 6∈ Sk, âd� Ak−1 = |Sk−1| > |Sk| = Ak, k = 2, 3, · · · , 8. –d, 1˜�K¼y. Ǒ )‰1��K,ELOŽA1, A2, A3. éAk�OŽØ ¡0��{ƒ ,„k˜«{Š �Ä: 4 EòUåü¶1k� 4a1. l , u = 6a1½12a1. �A = {a1, 5a2, 7a1, 11a1}½A = {a1, 11a1, 19a1, 29a1}. N´�y, þã�8Üþka1 + a2, a1 + a3, a1 + a4, a2 + a3Ñ�ØsA. nþ, �nAˆ�ŒŠ4ž, A = {a, 5a, 7a, 11a}½A = {a, 11a, 19a, 29a},Ù¥, a´��ê. 6. �X = {1, 2, · · · , 2001}. ����êm,·Ü‡�: éX�?ۘ‡m�f8W ,Ñ3u, v ∈ W (uÚvŒ±ƒÓ) , ��u+ v´2�˜. )‰ òX©¤±e5‡f8?1 : 2001 = 1024 + 977 ≥ x ≥ 1024− 977 = 47, 46 = 32 + 14 ≥ x ≥ 32− 14 = 18, 17 = 16 + 1 ≥ x ≥ 16− 1 ≥ 15, 14 = 8 + 6 ≥ x ≥ 8− 6 = 2, x = 1. Ǒ �E˜‡�K¥‡�Ø�÷v…q¹�ƒõ�~f, ù‡f8ØU¹2�?˜˜…zé ê{2r + a, 2r − a}¥Uk˜‡¹38¥. -Y = {2001, 2000, · · · , 1025} ∪ {46, 45, · · · , 33} ∪ {17} ∪ {14, 13, · · · , 9}, Kk|Y | = 998, …é?Ûu, v ∈ Y , u + vÑØ´2�˜. ¯¢þ, �u, v ∈ Yž, ؔ �u ≥ v…k2r < u ≤ 2r + a < 2r+1, Ù¥�r©O�Š10, 5, 4, 3ž, ƒA�aŠgǑ977, 14, 1, 6. (1) e2r < v ≤ u, K2r+1 < u+ v < 2r+2, u+ vØU´2�˜; (2) e1 ≤ v < 2r, K�2r < u ≤ 2r + a, 1 ≤ a < 2rž, 1 ≤ v ≤ 2r − a. u´2r < u+ v < 2r+1, ùL ²u+vǑØU´2�˜.¤±,f8Y¥?ÛüêƒÚÑØ´2�˜.�¤�����êm ≥ 999. 5 òXy©¤e�999‡p؃��f8: Ai = {1024 − i, 1024 + i}, i = 1, 2, · · · , 977; Bj = {32 − j, 32 + j}, j = 1, 2, · · · , 14; C = {15, 17}; Dk = {8− k, 8 + k}, k = 1, 2, · · · , 6; E = {1, 8, 16, 32, 1024}. éuS�?ۘ‡999�f8W ,eW ∩E 6= Ø,KlÙ¥?�˜‡�ƒ�2�Ñ´2�˜;eW ∩ E = Ø, KW¥�999‡�ƒ©áu ¡998‡2�f8. dÄT�KW¥7kØÓ�uÚv, áuӘ f8. w,, u+ vǑ2�˜. nþŒ, ¤�����êm = 999. 7. �n´‰½���ê, 8ÜS = {1, 2, · · · , n}, 隘�k¢ê8ÜAÚB, �|A∆S| + |B∆S| + |C∆S|��Š, Ù¥C = {a+ b|a ∈ A, b ∈ B}, X∆Y = {x|xT�áuXÚY¥�˜‡}, |X |L«k 8ÜX��ƒ‡ê. )‰ ¤���Š´n + 1. Äk, �A = B = S, Œ|A∆S|+ |B∆S|+ |C∆S| = n+ 1. e¡y²: l = |A∆S|+ |B∆S|+ |C∆S| ≥ n+ 1. PX\Y = {x|x ∈ X, x 6∈ Y }. w,, l = |A\S| + |B\S|+ |C\S|+ |S\A|+ |S\B|+ |S\C|. u´,  ‡y²: (I) |A\S|+ |B\S|+ |S\C| ≥ 1; (II) |C\S|+ |S\A|+ |S\B| ≥ n. ky(I) . ¯¢þ, e|A\S| = |B\S| = 0, KA,B ⊆ S. �1،U´C¥�ƒ, =S\C| ≥ 1. 2y(II) .eA∩S = Ø,K|S\A| ≥ n,(ؤá. eA∩S 6= Ø,KA∩S��ƒ¥Œ�˜‡´n−k (0 ≤ k ≤ n− 1) . K |S\A| ≥ k. (1) ,˜¡, éi = k + 1, k + 2, · · · , n, ½öi 6∈ B (dži ∈ S\B) , ½öi ∈ B (džn − k + i ∈ C, =n− k + i ∈ C\S) . ¤±, |C\S|+ |S\B| ≥ n− k. (2) dª(1) (2) =�(II) . nþŒ, l ≥ n+ 1. ¤±, ¤��Š´n+ 1. 