6.6南京学校集合讲义

êÆc�šŽùŒ êÆ¿m¥�8ܯK o d liqian.jmtlf@gmail.com 2010.06 8Ü¥�ƒ�5Ÿ 1. (1) �M = {1, 2, 3, · · · , 1995}, A´M�f8…÷v^‡: �x ∈ Až, 15x 6∈ A, KA¥�ƒ�‡ê õ´ . (2) ®A†B´8Ü{1, 2, 3, · · · , 100}�ü‡f8, ÷v: A†B��ƒ‡êƒÓ, …A ∩ B˜8. en ∈ Ažok2n+ 2 ∈ B, K8ÜA ∪B��ƒ‡êõ . )‰ (1) ·‚¦ŒU�Eј‡÷v^‡…¹�ƒõ�f8A, Ϗ‡¦�x ∈ Ažk15x 6∈ A, ‡15x > 1995, =x > 133, Œ„A¥Œ¹{134, 135, · · · , 1995}, ¿…‡¦�x ∈ Až15x 6∈ A,  ‡15x < 134, �AqŒ¹{1, 2, · · · , 8}. u´·‚�M�f8A = {1, 2, · · · , 8} ∪ {134, 135, · · · , 1995}, §÷vK8^‡, ùžA¥�ƒ‡ê|A| = 8 + (1995− 133) = 1870. ,˜¡, ?�M˜‡÷vK8^‡�f8A, Ϗx†15x (x = 9, 10, 11, · · · , 132, 133) ¥–�k ˜‡ØáuA, �A¥�ƒ‡ê|A| ≤ 1995− (133− 8) = 1870. nþA¥�ƒõk1870‡. (2) ky|A ∪B| ≤ 66, Iy|A| ≤ 33, dIyeA´{1, 2, · · · , 49}�?˜‡34�f8, K7 3n ∈ A, ¦�2n+ 2 ∈ A. y²Xe: ò{1, 2, · · · , 49}©¤Xe33‡8Ü: {1, 4}, {3, 8}, {5, 12}, · · · , {23, 48}�12‡; {2, 6}, {10, 22}, {14, 30}, {18, 38}�4‡; {25}, {27}, {29}, · · · , {49}�13‡; {26}, {34}, {42}, {46}�4‡.duA´{1, 2, · · · , 49}�34� f8, l dÄT�Kþã33‡8Ü¥–�k˜‡2�8Ü¥�êþáuA, =3n ∈ A, ¦�2n+ 2 ∈ A. X�A = {1, 3, 5, · · · , 23, 2, 10, 14, 18, 25, 27, 29, · · · , 49, 26, 34, 42, 46}, B = {2n+2|n ∈ A},KAÚB÷ vK�…|A ∪B| = 66. 2. òê8A = {a1, a2, · · · , an}¥¤k�ƒ�Žâ²þŠPP (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‡, Bi = {−i, i}, i = 1, 2, 3, 4, Ïdf(2) = 4. M ′�n�þïf8kü«œ¹: (1) ¹k�ƒ0�Bi ∪ {0} = {−i, 0, i}, i = 1, 2, 3, 4, �4‡; (2) ع�ƒ0�, du�ª3 = 1 + 2, 4 = 1 + 3Œ±L«−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. 1 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+3ŒL−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′´M ′�þïf8, �ö ˜˜éA. Ïd, f(9− k) = f(k), k = 1, 2, 3, 4. l M ′�þïf8‡ê 9∑ k=1 f(k) = f(9) + 2 4∑ k=1 f(k) = 1 + 2(1 + 4 + 8 + 12) = 51. =M�þï f8k51‡. 