De thi HSG lop 9 quan hong bang hai phong

PHoNGGrAo DVC vA DAo r4o

KY TrrrcHgt\ Hec srNHcrol cAp euAN

HONG BANG

. ZOt:
r,{AMHQC ZO1Z
DE THI MON: TOAN _ LOP 9

EE CHINH THTTC

Thdi gian." I50 philr (kh6rtg ki giao di1

(Ei rhi gim 0t rrang)
a -

Bii 1:(2.0cli6m).

a. Chirngminh ,irrg ---l--- * --=1,--- +...+
>24
Vl +V: 45+47
,19997
+.,19999
b. Cho ba s6 a, b, c + 0 th6a mdn:
I
I
I
I
- ; V .ar Vr -a + Vtbr - + { 6 =
t [ - ' vt lD

,- , zfva
vc J

6 + 2 J i-

-_t__r

chrmg minh ring trong cdcs6 a, b, c c6 it nh6t mot s6 bing 2l .
Bii 2: (2.5 cli€m)
a. Giii phucrngtrinh y3 = 2qf2y -1-1 .
b. chrmg minh khdng t6n tai cFrygiittrf nguyCn(x; y) thoamdn x, - 2y, = 2013.
Biri 3: (2.0 cti6m).Cho tri gi6c ABCD .o e = 400;D: g00 vir AD : BC. Ggi E, F
lAn luot lir trung di6m cua AB vd CD. Tfnh eFD f
Itdi 4: (2.5 eli6m).Tir di€m A nim ngodi duong tron tAm 0, ke hai ti6p tuy6n AB vd
AC (B vd c ld titip diem). Ducrngthing qua A cet (o) tqi D vd E (D nim gifra A vd
E), k6 day cung EN song songv6i BC, DN cit BC tai I. chirng minh rdng BI : cI.
Bii 5: (1.0 tti6m). cho a,b, c la dg ddi ba canhctram6t tam gi6cthoam6n:
C20t3

62013

* ----;-

a+b-c a-b+c
Hdy x5c dinh dang.cfiatam gidc do.

Az0t3

-a+b+c

-

A2012* 62012 + Qt"'

Gidm thi khorcggiai th{chgi th1m

.


PHoNGcrAo uUC,vA DAo rAo

oAp Ax va BrEUonim
UONroAru- Lope

HONG BANG

tt+Atu
Hec 2orz-2ar3

Bii 1: (2.0eli6m)
>24

'19997+ 19999
b. Cho ba s6 a,

1
;e0 thoa mdn - - L - I

1

^r_ |

1

1

- va

V a V b V c 3

4li*Vu*i6-Jo*
zJi-lze-r24s.
I

_

Chring minh rSng tron g citcs6 a, b, c c6 it nh6tm6t sd bing 27.
D6p rln
.

-

a)Edt{-

1

1

-f--L...+-

1

vl + V3 ,ls + Jt
1

B:.......-r

Js+Js Jt +Jt

Jgggt+ Jgggs

1- . l . , . - j - -

'

1

+Jrooor
Jeeee

Jz"n -Jz"n

Ta c6 -

^ l 2 n + 1+ r / 2 n + 3

n6nA*B:

-r
Jrooor

ViA>Bn6nAt

A*B

2

100-r 99
r2rzo

0,5d

0,5d

b) Ta c6
,^';+ ^r
_#b"
=i1 c>r(v*
*
*
vu"
v;)
#.#.#
1

1

1

e:(V"u*Vu.*V;)-Vabc=o (1)

0o25aI


-"rir(1)vd(z)taco:3(V"b+ Vui * VA)- V"b.+ e($ +
Vu. fiF 2 7: 0

o ( $ - r X V o - r x i 6 - 3 ) = o e V a : 3 h o i cV b = 3h o d{c6 = 3 .

0,541

YQy itnhAtm6t trong ba so a, b, c co mQts6 bing27.

a. Gi6i phuorrgtrinh y3 =2ll2a-y -1
b. chrmg minh kh6ng tdn tai cFrygi|tri nguydn(x; y) th6amdn x2 - 2y2= 2013.

