96 5, ' : G! H '(G) =H,, H G, 37 Z Z m a 2 Z m a a p Z m (p.90 ) p(a + b) =a + b = a + b = p(a)+p(b):, p {p(ab) =p(a)p(b){, p ( 95 ). 97. m, n, Z m Z
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1 95 5,,,,,,,, ( ) S 3, f1 2 3g f1 2 3g,,, 5.1,,, G H ' : G! H, '(ab) ='(a)'(b) for 8a b 2 G (5.1), (,, )., 1 1,, ' e 2 G e 0 2 H '(e) =e 0., g 2 G, '(g ;1 )='(g) ;1 : (5.2) 2, G, H, G H, ' : G! G 0, G 0 H, ' G H,., G 2, ' H 2, ( 1 ). ' : G! H G '(G), H
2 96 5, ' : G! H '(G) =H,, H G, 37 Z Z m a 2 Z m a a p Z m (p.90 ) p(a + b) =a + b = a + b = p(a)+p(b):, p {p(ab) =p(a)p(b){, p ( 95 ). 97. m, n, Z m Z n 38 S 4 S 3 S n n (4.1)., S , S ( 74), S 3,1=x 1 ; x., S 4 4 x = a ; c b ; c : a ; d b ; d a, b, c, d a, b, c, d,, x a b c d, ( i.e. a 7! b, b 7! c, c 7! d, b c d a d 7! a), x x x;1 b ; d c ; d : b ; a c ; a = x x ; 1 : 98. x, 4 a, b, c, d 24, x
3 , 6 f (x) f : 7! f, f = f f f (x) = (a) ; (c) (b) ; (c), f (x) =, f (x) =f (f (x)). (a) ; (c) (b) ; (c), : (a) ; (d) (b) ; (d) = y : (a) ; (d) (b) ; (d) = f (y): f, f = f f, 38, f, a $ b b $ c c $ d, S 4,, 1=x, 1 ; x, 1=x, 2 1=x, 1 ; x, , ;1,,, 2 f+1 ;1g 39 G 1 T, f(z) =pz + a(jpj =1) (2.3). p, f g(z) =qz + b g g f, '(gf) ='(g)'(f). g(f(z)) = q(pz + a)+b = qpz +(aq + b):, 2, 2, R ' AR ;' B 100, l m, R ' A (l) R' B (m),, l m R ' A (l) R' B (m), R ' A (m) R' B (m)
4 ABC B, C, E, F, O ABC, AO EF a b a b m n 101. ad ; bc c d c d p q am + bp an + bq, 0, cm + dp cn + dq,, Z m Z! Z m Z m, mz Z Z m mz Z, Z m = Z=mZ, G, H G H G, Z m eh, g 1 H, g 2 H, ::: He, Hg 1, Hg 2, :::? Z, Z gh = fghjh 2 Hg, Hg = fhgjh 2 Hg, g 2 G ; H G, gh 6= Hg, G = D 3, H = fid S a g, g = R 2 D 3 ; H, D 3 p. 67, RH = fr id R S a g = fr S b g HR = fid R S a Rg = fr S c g:, D 3 C 3 = fid R R 2 g, g D 3 ; C 3,, 2 C 3 D 3 C 3 17 G H, g 2 G, gh = Hg H G, (g 1 H)(g 2 H)=g 1 (Hg 2 )H = g 1 (g 2 H)H = g 1 g 2 H:, g 1 H g 2 H 1, 1 (g 1 g 2 )H,,, g 1 g 2 = g 1 g 2.,, : g = gh = Hg.,, H = eh, gh g ;1 H, H, g 1 H, g 2 H
5 G, H G G H, g 1 g 2 = g 1 g 2 H G=H G=H, G,, D 3 (p.67),, R, S, R S R R R S S R, Z 2 2, D 3 =C 3 = Z2., = 40 G,, N A G, K G, N K G, N K, G, gh = Hg ghg ;1 = H, H, H H (p.61 ),, (1 ),,, A, A, A, K N, 58(p. 171 ), K f A, f(a), N K G, G=K G=K, 39 G T ' G 2 ', 2 G K, '(f) ='(g) =p, p =1, f, g K p =cos + i sin 6= 1, f g, f = RA, g = R B, f = R A = R B (R ; B RA ) 2 gk.
