Date post: | 19-Feb-2018 |
Category: |
Documents |
Upload: | doi-don-doc |
View: | 227 times |
Download: | 0 times |
of 13
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$nh c/a m4t tam gi%c. Chng minh &6ng a2- 2bc * b2- c2
Bi 3: 7i8i c%c 9h;ng t&nh:
a)2 1
1
2##< 2## 2##>
x x x =
b) ,12-
7/23/2019 Mot so de thi HSG toan 8.doc
2/13
D(i c tUn t$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$n thIng BD ED. Chngminh CQ PD .
2
7/23/2019 Mot so de thi HSG toan 8.doc
3/13
Bi 5 ,2 !im): a) Chng minh bt !Ing thc au: 2+x
y
y
x
,+i + V cfng u)
b) Tm gi% t&' nh( nht c/a biu thc 0 =2 2
2 23 5
x y x y
y x y x
+ + +
,+i # V # )
Eha s!t ch@t "HIng hc sinh gii
Bi 1: ,? !im)
1 Cho ba X a b c th(a mn+ + =
+ + =
2 2 2
a b c 0
a b c 2009 Tnh = + +4 4 4A a b c .
2 Cho ba X V e th(a mn x y z 3+ + = . Tm gi% t&' n nht c/a B xy yz zx= + + .
Bi 2: ,2 !im)
Cho !a thc ( )= + +2f x x px q+i p Z,q Z . Chng minh &6ng tUn t$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
7/23/2019 Mot so de thi HSG toan 8.doc
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 !)
?
7/23/2019 Mot so de thi HSG toan 8.doc
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
7/23/2019 Mot so de thi HSG toan 8.doc
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.
#
7/23/2019 Mot so de thi HSG toan 8.doc
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 "
<
7/23/2019 Mot so de thi HSG toan 8.doc
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 $i X 1.
1##
1##
12Bi
1CMu 4i ung !im
1. 2#1.1 ,#
7/23/2019 Mot so de thi HSG toan 8.doc
9/13
Bi 1: ,? !im)a) i^u i_n: V@ V # ,1 !im)
b) A = 2,-V) ,2 !im)c) Cn chr &a gi% t&' n nht c/a A T ! tm !`c tt c8 c%c gi% t&' nguVZn ;ng c/a A
- T ,gt): 32- V2- 2 d 2V = 1 22- 2V - 2d 2V - V2- 2, d V) = 1 2, - V) - , d V)2- 2, d V) - 1 = 2 A - , d V - 1)2= 2 A = 2 d , d V - 1)2 2 ,o , d V - 1) 0 ,+i mi @ V) A 2. ,#5!)
- A = 2 hi ( )
x y 1 0
2x x y 2
x y%y 0
+ = + =
1x
2
3y
2
= =
- A = 1 hi ( )
2)x y 1* 1
2x x y 1
x y%y 0
+ = + =
T ! chr cn chr &a !`c m4t cw9 gi% t&' c/a + V chIng h$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= # ,#
7/23/2019 Mot so de thi HSG toan 8.doc
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. ,#
7/23/2019 Mot so de thi HSG toan 8.doc
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) ,#
7/23/2019 Mot so de thi HSG toan 8.doc
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 $i #5
Bi 5 : O !im~ a : 2 !im
12
7/23/2019 Mot so de thi HSG toan 8.doc
13/13
Phng gi! 34c N& 6& t7Eim 'Ong
Fim t&a cht `ng Dc inh gi(i Pm hc 2## d 2##>KWn To%n 9
ThHi gian 15# 9ht d FhWng thHi gian giao !^^ chnh thc
Bi 1 ,3 !im)Tnh gi% t&' biu thc
? ? ?
? ? ? ?
1 1 1 11- 3 5 .......... 2>
? ? ? ?A=
1 1 1 12 - ? O .......... 3#
? ? ? ?
+ + +
+ + +
Bi 2 ,? !im)
asi mi X a b c hWng !ng thHi b6ng nhau hV Chng minh
a2- b2- c2d ab d ac d bc #
b Cho a - b - c = 2##>. Chng minh &6ng
3 3 3
2 2 2
a - b - c " 3abc= 2##>
a - b - c " ab " ac " bc
Bi 3 ,? !im). Cho a # b # @ a + b th8o mn 2a - 3b O + 2a - b ?. Tm gi% t&' n nht + gi%
t&' nh( nht c/a biu thc A = a2
d 2a d b
Bi ? ,3 !im). 7i8i Bi To%nb6ng c%ch Q9 9h;ng t&nh
K4t W tW !i T A !Sn B . Cng m4t c W tW th hai !i T B !Sn A +; +yn tXc b6ng2
3+yn tXc
c/a W tW th nht . jau 5 giH chng gk9 nhau. D%i mi W tW !i c8 ung !Hng AB th mHt bao Mu[
Bi 5 ,O !im). Cho tam gi%c ABC c ba gc nhn c%c !im K th t t&ung !im c/a BC + AC.C%c !Hng t&ung t&c c/a BC + AC cJt nhau T$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