Daty :?112x= ta c6h6:
j[ " ' * l = 2 y- < + {[ * ' * 1 = 2 y

'0',25d

L y '* 1 : 2 x
L t ' y 3= 2 ( y " )
* I:Zy
[ x ,+ I
: [*'

- v ) ( *+' x y +y '* 2 ) -oe 1 ' = ,
t("

0,25d

( v ix 2 + x yy+' + 2 = [ * * + J ' .
!*2,
\

4/

--t

0,5d
2b. (1.0ili6m)

x' =2y' + 2013+ x16,d4tx = 2k+ i;(k e Z)thayvdotac6:

0"25a1

2y' - 4k' + 4k - 2012e y2= 2k2+ 2k-1006 (l)+ y chfin,y = 2t;(t e z)

02sd

Thayx;y viro(1) vd bi6nd6i: 2t2:k(k + t) 503 (2)
X6t thay vr cria (2) ludn chEn;vp cfia (2) td
t6 16vi k(k+t) ch84(Tfch 2 s6
nguydnli6n ti6p).vpy d6u":" crta(2)kh6ngth6xay
ra<+rhbng ta" * ,ap ,o

nguy6n(x; y) th6a mdn: xt - 2y' - 2013

Biri 3: 12,$ai6m).cho ttt giacABCD c6 e :400;fr=g00 ve AD: BC. Goi E. F

l6n lucrtld trung diOmcriaAB vd CD. Tinh gm Z


- VE hinh binh hinh ACKD ta c6:
DCK=ADC=80''
:+ BCK : BCD + DCK =1200
C6 CK: AD : BC n€n tam ei6c CKB cAn
^

tai C :> CKB = 30'

1,0d

Lai co F la trung di€m cria cD ncn F cfrngld trung di€m cua AK
n6n EF ld duong trung binh ctra tam giitc ABK
:> EF songsongv6i BK
^

-^

^

:> EFD = BID = CIK.

.z (+r")"f)

Tt d6 x6t tam gi6c KIC co EFD = 6lR. =70a

1,0t1

Bii 4: (2.5tti6m).Tt di6mA nim ngodidudngtrdn tAmO, ke hai ti6p tuy6nAB vd
AC (B vd c ld ti6p dicm).Dulng thing quaA cat 1o; t4i D vd E (D nim gifraA vd
E), ke d6ycungEN songsongv6i BC, DN cdtBC tai I. chirng minh ring BI : cI.
Dfp 6n
^

Ei6m

^

- Ta c6 DBC = DEC;EN / /BC + BN : cE =+ BDN : enp = ABID.- AECD
_ > -BI
=_

ID

EC
CD

0o5d

(1)

CI EB
Tucrngtu ta c6: - = _
II)BD

0,5d

lLl

Ta c6 AAEC* AACD n€n
EC = AE
EB AE
; tu'ng tu ta cung.o gp =
CD AC
AB;

M e A C : A B n cS
n ==

0,5d

(3)

CD

BD
Tt (1),(2) vd (3) ta c6 BI : CI.

l,Oil

./

Bii 5: (1.0iti6m).Cho a,b, c le d0 ddi ba canhcrjam6ttam gi6cth6am6n:
C2013

6- 2 0 1 3
+

a2013
+--a2012

a*b-c
a-b+c -a+b+c
Hdy x6c dinh d4ng cua tam giSc do.

+b20t2+C'ot'.


_ -'r <*_

Edp 6n
-Tac6:

{u'ott

-b20r')(u -b)>

06

u z o t+zb 2 0 i 2} u 2 0 t ' b + a b t o , '( t )

Tuong tu ta co:
b 2 0 t+2c 2 0 1 >
2 6 2 0 r r " + b c r o ,, ( z ) ;
>c2'lra + caron (:);
c2()12
+ a20t2

0,25d

6p dr,rngb6t ding thric CoSi cho hai s6 ducrngta c6:
a2ot3

I
a2or3
+ (, - a + b +
>
2
r
l
_
_
", _ (_u*b+c)aro"
-a+b+c
")urr,' v - a + b + c \

=

a20t3

_ + c + a t o t t b + a ' o " c > 3 a t o t ,( 4 )
-a+b

Tuong tU ta c6:
0,5d

;2013

a-b+c

+ abr'r,+ br.,rc> 36zotz (5)

^2013

+ a c r o '*r 6 a 2 o r>t 3 c 2 o t z ( 6 )
",
a+b-c
Tt (1), (2), (3), (4), (5), (6) ta c6
C2013

a+b-c

I

+

b2ot:
-

42013
) u20rz* 62012
+ -_: + crorr.p6, rc_':xhy rakhi vd chi
a
+
b
+
c
a-b+c

khi a: b : c tric ld ta c6 tam si6c dAu
0.25d