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thi vo THPT
s 1Cu 1 ( 3 im )
Cho biu thc :2
22
12
1.)
1
1
1
1( x
x
xxA
++=
1) Tm iu kin ca x biu thc A c ngha .2) Rt gn biu thc A .3) Gii
phng trnh theo x khi A = -2 .
Cu 2 ( 1 im )Gii phng trnh :
12315 = xxxCu 3 ( 3 im )
Trong mt phng to cho im A ( -2 , 2 ) v ng th
(D) : y = - 2(x +1) .a) im A c thuc (D) hay khng ?b)Tm a trong
hm s y = ax2 c th (P) i qua A .c) Vit phng trnh ng thng i qua A v
vung gc vi (D
Cu 4 ( 3 im )Cho hnh vung ABCD c nh , c di cnh l a .E l i
chuyn trn on CD ( E khc D ) , ng thng AE ct ng thnti F , ng thng
vung gc vi AE ti A ct ng thng CD ti K .
1) Chng minh tam gi c ABF = tam gic ADK t suy ra tam
AFK vung cn .2) Gi I l trung im ca FK , Chng minh I l tm ng
trqua A , C, F , K .
3) Tnh s o gc AIF , suy ra 4 im A , B , F , I cng nmmt ng trn
.
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thi vo THPT
s 2Cu 1 ( 2 im )
Cho hm s : y =2
2
1
x 1) Nu tp xc nh , chiu bin thin v v thi ca hm s2) Lp phng trnh
ng thng i qua im ( 2 , -6 ) c h
gc a v tip xc vi th hm s trn .Cu 2 ( 3 im )
Cho phng trnh : x2 mx + m 1 = 0 .1) Gi hai nghim ca phng trnh l
x1 , x2 . Tnh gi tr ca
thc .
2212
21
2
2
2
1 1
xxxx
xxM
+
+=
. T tm m M > 0 .2)Tm gi tr ca m biu thc P = 122
2
1 + xx t gi tr nh.
Cu 3 ( 2 im )Gii phng trnh :
a) xx = 44b) xx =+ 332
Cu 4 ( 3 im )Cho hai ng trn (O1) v (O2) c bn knh bng R ct
nhau
v B , qua A v ct tuyn ct hai ng trn (O1) v (O2) th t ti F , ng
thng EC , DF ct nhau ti P .1) Chng minh rng : BE = BF .2) Mt ct
tuyn qua A v vung gc vi AB ct (O1) v (O2) ln
C,D . Chng minh t gic BEPF , BCPD ni tip v BP vunvi EF .
3) Tnh din tch phn giao nhau ca hai ng trn khi AB =
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thi vo THPT
s 3Cu 1 ( 3 im )
1) Gii bt phng trnh : 42
+ xx
Cu 2 ( 2 im )Cho phng trnh : 2x2 ( m+ 1 )x +m 1 = 0a) Gii phng
trnh khi m = 1 .
b) Tm cc gi tr ca m hiu hai nghim bng tchchng .Cu3 ( 2 im )
Cho hm s : y = ( 2m + 1 )x m + 3 (1)a) Tm m bit th hm s (1) i
qua im A ( -2 ; 3 ) .b) Tm im c nh m th hm s lun i qua vi m
tr ca m .Cu 4 ( 3 im )
Cho gc vung xOy , trn Ox , Oy ln lt ly hai im A v B
cho OA = OB . M l mt im bt k trn AB .Dng ng trn tm O1 i qua M v
tip xc vi Ox ti A , trn tm O2 i qua M v tip xc vi Oy ti B , (O1) ct
(O2) ti imhai N .
1) Chng minh t gi c OANB l t gic ni tip v ON l phnca gc ANB
.
2) Chng minh M nm trn mt cung trn c nh khi M thay 3) Xc nh v tr
ca M khong cch O1O2 l ngn nht .
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thi vo THPT
s 4 .Cu 1 ( 3 im )
Cho biu thc :
+++
+
=1
2:)
1
1
1
2(
xx
x
xxx
xxA
a) Rt gn biu thc .b)Tnh gi tr ca A khi 324+=x
Cu 2 ( 2 im )Gii phng trnh :
xx
x
xx
x
x
x
6
1
6
2
36
22222 +
=
Cu 3 ( 2 im )
Cho hm s : y = - 22
1x
a)Tm x bit f(x) = - 8 ; -8
1; 0 ; 2 .
b) Vit phng trnh ng thng i qua hai im A v B nm th c honh ln lt l
-2 v 1 .
Cu 4 ( 3 im )Cho hnh vung ABCD , trn cnh BC ly 1 im M . ng
tr
ng knh AM ct ng trn ng knh BC ti N v ct cnh AD ti 1) Chng minh
E, N , C thng hng .2) Gi F l giao im ca BN v DC . Chng minh CDEBCF
=3) Chng minh rng MF vung gc vi AC .
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thi vo THPT
s 5Cu 1 ( 3 im )
Cho h phng trnh :
=+
=+
13
52
ym x
ym x
a) Gii h phng trnh khi m = 1 .b) Gii v bin lun h phng trnh theo
tham s m .c)Tm m x y = 2 .
Cu 2 ( 3 im )
1)Gii h phng trnh :
=
=+
yyxx
yx
22
221
2)Cho phng trnh bc hai : ax2 + bx + c = 0 . Gi hai nghimphng
trnh l x1 , x2 . Lp phng trnh bc hai c hai ngl 2x1+ 3x2 v 3x1 + 2x2
.
Cu 3 ( 2 im )Cho tam gic cn ABC ( AB = AC ) ni tip ng trn tm O
mt im chuyn ng trn ng trn . T B h ng thng vgc vi AM ct CM D .
Chng minh tam gic BMD cnCu 4 ( 2 im )
1)Tnh :25
1
25
1
+
+
2) Gii bt phng trnh :( x1 ) ( 2x + 3 ) > 2x( x + 3 ) .
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thi vo THPT
s 6Cu 1 ( 2 im )
Gii h phng trnh :
=
=+
+
41
2
1
5
71
1
1
2
yx
yx
Cu 2 ( 3 im )
Cho biu thc :xxxxxx
xA
+++
=2
1:
1
a) Rt gn biu thc A .b) Coi A l hm s ca bin x v thi hm s A .
Cu 3 ( 2 im )Tm iu kin ca tham s m hai phng trnh sau c ng
chung . x2 + (3m + 2 )x 4 = 0 v x2 + (2m + 3 )x +2 =0 .Cu 4 ( 3
im )
Cho ng trn tm O v ng thng d ct (O) ti hai A,B . T mt im M trn d
v hai tip tuyn ME , MF ( E , F lim ) .
1) Chng minh gc EMO = gc OFE v ng trn i qua 3 M, E, F i qua 2 im
c nh khi m thay i trn d .
2) Xc nh v tr ca M trn d t gic OEMF l hnh vun
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thi vo THPT
s 7
Cu 1 ( 2 im )Cho phng trnh (m2 + m + 1 )x2 - ( m2 + 8m + 3 )x 1
= 0
a) Chng minh x1x2 < 0 .b) Gi hai nghim ca phng trnh l x1, x2
. Tm gi tr ln n
nh nht ca biu thc :S = x1 + x2 .
Cu 2 ( 2 im )Cho phng trnh : 3x2 + 7x + 4 = 0 . Gi hai nghim ca
p
trnh l x1 , x2 khng gii phng trnh lp phng trnh bc hai m
hai nghim l :1
2
1
xx
v1
1
2
xx
.
Cu 3 ( 3 im )1) Cho x2 + y2 = 4 . Tm gi tr ln nht , nh nht ca x
+ y .
2) Gii h phng trnh :
=+
=
8
122
yx
yx
3) Gii phng trnh : x4 10x3 2(m 11 )x2 + 2 ( 5m +6)x +0
Cu 4 ( 3 im )
Cho tam gic nhn ABC ni tip ng trn tm O . ng gic trong ca gc A ,
B ct ng trn tm O ti D v E , gi giao hai ng phn gic l I , ng thng DE
ct CA, CB ln lt ti M , N
1) Chng minh tam gic AIE v tam gic BID l tam gic cn .2) Chng
minh t gic AEMI l t gic ni tip v MI // BC .3) T gic CMIN l hnh g
?
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thi vo THPT
s 8
Cu1 ( 2 im )Tm m phng trnh ( x2 + x + m) ( x2 + mx + 1 ) = 0
nghim phn bit .Cu 2 ( 3 im )
Cho h phng trnh :
=+
=+
64
3
ym x
m yx
a) Gii h khi m = 3b) Tm m phng trnh c nghim x > 1 , y > 0
.
Cu 3 ( 1 im )Cho x , y l hai s dng tho mn x5+y5 = x3 + y3 . Chng
min
+ y2 1 + xyCu 4 ( 3 im )
1)Cho t gic ABCD ni tip ng trn (O) . Chng minhAB.CD + BC.AD =
AC.BD
2)Cho tam gic nhn ABC ni tip trong ng trn (O) ng
AD . ng cao ca tam gic k t nh A ct cnh BC ti K vng trn (O) ti E
.a) Chng minh : DE//BC .b) Chng minh : AB.AC = AK.AD .c) Gi H l trc
tm ca tam gic ABC . Chng minh t gic BH
hnh bnh hnh .
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thi vo THPT
s 9Cu 1 ( 2 im )
Trc cn thc mu cc biu thc sau :
232
12
++=A ;
222
1
+=B ;
123
1
+=C
Cu 2 ( 3 im )Cho phng trnh : x2 ( m+2)x + m2 1 = 0 (1)a) Gi x1,
x2 l hai nghim ca phng trnh .Tm m tho mn
x2 = 2 .b) Tm gi tr nguyn nh nht ca m phng trnh c
nghim khc nhau .Cu 3 ( 2 im )
Cho32
1;
32
1
+=
= ba
Lp mt phng trnh bc hai c cc h s bng s v c cc nghi
x1 =1
;1
2+
=+ a
bx
b
a
Cu 4 ( 3 im )Cho hai ng trn (O1) v (O2) ct nhau ti A v B .
Mt
thng i qua A ct ng trn (O1) , (O2) ln lt ti C,D , gi I , J l im
ca AC v AD .
1) Chng minh t gic O1IJO2 l hnh thang vung .
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thi vo THPT
2) Gi M l giao dim ca CO1 v DO2 . Chng minh O1 , O2 , nm trn mt
ng trn
3) E l trung im ca IJ , ng thng CD quay quanh A . Tmhp im E.
