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Ngi tng hp:Nguyn Huy Thnh

TNG HP CC THI HSG LP


9 NM 2011-2012














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Li ni u:
Cho tt c cc bn! Mnh l Nguyn Huy Thnh hc sinh lp 8/1 Trng THCS Tn Xun.Nay
mnh quyt nh tng hp li tt c cc thi HSG lp 9 (nm 2011-2012) cho cc bn n thi
tuyn sinh lp 10 v chun b cho k thi hc sinh gii lp 9 ca tnh mnh.Sau y l hn 30
thi hc sinh gii lp 9 c mnh tng hp trn VMF (din n ton hc).Mnh mong n s gip
cc bn phn no v n tp HSG












Ngi bin son
Nguyn Huy Thnh










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THI HC SINH GII QUN NG A 2011-2012
MN: TON
NGY THI: 10 thng 12 nm 2012
THI GIAN: 120 pht (khng k thi gian giao )

Bi 1: (4,0 im)
Rt gn biu thc:


( ) ( )
3 3
2
2
2 4 2 2
4

4
x x x
A
x
(
+ +
(

=
+


vi 2 2 x s s .

Bi 2: (6,0 im)
1) Cho trc s hu t m sao cho
3
m l s v t. Tm cc s hu t a,b,c :

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3 2 3
0 a m b m c + + =

2) Tm s t nhin c 4 ch s (vit trong h thp phn) sao cho 2 iu kin sau ng thi tha
mn:
(i) Mi ch s ng sau ln hn ch s ng lin trc.
(ii) Tng p+q ly gi tr nh nht, trong p l t s ca ch s hng chc v ch s hng n v
cn q l t s ch s hng nghn v ch s hng trm.

Bi 3: (4,0 im)
1) Tm tt c cc s nguyn x tha mn:

| 10| | 11| | 101| | 990| | 1000| 2012 x x x x x + + + + + + + =

2) Chng minh rng c th chia mt tam gic vung c di 3 cnh l cc s nguyn thnh 6
phn din tch bng nhau v din tch mi phn l s nguyn.

Bi 4: (4,0 im)
1) Cho tam gic ABC c 3 gc nhn, trung tuyn AO c di bng di cnh BC. ng trn
ng knh BC ct AB,AC th t ti M,N (M khc B, N khc C). ng trn ngoi tip tam gic
AMN v ng trn ngoi tip tam gic ABC ct ng thng AO ln lt ti I,K. Chng minh
t gic BOIM ni tip c mt ng trn v t gic BICK l hnh bnh hnh.
2) Cho tam gic ABC, im M di chuyn trn cnh BC. Gi P,Q ln lt l hnh chiu vung gc
ca M trn AB,AC. Xc nh v tr M PQ c di nh nht.

Bi 5: (2,0 im)
Trong mt hnh vung cnh bng 7, ly 51 im. Chng minh rng c 3 im trong 51 im
cho cng nm trong 1 hnh trn c bn knh bng 1.

K thi chn hc sinh gii lp 9
Nm hc 2011-2012
______________________________________
Mn thi:Ton
Thi gian lm bi: 150 pht (khng tnh thi gian giao )
______________________________________
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Bi 1. (2,0 im)

a) Cho biu thc:
2 1 1 2 1
. 1
1 1 2

x x
A
x x x x
| |
+ | |
= + +
|
|
|
+ +
\ .
\ .
vi 0; 1 x x > = . Rt gn biu thc A
v tm cc gi tr nguyn ca x A l s nguyn.

b) Cho biu thc:


( )( )( )( )
1 2 1 2 1 2 2 1 M x x x x x x x x x x x x = + + + + + + + + + + + + + +


Vi x l s t nhin khc 0 . Chng minh M cng l s t nhin.

Bi 2. (2,0 im)

a) Tm x bit: 24 16 10 x x + + =

b) Gii h phng trnh:
9
4
1
x xy y
y yz z
z zx x
+ + =

+ + =

+ + =



Bi 3. (2,0 im)

Trn mt phng ta Oxy cho t gic ABCD c (0;1); (0; 4); (6; 4) A B C v (4;1) D . Gi d l
ng thng ct cc on thng AD,BC ln lt ti M,N sao cho ng thng d chia t gic
ABCD thnh 2 phn c din tch bng nhau, bit phng trnh ng thng d c dng
5
3
m
y mx = (vi 0 m= ).

a) Tm ta ca M v N

b)Tm ton im Q trn d sao cho khong cch t Q n trc Ox bng 2 ln khong cch t Q
n Oy.

Bi 4. (2,0 im)

Cho tam gic ABC u ni tip ng trn tm O, gi H l trung im BC. Trn cc cnh
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AB,AC ln lt ly hai im D,E sao cho 60
o
DHE = . Ly M bt k trn cung nh AB.

a) Chng minh ba ng phn gic ca ba gc , , BAC BDE DEC ng quy.

b) Cho AB c di 1 n v. Chng minh:
4
3
MA MB + <

Bi 5. (1,0 im)

Cho tam gic ABC khng cn, v phn gic trong Ax ca gc A. V ng thng d l trung trc
ca on thng BC. Gi E l giao ca Ax v d. Chng minh E nm ngoi tam gic ABC.

Bi 6. (1,0 im)

Cho x,y,z l ba s thc dng tha iu kin xyz=1. Chng minh rng:


3 3 3 3 3 3
1 1 1
1
1 1 1 x y y z z x
+ + s
+ + + + + +



*Lu : Th sinh khng c s dng my tnh cm tay khi lm bi thi.




----------------------HT----------------------




thi HSG vng 2 qun H ng - H Ni
Bi 1:
a)Gii pt:
2 2 2 3
2( 1) 7( 1) 13( 1) x x x x + + =
b)Cho pt :
2
2( 1) 3 0 mx m x m + =
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Tm m pt c 2 nghim phn bit
1 2
; x x m
2 2
1 2
x x + =3
Bi 2:
a)Tm x,y,z thuc N* sao cho xyz-x-y-z=5
b)Gii h:
1
2 (1 ) 3
1
2 (1 ) 1
x
x y
y
x y

+ =


Bi 3: Cho abc=2012, a,b,c >0
Tm max:
3 3 3 3 3 3
1 1 1
a b abc b c abc c a abc
+ +
+ + + + + +

Bi 4: Cho ng trn (O) .Dy BC c nh , A chuyn ng trn ng trn sao cho tam gic
ABC c ba gc nhn.K cc ng cao AD,BE,CF ct nhau ti H
a) CMR:
2 2 2
1 cos A cos B cos C + + <
b)Tm v tr im A din tch tam gic AEH max
c)CMR: ng trn ngoi tip tam gic DEF i qua 1 im c nh
d) CM:
2 2 2
4 BC AD EF + >













thi HSG ton 9 tnh Yn Bi nm hc 2011-2012
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Thi gian lm bi: 150 pht (Khng k thi gian giao )
Cu 1:<4 >
Tm hai s x,y nguyn tho mn
2
7 2 15 x xy x y =

Cu 2:<3 >
Gii h phng trnh:
2 2
2
1 1 3
1
( )(1 ) 6
x y
x y
x y
xy

+ =

+ +

+ + =



Cu 3:<5 >
Cho hnh thang ABCD(AB//CD). Trn y ln AB ly im M khng trng vi cc nh. Qua M
k cc ng thng song song vi AC v BD, cc ng thng ny ct hai cch BC, AD ln lt
ti E v F. on EF ct AC v BD ln lt ti I v J. Gi H l trung im ca IJ.
a. Chng minh rng: FH=HE
b. Cho AB=2CD. Chng minh rng: EJ=JI=IF

Cu 4:<3 >
Cho ng trn O v mt dy cung $AB(O\not\in AB)$. Cc tip tuyn ti A v B ca ng
trn ct nhau ti C. K dy cung CD ca ng trn ng knh $OC(D\neq A,B)$. Dy cung
CD ct cung AB ca ng trn (O) ti E (E nm gia C v D).
a. Chng minh: BED DAE =
b. Chng minh:
2
. DE DADB =

