.,.,..,? 2.,.?.,...,...,.,.,.,.,,..,..,,.,,.,.,..,..,....,.,.,.,?,...,,.... Dr.Hener, i
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1 2006 D r. H e n e r
2 .,.,..,? 2.,.?.,...,...,.,.,.,.,,..,..,,.,,.,.,..,..,....,.,.,.,?,...,,.... Dr.Hener, i
3 Day Camp vs ( ) RSA ii
4 A 66 A.1 2, A A B 77 B B B
5 Magic 1 ( ) E D C B A ( )
6 (1) (3) (2) (4) (1) (3) (2) (4) A B C D E F , 2, ( ) (16 ) (16 ) 3 2DA 16 (2 ) 4 AC8F 16 (10 ) (16 ) 2
7 1.2 2 ( ) Basic : CIRCLE(200,150),100,5 ( ) G R B G R B ,. 1.5 ( ) 1g, 2g, 3g, 4g, 1g 15g. 1 g,. g. X 40g g. 3
8 1.6 ( ) (1) g, 1 1 2g 1. 1, 1, (2) g, 1 2g 1. 1, 1, (1997 V(2)) 1. 1,, 1. 2,, 1. 3.,..,. 4
9 ( ) 1 2, , 15., 1.9 ( ) 20, ( ). 1.10,,.,...,, 1, , 3 4, 81, 2,.,.., (??? ). 5
10 1.11 ( ), 64,.,. 1, A B C 64. (1) 5? (2) 6? (3) 64,?,? /* */ #include<stdio.h> void hanoi(int k, char x, char y, char z); int cnt = 0; /* */ void main(void) { int n; } printf("? "); scanf("%d",&n); hanoi(n, A, B, C ); void hanoi(int k, char x, char y, char z) /* */ { if(k > 0){ hanoi( k - 1, x, z, y); printf("%5d: %d %c %c Y=n", ++cnt, k, x, y); hanoi( k - 1, z, y, x); } } 6
11 1.4 Magic 2 (, ) 1,. 2, ,. 2 5? K A Magic 3 ( )
12 Magic 4 ( )..,,.,. 3, ( ) ( ) 40,., 1, ( ) 7, , 8 1 ( ) ,, F 3 7F 2, 1.. 0F F 8
13 ( ),., 3. 10,. ( ) ( ) ( ),., 4, 11., 10,. 2.3 ( )
14 2.4 ( ) : : : : 2.5 ( ) 4 9 1, 2, 3, 5, ( ) hener/lecture/c1/loops/looptop.html cong : ( ).? 10
15 2.6 ( 2, 3 ; ) (1), ,. (2),.,. 2.7 ( 2, 3 ; ) (1),., ( ). (2),., ( ). 11
16 2.8 ( : ), ( ),,.,.,.,,,.,, ( ),,,,,,,.,,, 3, ( 1 ).,,.,.,. 12
17 2.11 ( ) 4 A, B, C, D. 4.. A B C D , ,. (1) A: 1, B: 2, C: 4, D: 8 (4 ) (2) A: 2, B: 6, C: 9, D: 10, E: 11, F: 12 (6 ) (3) A: 2, B: 6, C: 7, D: 9, E: 10, F: 11, G: 12 (7 ) 13
18 2.2.2 Day Camp 2.13 ( ) Day Camp. 10. A (8 ) B (2 ) C (3 ) D (3 ) F (7 ) H (18 ) G (7 ) E (2 ) I (8 ) J (8 ) (1), MRC, Hener 3. (a). MRC Hener ( ) (b) ( ) MRC Hener (2), MRC 2. MRC ( ) 14
19 ? ( ) 1.? 2.? 3.? 2.14 ( ), ( ) , , ( ) 7. 7,,,,.,.,., A B, A, B. 15
20 2.3 vs Fermat x, y, z? x n + y n = z n (n > = 3) 2.17 (Euler ) x, y, z, t? x 4 + y 4 + z 4 = t ( ) (1), 2, (1) (2) (1) 10. (3) (1) 15. (4) (1) 21. (2) (3) (4) 16
