# 年43屆國際數學奧林匹亞競賽試題與解答4.doc

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米高
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1 00 43 * ** *** **** ** * ** *** **** (IM) (Jury Meetg) ( ) 0 ~ 6 6 (Proble Selecto Cottee) IM IM 7 5

2 43 = d d + d d + L + d d 4 T y + y < R yzt (, y) T 3 ) (, y T Γ, Γ, L, Γ ' y' y ( ', y' ),, L, X- y Y- : X- Y- BC Γ A Γ o o 0 < AB < 0 C AB A AC JA Γ E a y F J CEF, a 0 a L, a b 0 b L, b 3 : aa La = bb Lb a a a + a + a 3 (a) : < (b) R R f ( f + f ( z))( f ( y) + f ( t)) = f ( y zt ) + f ( t + yz) < j ( ) π 4 j 43 b aa La = bb Lb a 0 a L, a b 0 b L, b 4 d, d, L, d = d < d < L < d = (, y) + y= 53

3 0 (colus) a a 0 L, a (rows) b 0 (, y+ ), b L, b (, y ) a 0 a L, a b 0 b L, a = b = b (, ) (, y ) ( +, y ) ( +, y ), ( ) (, ) (, y ) b a y (, y) < (, y) a0 a L, a b b + L, b a + a + L, a b 0 b L, a = = 0, a b L b 0 b L, b 54 a = b = 0 a b a > 0 (, y) ( colu) (, y ) (, y+ ) a b y ( ) ( y, ) b > 0 (y row) (row) (, y) a L, a b 0 y y b L, b (, y ) + y T (, y) a = by= y { X} { Y} a b y f a L, a b 0 b L, b a a b y b a y = b y a L, a X Y T b 0 b L, b a y y

4 X f :{ X } { Y} f :{ Y} { X} X 45 ( Y :flp) Y Y 45 X f ABC A B C I ABC φ A B C I ABC A B C A B C I A B C φ + ABC M ABC I A B C φ CA ECF A= C A= AB= AC ABC M X ( AJ = = E= AF 0 ) Y ( y 0 ) JFE= JFA EFA f = AJF ECA f ABC ABC I ABC φ JF EFC J CEF ABC M ABC I A B C φ Γ J 0 CEF ABC y0 0 M 55 E B A EAF JA AJ EAF = AJF JCF = JFC. AJ M EF Γ Γ M J A C EF J J 0 CA CA ECF A M A= E= EA= AF = F AEAF J F C

5 EF = 0 J CEF CEF EJ F = 90 + ECF = 90 + EF = 0 J J 0 Γ Γ CA J = J 0 EF A EJF= 0 EF = EJ + JF EJ JF cos0 A 3 (cos(70 60),s(70 60)) =, E= EA= F= FA= A 3 AE =AF π (cos( ),s( )) = F = ( ) (, ) 3 π π ECJ = FCJ = = JC 3 6 ECF ECF = 60 J CEF EJF = π JEF JFE = π + = π = 0. ( CEF CFE ) ( π ECF ) JC ECF EJF = 0 J CEF B E 0 EJF = JJ J JC J JA ()J JC 0 = EJA+ AJF > EJA + AJF = 0 G J J' F C ()J JA 0 = EJA+ AJF < EJA + AJF = 0 JJ J CEF EJF = 0 EJF EF = EJ + JF + EJ JF. EF = EJ + JF + EJ JF A (0, ) (0,0) AB = < 0 AE=AF= 60 EF EF = ( 3 ) = 3C ( 3 π cos,s( 3 π + π + π ) ) π π ( cos( ),s( ) ) ( s,cos ) = = AJ : JC = AJ : JC = A J s AJ : C J s JC = s AJ :s JC. // A J AJ = A= ( π ) A π π π = A= AB = 4 AJ + JC = AC = π JC = π

6 π π AJ : JC = s( ):s( ) = cos :cos ( 4 4 ) ( 4 ) = cos : 4cos 3cos = : 4cos 3 = : cos. J s + (cos )0cos + (cos )(), cos + cos + s cos + cos =, cos cos s cos cos + cos =, cos cos = s,cos. 3 s ( cos ) EJ = + + = + 3s cos 3 s ( cos ) JF = + = 3s cos EJ JF = (+ 3s cos ) (+ 3s cos ) = ( cos ) 3s = (cos ). 0< < 60 cos > cos > 0. EJ JF = cos + + = + = = EJ JF EJ JF 4 cos cos 3 EF. J CEF 0 < AB < 0 I AC Γ I A I A I = B I ABC 57 A BAC I = B BA= AC I = B = C Γ AB = ACB CB = CA = BAC AB = ACB IB = π BI ( I = B ) IBC = IB CB= ( π BI ) BAC C= ( π BI BAC) = ABC BI ABC I ABC AF AE E F A G A HI EF A ABC Γ BAC J AC AF = F B H F B AE = E F= A= E F = A= E = AF = AE A G A I J E I C C