4) Xc nh v tr ca dy CD dy CD c di ln nht
s 10Cu 1 ( 3 im )
1)V th ca hm s : y =2
2x
2)Vit phng trnh ng thng i qua im (2; -2) v (1 ; -4
3)Tm giao im ca ng thng va tm c vi th trCu 2 ( 3 im )a) Gii phng
trnh :
21212 =++ xxxx
b)Tnh gi tr ca biu thc22 11 xyyxS +++= vi ayxxy =+++ )1)(1(
22
Cu 3 ( 3 im )Cho tam gic ABC , gc B v gc C nhn . Cc ng trn
ng
AB , AC ct nhau ti D . Mt ng thng qua A ct ng trn knh AB , AC ln
lt ti E v F .
1) Chng minh B , C , D thng hng .2) Chng minh B, C , E , F nm
trn mt ng trn .3) Xc nh v tr ca ng thng qua A EF c di ln
.Cu 4 ( 1 im )
Cho F(x) = xx ++ 12
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thi vo THPT
a) Tm cc gi tr ca x F(x) xc nh .b) Tm x F(x) t gi tr ln nht
.
s 11Cu 1 ( 3 im )
1) V th hm s2
2xy =
2) Vit phng trnh ng thng i qua hai im ( 2 ; -2 ) v
- 4 )3) Tm giao im ca ng thng va tm c vi th trCu 2 ( 3 im )
1) Gii phng trnh :21212 =++ xxxx
2) Gii phng trnh :
512
412=
++
+x
x
x
x
Cu 3 ( 3 im )Cho hnh bnh hnh ABCD , ng phn gic ca gc BAD c
v BC theo th t ti M v N . Gi O l tm ng trn ngoi tip tamMNC .1)
Chng minh cc tam gic DAM , ABN , MCN , l cc tam gic c2) Chng minh B
, C , D , O nm trn mt ng trn .
Cu 4 ( 1 im )Cho x + y = 3 v y 2 . Chng minh x2 + y2 5
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thi vo THPT
s 12Cu 1 ( 3 im )
1) Gii phng trnh : 8152 =++ xx2) Xc nh a tng bnh phng hai nghim
ca phng t
x2 +ax +a2 = 0 l b nht .Cu 2 ( 2 im )
Trong mt phng to cho im A ( 3 ; 0) v ng thn2y = - 2 .a) V th ca
ng thng . Gi giao im ca ng th
trc tung v trc honh l B v E .b) Vit phng trnh ng thng qua A v
vung gc vi
thng x 2y = -2 .c) Tm to giao im C ca hai ng thng . Chng
rng EO. EA = EB . EC v tnh din tch ca t gic OACB .Cu 3 ( 2 im
)
Gi s x1 v x2 l hai nghim ca phng trnh :x2(m+1)x +m2 2m +2 = 0
(1)
a) Tm cc gi tr ca m phng trnh c nghim kp nghim phn bit .
b)Tm m 2221 xx + t gi tr b nht , ln nht .Cu 4 ( 3 im )
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thi vo THPT
Cho tam gic ABC ni tip ng trn tm O . K ng cao AH trung im ca AB
, BC theo th t l M , N v E , F theo th t l chiu vung gc ca ca B , C
trn ng knh AD .
a) Chng minh rng MN vung gc vi HE .
b) Chng minh N l tm ng trn ngoi tip tam gic HEF .
s 13Cu 1 ( 2 im )
So snh hai s :33
6;
211
9
=
= ba
Cu 2 ( 2 im )Cho h phng trnh :
=
=+
2
532
yx
ayx
Gi nghim ca h l ( x , y ) , tm gi tr ca a x 2 + y2 tr nh nht
.
Cu 3 ( 2 im )Gi h phng trnh :
=++
=++
7
5
22x yyx
x yyx
Cu 4 ( 3 im )
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thi vo THPT
1) Cho t gic li ABCD cc cp cnh i AB , CD ct nhau tiBC , AD ct
nhau ti Q . Chng minh rng ng trn ngoi tip ccgic ABQ , BCP , DCQ ,
ADP ct nhau ti mt im .
3) Cho t gic ABCD l t gic ni tip . Chng minh
BD
AC
DADCBCBA
CDCBADAB
=++
..
..
Cu 4 ( 1 im )Cho hai s dng x , y c tng bng 1 . Tm gi tr nh nht
c
xyyxS
4
3122+
+=
s 14Cu 1 ( 2 im )
Tnh gi tr ca biu thc :
322
32
322
32
+++
+=P
Cu 2 ( 3 im )1) Gii v bin lun phng trnh :(m2 + m +1)x2 3m = ( m
+2)x +3
2) Cho phng trnh x2 x 1 = 0 c hai nghim l x1 , x2 . H
phng trnh bc hai c hai nghim l :2
2
2
1
1;
1 x
x
x
x
Cu 3 ( 2 im )
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thi vo THPT
Tm cc gi tr nguyn ca x biu thc :2
32
+
=x
xP l nguy
Cu 4 ( 3 im )Cho ng trn tm O v ct tuyn CAB ( C ngoi ng tr
T im chnh gia ca cung ln AB k ng knh MN ct AB ti I ct ng trn ti
E , EN ct ng thng AB ti F .
1) Chng minh t gic MEFI l t gic ni tip .2) Chng minh gc CAE bng
gc MEB .3) Chng minh : CE . CM = CF . CI = CA . CB
s 15Cu 1 ( 2 im )
Gii h phng trnh :
=++
=
044
325
2
22
x yy
yx yx
Cu 2 ( 2 im )Cho hm s :
4
2x
y = v y = - x 1
a) V th hai hm s trn cng mt h trc to .
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thi vo THPT
b) Vit phng trnh cc ng thng song song vi ng th
= - x 1 v ct th hm s4
2xy = ti im c tung l
Cu 2 ( 2 im )Cho phng trnh : x2 4x + q = 0a) Vi gi tr no ca q th
phng trnh c nghim .b) Tm q tng bnh phng cc nghim ca phng trnh
.Cu 3 ( 2 im )
1)Tm s nguyn nh nht x tho mn phng trnh :413 =++ xx
2)Gii phng trnh :0113
22 = xx
Cu 4 ( 2 im )
Cho tam gic vung ABC ( gc A = 1 v ) c AC < AB , AH l cao k t
nh A . Cc tip tuyn ti A v B vi ng trn tm O tip tam gic ABC ct nhau
ti M . on MO ct cnh AB E , MC cng cao AH ti F . Ko di CA cho ct ng
thng BM D . ng thBF ct ng thng AM N .
a) Chng minh OM//CD v M l trung im ca on thng Bb) Chng minh EF
// BC .c) Chng minh HA l tia phn gic ca gc MHN .
s 16Cu 1 : ( 2 im )
Trong h trc to Oxy cho hm s y = 3x + m (*)1) Tnh gi tr ca m th
hm s i qua : a) A( -1 ; 3
B( - 2 ; 5 )2) Tm m th hm s ct trc honh ti im c hon
l - 3 .3) Tm m th hm s ct trc tung ti im c tung
- 5 .Cu 2 : ( 2,5 im )
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thi vo THPT
Cho biu thc :1 1 1 1 1
A= :1- x 1 1 1 1x x x x
+ + + + a) Rt gn biu thc A .b) Tnh gi tr ca A khi x = 7 4 3+
c) Vi gi tr no ca x th A t gi tr nh nht .Cu 3 : ( 2 im )Cho phng
trnh bc hai : 2 3 5 0x x+ = v gi hai nghim ca ptrnh l x1 v x2 .
Khng gii phng trnh , tnh gi tr ca ccthc sau :
a) 2 21 2
1 1
x x+ b) 2 21 2x x+
c) 3 31 2
1 1
x x+ d) 1 2x x+
Cu 4 ( 3.5 im )Cho tam gic ABC vung A v mt im D nm gia A v B
.trn ng knh BD ct BC ti E . Cc ng thng CD , AE ln lt cng trn ti cc
im th hai F , G . Chng minh :
a) Tam gic ABC ng dng vi tam gic EBD .b) T gic ADEC v AFBC ni
tip c trong mt ng trn .c) AC song song vi FG .d) Cc ng thng AC , DE
v BF ng quy .
s 17
Cu 1 ( 2,5 im )Cho biu thc : A =
1 1 2:
2
a a a a a
aa a a a
+ + +
a) Vi nhng gi tr no ca a th A xc nh .b) Rt gn biu thc A .c) Vi
nhng gi tr nguyn no ca a th A c gi tr nguyn .
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thi vo THPT
Cu 2 ( 2 im )Mt t d nh i t A n B trong mt thi gian nht
Nu xe chy vi vn tc 35 km/h th n chm mt 2 gi . Nu xevi vn tc 50
km/h th n sm hn 1 gi . Tnh qung ng AB v
gian d nh i lc u .Cu 3 ( 2 im )
a) Gii h phng trnh :
1 13
2 31
x y x y
x y x y
+ = + = +
b) Gii phng trnh : 2 2 25 5 25
5 2 10 2 50
x x x
x x x x x
+ + =
+ Cu 4 ( 4 im )
Cho im C thuc on thng AB sao cho AC = 10 cm ;CB cm . V v cng mt
na mt phng b l AB cc na ng trn knh theo th t l AB , AC , CB c tm ln
lt l O , I , K . ng vgc vi AB ti C ct na ng trn (O) E . Gi M , N
theo th t lim cuae EA , EB vi cc na ng trn (I) , (K) . Chng minh
:
a) EC = MN .b) MN l tip tuyn chung ca cc na ng trn (I) v (K) .c)
Tnh di MN .d) Tnh din tch hnh c gii hn bi ba na ng trn .
18Cu 1 ( 2 im )
Cho biu thc : A =1 1 1 1 1
1 1 1 1 1
a a
a a a a a
+ ++ +
+ + + +1) Rt gn biu thc A .2) Chng minh rng biu thc A lun dng vi
mi a .
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thi vo THPT
Cu 2 ( 2 im )Cho phng trnh : 2x2 + ( 2m - 1)x + m - 1 = 01) Tm m
phng trnh c hai nghim x1 , x2 tho mn 3x1
= 11 .
2) Tm ng thc lin h gia x1 v x2 khng ph thuc vo m3) Vi gi tr no
ca m th x1 v x2 cng dng .Cu 3 ( 2 im )
Hai t khi hnh cng mt lc i t A n B cch nhau 300 kt th nht mi gi
chy nhanh hn t th hai 10 km nn n B shn t th hai 1 gi . Tnh vn tc mi
xe t .Cu 4 ( 3 im )
Cho tam gic ABC ni tip ng trn tm O . M l mt im tcung AC ( khng
cha B ) k MH vung gc vi AC ; MK vung gc vi B
1) Chng minh t gic MHKC l t gic ni tip .
2) Chng minh AMB HMK=3) Chng minh AMB ng dng vi HMK .