21 2.19 ( )? 3 A, B, C., C, A B. C,.,,. 2 ( 2 ). ( ; ) ( ) ( 15) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 12) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.20 (Collatz ), ? 17
22 ( ) 2.1, km 4.0km 5.,., 5, km 1.0km 5.0km 1.5km 4.0km 2.1: A B C D E F G H I J km A B C D E F G H I J km km 1.0km 5.0km 1.5km 4.0km 18
23 3 3.1 ( ), 26.,. [ ] ( ) ( ) : KSDKS : aitai : aitai : KSDKS ˆ ˆ ( ), K = 9. (1). computer (2). WRQXW 19
24 A B C DE W X Y Z F GH T UV +10 S I R J Q K P O N M L K = 10 A B C DE W X Y Z T UV S R Q P O F GH I J K N M L 10 W X Y Z T UV S R Q T UV S R Q P O W X Y Z P O A B C DEFG H I J K N M L A B C DEFG H I J K N M L 3.2 ( ),. JDNXLQ , 3.1. a b c d e f g h i j k l m X N Y A H P O G Z Q W B T n o p q r s t u v w x y z S F L R C V M U E K J D I ( ), 3.1. (1). hener (2). AXZVUWZ ( : ) ( a b c d ) ( x y ) ( ) ( ) ( ) ( ax + by a b x z ax + by az + bw = = cx + dy c d y w cx + dy cz + dw ( ) 1 a b = c d 1 ( d b ad bc c a ) ) 20
25 3.2, = ( ). a b c d e f g h i j k l m n o p q r s t u v w x y z ( ) x = 24. ( ) ( 2 4 ) = ( ) = ( 26 8 ) (26, 8). ( ) (26, 8) ( , x. ( ) 1 ( ) = = 1 ( ) ) ( ) 26 = 1 ( ) = 1 ( ) ( ), 3.2, = (1) love. (2) (11, 12), (8, 11), (7, 4). ( ) =. ( 2 4 ) a.082 h.061 o.075 v.010 b.015 i.070 p.019 w.023 c.028 j.002 q.001 x.001 d.043 k.008 r.060 y.020 e.127 l.040 s.063 z.001 f.022 m.024 t.091 g.020 n.067 u (Beker, Piper) 21
26 ˆ (1) e : (2) t a o i n s h r : (3) d l : (4) c u m w f g y p b : (5) v k j x q z : ˆ 2 30 th he in er an re ed on es st en at to nt ha nd ou ea ng as or ti is et it ar te se hi of. ˆ 3 12 the ing and her ere ent tha nth was eth for dth. YIFQFMZRWQFYVECFMDZPCVMRZWNMDZVEJBTXCDDUMJ NDIFEFMDZCDMQZKCEYFCJMYRNCWJCSZREXCHZUNMXZ NZUCDRJXYYSMRTMEYIFZWDYVZVYFZUMRZCRWNZDZJJ XZWGCHSMRNMDHNCMFQCHZJMXJZWIEJYUCFWDJNZDIR A 0 H 4 O 0 V 5 B 1 I 5 P 1 W 8 C 15 J 11 Q 4 X 6 D 13 K 1 R 10 Y 10 E 7 L 0 S 3 Z 20 F 11 M 16 T 2 G 1 N 9 U A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Step 1 = Z e? Step C t, D a, F o, J i, M n, R s, Y h, N r. Step 3 22