6 100 5, R ; B RA, f g K, f = g h, '(f) ='(g)., ' G=K H f f, ' : G=K! H G=K 1, G K,, K ( ),, ' T, 10 ( ) ', G H, K G,, ' H G, G=K = H ' K, G, G=K, k 2 K, f(k) = e, f(gkg ;1 ) = f(g)f(k)f(g) ;1 = f(g)ef(g) ;1 = e gkg ;1 2 K., G, H, K, jgj = jhjjkj., G H, H G 102. (a) D 3 Z 2 (b) Z ' : S 4! K S 4 ' ',, 2 S 4 =K, jkj =24:6=4., 4, x x a b c d a b c d a b c d b a d c a b c d c d a b a b c d d c b a 4,, D 2, K = D2., S 4 =D 2 = D3 :
7 f (x) +f (;x) f : R! R F (f (x)) 104. S, C n n, S=C n = S 105., 104 S (R +) 5.3, 3 (p.68),,, 19 S S F (S), S (p.68 ), 2, s k s l = s k+l, s 0 =1, S 1, F (S) 20 R S ( r S r =1 ). R, F (S)=H(R), F (S) S, H(R) R F (S) (, H(R) R F (S) ). s 1 ::: s n, r 1 ::: r m h r 1 ::: r m j r 1 ::: r m i ( r, r =1 )., h a j a n =1i n 42 h a b j ab =1i, Z
8 102 5 F (a b), a k 1 b l 1 :::a kn b ln H(R), a = bh a = hb (h 2 H(R)), a b 1, a = (ab)b ;1, a b ;1, a b b, F (S)=H(R) b, H(R), a b., b n (n 6= 0) H(R), b, F (S)=H(R), a, b ba ab (i.e.ba = a ;1 (ab)a), ba 2 H(R) 20, H(R) R 106. h a b j a 2 =1 b n =1 aba= b ;1 i? 107. h a b j aba = bab i = h x y j x 2 = y 3 i : 5.4, 21 G, X 57 ) X G, G, X (p. T : G! Tr(X): T g, x 2 X T g (x) T g gx G, X, G! Tr(X), S 3,, 3 5.1, O, a, b, c D 3 (O), a, b, c 3 ( ),, D 3 6
9 : D 3 22 x 2 X O(x), x O(x) =ft g (x) j g 2 Gg: O(x), ,, 1, 5.1, 1 O 5.1 A = A 1, fa 1 A 2 A 3 g A 2 A 3,, x O(x), y 2 O(x), h 2 G y = hx, O(y) =fgyjg 2 Gg = fghxjg 2 Gg,, h 2 G, G = fghjg 2 Gg = G 2,, 2 O(x) O(y) z, O(x) =O(z) =O(y),, S, 1 ( 0),, 108. D 3,, 1 x, f0 1 ;1 2 ;2 i p 3 ;i p 3 4 ;4 2+2i p 3 2 ; 2i p 3 ;2 + 2i p 3 ;2 ; 2i p 3g D 3
10 : S 43, ( 74) 1,.,, 0 2 1, R 2 0, 1 ( x 1 ), 1, R 1 R, 1 F 1 F 2 F 3 F 4 F 5 F , R [1, 3 0, 1, 1,, 3 D , 3 ( ) 1,, F i 2, C [1,, 2, 2 2 3, 6 5.5
11 Q, G, Q Q (i.e. Q ), G, Q G G 2 (, 5.2 AA 0 ) (, BB 0 ) (, CC 0 ) , G : F G,,,, F, 6, 1 f 2 F, f 4, 3 6 2, G E V G, G, 24 ( ) 24
12 106 5,,, G 5.4, 4, 3, 2 D, M,, T 23 a b c 5.4:, G 112. F, E, V, D, M, a) b), (f e) G, f 2 F Q, e 2 E, F E ( F E ), 6 12 = 72, 3 e f ( 5.5a), e f 1 ( 5.5b) e f ( 5:5c), 3 1 2, ( ) 2 (f 1 e 1 ), (f 2 e 2 ), f 1 f 2,, 4 e 1 e 2 2, G, F E, 3, 5.5
13 a b c 5.5: { 113. G, (a) V F ( { ) (b) D F ( { ) (c) E E( )., 24 X G, x 2 X, x G st(x) St(x) =fg 2 G j T g (x) =xg:, G, g h 2 St(x), T gh (x) =T g (T h (x)) = T g (x) =x T g ;1(x) =T g ;1 (x) =x:, G, 5.1 O,, A, 2 ( ), B, ( ),, 44,, jo(x)jj St(x)j = jgj: (5.3) 5.3, G H =St(x) G = g 1 H [ g 2 H [[g k H:, x, x, h 2 H, T gh (x) = T g (T h (x)) = T g (x), g, k 2 G x, g k H, k = gh h 2 G, T h (x) =T g ;1 k(x) =T ;1 g T k (x) =x
14 108 5, h = g ;1 k 2 St(x)., x G H G= St(x), ,,,, G,, G X, O(x) x 2 X, 2 x, y 2O(x), St(x) St(y),, y = T g (x) g 2 G g St(x) St(y), h 2 St(x), T ghg ;1(y) =T g (T h (T g ;1 (y))) = T g (T h (x)) = T g (x) =y:, g St(x)g ;1 St(y). x y St(y) g St(x)g ;1, g St(x)g ;1 = St(y)., St(x) St(y) 46 p n,, p, 2,, (n p ) C p,, 2, 1 p, p 1,,, 1 n n p ; n, p, (n p ; 1)=p,,, (n p ; n)=p + n p, n p ; n p, p 114, 46,, Burnside g 2 G, X, T g, T g (x) =x x 2 M N(x)
15 , 1, 8, (180 ), 6, 9, 2, 3, 4, 5, 7, 180, (Burnside ) r, G, r = 1 jgj X g2g N(g):, x 2 M, P g2g N(g), g x, g,, j St(x)j O(x),,,, j St(x)jjO(x)j,, G jgj, jgj, jgj ( ) Burnside, 46, n p, n, r =(n p +(p;1)n)=p, ,,,,, 7 3 X, X D 10 Burnside, D 10 X, X ; 10 3 =120 (a) ( 5.6),, (b),, 24 =8, 1, 1, 4 1 (b) 5, Burnside r =(120+58)=20 = 8.