Cu 5 ( 1 im )
Tm nghim dng ca h :( ) 6
( ) 12
( ) 30
xy x y
yz y z
zx z x
+ = + = + =
19( Thi tuyn sinh lp 10 - THPT nm 2006 - 2007 - Hi dng -
pht - Ngy 28 / 6 / 2006Cu 1 ( 3 im )
1) Gii cc phng trnh sau :a) 4x + 3 = 0b) 2x - x2 = 0
2) Gii h phng trnh :2 3
5 4
x y
y x
= + =
Cu 2( 2 im )
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thi vo THPT
1) Cho biu thc : P = ( )3 1 4 4 a > 0 ; a 442 2
a a a
aa a
+ +
+a) Rt gn P .b) Tnh gi tr ca P vi a = 9 .
2) Cho phng trnh : x2
- ( m + 4)x + 3m + 3 = 0 ( m l tham a) Xc nh m phng trnh c mt
nghim bng 2 . Tmnghim cn li .
b) Xc nh m phng trnh c hai nghim x1 ; x2 tho m3 3
1 20x x+
Cu 3 ( 1 im )Khong cch gia hai thnh ph A v B l 180 km . Mt t
i
n B , ngh 90 pht B , ri li t B v A . Thi gian lc i n lv A l 10
gi . Bit vn tc lc v km vn tc lc i l 5 km/h . T
vn tc lc i ca t .Cu 4 ( 3 im )T gic ABCD ni tip ng trn ng knh AD
. Hai ng ch
AC , BD ct nhau ti E . Hnh chiu vung gc ca E trn AD l F . thng
CF ct ng trn ti im th hai l M . Giao im ca BD vl N
Chng minh :a) CEFD l t gic ni tip .b) Tia FA l tia phn gic ca gc
BFM .c) BE . DN = EN . BD
Cu 5 ( 1 im )Tm m gi tr ln nht ca biu thc 2
2
1
x m
x
++
bng 2 .
20Cu 1 (3 im )
1) Gii cc phng trnh sau :a) 5( x - 1 ) = 2b) x2 - 6 = 0
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thi vo THPT
2) Tm to giao im ca ng thng y = 3x - 4 vi hai tr .Cu 2 ( 2 im
)
1) Gi s ng thng (d) c phng trnh : y = ax + b .
Xc nh a , b (d) i qua hai im A ( 1 ; 3 ) v B ( - 1)2) Gi x1 ; x2
l hai nghim ca phng trnh x2 - 2( m - 1)x - 4
m l tham s )Tm m : 1 2 5x x+ =
3) Rt gn biu thc : P =1 1 2
( 0; 0)2 2 2 2 1
x xx x
x x x
+
+ Cu 3( 1 im)
Mt hnh ch nht c din tch 300 m2 . Nu gim chiu rng
3 m , tng chiu di thm 5m th ta c hnh ch nht mi c dinbng din tch
bng din tch hnh ch nht ban u . Tnh chu vhnh ch nht ban u .Cu 4 ( 3
im )
Cho im A ngoi ng trn tm O . K hai tip tuyn AB , Avi ng trn (B ,
C l tip im ) . M l im bt k trn cung nh( M B ; M C ) . Gi D , E , F
tng ng l hnh chiu vung gc ctrn cc ng thng AB , AC , BC ; H l giao
im ca MB v DF ; Kgiao im ca MC v EF .
1) Chng minh :a) MECF l t gic ni tip .b) MF vung gc vi HK .
2) Tm v tr ca M trn cung nh BC tch MD . ME ln nhCu 5 ( 1 im )
Trong mt phng to ( Oxy ) cho i-3 ; 0 ) v Parabol (P) c phng trnh y
= x2 . Hy tm to cim M thuc (P) cho di on thng AM nh nht .
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thi vo THPT
II, Cc thi vo ban t nhin
1
Cu 1 : ( 3 im ) ii cc phng trnha) 3x2 48 = 0 .b) x2 10 x + 21 =
0 .
c)5
203
5
8
=+
xx
Cu 2 : ( 2 im )a) Tm cc gi tr ca a , b bit rng th ca hm s y =
a
i qua hai im
A( 2 ; - 1 ) v B ( )2;2
1
b) Vi gi tr no ca m th th ca cc hm s y = mx + 33x7 v th ca hm s
xc nh cu ( a ) ng quy .
Cu 3 ( 2 im ) Cho h phng trnh .
=+=
nyx
nymx
2
5
a) Gii h khi m = n = 1 .
b)Tm m , n h cho c nghim
+==
13
3
y
x
Cu 4 : ( 3 im )Cho tam gic vung ABC ( C = 900 ) ni tip trong ng
trn t
. Trn cung nh AC ta ly mt im M bt k ( M khc A v C ) . V trn tm A
bn knh AC , ng trn ny ct ng trn (O) ti i( D khc C ) . on thng BM ct
ng trn tm A im N .
a) Chng minh MB l tia phn gic ca gc CMD .b) Chng minh BC l tip
tuyn ca ng trn tm A ni trnc) So snh gc CNM vi gc MDN .
d) Cho bit MC = a , MD = b . Hy tnh on thng MN theo a v
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thi vo THPT
s 2Cu 1 : ( 3 im )Cho hm s : y =
2
3 2x ( P )
a)Tnh gi tr ca hm s ti x = 0 ; -1 ;3
1 ; -2 .
b) Bit f(x) =2
1;
3
2;8;
2
9 tm x .
c) Xc nh m ng thng (D) : y = x + m 1 tip x(P) .
Cu 2 : ( 3 im )Cho h phng trnh :
=+=
2
2 2
yx
mmyx
a) Gii h khi m = 1 .b) Gii v bin lun h phng trnh .
Cu 3 : ( 1 im )Lp phng trnh bc hai bit hai nghim ca phng trnh l
:
2
321
=x 2
322
+=x
Cu 4 : ( 3 im )Cho ABCD l mt t gic ni tip . P l giao im ca
hai
cho AC v BD .a) Chng minh hnh chiu vung gc ca P ln 4 cnh ca t
g
4 nh ca mt t gic c ng trn ni tip .b) M l mt im trong t gic sao
cho ABMD l hnh bnh h
Chng minh rng nu gc CBM = gc CDM th gc ACD =BCM .c) Tm iu kin ca
t gic ABCD :
)..(2
1BCADCDABSABCD +=
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thi vo THPT
s 3
Cu 1 ( 2 im ) .Gii phng trnh
a) 1- x - x3 = 0b) 0322 = xx
Cu 2 ( 2 im ) .
Cho Parabol (P) : y = 22
1x v ng thng (D) : y = px + q .
Xc nh p v q ng thng (D) i qua im A ( - 1 ; 0tip xc vi (P) . Tm
to tip im .
Cu 3 : ( 3 im )
Trong cng mt h trc to Oxy cho parabol (P) : 24
1xy =
v ng thng (D) : 12 = mmxya) V (P) .b) Tm m sao cho (D) tip xc vi
(P) .c) Chng t (D) lun i qua mt im c nh .
Cu 4 ( 3 im ) .Cho tam gic vung ABC ( gc A = 900 ) ni tip ng trn
t
k ng knh AD .1) Chng minh t gic ABCD l hnh ch nht .
2) Gi M , N th t l hnh chiu vung gc ca B , C trn AD , ng cao ca
tam gic ( H trn cnh BC ) . Chng minhvung gc vi AC .
3) Xc nh tm ng trn ngoi tip tam gic MHN .4) Gi bn knh ng trn
ngoi tip v ng trn ni tip
gic ABC l R v r . Chng minh ACABrR .+
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thi vo THPT
s 4
Cu 1 ( 3 im ) .Gii cc phng trnh sau .
a) x2 + x 20 = 0 .
b)xxx
1
1
1
3
1 =
++
c) 131 = xx
Cu 2 ( 2 im )Cho hm s y = ( m2 ) x + m + 3 .
a) Tm iu kim ca m hm s lun nghch bin .b) Tm m th hm s ct trc
honh ti im c hn
l 3 .
c)Tm m th cc hm s y = - x + 2 ; y = 2x1v y =2 )x + m + 3 ng quy
.
Cu 3 ( 2 im )Cho phng trnh x2 7 x + 10 = 0 . Khng gii phng t
tnh .a) 22
2
1 xx +b) 2221 xx c) 21 xx +
Cu 4 ( 4 im )Cho tam gic ABC ni tip ng trn tm O , ng phn
trong ca gc A ct cnh BC ti D v ct ng trn ngoi tip ti I .a) Chng
minh rng OI vung gc vi BC .b) Chng minh BI2 = AI.DI .c) Gi H l hnh
chiu vung gc ca A trn BC .
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thi vo THPT
Chng minh gc BAH = gc CAO .
d) Chng minh gc HAO = B C
s 5
Cu 1 ( 3 im ) . Cho hm s y = x2 c th l ng Parabol (P) .
a) Chng minh rng im A( - )2;2 nm trn ng cong (P) .b)Tm m th (d )
ca hm s y = ( m 1 )x + m (
R , m 1 )ctng cong(P) ti mt im.c) Chng minh rng vi mi m khc 1 th
(d ) ca hm s y
1)x + m lun i qua mt im c nh .
Cu 2 ( 2 im ) .
Cho h phng trnh :
=+=+13
52
ymx
ymx
a) Gii h phng trnh vi m = 1b) Gii bin lun h phng trnh theo tham
s m .c)Tm m h phng trnh c nghim tho mn x2 + y2 = 1
Cu 3 ( 3 im )Gii phng trnh
5168143 =+++ xxxx
Cu 4 ( 3 im )Cho tam gic ABC , M l trung im ca BC . Gi s gcBAM
=
BCA.a) Chng minh rng tam gic ABM ng dng vi tam gic CBA .b) Chng
minh minh : BC2 = 2 AB2 . So snh BC v ng cho
vung cnh l AB .
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thi vo THPT
c) Chng t BA l tip tuyn ca ng trn ngoi tip tamAMC .
d) ng thng qua C v song song vi MA , ct ng thng D . Chng t ng
trn ngoi tip tam gic ACD tip xc vi
s 6 .Cu 1 ( 3 im )
a) Gii phng trnh : 231 =+ xxc) Cho Parabol (P) c phng trnh y =
ax2 . Xc nh a (
qua im A( -1; -2) . Tm to cc giao im ca (P) v trung trc ca on OA
.