3. (1) ¯: UÄò8Ü{1, 2, · · · , 96}L«§�32‡n�f8�¿8, …z‡n�f8��ƒƒÚÑ ƒ�. (2) ¯: UÄò8Ü{1, 2, · · · , 99}L«§�33‡n�f8�¿8, …z‡n�f8��ƒƒÚÑ ƒ�. )‰ (1) ØU. Ϗ 1 + 2 + · · ·+ 96 = 96× (96 + 1) 2 = 48× 97, 32 - 48× 97. (2) U. z‡n�f8��ƒÚ 1 + 2 + · · ·+ 99 33 = 99× (99 + 1) 33× 2 = 150. ò1, 2, · · · , 66zü‡˜|, ©¤33|, z|üêƒÚŒ±ü¤˜‡ú�1���ê�: 1 + 50, 3 + 49, · · · , 33 + 24, 2 + 66, 4 + 65, · · · , 32 + 51. �Xe33|ê: {1, 50, 99}, {3, 49, 98}, · · · , {33, 34, 83}, {2, 66, 82}, {4, 65, 81}, · · · , {32, 51, 67}. 4. �A = {1, 2, · · · , 2002}, M = {1001, 2003, 3005}. éA�?¿š˜f8B,�B¥?¿üêƒÚØá uMž, ¡BM–gd8. XJA = A1 ∪ A2, A1 ∩ A2 = Ø, …A1ÚA2þM–gd8, @o, ¡kS é(A1, A2)A�˜‡M–y©. Á¦A�¤kM–y©�‡ê. )‰ ém,n ∈ A, em+ n = 1001½2003½3005, K¡m†n “k'” . ´†1k'�ê=k1000Ú2002,†1001Ú2002k'�êÑ´1Ú1003,†1003k'�1000Ú2002. ¤±, 1, 1003, 1000, 10027L©Oü|{1, 1003}, {1000, 2002}. ÓnŒy©Ù¦ˆ|{2, 1004}, {999, 2001}; {3, 1005}, {998, 2000}; · · · ; {500, 1502}, {501, 1503}; {1001}, {1002}. ù�A¥�2002‡ê�y©¤501é, �1002|. du?¿ê†…†éA�,˜|k', ¤±, e˜é¥˜|3A1¥, ,˜|73A2¥. ‡ƒ½ ,, …A1†A2¥Ø2kk'�ê. �A�M–y©�‡ê2501. 5. �M´kê8, e®M�?Ûn‡�ƒ¥o3ü‡ê, §‚�ÚáuM , Á¯M¥õk õ�‡ê? )‰ ¤¦M¥�ƒ‡ê�ŒŠ7. 2 e¡y²Tê8¥–õk7‡�ƒ. Äky²Tê8¥õk3‡�ê. b�ŒUkØ�u4‡� �ƒ, Ù¥Œ�4‡ê©Oa1, 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. 6. �n (n ≥ 2) ´��ê, S´{1, 2, · · · , n}�˜‡f8, …S¥Ø3ù��êé: Ù¥˜‡ê½ö U�,˜‡ê�Ø, ½ö†,˜‡êpŸ. ¯8ÜSŒU¹��ƒõkõ�? )‰ �EN� f : S → T = {[n 2 ] + 1, [n 2 ] + 2, · · · , n } , é?¿�x ∈ S, k f(x) = 2kx ∈ (n 2 , n ] , Ù¥, k´šK�ê. w,, T¥?˜‡�ƒØU�Ø,˜‡�ƒ. ´f´ü�. ¯¢þ,e3šK�êk1Úk2,¦�2k1x = 2k2y,ؔ�k1 < k2,Ky | x,†S�½ Âgñ. qé?¿x, y ∈ S, k(x, y) > 1. u´, dx | f(x), y | f(y), �(f(x), f(y)) > 1. l , S¥Ø¹ kü‡ëY��ê. Ïd, |S| ≤  n2 + 1 2  = [n+ 2 4 ] . ,˜¡, �S = { k ∣∣∣k´óê, k > n 2 } ž, S÷vK�‡¦. nþ¤ã, |S|�ŒŠ [ n+ 2 4 ] . 7. �S´2002�8, N�ê, ÷v0 ≤ N ≤ 22002. y²: ŒòS�¤kf8/þçÚ½xÚ, ¦�e �^‡¤á: (a) ?ü‡xÚf8�¿8´x�; (b) ?