(, Goo Ishikawa, Go-o Ishikawa) ( ) 1

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1 (, Goo Ishikawa, Go-o Ishikawa) ( ) 1

2 ( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2

3 (geometry) ( ) ( ) 3

4 (?) (Topology) ( ) DNA ( ) 4

5 ( ) ( ) 5

6 ( ) H. 1 : , 1, 2, 3, 5, 8, 13, π ( ) 6

7 geometry geo ( ) ( ) ( ) =() 1 1 ( 1) ( 1) = 1 () 7

8 8

9 ( ) () ( ) ( ) ( ) 9

10 ( ) ABC AB < AC + CB 10

11 ( ) ABC AB < AC + CB ( ) R right angle, P Q P Q P Q P Q P Q P Q ( ) 11

12 ( ) () ( ) =0 0 12

13 ( ) ( ) geo geometry geo ( J P ) ( ) 13

14 ( ) ( ) 180 collinear R AC CB <AB A, B, C AB < AC + CB 14

15 AC CB <AB AC CB < AB (AC CB) <AB AC < AB + CB CB < AB + AC A, B, C AB < AC + CB ( ) ( ) <

16 (hypothesis) (conjecture) () ( ) ( ) ( ) ( ) 16

17 ABC A, B, C l BC,CA, AB D, E, F BD DC CE EA AF FB = 1 1: P, Q, R AB, BC, CA ( ) CD ( ) P, Q, R AB, BC, CA AB, BC, CA 17

18 ABC A, B, C P A, B, C D, E, F BD DC CE EA AF FB =1 ( ) ( ) ABC DEF AD, BE, CF BC EF P CA FD Q AB DE R P, Q, R A D B E C F A, C, E m B,D,F l AB DE P BC EF Q CD FA R P, Q, R AF DE ( ) P Q ( ) Q P P (Q ) ( ) (Q ) (P ) 18

19 ( ) P Q P ( ) Q ( ) ( ) ABC A B C AB = A B,AC = A C, ABC = A B C P ABC A B C Q AB = A B,AC = A C, ABC = A B C P Q Q P AB = A B,AC = A C, ABC = A B C ( ) () A A B A A B A B ( ) 19

20 ABC AB < AC + CB ABC AB < AC + CB π π e πi = 1 ( ) ( ) ( 180 ) (a 360 ) a () ( 180 ) (a 360 ) () 20

21 R n R n ( ) 3 2 = 6 ( ) =( 4 ) ( ) =1+ 1 2,( ) = = 2 5 ( ) ( ) ( ) 21

22 ( ) ( ) ( ) ( ) 22

23 ( ) 23

24 ( ) (x 2 + y 2 z 2 =0 ) ( ) () A, B, C, D, E, F AB DE P BC EF Q CD FA R P, Q, R 0 ( ) Pa( ) ([Pa] = [N/m 2 ]). 24

25 ( = ) = ( ) ( )

26 p p p p 150 ( ) =( ) ABCD AB CD + AD BC = AC BD A, B, C, D AB CD + AD BC = AC BD ABCD sin t, cos t ( ) (e πi = 1) ( ) () 4πr πr3 r 4πr 2 θ 2πr cos θ S =2 π 2 2πr cos θrdθ =2 π πr2 cos θdθ =4πr 2. π x 2 2=0 π e 26

27 π 10 = lim n ( n ) (x, y, z t x 2 + y 2 + z 2 + t 2 = r 2 ) 1 3= = = , 0.99, 0.999, , , = = A A A 5 A A A

28 1 2 A A A ( ) 15, 16C ( ) () ( ) 28

29 ( ) 2001 () 29

30 30

31 R 3 ( ) l m l, m ( ) ( ) xyz R 3 ( ) () R 3 RP 2 R 3 31

32 RP 2 R P R real P projective 2 ( ) ( ) ( ) ( )

33 X ( ) Y ( ) X Y X ( ) x Y ( ) y y = f(x) ( ) CAD ( ) CAD computer aided design 33

34 L =2πr π r ε ( ) (M. ) ( ) 1, 2, 3,... 34

35 () () () 1, 2, 3 ( ) ( ) ( ) = = 0 =1 0 1 =0 0 35

36 ( ) ( ) NHK ( ) ( ) 36

37 ( ) () ( ) ( ) R 3 Π (R 3 ) Π ( ) R 3 ( ) l m P l m l Q QP m R Q R () 0 l m 37

38 ( ) ( ) () ( ) l m A l B m B A A B B l m B B P l m P BB A A B B ( ) l m l m l m 38