27 ˆ DZ, ZW (4 ), WZ, W. ˆ NZ, ZU (3 ). ˆ RZ, HZ, XZ, FZ, ZR, ZV, ZC, ZD, ZJ (2 ). ˆ ZRW RZW, RW, R. W d, R n YIFQFMZRWQFYVECFMDZPCVMRZWNMDZVEJBTXCDDUMJ NDIFEFMDZCDMQZKCEYFCJMYRNCWJCSZREXCHZUNMXZ NZUCDRJXYYSMRTMEYIFZWDYVZVYFZUMRZCRWNZDZJJ XZWGCHSMRNMDHNCMFQCHZJMXJZWIEJYUCFWDJNZDIR ˆ,. A B C D E F G H I J K L M y a s p r b c u t v i N O P Q R S T U V W X Y Z h x f n k g w m d l o e 3.6 ( ) ( ) Our friend from Paris examined his empty glass with surprise, as if evaporation had taken place while he wasn t looking. I poured some more wine and he settled back in his chair, face tilted up towards the sun.,.,, ( ), ,. K (3, 8) O (7, 20),. 3.3 ˆ. ˆ,. 23
28 : muzui : kyhtb ( ) : a b c d e f g h i j k l m n o p q r s t u v w x y z Q R S T U V W X Y Z A B C D E F G H I J K L M N O P a b c d e f g h i j k l m n o p q r s t u v w x y z C D E F G H I J K L M N O P Q R S T U V W X Y Z A B a b c d e f g h i j k l m n o p q r s t u v w x y z T U V W X Y Z A B C D E F G H I J K L M N O P Q R S a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z 3.6 ( ). = modnar. AQKQOR a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z,., ( ).,.? ( ). 24
29 (1) ? (2) [ ] ( ) ( ) : : 68,24,70,24,100 : 17,40,49,40,53 H e n e r : H e n e r 17,40,49,40,53 : 68,24,70,24,100 ˆ. ˆ.., ( ). RSA 1,. 1 (Ronald L.Rivest), (Adi Shamir), (Leonald Adleman)
30 3.4.1 RSA ˆ. RSA A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z,. ;!? + - * / < > p, q. 2 n = pq( > = ). 3 p 1, q 1 E. 4 (E, n). : 2. 1 p = 7, q = n = 7 17 = 119 > = = 7 1, 5 16 = 17 1, E = 5. 4 (E, n) = (5, 119). 3.. = E % n % 119 = 68 ( H = 17) f(n) = (p 1)(q 1). ( ) 2 D. E D % f(n) = 1. 3 = D % n. D E p 1, q 1. 1 f(119) = (7 1)(17 1) = D = 77., 5 77 % 96 = % 119 = 17 (17 = H )
31 77 = % 119 = % 119 = % 119 = % 119 = % 119 = 51 2 % 119 = % 119 = % 119 = % 119 = 51 2 % 119 = % 119 = % 119 = % 119 = 17, 68 * 51 % 119 = 17, 17 * 102 % 119 = 68, 68 * 51 % 119 = 17 ( ) RSA, ( ) :(E, n) = (5, 119) ( ) :(E, n) = (11, 95) D : p = 7 q = 17 : 17,40,49,40,53 H e n e r p = 5 q = 19 63,70,64,70,2 : D : 105,49,64,49,32 27
32 ( ). A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 4.2 ( ),.. A B C D E F 4.3 ( )... A B C D E F
33 ( )..? (, 2. ) (1). (2). (3) 2. ˆ ˆ : ( ). : v.,. =. =. 1 ( ),. 2 ( ),. 3 ( ), ( ) (1),,,,.,.. (2). 29
34 ˆ ˆ :. :,. 4 ( ) (1) 4. (2) ( ) A, B, C,,, ( ), 2.,, ( ),.. A B C D D A C B 30
35 ( ) ( ) ( ) , ( ) A, B, C, D, E, F, G A G A G. 1 5 ( 1 B C E B C E, B C E ) (1999 ) BCE DEG BFG AEF ACG 4.11 ( ) A, 3 ( 4 )..,,., A,., (, 0 ), A? (1997 ) 31