ü‡çÚf8�¿8´ç�; (c) TÐ3N‡xÚ�f8. )‰ ÄS = Sn¥kn‡�ƒ�˜„œ/, ùžN÷v0 ≤ N ≤ 2n��ê, ¿…�Sn = {a1, a2, · · · , an}, énA^êÆ8B{y². �n = 1ž, eN = 0, KòØ9{a1}Ñ/¤çÚ, ÎÜK8‡¦; eN = 1, KòØ/¤çÚ, {a1}/ ¤xÚ, ÎÜK8‡¦; eN = 2, KòØ, {a1}Ñ/¤xÚ, ÎÜK8‡¦. �én�8ÜSn9�ê0 ≤ N ≤ 2n, 3÷vK¥n‡^‡�/ڐ{. Än + 1�8Sn+1 = Sn ∪ {an+1}. (i) e0 ≤ N ≤ 2n, Kd8Bb�, 3˜«/ڐYòSn�¤kf8/¤çÚ½xÚ¦�÷vK ¥�n‡^‡, ùž2òSn+1¥¤k¹an+1�f8�/¤çÚ, u´E÷vK8^‡. (ii) e2n < N ≤ 2n+1, ؔ�N = 2n + k (k = 1, 2, · · · , 2n) , Kd8Bb�3Sn�f8 �˜«/ڐ{¦�÷vK¥^‡(1) (2) …Tkk‡f8�/¤xÚ, 3Sn+1¥ò¹an+1�¤kf 8(�2n‡) /¤xÚ, u´K8^‡(1) (2) E,¤á, …˜�kN = 2n + k‡f8�/¤xÚ, =^ ‡(3) ÷v, ùÒ�¤ éSn�8By². AO/, �n = 2002B�K(ؤá. 8. ‰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. 3 )‰ ½ÂSk = {k2 − k + 1, · · · , k2}, Tk = {k2 + 1, · · · , k2 + k}, k = 1, 2, · · · , n− 1. -S = n−1⋃ k=1 Sk, T = n−1⋃ k=1 Tk. e¡y²SÚT=÷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¥ ��˜‡ak, 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=÷vK¥‡¦�ü‡f8. 9. �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, 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. 10. ¦¤k��ên (n ≥ 2) , ¦�3¢êa1, a2, · · · , an, ÷v {|ai − aj ||1 ≤ i < j ≤ n} = { 1, 2, · · · , n(n− 1) 2 } . )‰ a1, a2, · · · , ankXe5Ÿ: (i) a1, a2, · · · , anüüØ�; (ii) §‚��ýéŠüüØ�. u´, n = 2, a1 = 0, a2 = 1; n = 3, a1 = 0, a2 = 1, a3 = 3; n = 4, a1 = 0, a2 = 2, a3 = 5, a4 = 6. ey�n ≥ 5ž, Ø3a1, a2, · · · , an·Ü^‡. y{˜ -0 ≤ a1 < a2 < · · · < an, bi = ai+1 − ai, i = 1, 2, · · · , n − 1. K�i < jž, |ai − aj | = aj − ai = bi + bi+1 + · · ·+ bj−1, 1 ≤ i < j ≤ n. w,, max 1≤i n, i > 1. ®3Ii, bi+1 = 2. u ´, bi > n − 2. ¤±, bi = n − 1. ùíÑb3 = 2, b4 = n − 2. ùž, b1 + b2 = b3 + b4. �Ñgñ. ¤±, �n− 1 ≥ 4ž, =n ≥ 5žØ3a1, a2, · · · , an·ÜK�^‡. y{� -0 ≤ a1 < a2 < · · · < an, ´an = n(n− 1) 2 . ùž, 73,ü‡eIi < j, ¦�|ai − aj | = an − 1. ¤±, an − 1 = an−1 − a1 = an−1½an − 1 = an − a2, =a2 = 1. ¤±, Ñyan = n(n− 1) 2 , an−1 = an − 1, ½an = n(n− 1) 2 , a2 = 1. e¡©œ¹?