Cu 5:<2 >
Cho
1 1 1 1
... ... , ( ;1 2012)
1.2012 2.2011 (2012 1) 2012.1
S k k
k k
= + + + + + e s s
+

So snh S v
4024
2013


Cu 6:<3 >
Cho $x,y,z$ l ba s dng tho mn xyz=1.
Chng minh rng:
2 2 2
3
1 1 1 2
x y z
y z x
+ + >
+ + +

THI HSG TON TNH H TNH NM 2011-2012

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Bi 1a) Rt gn biu thc 5 3 29 12 5
b) Tm cc s nguyn a,b sao cho
3 2
7 20 3
3 3 a b a b
=
+


Bi 2a) Gii phng trnh
2
12 1 36 x x x + =
b) Gii h phng trnh


( 1)( 1) 10
( )( 1) 3
x y
x y xy
+ + =

+ =



Bi 3Cho ba s m,n,p tha mn:
2 2 2
2 2
2 2 2
2
m m m
m n
n n p
+ = + + = v
2 2 2 2
2 2
4
p p n n
n m p
+
+ + =
Tnh
2 3 4
Q m m p = + +

Bi 4Cho tam gic ABC c B nhn, trn cung nh AC ca (ABC) ly D khc A. K v H l hnh
chiu ca D trn cc ng thng BC,AB. I l giao im KH v AC.
a.CM DI vung gc vi AC v HK < AC
b.E l trung im AB . (HDE) ct IK ti F . CM IF=FK

Bi 5Cho hai s thc x,y khc 0 sao cho
2 2
( 1) x y xy x y + + = + Tm max ca
3 3
1 1
A
x y
= +







thi chn HSG tham d k thi cp TP H Ni
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Bi 1(6):
a) Cho : A=
1 1 1 1
1.2.3........2011.2012(1 ....... )
2 3 2011 2012
+ + + + +
CMR: A l 1 s t nhin v A chia ht cho 2013
b) Tm x tha mn:
3 2 3 2 3 3
3 2011 3 7 2012 6 2013 2012 x x x x x + + =
Bi 2 ( 3)
Gii h
2 2
2 2 2
2 5 2013
10 25 0
5 4 4 4 0
x y z t
z zt t
x y z xy zy
+ =

+ =

+ + =


Bi 3: Cho a,b,c thuc R , x,y,z>0 CM:
a)
2 2 2 2
( ) a b c a b c
x y z x y z
+ +
+ + >
+ +

b)Cho xy+yz+xz=671 CM:
2 2 2
1
2013 2013 2013
y z x
y xz z xy x yz x y z
+ + >
+ + + + +

Bi 4(5):
Cho ng trn ( O,R) . T im S ngoi ng trn k 2 tip tuyn SM, SN ti ng trn(
M,N l hai tip im), ng thng d qua S ct ng trn (O,R) ti A v B ( M thuc cung ln
AB). Qua A k ng thng Ax // SM. ng thng Ax ct MN ti E, ct MB ti C. ng
thng MN ct AB ti K . Gi I l trung im AB
a) CM: IS l phn gic MIN
b) CM:
SA SK
SI SB
=
c)CM: MA,SC,BE ng quy ti 1 im
Bi 5(2): Trong 1 cuc hi ngh c 100 i biu, trong mi ngi quen vi t nht 67 ngi
khc. CMR: trong hi ngh c t nht 4 ngi m mi ngi u quen vi 3 ngi cn li.







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S GIO DC V O TO
THI BNH
____________
CHNH THC


THI CHN HC SINH GII LP 9 THCS NM HC 2011-2012

Mn thi: TON
Thi gian lm bi: 150 pht, khng k thi gian giao

Cu 1: (3,0 im)
Cho tam gic vung c di cc cnh l nhng s nguyn v s o chu vi bng hai ln s o
din tch. Tm di cc cnh ca tam gic .
Cu 2: (3,0 im)
Cho biu thc:
2 2
1 (1 ) 1 1 (1 ) 1 P x x x x x x = + + vi [ 1;1] xe
Tnh gi tr biu thc P vi
1
2012
x

= .
Cu 3: (3,0 im)
Tm s thc x, y tha mn:
2 2 2 2 2 3 3
( 1) 16 2 9 8 8 x y x x x y x y xy + + + + = +
Cu 4: (3,0 im)
Trong mt phng ta Oxy, cho Parabol (P):
2
y x = v hai im A(-1;1). B(3;9) nm trn (P).
Gi M l im thay i trn (P) v c honh l m ($-1<m<3$). Tm m din tch tam gic
ABM ln nht.
Cu 5: (3,0 im)
Cho tam gic ABC ni tip (O;R). Gi I l im bt k trong tam gic ABC (I khng nm trn
cc cnh ca tam gic). Cc tia AI, BI, CI ct ln lt BC, CA, AB ti M, N v P.
a) Chng minh: 2
AI BI CI
AN BN CN
+ + = .
b) Chng minh:
2
1 1 1 4
. . . 3( ) AM BN BN CP CP AM R OI
+ + s

.
Cu 6: (3,0 im)
Cho tam gic ABC c gc A t, ni tip (O;R). Gi $x, y, z$ l khong cch t O n cc cnh
BC, CA, AB v r l bn knh ng trn ni tip tam gic ABC. Chng minh $y+z-x=R+r$.
Cu 7: (2,0 im)
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Cho x,y tha mn , x y R e v
1
0 ,
2
x y s s . Chng minh rng
2 2
1 1 3
y x
y x
+ s
+ +
.








thi HSG lp 9 tnh An Giang nm hc 2011 - 2012

Bi 1. (3 im)
Rt gn biu thc sau:
( )
3 3
3 2 31 12 3 3 : 5 2 7 5 2 7 A
| |
= + +
|
\ .

Bi 2. (3 im)
Chng minh rng nu hai phng trnh
2 2
0; 3 3 0 x bx c x bx c + + = + + = c nghim th phng
trnh
2
2 2 0 x bx c + + = c nghim.

Bi 3. (4 im)
Cho h phng trnh
( ) ( ) 1 1 37
2 3 1
m x m y m
x y m
=

+ = +

(m l tham s)
a. Vi m no th h phng trnh c nghim duy nht.
b. Tm m nguyn h phng trnh c nghim x,y nguyn v x+y b nht.

Bi 4. (4 im)
a.Chng minh rng vi mi s thc a,b th
4
4 4
2 2
a b a b + + | |
>
|
\ .

Du bng ca bt ng thc xy ra khi no.

b. Phn tch a thc sau thnh nhn t: ( )
4 2
8 12 P x x x x = +
Bi 5. (6 im)
Gi A',B',C' ln lt l trung im ca cc cung BC,CA,AB khng cha cc im A,B,C ca
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ng trn ngoi tip tam gic ABC. BC ct A'C' v A'B' ti M v N; CA ct A'B' v B'C' ti P
v Q; AB ct B'C v A'C' ti R v S.
a. Chng t rng AA',BB',CC' ng quy ti I.
b. Chng minh rng IQAR l hnh thoi.
c. Tm iu kin ca tam gic ABC MN=PQ=RS.





-------------HT-------------
thi HSG lp 9 tnh Vnh Long nm hc 2011 - 2012

Bi 1. (2 im)
Tm cc s t nhin c hai ch s, bit s chia cho tng cc ch s ca n c thng l 4
v s d l 3.

Bi 2. (6 im)
Gii cc phng trnh sau:
a.
( )
3 3
7 8 x x + =

b. 1 4 1 3x x + = + +

c. 2 2 1 5 x x + + =

Bi 3. (3 im)
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Cho Parabol
2
( ) : 2 P y x = . Trn (P) ly im A c honh bng 1, im B c honh bng 2.
Tm m v n ng thng
( ) : d y mx n = + tip xc vi parabol ( ) P v song song vi ng
thng AB.