7 AF ECF = EF A EF CA HI // A C AB A = B BH = IH HI // A H AIH = AI AJ= AC+ IC= BH + AC AC= CA AC = AB = B AJ = BH + B= H = AI = AJ A= AJ A= F = E EFJ = JFC AJ = E = F = AE = AF AJ J CEF B H E () J AC E AB F ACB AB B = A AC = AB= A = B M A AC// A J AJ Γ ra ga = 0 r R AJ = = R AB < 0 AC > 60 A= C J F I C 58 AC AC AC > 60 > AC AC > A= RAJ = R J AC () A M EF A AF = F = R = E = AE = AJ ( F AC )AE = AF AE=AF JCE = JCF AF = AJ AFJ = AJF AFE + EFJ = JFC + FCJ AE + EFJ = JFC + AF = AE AEJ = AJE AEF + FEJ = CEJ + JCE AF + FEJ = CEJ + AE FEJ = CEJ J CEF < f = + g () = + [ Z] g f g( ) = q + r g deg( r) < deg( g) r g 0 a f ( a) g ( a) a a( f a) ga

8 [,] [0,] f g(0,) g( ) f f() ( g), = +, = α α < h ( φ) = φ + φ = 0φ = 0.68 L f (0,) f ( φ) < h( φ ) = 0 = f ( α) α > φ αα ( )( α + α ) 0 α φ (od d ) = + = 3 5 = a 5 + a = ( a 3 + a )( a a+ ) (a) d, d, L, d (, ) = (5,3) = d < d < L < d = d = = ( + ) = ( + ) + ( )( + ) d = d , + α (0,) ( f ) ( g) [0,] α + + α = 0 + ( + + α + α = 0 59 = ) = α + α α + α = = 3 = + = 5 (a) d d = < α = α α = α d d + ( d d < + = ) < d d d < + (b) d = p, d =, d = p < p = p = (, ) = (5,3) = p a = d = g() = + 3 =. > > d d = p < ( ) 3 d (od d ) < (od d ) = < < p p = d 3 d = 3 3 = d 3 L 3 d dd = dd = L = dd = +

9 = dd + dd + L+ d d L+ ( ( )( ) L ) ( ) ( 3) L ( 3 ) ( ) < = = < (b) (b) = = d = d = = dd = p p = dd + dd + L+ d d d d = = d = pq = pd = q p p = p = d 3 p p dd + d d = p+ q > q d d L, d = p = p d = d < d < L < d = d = p = dd = p = d = p d = = d p = q > q= d > d > L > d > d = y = z = 0 < = d ( f + f ( z))( f ( y) + f ( t)) > L L (*) = f ( y zt) + f ( t + yz) Nd d L, d f (0)( f (0) + f ( t)) = f (0) f ( 0) = 0 = d < d < L < d = > > d d + d + L d + = =,, L, ; dd + = =,, L, d = d + =,, L, + (a) = dd + L+ dd + dd 3 + L+ + ( L ( )( ) ( )( ) ) ( ) L ( ) ( ) ( ) = = = < d d > ( < )d p d p p d = p 3 p d f ( t) = f (0) f ( 0) 0 f ( 0) = f (0) f ( 0) = f ( t) = 0 f ( 0) = z = t = 0 (*) f ( y) = f f ( y) L L(**) f = f () f = 0 f = 0 (**) f = f () f = 0 f = = 0, y = t = (*) f ( z) = f ( z) + f ( z), 60

10 f ( z) = f ( z) f u 0 u = v v (**) f ( u) = f ( v ) = f ( v) 0 f R f 0 t =, z = y (*)( f + f ( y)) = f ( + y ) a b 0 a b = s, b = r f ( a) = f ( r + s ) f ( r) = f ( r ) = f ( b) f ( a) f ( b) y = z = t = (*) f = (**) = f () = f ( ) = f f = f f = f = f ( r) = r (B) ff () = f() = 0 f = c f ( c) > c d f ( c) > d > c f ( c) > f ( d) > c f c d d > c f( )=(f()) N = f()=f() = f ( c) < c f f( )=(f()) =+ f = (B) f f = f( + )=f( )=f( ) f()=(f()) f()=(f()) + f = 0 f = f( ) = ( f ) N ( f + f ( z))( f ( y) + f ( t)) = f ( y zt) + f ( t + yz), y, z, t R y t 0 = = (A) (A) f (0) f + f ( z) = f(0) 6 f (0) 0 f + f() z =, z R z = f( ) = + + = + = (0) f = 0z = t = 0 (A) y = f f( y) = f( y), y R (B) (B) f f() = f R() f ( f ) = 0 R 0 0 = 0+ 0= 0 ( f + ) = f ( ) + f ( + ) f () = = y = 0 t = (A) f() z = f( z) z R, (C) f y = (B) f = ( f ) R f () 0 0 f f 0 R ( ) f() > 0 > 0 (),y,z,t >0 ( f ( )) = f (( ) ) = f ( ) = ( f ) = ( f ) r Q r f( ) r r > 0 r r ( f ) f ( ) = r r r f( ) = ( f ( )) = ( f) 0 0 f ( ) = f() = = ( f) (B) r r f( ) = ( f) r Q (E) = y= z= t= (A)