36 4.12 ( ) (1). 1 (2). 5 ( ). 1878, ,, ( ), 1,. c C d g A e D a B b f 1. 32
37 4.14 ( ) (1),. (2) (1),. 6 ( ) 4.15 ( ).,? (1) (2) (3) ? 33
38 4.17 ( ) 8. (, )., 1, ( ). v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v ( ),. (1) (2) (3) (4) ( ) V 1, V 2, 1,,.,. (1) V 1 = {,,,,,, ( )} (2) V 2 = {,,,,,,,,,,, } 34
39 ˆ : ( ) (2 ) (1) ,, 2 4, , 10, 01, 10 00, 01, 10, 11 ( ) ( ) (2) 0 1 8, 3 8, , 001, 010, 011, 100, 101, 110, 111 ( ) 35
40 ( ),,, ( ). 4, ( ). 4, 1., 5, 2, 6, 3, 7, 4 ( ).? A A A A
41 ( ) 1, 2. (1),,. P (2),. P (3),. P 37
42 5.2 ( ) 5.1 (1) (3), P? (1) (2) (3) 5.3 ( ) XY, A B 1, AB.,. XY, PQ. XY, PQ, M N., MN. MN AB,., P M N Q X A B Y 38
43 ( ) (1) (2) (3) (2) 3, 5. 39
44 ABC A B 5.7 ( ) C 1.,. 2.,. 5.8 ( ), ,. 40
45 OD = 5cm, DE = 4cm, ABCD CD. A 5.10,,. B O D E C , (1) (2) (3) ( ) 41
46 5.14 OA 4 1 AOB, OA, OB. OA = 2cm,. B A O cm, 1cm 4. (4 ) cm, ( ), 1 1cm,
47 ( )., (1). (2). (3). (4). (5). (6). (7). (8) 1 3 = 0.333, (9) , (10).. A, B, C,..., p, q, r, ,.,. 2 A, B, A B 43
48 ., A. A 1, B. B (1), (0)., (0), (1). A A. 2 ( ) A, B, A 1 B 0, A 0 B 1 A A 1 0, B A 6.2. (1). (2)., B A. 6.1 : (3) Dr.Hener. (4) Dr.Hener. (5). (6) ( ). p, p = ON = ( ) ( ) p. 6.1: NOT
49 ,,. (1) 1, (1 ) 0, (2) A 1 A, (2 ) A 0 A, (3) A B A B, (4) A, (5) A B A B. 6.3 P : Dr.Hener.. 2. p :. q : Dr.Hener. P p q,. (p q ). 3 ( ) A, B, A 1, B 1 1, 0 A B, A B. ( ) A B A B : 6.3 a a W, a S, J,. (1) a. (2) a. (3) a,. (4) a,. (5) a OL. p q ON ON ON OFF OFF ON OFF OFF p q 6.2: AND 45
50 ,,. (6) A B, (7) A B C A B C, (8) A A, (9) A 1, (10) A P :. Q :. P { p1 :. p 2 :. Q { q1 :. q 2 :. P p 1 p 2, Q q 1 q 2. p 1 p 2 P q 1 q 2 Q ( ) A, B, A 0, B 0 0, 1 A B, A B. ( ) 5 ( ) A, B, A B 0, 1 A B, A B. ( ) A B A B : A B A B : 46
51 6.1 ( 2!!),,. 6.4,,,. (1),. (2). (. = ) (3). (4) , 2,. 1., Dr.Hener. 2.,. 1000,. p q L R L R p q 6.3: OR 6.4: EOR p q ON(L) ON(L) ON(L) OFF(R) OFF(R) ON(L) OFF(R) OFF(R),,. (6 ) A B, (7 ) A B C A B C, (8 ) A A, (9 ) A 1, (10 ) A 0. 47
52 6.5,.,. P : Dr.Hener. Dr.Hener. 1 Dr.Hener.. 2 Dr.Hener.. Dr.Hener Dr.Hener 3 Dr.Hener.. 4 Dr.Hener.. A B Dr.Hener Dr.Hener A B. 6 ( ) A, B, A 1, B 0 0, 1 A B, A B. ( ) A B A B : A 1, 0 B. 48
53 6.6 (1) P :,.. P. 1,.. 2,.. 3,,. 4,,. (2) P :,. 1,. 2,. 3,. 4, (1). (2). (3). (4). ( ) B A B. 6.8 ( ) (1) X. (2),, A B A A B A B A B A B ( ),,,,. 49