Cu 2 ( 2 im )a) Gii h phng trnh
=
=+
11
3
2
2
22
11
1
xy
yx
1) Xc nh gi tr ca m sao cho th hm s (H) : y =
ng thng (D) : y = - x + m tip xc nhau .Cu 3 ( 3 im )
Cho phng trnh x2 2 (m + 1 )x + m2 - 2m + 3 = 0 (1a) Gii phng
trnh vi m = 1 .
b) Xc nh gi tr ca m (1) c hai nghim tri du .c)Tm m (1) c mt
nghim bng 3 . Tm nghim kia .
Cu 4 ( 3 im )Cho hnh bnh hnh ABCD c nh D nm trn ng trn ng
AB . H BN v DM cng vung gc vi ng cho AC .Chng minh :a) T gic
CBMD ni tip .
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thi vo THPT
b) Khi im D di ng trn trn ng trn th BMD BCD+ ki .
c) DB . DC = DN . AC
s 7Cu 1 ( 3 im )
Gii cc phng trnh :a) x4 6x2- 16 = 0 .b) x2 - 2 x - 3 = 0
c) 09
813
12
=+
xx
xx
Cu 2 ( 3 im )Cho phng trnh x2 ( m+1)x + m2 2m + 2 = 0(1)a) Gii
phng trnh vi m = 2 .b) Xc nh gi tr ca m phng trnh c nghim kp .
nghim kp .c) Vi gi tr no ca m th 22
2
1 xx + t gi tr b nht , ln nht Cu 3 ( 4 im ) .
Cho t gic ABCD ni tip trong ng trn tm O . Gi I l
im ca hai ng cho AC v BD , cn M l trung im ca cnhNi MI ko di ct
cnh AB N . T B k ng thng song song ving thng ct cc ng thng AC E .
Qua E k ng thsong song vi CD , ng thng ny ct ng thng BD F .
a) Chng minh t gic ABEF ni tip .b) Chng minh I l trung im ca on
thng BF v AI . IE =
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thi vo THPT
c) Chng minh2
2
NA IA=
NB IB
s 8
Cu 1 ( 2 im )
Phn tch thnh nhn t .a) x2- 2y2 + xy + 3y 3x .b) x3 + y3 + z3 -
3xyz .
Cu 2 ( 3 im )Cho h phng trnh .
=+
=
53
3
m yx
ym x
a) Gii h phng trnh khi m = 1 .b)Tm m h c nghim ng thi tho mn iu
k
13
)1(72
=+
+m
myx
Cu 3 ( 2 im )Cho hai ng thng y = 2x + m 1 v y = x + 2m .a) Tm
giao im ca hai ng thng ni trn .
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thi vo THPT
b) Tm tp hp cc giao im .Cu 4 ( 3 im )Cho ng trn tm O . A l mt im
ngoi ng trn , t A k
tuyn AM , AN vi ng trn , ct tuyn t A ct ng trn ti B
( B nm gia A v C ) . Gi I l trung im ca BC .1) Chng minh rng 5
im A , M , I , O , N nm trn mt trn .
2) Mt ng thng qua B song song vi AM ct MN v MC lnE v F . Chng
minh t gic BENI l t gic ni tip v E l im ca EF .
s 9Cu 1 ( 3 im )
Cho phng trnh : x2 2 ( m + n)x + 4mn = 0 .a) Gii phng trnh khi m
= 1 ; n = 3 .b) Chng minh rng phng trnh lun c nghim vi mi m ,n c)
Gi x1, x2, l hai nghim ca phng trnh . Tnh 2221 xx + th
,n .
Cu 2 ( 2 im )Gii cc phng trnh .a) x3 16x = 0b) 2=xx
c) 19
14
3
12
=
+ xx
Cu 3 ( 2 im )
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thi vo THPT
Cho hm s : y = ( 2m 3)x2 .1) Khi x < 0 tm cc gi tr ca m hm s
lun ng bin 2) Tm m th hm s i qua im ( 1 , -1 ) . V t
m va tm c .
Cu 4 (3im )Cho tam gic nhn ABC v ng knh BON . Gi H l trc tmtam
gic ABC , ng thng BH ct ng trn ngoi tip tam gic AM .
1) Chng minh t gic AMCN l hnh thanng cn .2) Gi I l trung im ca
AC . Chng minh H , I , N thng hn3) Chng minh rng BH = 2 OI v tam
gic CHM cn .
s 10 .Cu 1 ( 2 im )
Cho phng trnh : x2 + 2x 4 = 0 . gi x1, x2, l nghim ca ptrnh
.
Tnh gi tr ca biu thc :2
2
1
2
21
21
2
2
2
1
322
xxxx
xxxxA +
+=
Cu 2 ( 3 im)
Cho h phng trnh
=+
=
12
72
yx
yxa
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thi vo THPT
a) Gii h phng trnh khi a = 1b) Gi nghim ca h phng trnh l ( x ,
y) . Tm cc gi tr
x + y = 2 .Cu 3 ( 2 im )
Cho phng trnh x
2
( 2m + 1 )x + m
2
+ m 1 =0.a) Chng minh rng phng trnh lun c nghim vi mi m .b) Gi
x1, x2, l hai nghim ca phng trnh . Tm m sao cho : x2 )( 2x2 x1 ) t
gi tr nh nht v tnh gi tr nh nht
c) Hy tm mt h thc lin h gia x1 v x2 m khng ph tvo m .
Cu 4 ( 3 im )Cho hnh thoi ABCD c gc A = 600 . M l mt im trn
cnh
ng thng AM ct cnh DC ko di ti N .a) Chng minh : AD2 = BM.DN
.
b) ng thng DM ct BN ti E . Chng minh t gic BECD ni tc) Khi hnh
thoi ABCD c nh . Chng minh im E nm trn
cung trn c nh khi m chy trn BC .
thi vo 10 h THPT chuyn 1999 i hc khoa hc t nhin.Bi 1. Cho cc s
a, b, c tha mn iu kin:{ 2 2 20 14a b ca b c+ + =+ + = .Hy tnh gi tr
biu thc 4 4 41P a b c+ + + .Bi 2. a) Gii phng trnh 3 7 2 8x x x
=
b) Gii h phng trnh :1 1 9
21 5
2
x yx y
xyxy
+ + + = + =Bi 3. Tm tt c cc s nguyn dng n sao cho n2 + 9n 2
chia
cho n + 11.Bi 4. Cho vng trn (C) v im I nm trong vng trn. Dng
q
hai dy cung bt k MIN, EIF. Gi M, N, E, F l cc trung imIM, IN,
IE, IF.
a) Chng minh rng : t gic MENF l t gic ni tip.b) Gi s I thay i,
cc dy cung MIN, EIF thay i. Chng mirng vng trn ngoi tip t gic MENF
c bn knh khng c) Gi s I c nh, cc day cung MIN, EIF thay i nhng lun
vgc vi nhau. Tm v tr ca cc dy cung MIN, EIF sao cho t gMENF c din
tch ln nht.
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thi vo THPT
Bi 5. Cc s dng x, y thay i tha mn iu kin: x + y = 1
Tm gi tr nh nht ca biu thc : 2 22 21 1
P x yy x
= + +
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thi vo THPT
thi vo 10 h THPT chuyn ton 1992 i hc tng hpBi 1. a) Gii phng
trnh (1 + x)4 = 2(1 + x4).
b) Gii h phng trnh2 2
2 2
2 2
7287
x xy yy yz zz xz x
+ + = + + = + + =Bi 2. a) Phn tch a thc x5 5x 4 thnh tch ca mt a
tbc hai v mt a thc bc ba vi h s nguyn.
b) p dng kt qu trn rt gn biu thc4 4
2
4 3 5 2 5 12P=
+ Bi 3. Cho ABC u. Chng minh rng vi mi im M ta lu
MA MB + MC.Bi 4. Cho xOy c nh. Hai im A, B khc O ln lt chy
trn
v Oy tng ng sao cho OA.OB = 3.OA 2.OB. Chng minh rng
ng thng AB lun I qua mt im c nh.Bi 5. Cho hai s nguyn dng m, n
tha mn m > n v m khng ht cho n. Bit rng s d khi chia m cho n bng
s d khi chia m +
cho m n. Hy tnh t sm
n.
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thi vo THPT
thi vo 10 h THPT chuyn 1996 i hc khoa hc t nhin.Bi 1. Cho x >
0 hy tm gi tr nh nht ca biu thc
6 6
6
3 3
3
1 12
1 1
( ) ( )
( )
x xx xP
x xx x
+ + =+ + +
.
Bi 2. Gii h phng trnh
1 12 2
1 12 2
yx
xy
+ = + =Bi 3. Chng minh rng vi mi n nguyn dng ta c : n3 + 5n
6
Bi 4. Cho a, b, c > 0. Chng minh rng :3 3 3a b c
ab bc cab c a
+ + + .Bi 5. Cho hnh vung ABCD cnh bng a. Gi M, N, P, Q l cc
bt k ln lt nm trn cc cnh AB, BC, CD, DA.a) Chng minh rng 2a2 MN2
+ NP2 +PQ2 + QM2 4a2 .b) Gi s M l mt im c nh trn cnh AB. Hy xc nh
vcc im N, P, Q ln lt trn cc cnh BC, CD, DA sao cho MNPQ lmt hnh
vung.
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D
BAE
F
thi vo THPT
thi vo 10 h THPT chuyn 2000 i hc khoa hc t nhin
Bi 1. a) Tnh1 1 1
1 2 2 3 1999 2000....
. . .S= + + + .
b) GiI h phng trnh :
2
2
13
13
xx
y yxx
y y
+ + = + + =Bi 2. a) Gii phng trnh 3 2 44 1 1 1x x x x x+ + + + =
+
b) Tm tt c cc gi tr ca a phng trnh2 211
2 4 4 7 02
( ) x a x a+ + + = c t nht mt nghim nguyn.
Bi 3. Cho ng trn tm O ni tip trong hnh thang ABCD (ACD), tip xc
vi cnh AB ti E v vi cnh CD ti F nh hnh
a) Chng minh rngBE DF
AE CF .
b) Cho AB = a, CB = b (a < b), BE = 2AE. Tnhdin tch hnh thang
ABCD.
Bi 4. Cho x, y l hai s thc bt k khckhng.
Chng minh rng2 2 2 2
2 2 8 2 2
43( )
( )
x y x y
x y y x+ + . Du ng thc xy ra khi
no ?
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thi vo THPT
thi vo 10 h THPT chuyn 1998 i hc khoa hc t nhinBi 1. a) GiI phng
trnh 2 28 2 4x x+ = .
b) GiI h phng trnh :2 2
4 2 2 4
721
x xy yx x y y + + = + + =
Bi 2. Cc s a, b tha mn iu kin :3 2
3 23 193 98a abb ba = = Hy tnh gi tr biu thc P = a2 + b2 .