11 (+ )(+ ) = 0 + f () () f = 4 ( f + f ( z))( f ( y) + f ( t)) (E) r r r f = 4 = r Q (F) z = yt = (A) f + f y = f y y + f + y = f + y () f() + f() y = ( f) + ffy + ( f( y)) () f y f f y f f ( + ) = + > ( ) =, y 0 > y > 0 f > f( y) > 0 (G) f = > 0 0 w > w f ( w) = w w > w log w log w (log w log w ) > > f (0) = 0. log w log w > log w > > log w log w > > log w w > > w f( z) f( y) = f( yz) () (F) = f w = f( w) () t = y( f z) f ( y) = f( zy) w > f( w) > f (G) () f = f ( ) w > w > w! Z N < w w w > > f > f( w) > w w > f() = 0 0! > f f () f () = f (0) = 0f = R() f () f( ) = 0 R ( + z )( y + t ) = ( y zt) + ( t+ yz) : f( ) = R f( ) = 0 Rf = R f () = 4y = z= t= (A) 6 = f ( y zt) + f ( t + yz), y, z, t R (A) = y= z= t = 0 (A) f(0) f(0) = f(0) (0) f = 0 f (0) = f (0) = = z = 0 (A) f(0) + f(0) f( y) + f() t = f(0) + f(0), ( f y) + ft () = yt, R y = t f () t = t R + + = + = f( ) = R = y = 0 (A) f( z) f() t = f( zt) () = t = 0 (A) Rf ()( f a) f( b) = f( ab), ab, R () f f = f R 0 0 = 0+ 0= 0( f ) = 0 R () f () = = y= z= t= (A) f() f() = f(0) + f() f + f () = f ( ) + f( + )

12 f ( + ) = f + f ( ) R f = f( y zt) + f ( t+ yz) = y zt + t+ yz N =, =, = + f () f f ( ) ( ) + = + = + = + f = N = R + pq, N q p q ( p ) f () f ( p) = f p = f( q) f ( q) q f ( p) p f = = = f = R + z t = (A) = y f + f ( y) = f(0) + f ( + y ) = f( + y ), ( f) = f( ), y R + PQ = λ RS = µ f = ( f ) > 0 QSP = λ f( y) = ( f ( y) ) > 0 ( RPS = µ RXS = α = λ + µ R + f ( ) = 0 PQ+ RS = 4αρ f () = f f = 0 ) = y + t + z y + zt, = y + zt + t + y z f( ) = R f( ) = 0 Rf = R Ω ρ PR QS X PXQ= RXS = α PQ+ RS = 4αρ Ω f ( + y ) > f ( ), y R + f 0 Q X R r R + f() r r f ( r) f() r < r > r (a)f () r > r f () r = df () r = d d > r q Q + r < q < d f() r < f( q) = q < d f( r) < d (b)f ( r) < r, f( r) = d f () r = d d < r q Q +, d < q < r ( f q ) < f( r) q < d q < d f() r = r r R + f f = R ( f() + f() z )( f( y) + f() t ) = ( + z )( y + t ) 63 P ρ Ω Γ, Γ, L, Γ Γ Γ j j Γ Γ j PR QS α ( ) = csc α α sα = j j S

13 PQ+ RS = 4αρ 8 ρ PQ PQ+ + RS RS 8ρ j j j Ω < j j ( ) π 4 S R T Ω TU Ω T( ) T Γ Γ U Γ Γ Γ 3 Γ 4, L ( ) Γ Γ Γ 3 Γ L, Γ Γ P T Q T Ω ( ) πρ 64

SIK) 者, 需實施 1 年以上, 經體格檢查無後遺症者 5. 身體任何部分有刺青紋身穿耳洞者, 不得報考, 各項檢查結果須符合體位區分標準常備役體位二在校軍訓成績總平均 70 分

SIK) 者, 需實施 1 年以上, 經體格檢查無後遺症者 5. 身體任何部分有刺青紋身穿耳洞者, 不得報考, 各項檢查結果須符合體位區分標準常備役體位二在校軍訓成績總平均 70 分民國 102 年大專程度義務役預備軍官預備士官考選簡章壹依據 : 依民國 102 年大專程度義務役預備軍官預備士官考選計畫辦理貳考 ( 甄 ) 選對象 : 具中華民國國籍, 尚未履行兵役義務之役男, 年齡在 32 歲 ( 民國 70 年 1 月 1 日以後出