Bi 4. (3 im)
Cho phng trnh bc hai
( )
2
2 1 2 10 0 x m x m + + + = , vi m l tham s thc.
a.Tm m phng trnh c hai nghim
1 2
, x x
b. Tm m biu thc
2 2
1 2 1 2
6 P x x x x = + + t gi tr nh nht.

Bi 5. (4 im)
Cho tam gic ABC cn ti A. Cc cnh AB,BC,CA ln lt tip xc vi ng trn (O) ti
D,E,F.
a. Chng minh DF//BC v ba im A,O,E thng hng, vi O l tm ca ng trn (O).
b. Gi giao im th hai ca BF vi ng trn (O) l M v giao im ca DM vi BC l N.
Chng minh tam gic BFC ng dng vi tam gic DNB v N l trung im ca BE.
c. Gi (O') l ng trn qua ba im B,O,C. Chng minh AB,AC l cc tip tuyn ca ng
trn (O').

Bi 6. (2 im)
Cho tam gic ABC c , , BC a AC b AB c = = = . Gi , ,
a b c
h h h ln lt l cc ng cao ng vi
cc cnh a,b,c. Tnh s o cc gc ca tam gic ABC bit 9
a b c
h h h r + + = , vi r l bn knh
ng trn ni tip tam gic ABC.

-------------HT-------------


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thi HSG lp 9 tnh Tin Giang nm hc 2011 - 2012

Bi 1. (4,0 im)
1. Gii h phng trnh.
3 2
3 2
1 2( )
1 2( )
x x x y
y y y x
+ = +

+ = +


2. Cho phng trnh:
4 2
2 2 1 0(1) x mx m + =

a. Tm m (1) c 4 nghim
1 2 3 4
, , , x x x x tho
1 2 3 4
4 3 3 2 2 1
x x x x
x x x x x x
< < <

= =


b. Gii phng trnh (1) vi m tm c cu a.

Bi 2. (4,0 im)
Cho
2
( ) : ;( ) : P y x d y x m = = + . Tm m (P) v (d) ct nhau ti hai im phn bit A, B sao
cho: tam gic OAB l tam gic vung.

Bi 3. (4,0 im)
1. Cho 4 s a, b, c, d tho iu kin 2 a b c d + + + = . Chng minh:
2 2 2 2
1 a b c d + + + >
2. Cho v
3 2 2
3 3 ( 1) ( 1) 0 a a a m m + + + = . Hy tm gi tr nh nht (GTNN) ca $a$.

Bi 4. (3,0 im)
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Chng minh rng:
2 2 2 2
2 ( 1)(2 1)
2 4 6 ... (2 ) ; , 1
3
n n n
n n n
+ +
+ + + + = e

Bi 5. (5,0 im)

Cho tam gic ABC c cc phn gic trong ca cc gc nhn , , BAC ACB CBA theo th t ct cc
cnh i ti cc im M, P, N. t , , ; a BC b CA c AB = = = ,
MNP ABC
S S
A A
theo th t l din tch
ca tam gic MNP v ABC.
1. Chng minh rng:
( )( )( )
2
MNP
ABC
S abc
S a b b c c a
A
A
=
+ + +

2. Tm gi tr ln nht (GTLN) ca
MNP
ABC
S
S
A
A



-------------HT-------------

* Ghi ch: Th sinh khng c s dng my tnh cm tay.


thi HSG lp 9 tnh Lng Sn nm hc 2011 - 2012

Bi 1. (4 im)
Cho biu thc:
4 8 1 2
:

4 2 2
x x x
P
x x x x x
| | | |

= +
| |
| |
+
\ . \ .

a. Rt gn P
b. Tm m vi mi gi tr 9 x > ta c
( )
3 1 m x P x > +

Bi 2. (3 im)
Cho 1 abc = v
3
36 a > . Chng minh rng:
2
2 2
3
a
b c ab bc ca + + > + +
Bi 3. (4 im)
Cho phng trnh bc hai: ( ) ( )
2 2
2 2 7 0 1 x m m x m + + + = , (m l tham s)
a. Gii phng trnh (1) khi m = 1
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b. Tm m phng trnh (1) c hai nghim
1 2
; x x tha mn:
( )
1 2 1 2
2 4 x x x x =

Bi 4. (6 im)
Cho tam gic ABC c 5 ; 4 ; 3 BC a CA a AB a = = = , ng trung trc ca on AC ct ng
phn gic trong ca gc BAC ti K.
a. Chng minh tam gic ABC vung.
b. Gi (K) l ng trn c tm K v tip xc vi ng thng AB. Chng minh rng ng
trn (K) tip xc vi ng trn ngoi tip ca tam gic ABC.
c. Chng minh rng trung im ca on AK cng l tm ng trn ni tip ca tam gic ABC

Bi 5. (3 im)
Cho a,b,c l cc s nguyn t khc 0, a c = tha mn:
2 2
2 2
a a b
c c b
+
=
+
. Chng minh rng
2 2 2
a b c + + khng th l mt s nguyn t.



-------------HT-------------

thi HSG lp 9 tnh Hi Dng nm hc 2011-2012

Bi 1. (2,5 im)
1. Rt gn biu thc:
( )
2 2
2
5 6 3 6 8
3 12 3 6 8
x x x x
A
x x x x
+ + +
=
+ +

2. Phn tch thnh nhn t: ( )
3
3 3 3
a b c a b c + + + +
3. Tm x bit
( ) ( )
3
3
2 6
2 1 1 x x x x + + + = +

Bi 2. (2,0 im)
1. Gii h phng trnh:
2 2
2
2 0
3 3
x xy y
xy y x
+ =

+ + =


2. Gii phng trnh: ( )
3
3 3
3 16
2
x
x
x
| |
=
|

\ .


Bi 3. (2,0 im)
1. Tm nghim nguyn ca phng trnh:
2 2
8 23 16 44 16 1180 0 x y x y xy + + + =
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2. Cho n l s nguyn dng v $m$ l c nguyn dng ca
2
2n . Chng minh rng
2
n m +
khng l s chnh phng.

Bi 4. (3,0 im)
Cho ng trn (O;R) v AB l ng knh. Gi d l ng trung trc ca OB. Gi M v N l
hai im phn bit thuc ng thng d. Trn cc tia OM,ON ly ln lt cc im M' v N' sao
cho
2
. . OM OM ON ON R ' ' = = .

1. Chng minh rng bn im M,N,M',N' thuc mt ng trn.
2. Khi im M chuyn ng trn d, chng minh rng im M' thuc mt ng trn c
nh.
3. Tm v tr im M trn d nhng M khng nm trong ng trn (O;R) tng MO+MA
t gi tr nh nht.

Bi 5. (0,5 im)
Trong cc hnh bnh hnh ngoi tip ng trn (O;r), hy tm hnh bnh hnh c din tch nh
nht.