Ø: (i) �an = n(n− 1) 2 , an−1 = an − 1. Äan − 2, kan − 2 = an−2½an − 2 = an − a2, =a2 = 2. �an−2 = an − 2, Kan−1 − an−2 = 1 = an − an−1. ù�Ñgñ. ¤±, ka2 = 2. Äan − 3, kan − 3 = an−2½an−3 = an − a3, =a3 = 3. �an−2 = an − 3, Kan−1 − an−2 = 2 = a2 − a0. ùíÑgñ. �a3 = 3, Kan − an−1 = 1 = a3 − a2, qíÑgñ. ¤±ù«œ/ØÑy, ^‡ an−2 = a2, =n = 4. ��n ≥ 5, Ø3. (ii) �an = n(n− 1) 2 , a2 = 1. Äan − 2, kan − 2 = an−1½an − 2 = an − a3, =a3 = 2. ùža3 − a2 = a2 − a1, íÑgñ. ¤ ±, an−1 = an − 2. Äan − 3, kan − 3 = an−2½an − 3 = an − a3, =a3 = 3. u´, a3 − a2 = an − an−1. gñ. Ï d, an−2 = an − 3. ¤±, an−1 − an−2 = 1 = a2 − a1. ùqgñ. ¤±kan−2 = a2, =n = 4. � �n ≥ 5ž, Ø3. y{n Ä1¼êxa1 + xa2 + · · ·+ xan . dK� (xa1 + xa2 + · · ·+ xan)(x−a1 + x−a2 + · · ·+ x−an) = n− 1 + x−n(n−1)2 + · · ·+ x−1 + 1 + x+ · · ·+ xn(n−1)2 = n− 1 + x n(n−1) 2 +1 − x−n(n−1)2 x− 1 . �x = e2iθ = cos 2θ + i sin 2θ, x 6= 1. de−iα = eiα, � |e2ia1θ + e2ia2θ + · · ·+ eianθ|2 = n− 1 + e 2iθein(n−1)θ − e−in(n−1)θ e2iθ − 1 = n− 1 + sin(n 2 − n+ 1)θ sin θ . �(n2 − n+ 1)θ = 3pi 2 , Kθ = 3pi 2(n2 − n+ 1) . �n ≥ 5ž, 0 < θ < 3pi 2(52 − 5 + 1) = pi 14 < pi 2 . ùž,sin θ < θ, sin(n2 − n+ 1)θ = −1, “\, � |e2ia1θ + e2ia2θ + · · ·+ eianθ|2 = n− 1− 1 sin θ < n− 1− 1 θ = n− 1− 2(n 2 − n+ 1) 3pi < n− 1− 2(n 2 − n) 3pi = (n− 1) ( 1− 2n 3pi ) ≤ (n− 1) ( 1− 10 3pi ) < 0. 5 ùÒ�Ñgñ. ¤±, �n ≥ 5ž, Ø3a1, a2, · · · , an. 11. �S = {1, 2, · · · , 50}. ¦����êk, ¦S�?˜k�f8¥Ñ3ü‡ØÓ�êaÚb, ÷ v(a+ b) | ab. )‰ éu÷v^‡(a + b) | ab�a, b ∈ S, Pc = (a, b), u´a = ca1, b = cb1, Ù¥a1, b1 ∈ N+…(a1, b1) = 1. Ï k c(a1 + b1) = (a+ b) | ab = c2a1b1, =(a1 + b1) | ca1b1. Ϗ(a1 + b1, a1) = (a1 + b1, b1) = (a1, b1) = 1, ¤± (a1 + b1) | c. (1) qϏa, b ∈ S, ¤±a + b ≤ 99, =c(a1 + b1) ≤ 99. d(1)3 ≤ a1 + b1 ≤ 9. ddŒ, S÷v^ ‡(a+ b) | ab�¤kêéXe: a1 + b1 = 3 : (6, 3), (12, 6), (18, 9), (24, 12), (30, 15), (36, 18), (42, 21), (48, 24); a1 + b1 = 4 : (12, 4), (24, 8), (36, 12), (48, 16); a1 + b1 = 5 : (20, 5), (40, 10), (15, 10), (30, 20), (45, 30); a1 + b1 = 6 : (30, 6); a1 + b1 = 7 : (42, 7), (35, 14), (28, 21); a1 + b1 = 8 : (40, 24); a1 + b1 = 9 : (45, 36). �k23é. -M = {6, 12, 15, 18, 20, 21, 24, 35, 40, 42, 45, 48}, K|M | = 12…þã23ˆêé¥z‡êé¥Ñ–� ¹kM¥�˜‡�ƒ.