Bi 3. Cho cc s a, b, c [0,1]. Chng minh rng {M}Bi 4. Cho ng trn
(O) bn knh R v hai im A, B c nh
(O) sao cho AB < 2R. Gi s M l im thay i trn cung ln ABca ng
trn .a) K t B ng trn vung gc vi AM, ng thng ny ct AM v (O) ti N. Gi
J l trung im ca MN. Chng minh rng khi M
i trn ng trn th mi im I, J u nm trn mt ng c nh.b) Xc nh v tr ca
M chu vi AMB l ln nht.
Bi 5. a) Tm cc s nguyn dng n sao cho mi s n + 26 v nu l lp phng
ca mt s nguyn dng.b) Cho cc s x, y, z thay i tho mn iu kin x2 + y2
+z2 =Hy tm gi tr ln nht ca biu thc
( )2 2 2 2 2 212
( ) ( ) ( ) P xy yz zx x y z y z x z x y+ + + + + .
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thi vo THPT
thi vo 10 h THPT chuyn 1993-1994 i hc tng hp
Bi 1. a) GiI phng trnh 1 1 22 4
x x x+ + + = .b) GiI h phng trnh :
3 2
3 2
2 12 0
8 12
x xy y
y x
+ + = + =Bi 2. Tm max v min ca biu thc : A = x2y(4 x y) khi x
v
thay i tha mn iu kin : x 0, y 0, x + y 6.Bi 3. Cho hnh thoi
ABCD. Gi R, r ln lt l cc bn knh cc
trn ngoi tip cc tam gic ABD, ABC v a l di cnh hnh t
Chng minh rng2 2 2
1 1 4
R r a+ = .
Bi 4. Tm tt c cc s nguyn dng a, b, c I mt khc nhau
cho biu thc1 1 1 1 1 1
A
a b c ab ac bc
+ + + + + nhn gi tr nguyn dng.
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thi vo THPT
thi vo 10 h THPT chuyn 1991-1992 i hc tng hpBi 1. a) Rt gn biu
thc 3 62 3 4 2 44 16 6.A = + .
b) Phn tch biu thc P = (x y)5 + (y-z)5 +(z - x )5 thnh nhn
Bi 2. a) Cho cc s a, b, c, x, y, z tho mn cc iu kina b c x y z x
y
a b
+ + = + + + +hy tnh gi tr ca biu thc A = xa2 + yb2 + zc2.b) Cho
4 s a, b, c, d mi s u khng m v nh hn hoc b1. Chng minh rng0 a + b +
c + d ab bc cd da 2. Khi no ng thc xydu bng.
Bi 3. Cho trc a, d l cc s nguyn dng. Xt cc s c dng :a, a + d, a
+ 2d, , a + nd, Chng minh rng trong cc s c t nht mt s m 4 ch su tin
ca n l 1991.
Bi 4. Trong mt cuc hi tho khoa hc c 100 ngi tham gia. Gmi ngi u
quen bit vi t nht 67 ngi. Chng minh rng c tm c mt nhm 4 ngi m bt k
2 ngi trong nhm quen bit nhau.
Bi 5. Cho hnh vung ABCD. Ly im M nm trong hnh vunsao cho MAB =
MBA = 150 . Chng minh rng MCD
Bi 6. Hy xy dng mt tp hp gm 8 im c tnh cht : n
trung trc ca on thng ni hai im bt k lun I qua t nhai im ca tp hp
.
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thi vo THPT
thi vo 10 h THPT chuyn L 1989-1990
Bi 1. Tm tt c cc gi tr nguyn ca x biu thc22 3
2 3
x x
x
+ ++nguyn.
Bi 2. Tm gi tr nh nht ca biu thc P = a2
+ ab + b2
3a+ 3.Bi 3. a) Chng minh rng vi mi s nguyn dng m th biu th
m2 + m + 1 khng phI l s chnh phng.b) Chng minh rng vi mi s nguyn
dng m th m(m + 1) khth bng tch ca 4 s nguyn lin tip.
Bi 4. Cho ABC vung cn ti A. CM l trung tuyn. T A v
vung gc vi MC ct BC ti H. Tnh t sBH
HC.
Bi 5. C 6 thnh ph, trong c 3 thnh ph bt k th c tnht 2 thnh ph
lin lc c vi nhau. Chng minh rng trong 6 tph ni trn tn ti 3 thnh ph
lin lc c vi nhau.
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thi vo THPT
thi vo 10 h THPT chuyn nm 2004 i hc khoa hc t nhin(vBi 1. a) GiI
phng trnh 21 1 1 1x x x+ = +
b) Tm nghim nguyn cu h3 3
2 2
82 2 2 7x y x y
y x xy y x + + = + =
Bi 2. Cho cc s thc dng a v b tha mn a100 + b100 = a101 + ba102 +
b102 .Hy tnh gi tr biu thc P = a2004 + b2004 .Bi 3. Cho ABC c
AB=3cm, BC=4cm, CA=5cm. ng cao, n
phn gic, ng trung tuyn ca tam gic k t nh B chia tagic thnh 4
phn. Hy tnh din tch mi phn.
Bi 4. Cho t gic ABCD ni tip trong ng trn, c hai ng chAC, BD vung
gc vi nhau ti H (H khng trng vi tm cu ntrn ). Gi M v N ln lt l chn
cc ng vung gc h t H xucc ng thng AB v BC; P v Q ln lt l cc giao im
ca c
ng thng MH v NH vi cc ng thng CD v DA. Chng mrng ng thng PQ song
song vi ng thng AC v bn iN, P, Q nm trn cng mt ng trn .
Bi 5. Tm gi tr nh nht ca biu thc10 10
16 16 2 2 2
2 2
1 11
2 4( ) ( ) ( )x y
Q x y x yy x
+ + + + thi vo 10 h THPT chuyn nm 2004 i hc khoa hc t nhin(v
2)Bi 1. giI phng trnh 3 1 2x x+ =Bi 2. GiI h phng trnh
2 2
2 2 153( )( )( )( )x y x yx y x y + + = =Bi 3. Tm gi tr nh nht
ca biu thc
3 3 2 2
1 1
( ) ( )
( )( )
x y x yP
x y
+ += vi xcc s thc ln hn 1.
Bi 4. Cho hnh vung ABCD v im M nm trong hnh vunga) Tm tt c cc v
tr ca M sao cho MAB = MBC = MC MDA.b) Xt im M nm trn ng cho AC. Gi
N l chn ng vu
gc h t M xung AB v O l trung im ca on AM. Chng mrng t s
OB
CNc gi tr khng i khi M di chuyn trn ng c
AC.c) Vi gi thit M nm trn ng cho AC, xt cc ng trn (S)(S) c cc ng
knh tng ng AM v CN. Hai tip tuyn chung
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thi vo THPT
thi th vo THPT Chu Vn An 2004
Bi 1. Cho biu thc2 3 2 2 4
42 2 2 2( ) : ( )
x x x xP
xx x x x x
+ + = + +a) Rt gn P
b) Cho 23 114xx = . Hy tnh gi tr ca P.Bi 2. Cho phng trnh mx2 2x
4m 1 = 0 (1)
a) Tm m phng trnh (1) nhn x = 5 l nghim, hy tmnghim cn li.b) Vi
m 0
Chng minh rng phng trnh (1) lun c hai nghim x1, xphn bit.
Gi A, B ln lt l cc im biu din ca cc nghim x1, x
trn trc s. Chng minh rng di on thng AB khng (Khng chc lm)Bi 3.
Cho ng trn (O;R) ng knh AB v mt im M di
trn ng trn (M khc A, B) Gi CD ln lt l im chnh gia nh AM v BM.a)
Chng minh rng CD = R 2 v ng thng CD lun tip xcmt ng trn c nh.b) Gi
P l hnh chiu vung gc ca im D ln ng thng ng thng OD ct dy BM ti Q v
ct ng trn (O) ti giaoim th hai S. T gic APQS l hnh g ? Ti sao ?
c) ng thng I qua A v vung gc vi ng thng MC ct ng thng OC ti H.
Gi E l trung im ca AM. Chng minh rHC = 2OE.d) Gi s bn knh ng trn ni
tip MAB bng 1. Gi MK lng cao h t M n AB. Chng minh rng :
1 1 1 1
2 2 2 3MK MA MA MB MB MK + + + +
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thi vo THPT
thi vo 10 h THPT chuyn nm 2003 i hc khoa hc t nhin(v2)
Bi 1. Cho phng trnh x4 + 2mx2 + 4 = 0. Tm gi tr ca thamm phng
trnh c 4 nghim phn bit x1, x2, x3, x4 tha m
+ x24
+ x34
+ x44
= 32.Bi 2. Gii h phng trnh :
2 2
2 2
2 5 2 04 0
x xy y x yx y x y + + + = + + + =
Bi 3. Tm cc s nguyn x, y tha mn x2 + xy + y2 = x2y2 .Bi 4. ng
trn (O) ni tip ABC tip xc vi BC, CA, AB tng
ti D, E, F. ng trn tm (O) bng tip trong gc BAC ca tip xc vi BC v
phn ko di ca AB, AC tng ng ti P, M, N.a) Chng minh rng : BP = CD.b)
Trn ng thng MN ly cc im I v K sao cho CK // AB, BIAC. Chng minh rng
: t gic BICE v BKCF l hnh bnh hnh.c) Gi (S) l ng trn i qua I, K, P.
Chng minh rng (S) tip xvi BC, BI, CK.
Bi 5. S thc x thay i v tha mn iu kin : 2 23 5( )x x Tm min ca 4
4 2 23 6 3( ) ( )P x x x x+ + .
thi vo 10 h THPT chuyn nm 2003 i hc khoa hc t nBi 1. Gii phng
trnh 25 2 1 7 110 3( )( )x x x x + + + + = .Bi 2. Gii h phng
trnh
3 2
3 2
2 3 56 7
x yxy xy + = + =
Bi 3. Tm cc s nguyn x, y tha mn ng thc :2 2 22 1 2y x x y x y
xy+ + = + + .Bi 4. Cho na ng trn (O) ng knh AB = 2R. M, N l hai
trn na ng trn (O) sao cho M thuc cung AN v tng cc khcch t A, B n
ng thng MN bng 3Ra) Tnh di MN theo R.b) Gi giao im ca hai dy AN v
BM l I. Giao im ca ccng thng AM v BN l K. Chng minh rng bn im M, N,
I, Knm trn mt ng trn , Tnh bn knh ca ng trn theo
c) Tm gi tr ln nht ca din tch KAB theo R khi M, N thayi nhng vn
tha mn gi thit ca bi ton.Bi 5. Cho x, y, z l cc s thc tha mn iu kin
: x + y + z +
+ yz + zx = 6. Chng minh rng : x2 + y2 + z2 3.