-------HT-------



S GIO DC - O TO K THI HC SINH GII LP 9
THCS
NAM NH NM HC 2011-2012
Mn: Ton
(Thi gian lm bi: 150 pht, khng k thi gian giao )
Cu 1:
1) Cho cc s thc a, b, c khc nhau tng i mt vo tha mn iu kin:

2 2 2
a b b c c a = =
Chng minh rng: ( 1)( 1)( 1) 1 a b b c c a + + + + + + =
2) Cho ba s thc dng a, b, c tha mn: 1 ab bc ca + + =
Chng minh rng:
2
2 2
( ) 1
1
1 1
b c a
b c
+ +
=
+ +

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Cu 2:

1) Gii h phng trnh
2 2
3 8 5
( 3) ( 8) 13
y x x y
x x y y

+ + =

+ + =


2) Gii phng trnh:
2
1 3 3 4 2 x x x x + =
Cu 3: Tm tt c cc b ba s nguyn khng m (x;y;z) tha mn ng thc:
2012 2013 2014
x y z
+ =
Cu 4: Cho ng trn (O), AB l ng knh ca (O). im Q thuc on thng OB (Q khc
O; Q khc B). ng thng i qua Q, vung gc vi AB ct ng trn (O) ti hai im C v D
khc nhau (im D nm trong na mt phng b PS cha B). Gi G l giao im ca cc ng
thng CD v AP. Gi E l giao im ca cc ng thng CD v PS. Gi K l trung im ca
on thng AQ.
1) Chng minh rng tam gic PDE ng dng vi tam gic PSD
2) Chng minh rng EP=EQ=EG
3) Chng minh ng thng KG vung gc vi ng thng CD
Cu 5: Cho ba s thc dng a, b, c tha mn iu kin:

2 2 2
3 a b c + + =
Chng minh rng:
3 3 3
1 1 1
1
1 8 1 8 1 8 a b c
+ + >
+ + +


thi HSG lp 9 tnh Ph Th nm hc 2011 - 2012

Bi 1. (3 im)
Tm tt c cc s nguyn dng n hai s n + 26 v n 11 u l lp phng ca hai s
nguyn dng no .

Bi 2. (4 im)
Gi s $a$ l mt nghim ca phng trnh
2
2 1 0 x x + = . khng gii phng trnh, hy tnh
gi tr ca biu thc:
4 2
2 3
2(2 2 3) 2
a
A
a a a

=
+ +

Bi 3. (4 im)
a. Gii phng trnh:
2
8 1 1 3 x x x + = +
b. Gii h phng trnh:
2 2
2
2 1
2
x y
xy x
=

+ =



Bi 4. (7 im)
Cho ng trn (O;R) v im M nm ngoi ng trn. Qua im M v hai tip tuyn MA, MB
www.VNMATH.com
ti ng trn (A v B l cc tip im). Gi D l im di ng trn cung ln AB (D khng
trng A, B v im chnh gia ca cung) v C l giao im th hai ca ng thng MD vi
ng trn O;R).
a. Gi s H l giao im ca cc ng thng OM vi AB. Chng minh rng
. . MH MO MC MD = , t suy ra ng trn ngoi tip tam gic HCD lun i qua mt im c
nh.
b. Chng minh rng nu AD song song vi ng thng MB th ng thng AC i qua trng
tm G ca tam gic MAB.
c. K ng knh BK ca ng trn (O;R), gi I l giao im ca cc ng thng MK v AB.
Tnh bn knh ng trn ngoi tip tam gic MBI theo R, khi bit 2 OM R = .

Bi 5. (2 im)
Cho cc s thc dng a, b, c tho mn: 3 abc a b ab + + = . Chng minh rng:
3
1 1

1
ab b a
a b bc c ca c
+ + >
+ + + + + +



-------------HT-------------

THI CHN HSG LP 9 CP TNH, B RA VNG TU 2012

Cu 1. (3,0 im)
Gii cc phng trnh:
1.
2
2( 1) 0 x y x y + + = ( x, y l n )
2.
2 2
6 1 6 0 x x + + = .

Cu 2. (4,0 im)
Rt gn biu thc:
1.
6 2 2 6 2 2
2 4 2 3 2 4 2 3
A
+
=
+ +

2.
2 2 2 2 2
1 1 1 1 1
3 2 5 6 7 12 9 20
B
x x x x x x x x x x
= + + + +
+ + + + + + + + +


Cu 3. (4,0 im)
1. Tm tt c cc s t nhin n sao cho 3n+5 chia ht cho n-7
2. Tm nghim nguyn ca phng trnh:
2
( 1) 4 (1 ) x x x y y + + = + (vi x, y l n).

www.VNMATH.com
Cu 4. (2,0 im)
Cho a,b,c l di ba cnh tam gic. Chng minh:
2 2 2
2( ) ab bc ca a b c ab bc ca + + s + + < + +
Cu 5. (3,0 im)
Cho ng trn (O) c hai ng knh AB v CE vung gc vi nhau. Gi P l mt im di
ng trn cung nh AE (P khc A v E). CP ct OA ti M v BP ct OE ti N.
1. Chng minh tam gic CAM ng dng vi tam gic CPA v
.
.
OM PE OC
MA APCA
= .
2. Chng minh .
OM ON
MA NE
l mt hng s.

Cu 6. (4,0 im)
Cho tam gic ABC u c ng cao AH (H thuc BC). M l im di ng trn cnh BC(M
khc B v C).
Dng MP vung gc vi AB ti P v MQ vung gc vi AC ti Q, AM ct ng trn ngoi
tip tam gic ABC ti D (D $khc A).
1. Chng minh t gic APMQ ni tip trong mt ng trn.
2. Gi $O$ l tm ng trn ngoi tip t gic APMQ, chng minh H vung gc vi PQ.
3. Khi $M$ di ng trn cnh BC (M khc B v C), tm tp hp trung im E ca on AD.



------- Ht -------
thi hc sinh gii TP.HCM cp THCS nm hc 2011 - 2012
Bi 1: (4 im)
Cho phng trnh
2
2( 2) 3 0 mx m x m + + = (x l n s)
a) Tm m phng trnh c hai nghim tri du.
b) Tm m phng trnh c hai nghim tri du v nghim m c gi tr tuyt i ln hn
nghim dng.
Bi 2: (4 im)
Gii cc phng trnh:
a)
4
2 0
2
x x
x
+ + =
+

b) 1 2 x x x + =
Bi 3: (4 im)
a) Chng minh rng:
2 2 2 2 2
( )( ) ( ) a b c d ac bd s vi a, b, c, d l cc s thc.
b) Cho 1, 1 a b > > . Chng minh rng: 1 1 a b b a ab + s
Bi 4: (2 im)
Tm gi tr nh nht ca biu thc 2 3 A x y z = + bit x,y,z khng m v tha h phng trnh:
www.VNMATH.com
2 4 3 8
3 3 2
x y z
x y z
+ + =

+ =


Bi 5: (2 im)
Chng minh rng phng trnh
2 3 2
4 4 8 2 4 x x y z + = + khng c nghim nguyn.
Bi 6: (4 im)
Cho ng trn (O) ng knh AB, bn knh R. Tip tuyn ti M bt k thuc ng trn (O)
ct cc tip tuyn ca ng trn ti A v B ln lt ti C v D.
a) Chng minh rng:
2
. AC BD R =
b) Gi I v J ln lt l giao im ca OC vi AM v OD vi BM.
Chng minh IJ song song vi AB.
c) Xc nh v tr ca M ng trn ngoi tip t gic CIJD c bn knh nh nht.

thi HSG lp 9 tnh Qung Ninh 2011-2012

Cu 1 (2): cho
3 3
1 2 4 x = + + , chng minh rng P=
3 2
3 3 3 x x x + l mt s chnh phng.
Cu 2 (6):
- Gii h phng trnh:
-
2 2
4 5
4 2 7
x y
xy x y
+ =

+ + =


- Gii phng trnh

2 2 2
2 1 1 6 9 9
4
x y z
x y z

+ + =
Cu 3 (3) Tm tham s m tp nghim phng trnh sau c ng mt phn t:

2 2
(2 5) 1
0
1
m x m x
x
+ +
=


Cu 4 (7)
Cho (O) v (O') ct nhau ti A v B. Trn tia i ca tia AB ly M khc A. Qua M k tip
tuyn MC, MD vi ng trn (O') ( C,D l cc tip im, C nm ngoi (O)). ng thng AC
ct (O) tai P khc A, ng thng AD ct (O) ti Q khc A. ng thng CD ct PQ ti K.
Chng minh:
- Tam gic BCD ng dng vi tam gic BPQ
- ng trn ngoi tip tam gic KCP lun i qua mt im c nh khi M thay i.
- K l trung im PQ
www.VNMATH.com
Cu 5 (2)Vi a,b,c l ba s thc dng, chng minh bt ng thc:
3 3 3
2 2 2
a b c
a b c
b c a
+ + > + +