16 : , 6, , 1 6 (2, ) (i.e. 2 ) (a) (b), (,2 ). 5.6, Burnside, 3, 3 m, n, k, m, n, k 0 7, m + n + k =7, (m n k), m, n, k, (4!), m n k, 3, (0 0 7), (0 1 6), (0 2 5), (0 3 4), (1 1 5), (1 2 4), (1 3 3), (2 2 3)., 3 (m n k), 2 2 ( ), 3 2 3, 2
17 m, n, k 25 G X, X N ',,, ', N, 3 3 ( m ),,, 2, 5.7 2,, m, 26 ' : X! N, 5.7: M, M, M,, M, 2, 2., 121. M (a) (b) H M, H H,,, H A, A A A (p.25
18 112 5 ), A r, r f(r), 2 D 3 ( 5.1 ) O, OM, r,, r 1 cos '., cos ' D 3, D 3 ' 7! ;' ' 7! ' +120, cos 3', (r cos 3'),, cos 3' = b r = c, c>0 jbj1, 1 3, 6 5.7, (p.5 ). 2, b G, G, A, A, A, 1, 2,, A, A 122., 123.
19 , 29 G, 2 1 F, x 2 F ( ) F 2, 2, F ( ),, F, ( 27), 27, 124. C n D n,,, 3, p3m1 p1, 2 a, b 5.8a, p1 b a a b c d 5.8:, a b 5.8b, p1. 2, 1 1 2, b;a b 2 2, a +2b 2a +3b 125. ka + lb, ma + nb 2,, jkn ; lmj =1, k, l, m 1,
20 114 5 p1, p1, p1,, ( 5.8c)., p1,,,, 5.8d p1,, 1., p1 (i.e. 2 ) 1, G, g 2 G, g l, G, G,, G l 2,, 180, R ' 2 G, ' 6= 180,,, T a 2 GT a R ' T a R ;' ( 58 ) ,, G, l l,, G 126. G, l, l, 2 l, 1, 1, G, G, G, 2 p1 2. G p1, 127., 2
21 , p1 G, H G (H p1 )., H G, ( 8 ). 3. G, G=H, 10 C n, D n (n = ) 1, ( 130). G=H G G H G, g 1 g k G=H,, = k[ i=1 H,, g i,, ( ) ( ) G H (i.e. G=H ) G=H, 48 p3m1, H G g i 5.9a, p3m1,, 3 3 ( 5.9a, 3.1b )., MNB ( ), A B D, ABCD 2 ;! ;! AB, AD G H, S ABCD : S MNB =6:1 ABCD, 6 (, 6 )., H, BQCLDM, 6 G=H, H (` ' ), g h 1 g h 2 (h 1 h 2 2 H)
22 : p3m1, h 1 h 2,, 6, G=H 6 5.9b, 1 6,, gh,, g 2 G 1 $ 6, 2 $ 5, 3 $ 4. G=H D 3,, 3 3, G=H = D3., G D 3 p3m1=p1 p3m1 p G=K, G, K H ;! AC ;! AR ( 5.9a ).,, C n, D n 1 M p1?, 2 a, b,, 2 p1 (p.123 ): H T a, T b, K T c, T d, LHL ;1 = K, L L(a) =c, L(b) =d 30 2,
23 , (Fedorov{Schoniess),, ,,,, 1., 2, 3, 4, 6 2. G + G G ( ), G +, (i.e. G=G + ) G + = G(i.e. G ), G, p1, p2, p3, p4, p6, G 4. G + 6= G, G G + G n G + f, G +, 5 (f,, ), (a) G + = p1, G = pm, pg, cm. (b) G + = p2, G = pmm, pmg, pgg, cmm. (c) G + = p3, G = p31m p3m1. (d) G + = p4, G = p4m p4g. (e) G + = p6, G = p6m. ( ) : : (,, 4,,, 2, 3, 4, 6 ). : : 1 12, 13 17,, p1, p2, pm, pg, pmm, pmg, pgg, cm, cmm,,
24 118 5 p1 T 1, T 2 T 1 T 2 = T 2 T 1 R 1, R 2, R 3 p2 R 2 1 = R2 2 = R3 3 =id, (R 1 R 2 R 3 ) 2 =id pm S 1, S 2 T S 1 T = TS 1, S 2 T = TS 2, S 2 1 = S2 2 =id pg U 1, U 2 U 2 1 = U 2 2 cm S U S 2 = id, SU 2 = U 2 S S 1, S 2, S 3, S 4 pmm S 2 1 = S2 2 = S2 3 = S2 4 = id, (S 1 S 2 ) 2 =(S 2 S 3 ) 2 =(S 3 S 4 ) 2 =(S 4 S 1 ) 2 =id S R 1, R 2 pmg S 2 = R 2 1 = R2 2 =id, R 1 SR 1 = R 2 SR 2 pgg U 1, U 2 (U 1 U 2 ) 2 =(U ;1 1 U 2 ) 2 =id