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thi vo THPT
thi vo 10 h THPT chuyn nm 2002 i hc khoa hc t nBi 1. a) Gii phng
trnh : 2 23 2 3 2 3 2x x x x x x+ + + = + + .
b) Tm nghim nguyn ca phng trnh : x + xy + y = 9
Bi 2. Gii h phng trnh :2 2
3 3
1
3
x y xy
x y x y
+ + = + = + {M}Bi 3. Cho mi s nguyn dng 1, 2, , 10. Sp xp 10 s
m
cch ty vo mt hng. Cng mi s vi s th t ca n trong hta c 10 tng.
Chng minh rng trong 10 tng tn ti t nhhai tng c ch s tn cng ging
nhau.
Bi 4. Tm gi tr nh nht ca biu thc :4 3 1or 5ba b
Pb c a a c b a
= + ++ + +
Trong a, b, c l di ba cnh ca mt tam gic.Bi 5. ng trn (C) tm I ni
tip ABC tip xc vi cc cnh BC
CA, AB tng ng ti A, B, C .a) Gi cc giao im ca ng trn (C) vi cc
on IA, IB, IC lti M, N, P. Chng minh rng cc ng thng AM, BN, CP
ngquy.b) Ko di on AI ct ng trn ngoi tip ABC ti D (khc A
Chng minh rng.IB IC
rID
= trong r l bn knh ng trn (C) thi vo 10 h THPT chuyn nm 2002 i
hc khoa hc t nBi 1. a) Gii phng trnh : 8 5 5x x+ =
b) Gii h phng trnh : { 1 1 81 1 17( )( )( ) ( )x yx x y y xy+ +
=+ + + + =Bi 2. Cho a, b, c l di ba cnh ca mt tam gic. Chng min
rng phng trnh x2 + (a + b + c)x + ab + bc + ca = 0 v nghiBi 3.
Tm tt c cc s nguyn n sao cho n2 + 2002 l mt s ch
phng.
Bi 4. Tm gi tr nh nht ca bit thc:1 1 1
1 1 1S
xy yz zx+ ++ + + Tr
x, y, z l cc s dng thay i tha mn iu kin x2 + y2
3.Bi 5. Cho hnh vung ABCD. M l im thay i trn cnh BC
khng trng vi B) v N l im thay i trn cnh CD (N khngtrng D) sao
cho MAN = MAB + NAD.a) BD ct AN, AM tng ng ti p v Q. Chng minh rng
5 imQ, M, C, N cng nm trn mt ng trn.b) Chng minh rng ng thng MN lun
lun tip xc vi mt
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thi vo THPT
ng trn c nh khi M v N thay i.c) K hiu din tch ca APQ l S v din
tch t gic PQMN
Chng minh rng t s'
S
Skhng i khi M, N thay i.
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thi vo THPT
thi vo 10 h THPT chuyn nm 2001 i hc khoa hc t nBi 1. Tm cc gia
tr nguyn x, y tha mn ng thc: (y + 2)x
= y2 .Bi 2. a) Gii phng trnh : 23 1 1 2( ) ( )x x x x x = .
b) Gii h phng trnh :2
2 2 2 32x xy x yx y + + = + + =
Bi 3. Cho na vng trn ng knh AB=2a. Trn on AB ly M. Trong na mt
phng b AB cha na vng trn, ta k 2 tia Mv My sao cho AMx = BMy =300 .
Tia Mx ct na vng trn tia My ct na vng trn F. K EE, FF vung gc vi
AB.a) Cho AM= a/2, tnh din tch hnh thang vung EEFF theo ab) Khi M
di ng trn AB. Chng minh rng ng thng EF lunxc vi mt vng trn c
nh.
Bi 4. Gi s x, y, z l cc s thc khc 0 tha mn :
3 3 3
1 1 1 1 1 12
1
( ) ( ) ( )x y zy z z x x y
x y z
+ + + + + = + + = .Hy tnh gi tr ca1 1 1
Px y z
+ + .Bi 5. Vi x, y, z l cc s thc dng, hy tm gi tr ln nht ca
thc:
( )( )( )
xyzM
x y y z z x+ + +
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thi vo THPT
thi vo 10 nm 1989-1990 H Ni
Bi 1. Xt biu thc ( )2 2
2 5 1 11
1 2 4 1 1 2 4 4 1:
x xA
x x x x x
= + + +a) Rt gn A.
b) Tm gi tr x A = -1/2 .Bi 2. Mt t d nh i t A n B vi vn tc 50
km/h. Sau i c 2/3 qung ng vi vn tc , v ng kh i nn ngxe phi gim vn
tc mi gi 10 km trn qung ng cn li. Do t n B chm 30 pht so vi d nh.
Tnh qung ng AB.
Bi 3. Cho hnh vung ABCD v mt im E bt k trn cnh BCAx AE ct cnh CD
ko di ti F. K trung tuyn AI ca AEF vko di ct cnh CD ti K. ng thng
qua E v song song vi ABAI ti G.a) Chng minh rng AE = AF.
b) Chng minh rng t gic EGFK l hnh thoi.c) Chng minh rng hai tam
gic AKF , CAF ng dng v AF2 =KF.CF.d) Gi s E chy trn cnh BC. Chng
minh rng EK = BE + iuv chu vi ECK khng i.
Bi 4. Tm gi tr ca x biu thc2
2
2 1989x xy
x
+= t gi tr nnht v tm gi tr .
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thi vo THPT
thi tuyn sinh vo lp 10 chuyn nm hc 2000-2001. (Bi 1. Tm n nguyn
dng tha mn :
1 1 1 1 1 20001 1 1 1
2 1 3 2 4 3 5 2 2001( )( )( )......( )
. . . ( )n n+ + + =+
Bi 2. Cho biu thc2
4 4 4 416 8
1
x x x xA
x x
+ + = +a) Vi gi tr no ca x th A xc nh.b) Tm x A t gi tr nh
nht.c) Tm cc gi tr nguyn ca x A nguyn.
Bi 3. Cho ABC u cnh a. im Q di ng trn AC, im P ng trn tia i ca
tia CB sao cho AQ. BP = a2 . ng thng Act ng thng BQ ti M.
a) Chng minh rng t gic ABCM ni tip ng trn .b) Tm gi tr ln nht ca
MA + MC theo a.Bi 4. Cho a, b, c > 0. Chng minh rng
a b c a b c
b a c b a c b c c a a b+ < + ++ + + + + +
Bi 5. Chng minh rng sin750 = 6 24
+
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thi vo THPT
thi tuyn sinh vo lp 10 chuyn nm hc 2000-2001. (
Bi 1. Cho biu thc2
1 1 1 2
1 1 1 1 1( ) : ( )x x x
Px x x x x
+= + + .a) Rt gn P.
b) Chng minh rng P < 1 vi mi gi tr ca x 1.Bi 2. Hai vi nc cng
chy vo b th sau 4 gi 48 pht th Nu chy cng mt thi gian nh nhau th
lng nc ca vi II bnglng nc ca vi I chy c. Hi mi vi chy ring th sau
bao ly b.
Bi 3. Chng minh rng phng trnh : 2 6 1 0x x + = c hai nghimx1 = 2
3 v x2 = 2 3 .
Bi 4. Cho ng trn tm O ng knh AB = 2R v mt im ng trn mt na ng trn
( M khng trng vi A, B). Ngi ta v
mt ng trn tm E tip xc vi ng trn (O) ti M v tip xng knh AB. ng
trn (E) ct MA, MB ln lt ti cc im thl C, D.a) Chng minh rng ba im C,
E, D thng hng.b) Chng minh rng ng thng MN i qua mt im c nv tch
KM.KN khng i.c) Gi giao im ca cc tia CN, DN vi KB, KA ln lt l P v
Q. Xnh v tr ca M din tch NPQ t gi tr ln nht vchng t khi chu vi NPQ
i gi tr nh nht.d) Tm qu tch im E.
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thi vo THPT
thi vo 10 h THPT chuyn nm 2001 i hc khoa hc t nBi 1. a) Cho f(x)
= ax2 + bx + c c tnh cht f(x) nhn gi tr ng
khi x l s nguyn hi cc h s a, b, c c nht thit phi l cc nguyn hay
khng ? Ti sao ?
b) Tm cc s nguyn khng m x, y tha mn ng thc :2 2 1x y y+ Bi 2.
Gii phng trnh 24 1 5 14x x x= +Bi 3. Cho cc s thc a, b, x, y tha mn
h :
2 2
3 3
4 4
35917
ax byax byax byax by
+ = + = + = + =Tnh gi tr ca cc biu thc 5 5A ax by+ v 2001 2001B
ax by+
Bi 4. Cho on thng Ab c trung im l O. Gi d, d l cc thng vung gc
vi AB tng ng ti A, B. Mt gc vung nh
mt cnh ct d M, cn cnh kia ct d N. k OH MN. Vng tngoi tip MHB ct
d im th hai l E khc M. MB ct NA ng thng HI ct EB K. Chng minh rng K
nm trn mt trn c inh khi gc vung uqay quanh nh O.
Bi 5. Cho 2001 ng tin, mi ng tin c sn mt mt m v mt mt mu xanh.
Xp 2001 ng tin theo mt vtrn sao cho tt c cc ng tin u c mt xanh nga
ln ptrn. Cho php mi ln i mt ng thi 5 ng tin lin ticnh nhau. Hi vi
cnh lm nh th sau mt s hu hn ln ta c
lm cho tt c cc ng tin u c mt nga ln pha trc hay khng ? Ti sao
?
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thi vo THPT
thi tuyn sinh vo lp 10 chuyn Ton Tin nm 2003-2004 hc s phm
HN
Bi 1. Chng minh rng biu thc sau c gi tr khng ph thc
x 3 6
4
2 3 7 4 3
9 4 5 2 5
.
.
xA x
x
+ = + + +Bi 2. Vi mi s nguyn dng n, t Pn = 1.2.3.n. Chng
minh
rnga) 1 + 1.P1 + 2.P2 + 3.P3 +.+ n.Pn = Pn+1 .
b)1 2 3
1 2 3 11.....
n
n
P P P P
+ + + + 0 .
4)Tm gi tr ca m biu thc P = 1222
1 + xx t gi tr nh
.Cu 3 ( 2 im )Gii phng trnh :
c) xx = 44d) xx =+ 332
Cu 4 ( 3 im )
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thi vo THPT
Cho hai ng trn (O1) v (O2) c bn knh bng R ct nhauv B , qua A v
ct tuyn ct hai ng trn (O1) v (O2) th t ti F , ng thng EC , DF ct
nhau ti P .
4) Chng minh rng : BE = BF .