THI CHN HSG LP 9 THCS
NM HC 2011-2012
Thi gian lm bi: 150 pht (khng k thi gian giao )
CU 1: (5)
Cho biu thc :
2
3 2 4
1 2
a a a a a
P
a a a a

= +
+ +

1. Rt gn P.
2. Tm GT nh nht ca P.
CU 2: (5)
Gii cc phng trnh sau:
1.
3 2 3 3 3 2
2 2 3 1 3 1 2 x x x x x x + + = + + +
2.
4 4 2 2 2 2
2 4 7 5 0 x y x y x y = ; (vi x;y nguyn)
CU 3: (4)
Cho ng trn ( ) ; O R . ng thng d khng i qua O ct ng trn ( ) O ti hai im A
v B . T mt im M tu trn ng thng d v ngoi ng trn (O), v hai tip tuyn
MN v MP vi ng trn (O), ( N,P l hai tip im).
1. Dng v tr im M trn ng thng d sao cho t gic $MNOP$ l hnh vung.
2. CMR tm ca ng trn i qua 3 im $M, N, P$ lun chy trn ng thng c nh khi
M di chuyn trn ng thng d .
.CU 4:(4)
www.VNMATH.com
1. a) Tm Max ca :
2
9 y x x =
b) GT $x, y, z$ l nhng s dng tho mn k: 1 xyz = .
Tm min: ( )
( ) ( ) ( )
3 3 3
1 1 1
f x
x y z y x z z x y
= + +
+ + +
.
2. Cho 3 s $a,b,c$ tho mn: 1 a b c + + = ;
2 2 2 3 3 3
1; 1 a b c a b c + + = + + = .
CMR:
2 1 2 1 2 1
1
n n n
a b c
+ + +
+ + = vi * n .
CU 5: (2)
Cho ABC thay i c 6 AB = v 2 AC BC = . Tm gi tr ln nht ca
ABC
S .

thi HSG lp 9 tnh Qung Ngi nm hc 2011 - 2012

Ngy thi: 29/03/2012
Thi gian: 150'

Bi 1: a) Tm x, y nguyn dng sao cho 6 5 18 2 x y xy + + =
b) Chng minh A l s t nhin vi mi a thuc N:
5 4 3 2
7 5
120 12 24 12 5
a a a a a
A = + + + +

Bi 2: a) Gii phng trnh:
2
4 1 5 14 x x x + = +
b) Gii h phng trnh:
2
1 1 6
y x
z xy
x y z



Bi 3: a) Cho
1 2
2
a

= . Tnh gi tr biu thc
8
16 51 a a
b) Cho a, b l cc s thc dng. Chng minh:
www.VNMATH.com

2
( ) 2 2
2
a b
a b a b b a
+
+ + > +

Bi 4: Cho im M thuc ng trn (O) ng knh AB. T 1 im C trn on OB, k CN
vung gc vi AM ti N. Tia phn gic ca gc MAB ct CN ti I, ct (O) ti P. Tia MI ct
ng trn (O) ti Q.
a) Chng minh P, C, Q thng hng.
b) Khi AM = BC, chng minh tia MI i qua trung im ca AC.

Bi 5: Cho tam gic ABC nhn, ng cao AH. Trn AH, AB, AC ln lt ly cc im D, E, F
sao cho 90
o
EDC FDB = = . Chng minh rng: EF // BC.

THI HSG LP 9 TINH NG NAI 2011-2012
Cu 1 (4)
Cho ac=bd v ab>0 chng minh
2 2 2 2 2 2
( ) ( ) a b c d a d b c + + + = + + +

Cu 2 (4)
GHPT
2 2
4 x y =

3 3
8 x y =

Cu 3 (4)
Cho m,n,k l cc s nguyn tha
2 2 2
m n k + =

Chng minh tch mn 12
Cu 4(3,5)
Trong mt phng ta Oxy mi im vi honh v tung u nguyn c gi l 1 im
nguyn
Trong mt phng vi h ta Oxy cho cc im M(p,q) E(p,0) F(0,q)
Bit p,q l hai s t nhin v nguyn t cng nhau p>1 q>1
1) tnh p v q theo s im nguyn bn trong hnh ch nht OEMF
2) Chng minh rng ch c 2 im nguyn thuc on OM
www.VNMATH.com
Cu 5(4,5)
Cho (O;R) tm O bn knh R gi A,B l hai im c nh thuc (O;R) A= B Gi C l im thay
i thuc (O;R) vi C= A C= B V (
1
O
) i qua A tip xc vi BC ti C . V
2
( ) O
i qua B
v tip xc ci AC ti C.
1
( ) O
v
2
( ) O ct nhau ti D = C
1) Chng minh
1 2
OOCO
l hnh bnh hnh
2) Xc nh v tr im C tha iu kin cho di on CD ln nht





thi HSG lp 9 tnh Thanh Ha nm hc 2011 - 2012

Bi 1. (4,0 im)
Cho biu thc

1 8 3 1 1 1
:
10 3 1 3 1 1 1
x x x
P
x x x x x
| | | |
+ +
= +
| |
| |
+
\ . \ .

1. Rt gn P
2. Tnh gi tr ca P khi
4 4
3 2 2 3 2 2
3 2 2 3 2 2
x
+
=
+


Bi 2. (4,0 im)
Trong cng mt h ta , cho ng thng - : 2 d y x = .v parabol ( )
2
: P y x = . Gi A v
B l giao im ca d v ( ) P
1. Tnh di AB
2. Tm m ng thng ' : d y x m = + ct ( ) P ti hai im C v D sao cho CD AB =

Bi 3. (4,0 im)
1. Gii h phng trnh:
2
2
2
1
2
x
x
y
y
y
x

+ =

+ =


2. Tm nghim nguyn ca phng trnh:
6 2 3
2 2 320 x y x y + =
www.VNMATH.com

Bi 4. (6,0 im)
Cho tam gic nhn ABC c AB AC > . Gi M l trung im ca BC; H l trc tm; AD,BE,CF
l cc ng cao ca tam gic ABC. K hiu
1
( ) C v
2
( ) C ln lt l ng trn ngoi tip tam
gic AEF v DKE, vi K l giao im ca EF v BC. Chng minh rng:
1. ME l tip tuyn chung ca
1
( ) C v
2
( ) C
2. KH AM

Bi 5. (2,0 im)
Vi 0 , , 1 x y z s s . Tm tt c cc nghim ca phng trnh:
3
1 1 1
x y z
y zx z xy x zy x y z
+ + =
+ + + + + + + +


S GIO DC VO O TO BC GIANG
THI CHN HC SINH GII CP TINH
Cu 1:
1. Tnh gi tr ca biu thc sau:
1 4 1 4
1 1 4 1 1 4
x x
x x
+
+
+ +
bit
2
9
x =
2. Tm cc gi tr ca tham s m phng trnh
2
( 1) (2 1) 1 0 m x m x m + + + = c 2 nghim
phn bit
1 2
, x x tha mn
2 2
1 2 1 2
2009 2012 x x x x + =
Cu 2:
1. Gii phng trnh
2
(2 2 4 1)(2 3 4 9 2) 7 x x x x x + + + + + + =
2. Gii h phuong trnh sau
2 4 2
2 4 2
2 4 2
x y z
y z x
z x y

+ =

+ =

+ =


Cu 3:
1. Tm gi tr ln nht, nh nht ca x bit x, y l 2 s tha mn ng thc
2 2
3( ) y xy y x x = +
2. Tm cc s nguyn k biu thc
4 3 2
8 23 26 10 k k k k + + l s chnh phng.
Cu 4: Cho ng trn ng knh AB. Trn on thng AO ly im H bt k khng trng vi
A v O, k ng thng d vung gc vi AB ti H, trn d ly im C nm ngoi ng trn, t C
k 2 tip tuyn CM v CN vi ng trn (O) vi M v N l cc tip im, (M thuc na mt
phng b d c cha im A). Gi P v Q ln lt l giao im ca CM, CN vi ng thng AB.
1, Chng minh rng HC l tia phn gic MHN
2. ng thng i qua O vung gc vi AB ct MN ti K v ng thng CK ct ng thng
AB ti I. Chng minh I l trung im ca PQ
3. Chng minh rng ba ng thng PN, QM, CH ng quy.
www.VNMATH.com
Cu 5:
Cho 3 s dng x, y, z tha mn x+y+z=6. Chng minh rng
2 2 2
8 x y z xy yz zx xyz + + + >

thi HSG lp 9 Bnh Thun nm 2011-2012

Bi 1: (4 im)
1/ Chng minh rng nu a+b+c+d = 0 th
3 3 3 3
3( )( ) a b c d ac bd b d + + + = +
2/ Tm mt s gm hai ch s sao cho t s gia s vi tng hai ch s ca n l ln nht.