8. �T´d2004100�¤k�ê|¤�8Ü, S´T�˜‡f8, Ù¥vk˜‡ê´,˜‡ê��ê. �oSõ¹kõ�‡�ƒ? )‰ 5¿�2004 = 22 × 3× 167,KT = {2a3b167c|0 ≤ a ≤ 200, 0 ≤ b, c ≤ 100}. �S = {2200−b−c3b167c|0 ≤ b, c ≤ 100}. éu0 ≤ b, c ≤ 100, k0 ≤ 200 − b − c ≤ 200, ¤±, S¹ k1012‡�ƒ. e¡y²: S¥vk˜‡ê´,˜‡ê��ê. b�2200−b−c3b167c´2200−i−j3i167j��ê, K 200− b− c ≥ 200− i− j, b ≥ i, c ≥ j, =b = i, c = j. �S¥vk˜‡ê´,˜‡ê��ê. 2^‡y{y²: ÷v^‡�Sõ¹k1012‡�ƒ. �U´T�˜‡‡L1012‡�ƒ�f8. Ï Ǒk1012‡pÉ�(b, c), ¤±, dÄT�K7kü‡�ƒu1 = 2 a13b1167c1, u2 = 2 a23b2167c2� �b1 = b2, c1 = c2, a1 6= a2. u´, �a1 > a2ž, u1´u2��ê; �a1 < a2ž, u2´u1��ê. Ïd, f 8UØ÷v^‡. ¤±, Sõ¹k1012‡�ƒ. 6 9. A, B, CnI?1ŒÚ_�m,zè9<. 5KXe: z|düèˆÑ1<'m,‘öÅ_,Kö�= �, ¿d,˜è�1<�_. ÄkdA, Büèˆ�1 t+ 1, �o C3s − C 3 s−1 = C 2 s−1 > C 2 t = C 3 t+1 − C 3 t , =C3s +C 3 t > C 3 s−1 +C 3 t+1. Šâ±þ&?, ÏLN�{Œ±ä½ |S| = n∑ i=1 C3si ≥ nC 3 m, Ù¥, m = 1 n n∑ i=1 si = 1 n (C2n − n) = n− 3 2 . âd�OpÏoÕ|oê�þ.ǑC4n − |S| − |T | ≤ C 4 n − nC 3 m. (iii) XJU�O˜‡Y, ��SaØpÏoÕ|�ê8ü��ŠnC3m , ¿…TaØpÏ�oÕ |�ê8Ǒü��(¢Sü�0) , �o, TY�pÏoÕ|�ê8ˆ�Œ. 7 Ǒd8�,Äkò?ÒǑ1, 2, · · · , n�˜m՝^ž�gSSü3˜‡�±þ. e¡ò‰Ñ÷v‡ ��ü«Y. 1˜Y Äkò÷�±ƒ��˜mÕéƒm�Ï�½ǑÌZ�, ù��½ n^ÌZ�: {1, 2}, {2, 3}, · · · , {n− 1, n}, {n, 1}. éui, j ∈ {1, 2, · · · , n}, i 6= j, XJ�±þ÷^ž�•li�j�l²LÛꇥmÕ, �o, 5½iÒ Õ†jÒՃm�Ï�Ǒi→ jü1�. ÏǑn´Ûê, lk�l�^ž��lÚll�k�^ž��l�¥, Tk˜^²LÛꇥmÕ, ¤±þãü1�½Ø¬�—gñœ/. UìdY, lz‡˜mÕuÑ�ü1�ÑǑm = n− 3 2 ^. Ïd, SaØpÏoÕ|oêü– �|S| = nC3m. e¡òÑ, UìdYk|T | = 0. XJoÕ|¥,üÕmkÌZ�ƒë, �o, oÕ|¥Ù{?˜ÕцùüÕpÏ. Ïd, ù�� oÕ|ǑpÏoÕ|.  loÕ|�,nÕ�e˜Õ�n^Ï�Ñü•Ï ùe˜Õ�œ/. �3Ø�e˜ ÕD��±þ, ¤ã�nÕU^ž�•gǑA, B, C. ÏǑA→ D, B → D, C → D, ŠâY�ü 15½Œ±�äA†BƒmÚA†Dƒm�^ž��lþˆ²LÛꇥmÕ. ·‚�²Ï�A → B, A→ C, A→ D�ü1•, Ïd, ù��ØpÏ�oÕ|{A,B,C,D}A8\Sa. Šâ±þ�?Ø, Œ±ä½|T | = 0. �, ŽÑpÏoÕ|ê�ŒŠ C4n − nC 3 m = n(n− 3) 48 (n2 + 6n− 31). éun = 99, pÏoÕ|�ê8�ŒŠǑ99 × 96 48 × (9801 + 594− 31) = 99× 2× 10364 = 2052072. 1�Y (Ó�kò?ÒǑ1, 2, · · · , n�˜mÕU^ž�gSSüu˜‡�±þ. ) XJlaҘmÕÚbҘmÕ�^ž��lT²L n− 3 2 ‡½ö n− 1 2 ‡¥mÕ,�o,5½a†bƒ m�Ï�ǑV1ÌZ�. XJliҘmÕ�jҘmÕ�^ž��l²L�¥mÕê�u n− 3 2 ,K5 ½i†jƒm�Ï�ü•liÏ j. UìdY, lz‡˜mÕuÑ�ü1Ï�êÑǑm = n− 3 2 ^. Ïd, SaØpÏ�oÕ|êü –�Š|S| = nC3m. UìdY, Ó�Œy|T | = 0. ¯¢þ, †1˜Yaq/�y?Ø, Œ±�½: XJoÕ|¥ küÕm�Ï�´ÌZ�, �o, ùoÕ|´pÏ�. „Œ±�½: XJloÕ|¥�,nÕ�e ˜ÕD�Ï�Ñü•Ï TÕ,�o, ùnÕ3Ø�D:��±þ^ž�•üÞ�˜ÕAÏ Ù�n ÕB, C, D�Ï�Ñü•uÑ: A→ B, A→ C, A→ D. Ïd, ùaoÕ|{A,B,C,D}A8\Sa. Ïd, UìdYïE��˜¢, vkTapÏoÕ|, ¿…pÏoÕ|êˆ�Œ. e�OŽ Ó1˜Y. 8