5) Mt ct tuyn qua A v vung gc vi AB ct (O1) v (O2) lnC,D . Chng
minh t gic BEPF , BCPD ni tip v BP vunvi EF .
6) Tnh din tch phn giao nhau ca hai ng trn khi AB =
s 13Cu 1 ( 3 im )
3) Gii bt phng trnh : 42
+ xx
Cu 2 ( 2 im )Cho phng trnh : 2x2 ( m+ 1 )x +m 1 = 0c) Gii phng
trnh khi m = 1 .d) Tm cc gi tr ca m hiu hai nghim bng tch
chng .Cu3 ( 2 im )
Cho hm s : y = ( 2m + 1 )x m + 3 (1)c) Tm m bit th hm s (1) i
qua im A ( -2 ; 3 ) .d) Tm im c nh m th hm s lun i qua vi m
tr ca m .Cu 4 ( 3 im )
Cho gc vung xOy , trn Ox , Oy ln lt ly hai im A v Bcho OA = OB .
M l mt im bt k trn AB .
Dng ng trn tm O1 i qua M v tip xc vi Ox ti A , trn tm O2 i qua M
v tip xc vi Oy ti B , (O1) ct (O2) ti imhai N .
4) Chng minh t gi c OANB l t gic ni tip v ON l phnca gc ANB
.
5) Chng minh M nm trn mt cung trn c nh khi M thay 6) Xc nh v tr
ca M khong cch O1O2 l ngn nht .
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thi vo THPT
s 14 .Cu 1 ( 3 im )
Cho biu thc :
++ ++= 12
:)1
1
1
2(
xx
x
xxx
xxA
c) Rt gn biu thc .d)Tnh gi tr ca A khi 324+=x
Cu 2 ( 2 im )
Gii phng trnh :xx
x
xx
x
x
x
6
1
6
2
36
22222 +
=
Cu 3 ( 2 im )
Cho hm s : y = - 22
1x
c)Tm x bit f(x) = - 8 ; - 81 ; 0 ; 2 .d) Vit phng trnh ng thng i
qua hai im A v B nm
th c honh ln lt l -2 v 1 .Cu 4 ( 3 im )
Cho hnh vung ABCD , trn cnh BC ly 1 im M . ng trn ng knct ng trn
ng knh BC ti N v ct cnh AD ti E .
4) Chng minh E, N , C thng hng .5) Gi F l giao im ca BN v DC .
Chng minh CDEBCF =6) Chng minh rng MF vung gc vi AC .
s 15Cu 1 ( 3 im )
Cho h phng trnh :
=+
=+
13
52
ym x
ym x
d) Gii h phng trnh khi m = 1 .e) Gii v bin lun h phng trnh theo
tham s m .f) Tm m x y = 2 .
Cu 2 ( 3 im )
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thi vo THPT
3)Gii h phng trnh :
=
=+
yyxx
yx
22
221
4)Cho phng trnh bc hai : ax2 + bx + c = 0 . Gi hai nghimphng
trnh l x1 , x2 . Lp phng trnh bc hai c hai ngl 2x1+ 3x2 v 3x1 + 2x2
.
Cu 3 ( 2 im )Cho tam gic cn ABC ( AB = AC ) ni tip ng trn tm
O
mt im chuyn ng trn ng trn . T B h ng thng vgc vi AM ct CM D
.
Chng minh tam gic BMD cnCu 4 ( 2 im )
3)Tnh : 251
25
1
++
4) Gii bt phng trnh :( x1 ) ( 2x + 3 ) > 2x( x + 3 ) .
s 16Cu 1 ( 2 im )
Gii h phng trnh :
=
=
+
+
4
1
2
1
5
7
1
1
1
2
yx
yx
Cu 2 ( 3 im )
Cho biu thc :xxxxxx
xA
+++
=2
1:
1
c) Rt gn biu thc A .d) Coi A l hm s ca bin x v thi hm s A .
Cu 3 ( 2 im )Tm iu kin ca tham s m hai phng trnh sau c ngchung
.
x2 + (3m + 2 )x 4 = 0 v x2 + (2m + 3 )x +2 =0 .Cu 4 ( 3 im )
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thi vo THPT
Cho ng trn tm O v ng thng d ct (O) ti hai A,B . T mt im M trn d
v hai tip tuyn ME , MF ( E , F lim ) .
3) Chng minh gc EMO = gc OFE v ng trn i qua 3
M, E, F i qua 2 im c nh khi m thay i trn d .4) Xc nh v tr ca M
trn d t gic OEMF l hnh vun
s 17
Cu 1 ( 2 im )Cho phng trnh (m2 + m + 1 )x2 - ( m2 + 8m + 3 )x 1
= 0c) Chng minh x1x2 < 0 .
d) Gi hai nghim ca phng trnh l x1, x2 . Tm gi tr ln nnh nht ca
biu thc :S = x1 + x2 .
Cu 2 ( 2 im )Cho phng trnh : 3x2 + 7x + 4 = 0 . Gi hai nghim ca
p
trnh l x1 , x2 khng gii phng trnh lp phng trnh bc hai m
hai nghim l :1
2
1
xx
v1
1
2
xx
.
Cu 3 ( 3 im )
4) Cho x2
+ y2
= 4 . Tm gi tr ln nht , nh nht ca x + y .
5) Gii h phng trnh :
=+
=
8
122
yx
yx
6) Gii phng trnh : x4 10x3 2(m 11 )x2 + 2 ( 5m +6)x +0
Cu 4 ( 3 im )Cho tam gic nhn ABC ni tip ng trn tm O . ng
gic trong ca gc A , B ct ng trn tm O ti D v E , gi giao hai ng
phn gic l I , ng thng DE ct CA, CB ln lt ti M , N4) Chng minh tam
gic AIE v tam gic BID l tam gic cn .5) Chng minh t gic AEMI l t gic
ni tip v MI // BC .6) T gic CMIN l hnh g ?
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thi vo THPT
s 18
Cu1 ( 2 im )Tm m phng trnh ( x2 + x + m) ( x2 + mx + 1 ) = 0
nghim phn bit .Cu 2 ( 3 im )
Cho h phng trnh :
=+
=+
64
3
ym x
m yx
c) Gii h khi m = 3d) Tm m phng trnh c nghim x > 1 , y > 0
.
Cu 3 ( 1 im )Cho x , y l hai s dng tho mn x5+y5 = x3 + y3 . Chng
min
+ y2 1 + xyCu 4 ( 3 im )
4)Cho t gic ABCD ni tip ng trn (O) . Chng minhAB.CD + BC.AD =
AC.BD
5)Cho tam gic nhn ABC ni tip trong ng trn (O) ng AD . ng cao ca
tam gic k t nh A ct cnh BC ti K vng trn (O) ti E .
d) Chng minh : DE//BC .e) Chng minh : AB.AC = AK.AD .f) Gi H l
trc tm ca tam gic ABC . Chng minh t gi c BH
hnh bnh hnh .
s 19Cu 1 ( 2 im )
Trc cn thc mu cc biu thc sau :
232
12
++
=A ;222
1
+=B ;
123
1
+=C
Cu 2 ( 3 im )Cho phng trnh : x2 ( m+2)x + m2 1 = 0 (1)c) Gi x1,
x2 l hai nghim ca phng trnh .Tm m tho mn
x2 = 2 .
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thi vo THPT
d) Tm gi tr nguyn nh nht ca m phng trnh cnghim khc nhau .
Cu 3 ( 2 im )
Cho32
1;
32
1
+=
= ba
Lp mt phng trnh bc hai c cc h s bng s v c cc nghi
x1 =1
;1
2+
=+ a
bx
b
a
Cu 4 ( 3 im )Cho hai ng trn (O1) v (O2) ct nhau ti A v B .
Mt
thng i qua A ct ng trn (O1) , (O2) ln lt ti C,D , gi I , J l im
ca AC v AD .
5) Chng minh t gic O1IJO2 l hnh thang vung .6) Gi M l giao dim
ca CO1 v DO2 . Chng minh O1 , O2 ,
nm trn mt ng trn7) E l trung im ca IJ , ng thng CD quay quanh A
. Tmhp im E.
8) Xc nh v tr ca dy CD dy CD c di ln nht
s 20Cu 1 ( 3 im )
1)V th ca hm s : y =2
2x
2)Vit phng trnh ng thng i qua im (2; -2) v (1 ; -46)Tm giao im
ca ng thng va tm c vi th tr
Cu 2 ( 3 im )a) Gii phng trnh :
21212 =++ xxxx
b)Tnh gi tr ca biu thc22 11 xyyxS +++= vi ayxxy =+++ )1)(1(
22
Cu 3 ( 3 im )Cho tam gic ABC , gc B v gc C nhn . Cc ng trn ng
knh AB , Act nhau ti D . Mt ng thng qua A ct ng trn ng knh AB , AC
ln lv F .
4) Chng minh B , C , D thng hng .5) Chng minh B, C , E , F nm
trn mt ng trn .
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6) Xc nh v tr ca ng thng qua A EF c di ln.
Cu 4 ( 1 im )Cho F(x) = xx ++ 12
c) Tm cc gi tr ca x F(x) xc nh .d) Tm x F(x) t gi tr ln nht
.
s 21Cu 1 ( 3 im )
4) V th hm s 2
2xy
=5) Vit phng trnh ng thng i qua hai im ( 2 ; -2 ) v
- 4 )6) Tm giao im ca ng thng va tm c vi th tr
Cu 2 ( 3 im )3) Gii phng trnh :
21212 =++ xxxx
4) Gii phng trnh :
512
412=
++
+x
x
x
x
Cu 3 ( 3 im )Cho hnh bnh hnh ABCD , ng phn gic ca gc BAD c
v BC theo th t ti M v N . Gi O l tm ng trn ngoi tip tamMNC .
3) Chng minh cc tam gic DAM , ABN , MCN , l cc tam gic c4) Chng
minh B , C , D , O nm trn mt ng trn .
Cu 4 ( 1 im )Cho x + y = 3 v y 2 . Chng minh x2 + y2 5
s 22Cu 1 ( 3 im )
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4) Gii phng trnh : 8152 =++ xx5) Xc nh a tng bnh phng hai nghim
ca phng t
x2 +ax +a2 = 0 l b nht .Cu 2 ( 2 im )
Trong mt phng to cho im A ( 3 ; 0) v ng thn2y = - 2 .d) V th ca
ng thng . Gi giao im ca ng th
trc tung v trc honh l B v E .e) Vit phng trnh ng thng qua A v
vung gc vi
thng x 2y = -2 .f) Tm to giao im C ca hai ng thng . Chng
rng EO. EA = EB . EC v tnh din tch ca t gic OACB .Cu 3 ( 2 im
)
Gi s x1 v x2 l hai nghim ca phng trnh :
x2(m+1)x +m2 2m +2 = 0 (1)c) Tm cc gi tr ca m phng trnh c nghim
kp
nghim phn bit .d)Tm m 2221 xx + t gi tr b nht , ln nht .