Bi 2: (4 im)
1/ Gii phng trnh
3
1 2 x x = 5
2/ Trong mt lp hc ch c hai loi hc sinh l gii v kh. Nu c 1 hc sinh gii chuyn i th
1
6
s hc sinh cn li l hc sinh gii. Nu c 1 hc sinh kh chuyn i th
1
5
s hc sinh cn li
l hc sinh gii. Tnh s hc sinh ca lp.

Bi 3:(4 im)
1/ Cp s (x, y) l nghim phng trnh:
2
2 4 0 x y xy x y + + = . Tm gi tr ln nht ca y.
2/ Cho ba s thc $a, b, c$ 0 =/ tha 0 a b c + + =/ v
1 1 1 1
a b c a b c
+ + =
+ +
.
Chng minh rng trong ba s a, b, c c hai s i nhau.

Bi 4: (5 im)
Cho (O; R) c ng knh AB c nh; mt ng knh CD thay i khng vung gc v khng
trng AB. V tip tuyn (d) ca ng trn (O) ti B. Cc ng thng AC, AD ln lt ct (d)
ti E v F.
1/ Chng minh t gic CEFD ni tip c trong ng trn.
2/ Gi I l tm ng trn ngoi tip tam gic CDE. Chng minh rng I di ng trn mt ng
thng c nh.

Bi 5: (3 im)
Cho tam gic ABC c cc ng phn gic trong BD v CE ct nhau ti G. Chng minh rng
nu GD =GE th tam gic ABC cn ti A hoc gc A bng 60
o

--------HT--------



www.VNMATH.com


















www.VNMATH.com

www.VNMATH.com

S GIO DC V O TO K THI CHN HC SINH GII LP 9 CP TNH
GIA LAI Nm hc 2011 2012
------------------------------------- MN: Ton
CHNH THC Thi gian: 150 pht (khng k thi gian pht
)
-------------------------------------------------------------
Cu 1. (3,0 im)
a) Cho
2
1 1
2 1 1 2 1 1
x =

+ + +


. Tnh gi tr ca biu thc
( )
2012
4 3 2
1 2 A x x x x = +

b) Chng minh biu thc

3 2 2
( 7) 36 P n n n =
chia ht cho 7 vi mi s nguyn n.
Cu 2. (3,0 im)
a) Trong mt phng, h ta Oxy cho ng thng A c phng trnh y = x + 1.
Tm trn ng thng A cc im M (x; y) tha mn ng thc
2
3 2 0 y y x x + =
b) Trong mt phng, h ta Oxy cho ng thng d c phng trnh y = ax + b.
Tm a, b d i qua im B(1;2) v tip xc vi Parabol (P) c phng trnh: y = 2x
2

Cu 3. (4,0 im)
a) Gii h phng trnh


5
1
2 x y
x y
+ =

+ =


b) Gi
1 2
; x x l hai nghim ca phng trnh
2
2012 (20 11) 20 2 1 0 x a x = (a l s
thc)
Tm gi tr nh nht ca biu thc ( )
2
2
1 2
1 2
1 2
3 1 1
2
2

2
x x
P x x
x x
| |
= + +
|
\ .

Cu 4. (4,0 im)
a) Cho cc s thc $a,b,c$ sao cho 1 , , 2 a b c s s
. Chng minh rng: ( )
1 1 1
10 a b c
a b c
| |
+ + + + s
|
\ .

b) Trong hi tri ngy 26 thng 3, lp 9A c 7 hc sinh tham gia tr chi nm bng vo
r. 7 hc sinh ny nm c tt c 100 qu bng vo r. S qu bng nm c vo r ca
mi hc sinh u khc nhau. Chng minh rng c 3 hc sinh nm c tng s qu
bng vo r khng t hn 50 qu.
www.VNMATH.com
Cu 5. (6,0 im)
Cho tam gic ABC vung ti A (AB < AC) c ng cao AH v trung tuyn
AM (H, M thuc BC). ng trn tm H bn knh HA, ct ng thng AB v ng
thng AC ln lt ti D v E (D v E khc im A)
a) Chng minh D, H, E thng hng v MA vung gc vi DE
b) Chng minh 4 im B, E, C, D cng thuc mt ng trn. Gi O l tm ca ng
trn i qua 4 im B, E, C, D. T gic AMOH l hnh g?
c) t

; ACB AMB o | = = . Chng minh rng:

( )
2
sin o s 1 sin c o o | + = +


----------------------Ht----------------------------

thi HSG lp 9 tnh Bnh nh nm hc 2011 - 2012

Bi 1. (4 im)
a. Rt gn biu thc sau:
8 15 8 15
2 2
A
+
= +

b. Gii phng trnh:
3
2
2
16 0
16
x
x
x
+ =



Bi 2. (4 im)
a. Chng minh rng
3
n n chia ht cho 24 vi mi s t nhin n l.

b. Cho $a, b, c$ l cc s thc dng tha mn iu kin:
( ) ( ) ( )
2 2 2
2 2 2
a b c a b b c c a + + = + +
Chng minh rng nu c a v c b th c a b +

Bi 3. (3 im)
Cho phng trnh ( )
2
1 6 0 x m x + = . Tm m phng trnh c 2 nghim phn bit
1
x v
2
x
sao cho biu thc
( )( )
2 2
1 2
9 4 A x x = t gi tr ln nht.

Bi 4. (6 im)
a. Cho tam gic ABC cn ti A c
0
20 , BAC AB AC b = = = v BC a = . Chng minh rng:
3 3 2
3 a b ab + =
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b. Cho hai im $A, B$ thuc ng trn ( ) O ($AB$ khng qua O) v c hai im C, D di
ng trn cung ln AB sao cho / / AD BC (C, D khc A, B v AD BC > ). Gi M l giao im
ca BD$ v $AC. Hai tip tuyn ca ng trn ( ) O ti A v D ct nhau ti I

b.1. Chng minh ba im I, O, M thng hng.
b.2. Chng minh bn knh ng trn ngoi tip tam gic MCD khng i.
Cho x, y l cc s thc dng tha mn 1 xy = . Chng minh rng
( )( )
2 2
4
1 8 x y x y
x y
+ + + +
+

thi HSG lp 9 tnh Ngh An nm hc 2011 - 2012
Bi 1. (5,0 im)
a. Cho a v b l cc s t nhin tha mn iu kin
2 2
7 a b + . Chng minh rng a v b u
chia ht cho 7.
b. Cho
2012 2011
1 A n n = + + . Tm tt c cc s t nhin n A nhn gi tr l mt s nguyn t.