25 S 1, S 2 R cmm S 2 1 = S 2 2 = R 2 =id, (S 1 S 2 ) 2 =(S 1 RS 2 R) 2 =id p4 R 90 R 1 R 2 = R 4 1 =(R 1 R) 4 =id S 1, p4m S 2, S 3 S 2 1 = S 2 2 = S 2 3 =id, (S 1 S 2 ) 2 =(S 2 S 3 ) 4 =(S 3 S 1 ) 4 =id p4g S 90 R S 2 = R 4 =(R ;1 SRS) 2 =id p R 1, R 2, R 3 R 3 1 = R 3 2 = R 3 3 = R 1 R 2 R 3 =id p31m R 120 R R 3 = S 2 =(R ;1 SRS) 3 =id S 1, S 2, S 3 p3m1 S 2 1 = S2 2 = S2 3 =id, (S 1 S 2 ) 3 =(S 2 S 3 ) 3 =(S 3 S 1 ) 3 =id p6 R 120 R 1 R 2 = R 3 1 =(R 1 R) 6 =id ( ) p6m S 1, S 2, S 3 S 2 1 = S 2 2 = S 2 3 =id, (S 1 S 2 ) 2 =(S 2 S 3 ) 3 =(S 3 S 1 ) 6 =id
26 d 130., 3 (p. 115) 131.
70 : 20 : A B (20 ) (30 ) 50 1
70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
熊本県数学問題正解
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76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C(
3 3.1 3.1.1 1 1 A P a 1 a P a P P(a) a P(a) a P(a) a a 0 a = a a < 0 a = a a < b a > b A a b a B b B b a b A a 3.1 A() B(5) AB = 5 = 3 A(3) B(1) AB = 3 1 = A(a) B(b) AB AB = b a 3.1 (1) A(6) B(1) () A(
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A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin
121 6,.,,,,,,. 2, 1. 6.1,.., M, A(2 R).,. 49.. Oxy ( ' ' ), f Oxy, O 1 x 1 y 1 ( ' ' ). A (p q), A 0 (p q). y q A q q 0 y 1 q A O 1 p x 1 O p p 0 p x 6.1: ( ), 6.1, 122 6 A 0 (p 0 q 0 ). ( p 0 = p cos
2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l
ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE
1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx
4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan
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18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 1 月 17 日 CPC 版のプレ リリースが公開されました 原文及び詳細はCPCホームページの C
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 1 月 17 日 CPC 2019.02 版のプレ リリースが公開されました 原文及び詳細はCPCホームページの CPC Revisions(CPCの改訂 ) 内のPre-releaseをご覧ください http://www.cooperativepatentclassification.org/cpcrevisions/prereleases.html
1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
III
III http://www.manabino-academ.com . = k...................................... = k p + q................................. = a + b c + d.................................. 4.4..........................................
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 7 月 31 日 CPC 版が発効します 原文及び詳細はCPCホームページのCPC Revision
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 7 月 31 日 CPC 2019.08 版が発効します 原文及び詳細はCPCホームページのCPC Revisions(CPCの改訂 ) をご覧ください https://www.cooperativepatentclassification.org/cpcrevisions/noticeofchanges.html
IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
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