54 6.1. (1) A B (2) ((A B) C) ( A B) A B A B A B C ((A B) C) ( A B) 6.9. (1) (A B) (2) A A (3) A (A B) (4) A B (5) A B (6) (A B) (7) A A (8) A (A B) A B (A B) A A A (A B) A B A B A B (A B) A A A (A B) (1) A (B C) (2) A (B C) (3) (A B) (A C) (4) (A B) (A C) A B C A (B C) A (B C) (A B) (A C) (A B) (A C) 50
55 7 ( ) ( ) A, B,. A A, A A, (A B) A B, (A B) A B, A (A B) A, A (A B) A, A (B C) (A B) (A C), A (B C) (A B) (A C). 6.1,. A B C A B ((A B) C) ( A B), A B ((A B) C) ( A B) (1). (2). 1 A B 2 A B 3 A B 4 (A B) 5 A B 6 B A 7 B A A B A B A B A B (A B) A B B A B A A B A B (A B) A B 51
56 6.3,, Q :, R : A B 2 A B 3 (A B) B 4 (A B) (A B) 5 (B A) A 6 A (A B) B 7 A A B 8 A (B C) 9 A B C 10 (A B) ((A C) (A B C)) A B A B A B (A B) B (A B) (A B) A B (B A) A A (A B) B A A B A B C A (B C) A B C (A B) ((A C) (A B C))
57 7 7.1 p q p p p q p q p q p q p q L R L R p q 7.1, x. p : 1100 q : 1010 (1) (2) (3) p p p q x q x q x 53
58 7.2, x. p : 1100 q : 1010 (1) (2) (3) p p q p q x q x x 7.1 L, x. p q r x p : x : q : r : L, L = 6.1, L. p q x 7.3,, 7.2 ( ). p q x 54
59 ( ) 3.,,, 3., , 1, 1, 1. (1),. (2),. (3),. (4),. (5),. 4? 7.6 A, B, C, D 4. 4, : B A. 2: A C 1. 3: A B 2. 4: B D 3. A B C D
60 7.7, A, B, C, D, E , 1, 1., ) A, B, C ) A, B, D 3 1. ) A, C, E ,. A B C D E 7.8 (1995 ) A E,,,,., A E. ) A 3,. ) B E. 2. ) D.. ),. ( ).. A B C D E 56
61 7.3,.,,., ( ).,,.,,,,.. ( ) N(x) : x. : x.,,,. 2, 2. 2 a, b. a, a.,, a. a : : ,
62 7 ( ) x P. ( P? ) Proof. [1] x P. N(x) P (i) x. (ii) x. [2] N(x) P x P. (i) x. (ii) x.,,, a, b. 58
63 . b, g : : :, ,....,.,,.,. h, K. : : : ,. 59
64 7.11,,., (a) (b),. (p). 1. 2,,.,,., , (1998 ) c, d..,.,. c. d c, d,. 60
65 7.4, A : ( )Dr.Hener. B : ( ). Dr.Hener (Dr.Hener ).., ( ) Dr.Hener..... [ D] A B (Dr.Hener ), A (Dr.Hener ). B ( ).. [ D] A A B B 9 ( ) A 1, A 2,..., A n B, A 1, A 2,..., A n, B,, A 1 A 2... A n 1 B 1, A 1, A 2,..., A n B. A 1 A 2... A n B A 1, A 2,..., A n, B., [ D]? 10 ( ) A 1 A 2... A n B. (A 1 A 2... A n ) B [ D]? A B
66 7.14. (1) A B B A (2) A B B C A C (3) A B C A C (4) A B C A C A B A B C A B C A B C (1)? 1. Dr.Hener.. 2 Dr.Hener... (2) ( ) (1),. A B A B