Bi 2. (4,5 im)
a. Gii phng trnh:
4 1 5
2 x x x
x x x
+ = +

b. Cho $x,y,z$ l cc s thc khc 0 tha mn 0 xy yz zx + + = . Tnh gi tr ca biu thc:
2 2 2
yz zx xy
M
x y z
= + +
Bi 3. (4,5 im)
a. Cho cc s thc $x,y,z$ tha mn iu kin: 6 x y z xy yz zx + + + + + = . Chng minh rng:
2 2 2
3 x y z + +
b. Cho $a,b,c$ l cc s thc dng tha mn iu kin: 3 a b c + + = . Tm gi tr nh nht ca
biu thc:
3 3 3
2 2 2 2 2 2
a b c
P
a b b c c a
= + +
+ + +

Bi 4. (6,0 im)
Cho ng trn (O;R) v mt dy BC c nh khng i qua O. T mt im A bt k trn tia i
ca tia BC v cc tip tuyn AM,AN vi ng trn (M v N l cc tip im, M nm trn cung
nh BC). Gi I l trung im ca dy BC, ng thng MI ct ng trn (O) ti im th hai l
P.
a. Chng minh rng NP//BC.
b. Gi giao im ca ng thng MN v ng thng OI l K. Xc nh v tr ca im A trn
tia i ca tia BC tam gic ONK c din tch ln nht.
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.


-------------HT-------------
thi hc sinh gii lp 9 tnh hng yn 2011-2012
cu I (2.0 im):
1. cho hm s
4 2012
( ) ( 2 7) f x x x = + . tnh ( ) f a vi
(4 15)( 5 3). 4 15 a = + .
2. cho parabol (P):
2
y x = .
trn (P) ly 2 im
1 2
, A A sao cho

1 2
90
o
AOA Z = (O l gc ta ).
hnh chiu vung gc ca
1 2
, A A lm trc honh ln lt l

1 2
, B B .
CMR
1 2
. 1 OB OB = .
cu II (2im):
1. cho PT
2
3 0 x mx m = (m l tham s khc 0) c 2 nghim phn bit
1 2
, x x tm gi tr nh
nht ca:

2 2
1 2
2 2
2 1
3 3
3 3
x mx m m
A
x mx m m
+ +
= +
+ +
.
2. tm nghim nguyn ca PT:

4 4 2 2 2 2
2 4 7 5 0 x y x y x y =
cu III: (2.0 im):
1. gii h:

2 2
2
2

xy
x y
x y
x y x y

+ + =

+ =


2. gii PT:

2 2
3
(3 1) 2 1 5 3
2
x x x x + = +
cu IV: (3.0 im)
1. cho ng trn tm O c ng knh CD l ng cao ca tam gic ABC vung ti C.
ng trn (O) ct cc cnh AC, BC ln lt ti E v F, gi M l giao im ca ng trn tm
O vi BE (M khc E). hai ng thng AC, MF ct nhau ti K, EF v BK ct nhau ti P
a. CMR B,M,F,P cng thuc 1 ng trn
b. tnh cc gc ca tam gic ABC khi 3 im D,M,P thng hng.
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2.cho tam gic ABC vung ti C, gc BAC bng 60
o
v trung tuyn
3
4
BD a = . tnh din tch
tam gic ABC theo a.
cu V:(1 im ): trn mt phng cho 6 ng trn c bn knh bng nhau v c im chung.
CMR t nht 1 trong nhng ng trn ny cha tm ca 1 ng trn khc trong chng.




thi HSG lp 9 tnh Qung Bnh nm hc 2011 - 2012
Cu1: Gii h phng trnh sau:
( )
2
2
2
1
3
2
x y
x
x y
x

+ =

+ + =



Cu2: Cho pt: x
2
-2mx +1 =0 (n x)
a) Tm m pt c hai nghim dng.
b)Gi x1 ,x2 (
1 2
x x s ) l hai nghim ca pt
Tnh
1 2
P x x = theo m v tm GTNN ca biu thc
1 2
1 2
2
Q x x
x x
= + +
+


Cu3: Cho tam gic ABC c gc u nhn v H l trc tm. Gi M,N,P ln lt l giao im th
hai ca cc dng thng AH,BH,CH vi dng trn ngoi tip tam gic ABC; D,E,F ln lt l
chn cc dng cao h t A,B,C ca tam gic ABC
a) Chng minh tam giac CHM cn
Tnh tng
AM
AD
+
BN
BE
+
CP
CF


Cu 4: Khng s dng my tnh hy chng minh:

1 1 1
... 2
2 1 3 2 2012 2011
+ + + <
Cu5: Tm s nguyn t p 4p
2
+ 1 va 6p
2
+1 cng l s nguyn t


S GIO DC V O TO K THI CHN HC SINH GII TNH
K LK NM HC 2011 - 2012
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CHNH THC MN TON 9 - THCS
(Thi gian lm bi 150 pht, khng k thi gian giao )
Ngy thi: 20/03/2012




Bi 1. (4 im)

Cho biu thc:
15 11 3 2 2 3
2 3 1 3
x x x
P
x x x x
+
= +
+ +

1/ Tnh gi tr ca P.
2/ Tm gi tr ln nht ca P.

Bi 2. (4 im)
1/ Tm tt c s thc m h phng trnh:
2
3 5
mx y
x my
=

+ =

c nghim (x;y) tha mn x > 0 v y >


0

2/ Cho x, y l hai s thc dng tha mn:
3 3
x x x y + = . Chng minh rng
2 2
1 x y + < .

Bi 3. (4 im)

1/ Chng t rng khng c hai s nguyn x, y tha mn ng thc
2 2
2012 x y + = .
2/ Tm tt c s nguyn n s
4 3 2
3 4 5 2 1 A n n n n = + + l mt s nguyn t.

Bi 4. ( 4 im)
Cho tam gic ABC vung ti A, c AH l ng cao.
1/ Chng minh rng
4 4 4 2 2
1 1 1 1 1
AB AC BC AH BC
+ + =
2/ Tam gic ABC c c im g nu c
4 4 4 2
1 1 1 3
4 AB AC BC AH
+ + =

Bi 5. (4 im)

Cho tam giac ABC cn ti A, mt im F di ng trn cch AC v F khng trng vi im A.
1/ Xc nh im E nm trn ng thng AB sao cho trung im I ca on thng EF nm trn
cnh BC.
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2/ Chng minh rng vi mi im E xc nh trn th tm ng trn ngoi tip tam gic AEF
nm trn mt ng thng c nh.

thi HSG khi 9 thnh ph Hi Phng 2011-2012 Bng A

Bi 1: (2.0 im)
a. Cho A =
3
7 5 2 + ; B =
3
20 14 2 . Tnh A + B
b. Cho a, b, c l cc s khc 0 tho mn 0 a b c + + = . Chng minh rng:
4 4 4
4 2 2 2 4 2 2 2 4 2 2 2
3
( ) ( ) ( ) 4
a b c
a b c b c a c a b
+ + =



Bi 2: (2.0 im)
a. Gii h phng trnh
2 2 4
7 7 6
x y
x y

+ + + =

+ + + =


b. Cho x, y l hai s nguyn khc -1 sao cho
4 4
1 1
1 1
x y
y x

+
+ +
l s nguyn.
Chng minh rng
2012
1 x chia ht cho 1 y +

Bi 3: (1.0 im)
Tm nghim nguyn ca phng trnh
6 6 6 6
32 16 4 x y z t + + =

Bi 4: (2.0 im)
Cho t gic li ABCD bit , 30 , 150
o o
AB BD BAC ADC = = = . Chng minh rng CA l tia phn
gic ca gc BCD

Bi 5: (2.0 im)
Cho ng trn (O) ni tip tam gic ABC, gi K, P, Q ln lt l cc tip im ca cc cnh
BC, AC v AB. Gi R l trung im ca on thng PK. Chng minh rng PQC KQR =