39 I, II, III, IV ( ) T T (x + x )= T (x)+t(x ),T(cx) =ct (x) ( ) ( ) ( ) ( x 2 + y 2 =0) ( ) 39

40 () TEX Povray = (x, y, z, t) C 2 n n ( ( ) ) 0 =1 0 1=0 0 =1 0 (0 )=(0 0) =0 =1 0 1=0 40

41 n n n n n n n n π 1 2 Mac Cayley ( ) (?) ( ) ( ) http// 41

42 () (a 0 : a 1 : a 2 ) a 0 x + a 2 y + a 2 z =0 (a 0 : a 1 : a 2 ) (a 0,a 1,a 2 ) a 0 x + a 2 y + a 2 z =0 ( ) ( ) ( ) P P 42

43 P P P P ( ) RP 2 CP 1 RP 1 RP 2 CP 0 CP 1 ( ) ( ) () 43

44 ( ) n r n r 1 P l m l Q PQ m n ( ) 44

45 ( ) CB FE ABC DEF P, Q, R ( ) ( ) R 4 R 3 ( ) 45

46 x + iy x y C 2 x + iy, u + iv x, u y, v n n e e =2.718 (e x ) = e x e e e = ( ) ( ) ( ) ( ) 46

47 sin 270 = ( ) () ( ) () ( ) Povray Shade Shade ( ) ( ) 47

48 ( ) (?) ( ) 48

49 CM ( ) ( ) ( ) ax 2 + by 2 + cxy + ex + fy + d =0 ( ) ( ) (ax + by + c)(a x + b y + c )=0 49

50 ( ) C C 2 ( CP 2 ) a =0,b=0,c=0 y = Ax + B ( ) x 2 + y2 4 =1 ax 2 + by 2 + cxy + ex + fy+ d =0 a, b, c, e, f, d 1, 1, 0, 0, 0, 1 4 n n (range, ) P, Q, R l, m (1) P (2) l (3) Q (4) 50

51 m (5) R OO OO O O O ( ) O ( ) O O OO OO l, m, n P l,m,n Q l l a, n m b m l c n n d m m e l n f ab, cd, ef x 2 + y 2 z 2 =0 (x, y) (ax + by + c, a x + b y + c,a x + b y + c ) xy (ax + by + c) 2 +(a x + b y + c ) 2 (a x + b y + c ) 2 =0 ( ) xy x 2 =0 x 2 + y 2 = 3 x, y ax 2 +by 2 +cxy +ex+fy+d =0 a >0.b > 0,c= e = f =0,d>0 51

52 l m l m l m?! ( ) ( )!! n n n 1 R n n 1 RP n 1 (2 3 4 ) R n (R n ) R n (R n ) 0 R n (n 1) () 52

53 sin, cos, tan N N x 2 + y 2 =4 x 2 4x + y 2 4y =16 (x 2 + y 2 4) + k(x 2 4x + y 2 4y 16) = 0 k a + bi + cj + dk (a, b, c, d ) i, j, k i 2 = 1,j 2 = 1,k 2 = 1,ij = ji = k, jk = kj = i, ki = ik = j Hamilton CG CAD (isometry) (Euler) (Gauss) () 53

55 ( ) () () 55

56 T V Shade3D-CG Mac Shadepersonal Pro 3D Shade CG TV CM () ( ) 56

57 ( ) ( ) ( ) ( ) t x(t),y(t) (x(t),y(t)) x(t) =a sin t, y(t) =a cos t (x (t),y (t)) (0, 0) ( ) (x(t),y(t)),a t b, (x(a),y(a)) =(x(b),y(b)), (x (t),y (t)) (0, 0) θ(t) (cos θ(t)m sin θ(t)) = (x (t),y (t)) (x (t),y (t)) 1 b 2π a θ (t)dt 0, ±1, ±2, ±3,..., ( ) ( ) ( ) (u, v) (x(u, v),y(u, v),z(u, v)) (3 2 ) 1 0 ( ) 57

58 ( ) ( ) R 3 ax + by + cz =0 z =1 z =1 ax + by + cz =0 ax + by c =0 ax 2 + by 2 + cxy + dx + ey + f =0 ax 2 + by 2 + cxy + dxz + eyz + fz 2 =0 ( ) () ( ) 58

59 ( ) y = x 2 xy y (y ) ( ) ( ) y = tan x y = z tan x z x 2 +y 2 = 1 x, y x = i, y =0 x 2 +y 2 = 1 curve () line () (streight line) (curved line) 59

60 ( n ) ( ) ( ) ( ) ( ) () 60

61 (3+3) 2=12 13 ( (3+3+1) 2 =14 ) ( ) () ( ) CG 3D 1, 2,... 61

62 () ( ) (break through) CG ( ) 62

63 e e iπ = 1 [x] x [ ] CUBE No.1 (? ) () ( ) () 63

64 ( ( ) ) ( ) 64

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (