Mot so de thi HSG toan 8.doc

7/23/2019 Mot so de thi HSG toan 8.doc

1/13

Phng GD- T thi hc sinh gii nm hc 2008 - 2009 Can Lc Mn: T!n "#$ 8

Th%i gian "&m '&i (20 $h)t

Bi 1. Cho biu thc: A =5 2

3 2

x x

x x x

+ +

a) Rt gn biu thc Ab) Tm ! A " #A=c) Tm ! A !$t gi% t&' nh( nht.Bi 2: a) Cho a * b * # + 2, a2- b2) = 5ab

Tnh gi% t&' c/a biu thc: 0 =3

2

a b

a b

+

b) Cho a b c 4 i 3 c$nh c/a m4t tam gi%c. Chng minh &6ng a2- 2bc * b2- c2

Bi 3: 7i8i c%c 9h;ng t&nh:

a)2 1

1

2##< 2## 2##>

x x x =

b) ,12-


2/13

D(i c tUn t$i haV hWng c%c X c hai chY X c th +iSt cPn bQc hai c/a chng i $ng nht&Zn + m4t X nguVZn[

2. Tm X t&ong 9hL9 chia c/a biu thc ( ) ( ) ( ) ( )2 ? O 2##x x x x+ + + + + cho !a thc2 1# 21x x+ + .

Bi ?: ,? !im)

Cho tam gi%c ABC +uWng T$i A ,AC * AB) !Hng cao AD ,D BC). T&Zn tia DC V !im E aocho DE = DA. Hng +uWng gc +i BC T$i E cJt AC T$i \.

1. Chng minh &6ng hai tam gi%c B\C + AEC !Ung $ng. Tnh 4 i o$n B\ th]o m AB= .2. 7i K t&ung !im c/a o$n B\. Chng minh &6ng hai tam gi%c BDK + B\C !Ung $ng.

Tnh X !o c/a gc ADK

3. Tia AK cJt BC T$i 7. Chng minh:GB HD

BC AH HC=

+.

DST

Phng Gi! 34c - & t7T,.

?????

thi chn hc sinh gii c@$ h/ABn,m hc 2008 - 2009

Mn: T!n8Th%i gian "&m '&i: (20 $h)t 1hng EF th%i gian gia 6

Bi 1 ,? !im): Cho biu thc

+++

=

222222 2

11:V

?VAxxyyxyx

a) Tm !i^u i_n c/a V ! gi% t&' c/a A !`c %c !'nh. b) Rt gn A. c) Zu @ V c%c X thc m cho A %c !'nh + tho8 mn: 32- V2- 2 d 2V = 1 hV tm tt c8 c%cgi% t&' nguVZn ;ng c/a A[Bi 2 ,? !im):

a) 7i8i 9h;ng t&nh :

=2

??

>3

33

1#?

22

115

11 ++

+=

++

+ xxxx

b) Tm c%c X V e biSt : 2- V2- e2= V - Ve - e+ 2#1#2##>2##>2##> 3=++ zyx

Bi 3 ,3 !im): Chng minh &6ng +i mi n th n5+ n uWn c chY X tQn cfng giXng nhau.Bi ? ,< !im): Cho tam gi%c ABC +uWng t$i A. V m4t !im K bt t&Zn c$nh AC. T C +G m4t!Hng thIng +uWng gc +i tia BK !Hng thIng nV cJt tia BK t$i E cJt tia BA t$i \. a) Chng minh: \A.\B = \E.\C + EAD ECB= b) Cho #12#BMC= +

23OAED

S cm= . Tnh j\BC[ c) Chng minh &6ng hi !im K i chuVn t&Zn c$nh AC th tkng BK.BE - CK.CA c gi% t&' hWng!ki.

) FDH BC ( )H BC . 7i 0 l n `t t&ung !im c/a c%c !o$n thIng BD ED. Chngminh CQ PD .

2


3/13

Bi 5 ,2 !im): a) Chng minh bt !Ing thc au: 2+x

y

y

x

,+i + V cfng u)

b) Tm gi% t&' nh( nht c/a biu thc 0 =2 2

2 23 5

x y x y

y x y x

+ + +

,+i # V # )

Eha s!t ch@t "HIng hc sinh gii

Bi 1: ,? !im)

1 Cho ba X a b c th(a mn+ + =

+ + =

2 2 2

a b c 0

a b c 2009 Tnh = + +4 4 4A a b c .

2 Cho ba X V e th(a mn x y z 3+ + = . Tm gi% t&' n nht c/a B xy yz zx= + + .

Bi 2: ,2 !im)

Cho !a thc ( )= + +2f x x px q+i p Z,q Z . Chng minh &6ng tUn t$i X nguVZn !

( ) ( ) ( )=f k f 2008 .f 2009 .

Bi 3: ,? !im)

1 Tm c%c X nguVZn ;ng V th(a mn 3xy x 15y 44 0+ + = .

2 Cho X t nhiZn ( )= 20099a 2 b tkng c%c chY X c/a a c tkng c%c chY X c/a b

tkng c%c chY X c/a c. Tnh .Bi ?: ,3 !im)

Cho 9h;ng t&nh2x m x 1

3x 2 x 2

+ =

+ Tm m ! 9h;ng t&nh c nghi_m ;ng.

Bi 5: ,3 !im)

Cho hnh thoi ABCE c c$nh b6ng !Hng chLo AC t&Zn tia !Xi c/a tia AE V !im \ !Hng

thIng \B cJt !Hng thIng EC T$i C\ cJt T$i p. Chng minh AEC !Ung $ng CAF Tnh

EOF .

Bi O: ,3 !im)

Cho tam gi%c ABC 9hMn gi%c t&ong gc A cJt BC T$i E t&Zn c%c o$n thIng EB EC n

`t V c%c !im \ + ao cho EAD FAD= . Chng minh &6ng: =2

2

BE BF AB

CE CF AC.

Bi


4/13

T&Zn b8ng c c%c X t nhiZn t 1 !Sn 2## ngHi ta m nh au V &a hai X bt + thaV

b6ng hi_u c/a chng c m nh +QV !Sn hi cqn m4t X t&Zn b8ng . C th m ! t&Zn b8ng chr cqn $i

X 1 !`c hWng[ 7i8i thch.

..........................................DST...........................................

Mn T!n (J0 $h)t 1hng EF th%i gian gia 6

CMu 1,5!im) Tm X t nhiZn n ! :a) A=n3"n2-n"1 X nguVZn tX.

b) B= 22O23

2

23?

+ +++ nnnnn

c gi% t&' m4t X nguVZn .

c) E=n5"n-2 X chnh 9h;ng . ,n )2CMu 2: ,5 !im) Chng minh &6ng :

a) 1111

=++

+++

+++ cac

c

bbc

b

aab

abiSt abc=1

b) si a-b-c=# th a?-b?-c?=2,ab-bc-ca)2

c)c

a

a

b

b

c

a

c

c

b

b

a++++

2

2

2

2

2

2

CMu 3: ,5 !im) 7i8i c%c 9h;ng t&nh au:

a) O=25?

=?

132

=O

21?

=

+

+ xxx

b) 2,"1)2,?"1)=>c) 2"V2-2"?V"1#=# +i V nguVZn ;ng.

CMu ?: ,5 !im).Cho hnh thang ABCE ,ABCE) p giao !im hai !Hng chLo. lua p !HngthIng ong ong +i AB cJt EA T$i \ cJt BC T$i .

a) Chng minh &6ng : i_n tch tam gi%c ApE b6ng i_n tch tam gi%c BpC.

b) Chng minh :EFCDAB

211=+

c) 7i F !im bt thu4c p\.Zu c%ch ng !Hng thIng !i ua F + chia !Wv i_n tch tam gi%cE\.

Mn : T!n (20 $h)t 1hng EF th%i gian gia 6

Bi 1: ,1 !)Cho biSt a"b=< Tnh gi% t&' c/a biu thc: a,a-2)-b,b"2)"2abBi 2: ,1 !)Chng minh &6ng biu thc au uWn uWn ;ng ,howc Mm) +i mi gi% t&' c/a biSn ! cho : "a2-a"3Bi 3: ,1 !) Chng minh &6ng Zu m4t t gi%c c tMm !Xi ng th t gi%c ! hnh bnh hnh.

Bi ?: ,2 !)

?


5/13

Tm gi% t&' nh( nht c/a biu thc au:5=?

22 + xx

Bi 5: ,2 !)Chng minh &6ng c%c X t nhiZn c $ng 29-1 t&ong ! 9 X nguVZn tX chr c m4t X Q9 9h;ng

c/a m4t X t nhiZn h%c.Tm X !.

Bi O: ,2 !)Cho hnh thang ABCE c !%V n AE !Hng chLo AC +uWng gc +i c$nh bZn CE CADBAC= .Tnh AE Zu chu +i c/a hnh thang b6ng 2# cm + gc E b6ng O##.Bi


6/13

Eu = 8V &a hi

y 1 0

y 3x 0 x y z 1

2

x y z 0

= + = = = =

+ + =

sQV gi% t&' n nht c/a B 3 hi = V = e = 1

#25

2 Cho !a thc ( )= + +2f x x px q+i p Z,q Z . Chng minh &6ng tUn t$i X nguVZn !

( ) ( ) ( )=f k f 2008 .f 2009 .

2##

( ) ( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( )

2

2 2

2

2

2

x x f x x p f x x q

f x 2.x.f x x p.f x p.x q

f x f x 2x p x px q

f x x px q 2x p 1

f x x 1 p x 1 q f x f x 1

+ = + + + + = + + + + +

= + + + + + = + + + + + = + + + + = +

si = 2## chn ( )k f 2008 2008= + !

juV &a ( ) ( ) ( )f k f 2008 .f 2009=

125

#5#

#253.1 Tm c%c X nguVZn ;ng V th(a mn 3xy x 15y 44 0+ + = . 2##

( ) ( )3xy x 15y 44 0 x 5 3y 1 49+ + = + + = V nguVZn ;ng o +QV - 5 3V - 1 nguVZn ;ng + n h;n 1.

Th(a mn VZu cu Bi To%n hi - 5 3V - 1 c n h;n 1 c/a ?> nZn c:x 5 " x 2

3y 1 " y 2

+ = =

+ = = sQV 9h;ng t&nh c nghi_m nguVZn = V = 2.

#


7/13

0h;ng t&nh c nghi_m ;ng

2m 142

1 mm 42m 14

21 m 1 m "

2m 14

01 m

<

sQV th(a mn VZu cu Bi To%nhim 4

1 m "

<


8/13

thaV b6ng hi_u c/a chng c m nh +QV !Sn hi cqn m4t X t&Zn b8ng th ng $i. C th

m ! t&Zn b8ng chr cqn $i X 1 !`c hWng[ 7i8i thch.Fhi thaV hai X a b byi hi_u hi_u hai X th Tnh chtt chIn c/a tkng c%c X c t&Zn b8nghWng !ki.

K( )2008. 2008 1

( 1 2 3 ... 2008 1004.2009 0m$#22

+= + + + + = = @ 1 1m$#2 o

+QV t&Zn b8ng hWng th chr cqn $i X 1.

1##

1##

12Bi

1CMu 4i ung !im

1. 2#1.1 ,#


9/13

Bi 1: ,? !im)a) i^u i_n: V@ V # ,1 !im)

b) A = 2,-V) ,2 !im)c) Cn chr &a gi% t&' n nht c/a A T ! tm !`c tt c8 c%c gi% t&' nguVZn ;ng c/a A

- T ,gt): 32- V2- 2 d 2V = 1 22- 2V - 2d 2V - V2- 2, d V) = 1 2, - V) - , d V)2- 2, d V) - 1 = 2 A - , d V - 1)2= 2 A = 2 d , d V - 1)2 2 ,o , d V - 1) 0 ,+i mi @ V) A 2. ,#5!)