67 . A B C A C 1 A B B C A C 2,? A B C C D A D (A B C) (C D) (A D),,. A B C A C 1 A D C D A C.. A :. B :,,. C : p :. q :. r :. s :. t :., A C. ( I) ( ),. X : Y :,.,. A, B,. A : B :,.,. 63
68 (1) X, Y A, B. X : Y : (2) X, Y.. 1,. 2,. 3,.... A B (1) (4).. (1),. (2),. (3),. (4),. 64
69 7.19 1, 2, 3. (1). 1 : 2 : 3 : (2) : 1 : 2 : 3 : 7.20.( ) (1),. (2),. (3),. (4),...?. 65
70 A A.1 2, Magic 5 ( ) 1 ( 16 ). 2 1, (15 ) ,.. 9,, 2, 3,,,. 10 ( ). 11, ? 13 1,. A.1 7,,.,. 66
71 Magic 6 ( 1) 1 10, (i),. (ii),.,. (iii),.. 3. (10 ) ,., ( ) 7.. (i),. (ii),. (iii),,. (iv),,. 8,., 10. Magic 7 ( 2) 1, , ,. 67
72 5 10., 2,.. 6 1,. Magic 8 ( 3) 2 6, 2,. 3., (i) 2 (1 2 1 ), 20. (ii) 2, ,. 4. (i) 1. (ii). (iii) 1, , , ( 1.) 9., 1, ?, Bob Hummer. 68
73 A.2 Magic 8 4, 1, 1,.,, 1..,, A.3,, 2, 3,. A.2,,
74 A.2 Magic 9 ( ) ?.. A.4? Magic 10 ( ) ,.. 3.,. 4,. 5,. 6,, , ,
75 12... A.5. Magic 11 ( ) 1 ( 12 ) , (J, Q, K) 10 2, ,. 7. A.6, 4,,? Magic 12 ( ) 1 ( 9 ) ,.,. 4, 1, 10, 9, 8, 7, 1.,., 1,,,.. 5 4,. 6, 4.. 7,
76 Magic 13 (13 ) , , , 3,. 6, 3, 4 13 ( 10 ) ,. 8 ( 2 ). 9, 1. A.7. Magic 14 ( ) , , , 34. A.8,. 72
77 Magic 15 ( ) 1. Magic 14. 2, , , , ? 11,. A.9. 73
78 A.3 Magic 16 (,?) 1., , 6,?,,.?? Magic 17 ( ) 1, ( A.1) 3.,. ( A.2) 4,, 7! 1!! ( A.3) 5.,. ( A.4) 6,, 7! 1!! ( A.5) A.1 A.2 A.3 A.4 A.5 74
79 Magic 18 ( ) ( ),. ( ),. 4,. 5, ( ). 6. ( ) Magic 19 ( ) 1 3,. 2,. 3. 4,.,!,. A.10.,,. 75
80 Magic 20 ( ) ,.,. 3,. 4, 1. 5,. (,.) 6 6, 1! A
81 B B.1 2,
82 B.2 (1) 6 2. (2)
83 B.3 79
84 B.1 ( ),. 80
85 Hener,.,,...,,.,.,.,,,.,.,.,..,.... Albert Kurt Hener (Dr.Hener) God Door university graduate course 81
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
2008 (2008/09/30) 1 ISBN 7 1.1 ISBN................................ 7 1.2.......................... 8 1.3................................ 9 1.4 ISBN.............................. 12 2 13 2.1.....................
熊本県数学問題正解
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17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
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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...............................
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1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z
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