Ïd,e-T = S−M ,K|T | = 38…T¥�?ÛüêÑØ÷vK¥�‡¦. ¤±, ¤¦����êk ≥ 39. 5¿�e�12‡êé (6, 3), (12, 4), (20, 5), (42, 7), (24, 8), (18, 9), (40, 10), (35, 14), (30, 15), (48, 16), (28, 21), (45, 36) p؃�…Ñ÷vK¥�‡¦. ¤±,éuS�?˜39�f8,§'S�11‡�ƒ, ù11‡�ƒ–õ áuþã12‡êé¥�11é, l 7k12é¥�˜éáuù‡39�f8. nþŒ, ¤¦����êk = 39. 8Ü�$Ž 12. �S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A1, A2, · · · , Ak´S�f8, ÷v (1) |Ai| = 5, i = 1, 2, · · · , k; (2) |Ai ∩Aj | ≤ 2, 1 ≤ i < j ≤ k. ¦k�ŒŠ. )‰˜ Š10× k�L‚, Ù¥1i1, 1j�?��ƒ aij = { 1, ei ∈ Aj , 0, ei 6∈ Aj . i = 1, 2, · · · , 10; j = 1, 2, · · · , k. 6 u´L¥1i1��ƒƒÚli = k∑ j=1 aijL«iáuA1, A2, · · · , Ak¥li‡8Ü, 1j��ƒƒÚ 10∑ i=1 aij = |Aj |L«8ÜAj¥�ƒ�‡ê, d®^‡(1) k 10∑ i=1 aij = |Aj | = 5, ¤± 10∑ k=1 li = 10∑ i=1 k∑ j=1 aij = k∑ j=1 10∑ i=1 aij = k∑ j=1 |Aj | = 5k. (1) er ∈ Ai∩Aj ,Kò{Ai, Aj , r}|¤3�|,ù«n�|�‡êPS. ˜¡,ÏráuAi, A2, · · · , Ak¥ �lr‡8Ü, Œ/¤C2lr‡¹r�n�|, ¤±S = 10∑ r=1 C2lr . ,˜¡, é?¿Ai, Aj (1 ≤ i < j ≤ k) k|Ai ∩Aj |‡�ƒáuAi ∩Aj , Œ/¤|Ai ∩Aj |‡¹AiÚAj�n�|, ¤±S = ∑ 1≤i 90, Ã{¦(3) ¤á. ¤±¦^‡(1) – (3) ¤á, S¥�z‡êі�áu7‡f8. ù�˜5, 7kn ≥ 15. ^i;ü�{Œ±�Ñ÷vK¥‡¦�15‡7�f8Xe: {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 8, 9, 10, 11}, {1, 2, 3, 12, 13, 14, 15}, {1, 4, 5, 8, 9, 12, 13}, {1, 4, 5, 10, 11, 14, 15}, {1, 6, 7, 8, 9, 14, 15}, {1, 6, 7, 10, 11, 12, 13}, {2, 4, 6, 8, 10, 12, 14}, {2, 4, 6, 9, 11, 13, 15}, {2, 5, 7, 8, 10, 13, 15}, {2, 5, 7, 9, 11, 12, 14}, {3, 4, 7, 8, 11, 12, 15}, {3, 4, 7, 9, 10, 13, 14}, {3, 5, 6, 8, 11, 13, 14}, {3, 5, 6, 9, 10, 12, 15}. nþŒ, f8‡ên��Š15. 15. �A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, · · · , n}, 3A¥?�3‡ê, 3B¥�ü‡ê, |¤¹k5‡�ƒ �8ÜAi (i = 1, 2, 3, · · · , 20), ¦�|Ai ∩Aj | ≤ 2, 1 ≤ i < j ≤ 20. ¦n��Š. )‰ n��Š´16. �B¥z‡ê3¤kAi¥õ­EÑykg, 7kk ≤ 4. eØ,, êmÑykg, k > 4, 3k > 12, 3mÑy�¤kAi¥,–�k˜‡A�êÑy3g. ؔ�§´1,Òk8Ü{1, a1, a2,m, b1}, {1, a3, a4,m, b2}, {1, a5, a6,m, b3}, Ù¥ai ∈ A, 1 ≤ i ≤ 6.  ÷vK¿, ai7LˆØƒÓ, �U´2, 3, 4, 5, 6ʇê. ù´ØŒU�. k ≤ 4, 20‡Ai, B¥êk40‡, Ïd–�´10‡ØÓ�, 6 + 10 = 16, kn ≥ 16. �n = 16ž, ŒŠÑ Xe20‡8Ü: {1, 2, 3, 7, 8}, {1, 2, 4, 12, 14}, {1, 2, 5, 15, 16}, {1, 2, 6, 9, 10}, {1, 3, 4, 10, 11}, {1, 3, 5, 13, 14}, {1, 3, 6, 12, 15}, {1, 4, 5, 7, 9}, {1, 4, 6, 13, 16}, {1, 5, 6, 8, 11}, {2, 3, 4, 13, 15}, {2, 3, 5, 9, 11}, {2, 3, 6, 14, 16}, {2, 4, 5, 8, 10}, {2, 4, 6, 7, 11}, {2, 5, 6, 12, 13}, {3, 4, 5, 12, 16}, {3, 4, 6, 8, 9}, {3, 5, 6, 7, 10}, {4, 5, 6, 14, 15}. 16. �X´˜‡56�8Ü. ¦����ên, ¦�éX�?¿15‡f8, ‡§‚¥?Û7‡�¿� �ƒ‡êÑØ�un, Kù15‡f8¥˜½33‡, §‚��š˜. )‰ n��Š41. Äk, y²n = 41Ü�‡¦. ^‡y{. b½3X�15‡f8, §‚¥?Û7‡�¿Ø�u41‡�ƒ, ?Û3‡��я˜8. Ïz‡ �ƒ–õáu2‡f8, ؔ�z‡�ƒTÐáu2‡f8(ÄK3˜ f8¥V\˜ �ƒ, þã^‡ 8 E,¤á) . dÄT�n, 7k˜‡f8, �A, –�¹k [ 2× 56 15 ] + 1 = 8‡�ƒ, q�Ù¦14‡ f8A1, A2, · · · , A14.  عA�?Û7‡f8, ÑéAX¥�41‡�ƒ, ¤kعA�7–f8|˜ �–�éA41C714‡�ƒ. , , éu�ƒa, ea 6∈ A, KA1, A2, · · · , A14¥k2‡¹ka. �a�OŽ C714 − C712g. ea ∈ A, KA1, A2, · · · , A14¥k1‡¹ka. �a�OŽ C714 − C713g. l , 41C714 ≤ (56− |A|)(C714 − C712) + |A|(C714 − C713) = 56(C714 − C712)− |A|(C713 − C712) ≤ 56(C714 − C712)− 8(C713 − C712). ddŒ�196 ≤ 195, gñ. Ùg, y²n ≥ 41, ^‡y{. b½n ≤ 40. �X = {1, 2, · · · , 56}. - Ai = {k ≡ i (mod 7), k ∈ X}, i = 1, 2, · · · , 7, Bj = {k ≡ j (mod 8), k ∈ X}, j = 1, 2, · · · , 8. w,, |Ai| = 8 (i = 1, 2, · · · , 7) , |Ai ∩Aj | = 0 (1 ≤ i < j ≤ 7) , |Bj | = 7 (j = 1, 2, · · · , 8) , |Bi ∩Bj | = 0 (1 ≤ i < j ≤ 8) , |Ai ∩ Bj | = 1 (1 ≤ i ≤ 7, 1 ≤ j ≤ 8) . u´, éÙ¥?Û3‡f8, 7k2‡ÓžAi, ½öӞBj , Ù�˜8. éÙ¥?Û7‡f8Ai1 , Ai2 , · · · , Ais , Bj1 , Bj2 , · · · , Bjt (s+ t = 7) , k |Ai1 ∪Ai2 ∪ · · · ∪Ais ∪Bj1 ∪Bj2 ∪ · · · ∪Bjt | = |Ai1 |+ |Ai2 |+ · · ·+ |Ais |+ |Bj1 |+ |Bj2 |+ · · ·+ |Bjt | − st = 8s+ 7t− st = 8s+ 7(7− s)− s(7− s) = (s− 3)2 + 40 ≥ 40. ¤±, n ≥ 41. nþ¤ã, n��Š41. 9