- A = 2 hi ( )

x y 1 0

2x x y 2

x y%y 0

+ = + =

1x

2

3y

2

= =

- A = 1 hi ( )

2)x y 1* 1

2x x y 1

x y%y 0

+ = + =

T ! chr cn chr &a !`c m4t cw9 gi% t&' c/a + V chIng h$n:

2 1x

2

2 3y

2

=

+ =

- sQV A chr c th c 2 gi% t&' nguVZn ;ng : A = 1@ A = 2 ,#5 !im)

Bi 2: ,? !im)

a)x 11 x 22 x 33 x 44

115 104 93 82

+ + + ++ = +

x 11 x 22 x 33 x 44) 1* ) 1* ) 1* ) 1*

115 104 93 82

+ + + + + + + = + + ,1 !im)

x 126 x 126 x 126 x 126

115 104 93 82

+ + + + + = +

x 126 x 126 x 126 x 1260

115 104 93 82

+ + + + + = ,#5 !im)

...x 126 0 + =x 126 = ,#5 !im)

b) 2- V2 - e2= V - Ve - e 22-2V2- 2e2d 2V d 2Ve d 2e = #

,"V)2- ,V"e)2- ,e")2= # ,#


10/13

x y z = = 2##>= V2##>= e2##> ,#= 32#1#

e2##>= 32##>

e = 3sQV = V = e = 3 ,#5 !im)

Bi 3 ,3 !im)Cn Chng minh: n5d n 1#" Chng minh : n5 " n 2 n5d n = n,n2d 1),n2- 1) = n,n d 1),n - 1),n2- 1) 2 ,+ n,n d 1) tch c/a hai X nguVZn iZn tiS9)

,1 !im)" Chng minh: n5d n 5n5 " n = ... = n, n " 1 ), n - 1), n 2d ? - 5) = n, n d 1 ) ,n - 1),n d 2) , n - 2 ) - 5n, n d 1), n - 1 )

~ uQn }n !Sn tkng t&Zn chia DST cho 5 ,125 !im)" s , 2 @ 5 ) = 1 nZn n5d n 2.5 tc n5d n 1#juV &a n5+ n c chY X tQn cng giXng nhau. ,#


11/13

" T BMC= 12#o AMB = O#

o ABM = 3#o #5 !im

" xzt \EB +uWng T$i E c ,B = 3#o

\E =1

2

\B 1

2

ED

EB

= #5 !im

" ~ uQn cho

2

EAD

ECB

S ED

S EB

=

T ! j\CB= 1?? cm2 #5 !im

CMu c: 15 !im" Chng minh BKv !Ung $ng +i BCE ,gg) #5 !im" Chng minh CK.CA = Cv.BC #5 !im" Chng minh BK.BE - CK.CA = BC2c gi% t&' hWng !ki #5 !imC%ch 2: C th biSn !ki BK.BE - CK.CA = AB2- AC2= BC2

CMu : 2 !im" Chng minh BDE !Ung $ng +i EDC ,gg) #5 !im

2

2

BH BD BP BD BP BD

DH DC DQ DC DQ DC = = = #5 !im

" Chng minh E0B !Ung $ng +i ClE ,cgc)

>#o

BDP DCQCQ PD

ma BDP PDC

= + =

1 !im

Bi 5: ,2 !im)

a) + V cfng u nZn V * # o ! + x y

2y x

,|) + 2 2x y 2xy

2)x y* 0 ,||). Bt !Ing thc ,||) uWn !ng uV &a b!t ,|) !ng ,!9cm) ,#


12/13

Mn T!n84i ung !im

Bi 1 ,3 !im)

C a?-1

?