Bi 6: (1.0 im)
Cho ba s dng a, b, c. Chng minh rng

Du ng thc xy ra khi no?
thi chn hc sinh gii lp 9 tnh Hi Phng. Mn thi: Ton - Bng B
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. THI MN: TON - BNG B
Thi gian: 150 pht ( khng k thi gian giao )
Ngy thi: 06/04/2012
( thi gm 1 trang)
Bi 1: (2.0 im)
a. Cho
3 3
7 5 2; 20 14 2 A B = + = . Tnh A B + .
b. Cho a,b,c l cc s khc ) tha mn 0 a b c + + = . CMR:
2 2 2
2 2 2 2 2 2 2 2 2
3
2
a b c
a b c b c a c a b
+ + =


Bi 2:(2.0 im)
a. Gii hpt:
2 2 4
7 7 6
x y
x y

+ + + =

+ + + =


b. Cho $x,y,z$ l nhng s nguyn tha mn iu kin
4 4 4
x y z + + chia ht cho 4. CMR: c
x,y,x u chia ht cho 4.
Bi 3:(1.0 im). Tm cc nghim nguyn ca pt:
4 3 2 2
4 7 6 4 x x x x y + + + + =
Bi 4:(2.0 im). Cho tam gic ABC ni tip (O). Tip tuyn ti A & C vs ng trn ct tip
tuyn v t im B ca ng trn ln lt ti P & Q. Trong tam gic ABCv ng cao BH (H
nm gia A & C). CMR: HB l tia phn gic ca PHQ.
Bi 5:(2.0 im). Cho tam gic ABC ni tip (O). ng phn gic ca cc gc BAC & ACB ct
nhau ti I & ct ng trn taam O ln lot ti E & D. CMR: DE vung gc vi BI.
Bi 6:(1.0 im). Cho a,b,c l cc s thc dng. CMR:
2 2 2
1
( 2 ) ( 2 ) ( 2 )
a b c
b c a c a b a b c
+ + >
+ + +

Du ng thuc xy ra khi no?
..........HT..................

thi hc sinh gii tnh Vnh Phc 2011-2012 .
Cu 1 (3 im )

1. Cho
3
2
( )
1 3 3
x
f x
x x
=
+
. Hy tnh gi tr ca biu thc sau :

1 2 2010 2011
( ) ( ) ... ( ) ( )
2012 2012 2012 2012
A f f f f = + + + +

2.Cho biu thc :
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2
2 1 1 2 2
1
x x x x x
P
x x x x x x x x
+ +
= + +
+ +

Tm tt c cc gi tr ca $ x$ sao cho gi tr ca P l mt s nguyn.

Cu 2 (1,5 im )
Tm tt c cc cp s nguyn duong ( ; ) x y tha mn
3 2
( ) ( 6) x y x y + = .

Cu 3 (1,5 im )
Cho , , , a b c d l cc s thc tha mn iu kin :
2012 abc bcd cda dab a b c d + + + = + + + +
CMR :
2 2 2 2
( 1)( 1)( 1)( 1) 2012 a b c d + + + + > .

Cu 4 (3 im )

Cho 3 ung trn (
1
O ), (
2
O ) v ( O) . Gi s (
1
O ) v (
2
O ) tip xc ngoi vs nhau ti I v cng
tip xc trong vs ( O) ti
1
M ,
2
M . Tip tuyn ca (
1
O ) ti I ct ( O) ti A , $A'$. A
1
M ct
li (
1
O ) ti im
1
N , A
2
M ct li (
2
O ) ti im
2
N .

1. CMR : t gic
1
M
1
N
2
N
2
M ni tip v O A vung gc vs
1
N
2
N .

2. K ung knh P Q ca ( O) sao cho P Q vung gc vs I A ( im P nm trn cung A
1
M
ko cha im
2
M ) .CMR : Nu P
1
M v Q
2
M khng song song th A I , P
1
M v Q
2
M
ng quy .

Cu 5 ( 1 im )

Tt c cc im trn mt phng u c t mu , trong mi im c t bi 1 trong 3 mu xanh,
, tm. CMR : lun tn ti t nht mt tam gic cn, c 3 nh thuc cc im ca mt phng m
3 nh ca tam gic i mt cng mu hoc khc mu .

thi HSG lp 9 thnh ph H Ni nm 2011-2012
Thi gian lm bi: 150 pht (khng k thi gian giao )
Ngy thi: 04/4/2012

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Cu 1:
1/Cmr: A=
2012 2012 2012 2008 2008 2008
( ) ( ) 30 a b c a b c + + + + mi a;b;c nguyn dng
2/Cho
3 2012
( ) (2 21 29) f x x x =
Tnh f(x) khi
3 3
49 49
7 7
8 8
x = + +

Cu 2:
1/Gii phng trnh :
2 2
5 3 12 5 x x x + + = + +

2/Gii h phng trnh :
2 2
2 0 x xy x y y + + = v
2 2
6 x y x y + + =

Cu 3: Gii phng trnh nghim nguyn dng:
2 2
2 5 3 3 4 0 x xy y x y + + =

Cu 4: Cho na ng trn tm O ng knh BC v A bt k nm trn ng trn.T A h AH
vung gc BC v v ng trn ng knh HA ct AB;AC M v N.
a/Cmr: OA vung gc MN
b/Cho 2; 7 AH BC = = Tnh bn knh ng trn ngoi tip tam gic CMN

Cu 5:
1/Chng minh rng: iu kin cn v 1 tam gic c cc ng cao
1 2 3
; ; h h h v bn knh
ng trn ni tip r l tam gic u l:
1 2 2 3 3 1
1 1 1 1
2 2 2 3 h h h h h h r
+ + =
+ + +

2/Cho 8045 im trn 1 mt phng sao cho c 3 im bt k th to thnh 1 tam gic c din tch
<1.Chng minh rng: Lun c th c t nht 2012 im nm trong tam gic hoc trn cnh ca 1
tam gic c din tch <1
S GIO DC V O TO
THA THIN HU

K THI CHN HC SINH GII LP 9 NM HC 2012
Mn: TON
Ngy thi: 11/04/2012

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Bi 1. (2,5 im) Cho biu thc:
2
4( 1) 4( 1)
1
. 1
1
4( 1)
x x x x
Q
x
x x
+ +
| |
=
|

\ .


1. Rt gn Q.
2. Tnh gi tr ca Q vi 2013 x =

Bi 2. (4,0 im) Cho phng trnh:
2
2( 1) 2 5 0 (1) x m x m + =
1. Tm m phng trnh c nghim dng.
2. Gi
1 2
, x x l hai nghim ca phng trnh (1) . Tm m nguyn dng
2 2
1 2
2 1
x x
A
x x
| | | |
= +
| |
\ . \ .

c gi tr nguyn.

Bi 3. (4,0 im)
1. Gii phng trnh sau:
2 2
1 1
3 1 2
( 1)
x x
x x
+ + = + +


2. Tm tt c cc cp s nguyn ( ; ) x y sao cho x y < v 2012. x y + =

Bi 4. (5,0 im)
Cho hai ng trn ( ; ) O R v ( ; ), ( ) O R R R ' ' ' > ct nhau ti A v $B.$ Mt tip tuyn chung
tip xc vi ng trn ( ) O ti $C,$ tip xc vi ng trn ( ) O' ti D. Gi I l giao im
ca AB v CD. B' l im i xng ca B qua I, C' l im i xng ca B qua CD. Qua A k
ct tuyn song song vi CD ct ng trn ( ) O ti P , ct ng trn ( ) O' ti Q. Gi M,N ln
lt l giao i ca DB,CB vi PQ.



1. Chng minh rng A l trung im ca $MN$.
2. Chng minh rng A,C,B',C',D cng thuc mt ung trn.

Bi 5. (2,5 im) Cho ung trn tm O ni tip tam gic ABC vung ti A. ng trn
( ) O ni tip tam gic tip xc vi BC ti D. Chng minh rng .
ABC
S BDCD = .

Bi 6. (2,0 im) C hay khng s t nhin n tho
2
2012 n + l s chnh phng? Tm n .

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----HT----
...
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