=2

2 2 2 21 1 1a a

2 2 2

a a a a + = + + +

1#

Fhi cho a c%c gi% t&' T 1 !Sn 3# th:TN thc +iSt !`c thnh

,12-1-1

2),12"1-

1

2),32-3-

1

2),32"3-

1

2).,2>2-2>-

1

2),2>2"2>-

1

2)

#5

K}u thc +iSt !`c thnh

,22-2-1

2),22"2-

1

2),?2-?-

1

2),?2"?-

1

2),3#2-3#-

1

2),3#2"3#-

1

2)

#5

Kwt h%c ,-1)2",-1)-1

2=.=2--

1

2

#5

Zn A=

2

2

11 1 121 1O13# 3#2

+=

+ +

#5

Bi 2: ? !im~ a: 2 !im"C ~ tyng t%ch thZm bt howc th hi_n !`c nh +QV ! N {ng bc au #5"siSt !`c $ng bnh 9h;ng c/a m4t hi_u #5" siSt !`c bnh 9h;ng c/a m4t hi_u #5" Q9 uQn + St uQn !`c #5

~ b: 2 !im0hMn tch !`c tN thc thnh nhMn TN 1#Rt gn + St uQn !`c 1#Bi 3 : ? !im|T 2a - b ? + b # ta c 2a ? haV a 2 1# Eo ! A=a2 " 2a " b # #5

Zn gi% t&' n nht c/a A # hi a=2+ b=# #5

| T 2a - 3b O uV &a b 2 "2

3a

1#

Eo ! A a2d 2a d 2 -2

3

a = ,2

3

a )2 "22

>

"22

>

#5

sQV A c gi% t&' nh( nht "22

>hi a =

2

3+ b =

2

3

#5

Bi ? : 3 !im" Chn Xn + !$t !i^u i_n !`c #25

" Biu th' !`c mi !$i `ng th]o Xn + X i_u ! biSt,? !$i `ng) #25 ?" Q9 !`c 9h;ng t&nh #25" 7i8i !`c 9h;ng t&nh #5" !Xi chiSu + t&8 Hi !`c thHi gian c/a 1 W tW #5" Q9 uQn Tnh + t&8 Hi !`c thHi gian c/a W tW cqn $i #5

Bi 5 : O !im~ a : 2 !im

12


13/13

Phng gi! 34c N& 6& t7Eim 'Ong

Fim t&a cht `ng Dc inh gi(i Pm hc 2## d 2##>KWn To%n 9

ThHi gian 15# 9ht d FhWng thHi gian giao !^^ chnh thc

Bi 1 ,3 !im)Tnh gi% t&' biu thc

? ? ?

? ? ? ?

1 1 1 11- 3 5 .......... 2>

? ? ? ?A=

1 1 1 12 - ? O .......... 3#

? ? ? ?

+ + +

+ + +

Bi 2 ,? !im)

asi mi X a b c hWng !ng thHi b6ng nhau hV Chng minh

a2- b2- c2d ab d ac d bc #

b Cho a - b - c = 2##>. Chng minh &6ng

3 3 3

2 2 2

a - b - c " 3abc= 2##>

a - b - c " ab " ac " bc

Bi 3 ,? !im). Cho a # b # @ a + b th8o mn 2a - 3b O + 2a - b ?. Tm gi% t&' n nht + gi%

t&' nh( nht c/a biu thc A = a2

d 2a d b

Bi ? ,3 !im). 7i8i Bi To%nb6ng c%ch Q9 9h;ng t&nh

K4t W tW !i T A !Sn B . Cng m4t c W tW th hai !i T B !Sn A +; +yn tXc b6ng2

3+yn tXc

c/a W tW th nht . jau 5 giH chng gk9 nhau. D%i mi W tW !i c8 ung !Hng AB th mHt bao Mu[

Bi 5 ,O !im). Cho tam gi%c ABC c ba gc nhn c%c !im K th t t&ung !im c/a BC + AC.C%c !Hng t&ung t&c c/a BC + AC cJt nhau T$i p . lua A !Hng thIng ong ong +i pK ua B !Hng thIng ong ong +i p chng cJt nhau T$i D

a) Xi K ADB !Ung $ng +i tam gi%c no [b) 7i 7 t&ng tMm ABC Chng minh AD7 !Ung $ng +i Kp7 [c) Chng minh ba !im K p 7 thIng hng [

13

Mot so de thi HSG toan 8.doc

Documents