Cu 4 ( 3 im )Cho tam gic ABC ni tip ng trn tm O . K ng cao
AH
trung im ca AB , BC theo th t l M , N v E , F theo th t l chiu
vung gc ca ca B , C trn ng knh AD .
c) Chng minh rng MN vung gc vi HE .
d) Chng minh N l tm ng trn ngoi tip tam gic HEF .
s 23Cu 1 ( 2 im )
So snh hai s :33
6;
211
9
=
= ba
Cu 2 ( 2 im )Cho h phng trnh :
=
=+
2
532
yx
ayx
Gi nghim ca h l ( x , y ) , tm gi tr ca a x 2 + y2 tr nh nht
.
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Cu 3 ( 2 im )Gi h phng trnh :
=++
=++
7
5
22 x yyx
x yyx
Cu 4 ( 3 im )1) Cho t gic li ABCD cc cp cnh i AB , CD ct nhau ti
P v BC , AD
nhau ti Q . Chng minh rng ng trn ngoi tip cc tam gic ABQ , BCP ,
DCQADP ct nhau ti mt im .
6) Cho t gic ABCD l t gic ni tip . Chng minh
BD
AC
DADCBCBA
CDCBADAB =++
..
..
Cu 4 ( 1 im )
Cho hai s dng x , y c tng bng 1 . Tm gi tr nh nht cxyyx
S4
3122+
+=
s 24Cu 1 ( 2 im )
Tnh gi tr ca biu thc :
322
32
322
32
+
+++=P
Cu 2 ( 3 im )3) Gii v bin lun phng trnh :
(m2 + m +1)x2 3m = ( m +2)x +34) Cho phng trnh x2 x 1 = 0 c hai
nghim l x1 , x2 . H
phng trnh bc hai c hai nghim l :2
2
2
1
1;
1 x
x
x
x
Cu 3 ( 2 im )
Tm cc gi tr nguyn ca x biu thc : 232 += xxP l nguyCu 4 ( 3 im
)
Cho ng trn tm O v ct tuyn CAB ( C ngoi ng trT im chnh gia ca
cung ln AB k ng knh MN ct AB ti I ct ng trn ti E , EN ct ng thng AB
ti F .
4) Chng minh t gic MEFI l t gic ni tip .
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5) Chng minh gc CAE bng gc MEB .6) Chng minh : CE . CM = CF . CI
= CA . CB
s 25Cu 1 ( 2 im )
Gii h phng trnh :
=++
=
044
325
2
22
x yy
yx yx
Cu 2 ( 2 im )
Cho hm s :4
2xy = v y = - x 1
c) V th hai hm s trn cng mt h trc to .d) Vit phng trnh cc ng
thng song song vi ng th
= - x 1 v ct th hm s4
2xy = ti im c tung l
Cu 2 ( 2 im )Cho phng trnh : x2 4x + q = 0c) Vi gi tr no ca q th
phng trnh c nghim .
d) Tm q tng bnh phng cc nghim ca phng trnh .
Cu 3 ( 2 im )3)Tm s nguyn nh nht x tho mn phng trnh :
413 =++ xx
4)Gii phng trnh :0113
22 = xx
Cu 4 ( 2 im )Cho tam gic vung ABC ( gc A = 1 v ) c AC < AB ,
AH l
cao k t nh A . Cc tip tuyn ti A v B vi ng trn tm O tip tam gic
ABC ct nhau ti M . on MO ct cnh AB E , MC cng cao AH ti F . Ko di
CA cho ct ng thng BM D . ng thBF ct ng thng AM N .
d) Chng minh OM//CD v M l trung im ca on thng Be) Chng minh EF
// BC .f) Chng minh HA l tia phn gic ca gc MHN .
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s 26Cu 1 : ( 2 im )
Trong h trc to Oxy cho hm s y = 3x + m (*)
1) Tnh gi tr ca m th hm s i qua : a) A( -1 ; 3 B( - 2 ; 5 )
2) Tm m th hm s ct trc honh ti im c honl - 3 .
3) Tm m th hm s ct trc tung ti im c tung - 5 .Cu 2 : ( 2,5 im
)
Cho biu thc :1 1 1 1 1
A= :1- x 1 1 1 1x x x x
+ + + +
a) Rt gn biu thc A .b) Tnh gi tr ca A khi x = 7 4 3+c) Vi gi tr
no ca x th A t gi tr nh nht .
Cu 3 : ( 2 im )Cho phng trnh bc hai : 2 3 5 0x x+ = v gi hai
nghim ca ptrnh l x1 v x2 . Khng gii phng trnh , tnh gi tr ca ccthc
sau :
a) 2 21 2
1 1
x x+ b) 2 21 2x x+
c) 3 31 2
1 1x x
+ d) 1 2x x+
Cu 4 ( 3.5 im )Cho tam gic ABC vung A v mt im D nm gia A v B
.
trn ng knh BD ct BC ti E . Cc ng thng CD , AE ln lt cng trn ti
cc im th hai F , G . Chng minh :
a) Tam gic ABC ng dng vi tam gic EBD .
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b) T gic ADEC v AFBC ni tip c trong mt ng trn .c) AC song song
vi FG .d) Cc ng thng AC , DE v BF ng quy .
s 27Cu 1 ( 2,5 im )
Cho biu thc : A =1 1 2
:2
a a a a a
aa a a a
+ + +
a) Vi nhng gi tr no ca a th A xc nh .b) Rt gn biu thc A .c) Vi
nhng gi tr nguyn no ca a th A c gi tr nguyn .
Cu 2 ( 2 im )Mt t d nh i t A n B trong mt thi gian nht nh . Nu
xe
vi vn tc 35 km/h th n chm mt 2 gi . Nu xe chy vi vn tc 50 km/h
tn sm hn 1 gi . Tnh qung ng AB v thigian d nh i lc u .Cu 3 ( 2 im
)
a) Gii h phng trnh :
1 13
2 31
x y x y
x y x y
+ = + =
+ b) Gii phng trnh : 2 2 2
5 5 25
5 2 10 2 50
x x x
x x x x x
+ + =
+ Cu 4 ( 4 im )
Cho im C thuc on thng AB sao cho AC = 10 cm ;CB cm . V v cng mt
na mt phng b l AB cc na ng trn knh theo th t l AB , AC , CB c tm ln
lt l O , I , K . ng v
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gc vi AB ti C ct na ng trn (O) E . Gi M , N theo th t lim cuae
EA , EB vi cc na ng trn (I) , (K) . Chng minh :
a) EC = MN .b) MN l tip tuyn chung ca cc na ng trn (I) v (K)
.
c) Tnh di MN .d) Tnh din tch hnh c gii hn bi ba na ng trn .
28Cu 1 ( 2 im )
Cho biu thc : A =1 1 1 1 1
1 1 1 1 1
a a
a a a a a
+ ++ +
+ + + +1) Rt gn biu thc A .2) Chng minh rng biu thc A lun dng vi
mi a .
Cu 2 ( 2 im )Cho phng trnh : 2x2 + ( 2m - 1)x + m - 1 = 01) Tm m
phng trnh c hai nghim x1 , x2 tho mn 3x1
= 11 .2) Tm ng thc lin h gia x1 v x2 khng ph thuc vo m3) Vi gi
tr no ca m th x1 v x2 cng dng .
Cu 3 ( 2 im )Hai t khi hnh cng mt lc i t A n B cch nhau 300
k
t th nht mi gi chy nhanh hn t th hai 10 km nn n B s
hn t th hai 1 gi . Tnh vn tc mi xe t .Cu 4 ( 3 im )
Cho tam gic ABC ni tip ng trn tm O . M l mt im trn cung AC( khng
cha B ) k MH vung gc vi AC ; MK vung gc vi BC .
1) Chng minh t gic MHKC l t gic ni tip .2) Chng minh AMB HMK=3)
Chng minh AMB ng dng vi HMK .
Cu 5 ( 1 im )
Tm nghim dng ca h :
( ) 6
( ) 12( ) 30
xy x y
yz y z zx z x
+ =
+ = + =
29
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( Thi tuyn sinh lp 10 - THPT nm 2006 - 2007 - 120 pht - N28 / 6
/ 2006
Cu 1 ( 3 im )1) Gii cc phng trnh sau :
a) 4x + 3 = 0b) 2x - x2 = 0
2) Gii h phng trnh :2 3
5 4
x y
y x
= + =
Cu 2( 2 im )
1) Cho biu thc : P = ( )3 1 4 4 a > 0 ; a 442 2
a a a
aa a
+ +
+a) Rt gn P .b) Tnh gi tr ca P vi a = 9 .2) Cho phng trnh : x2 -
( m + 4)x + 3m + 3 = 0 ( m l tham a) Xc nh m phng trnh c mt nghim
bng 2 . Tm
nghim cn li .b) Xc nh m phng trnh c hai nghim x1 ; x2 tho m
3 3
1 20x x+
Cu 3 ( 1 im )Khong cch gia hai thnh ph A v B l 180 km . Mt t i t
A n
ngh 90 pht B , ri li t B v A . Thi gian lc i n lc tr v A l 10 gi
vn tc lc v km vn tc lc i l 5 km/h . Tnh vn tc lc i ca t .Cu 4 ( 3
im )
T gic ABCD ni tip ng trn ng knh AD . Hai ng chAC , BD ct nhau ti
E . Hnh chiu vung gc ca E trn AD l F . thng CF ct ng trn ti im th
hai l M . Giao im ca BD vl N
Chng minh :a) CEFD l t gic ni tip .b) Tia FA l tia phn gic ca gc
BFM .c) BE . DN = EN . BD
Cu 5 ( 1 im )
Tm m gi tr ln nht ca biu thc 22
1
x m
x
++ bng 2 .
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29( Thi tuyn sinh lp 10 - THPT nm 2006 - 2007 - 120 pht - N30 /
6 / 2006
Cu 1 (3 im )1) Gii cc phng trnh sau :
a) 5( x - 1 ) = 2b) x2 - 6 = 0
2) Tm to giao im ca ng thng y = 3x - 4 vi hai tr .Cu 2 ( 2 im
)
1) Gi s ng thng (d) c phng trnh : y = ax + b .Xc nh a , b (d) i
qua hai im A ( 1 ; 3 ) v B ( -
1)2) Gi x1 ; x2 l hai nghim ca phng trnh x2 - 2( m - 1)x - 4
m l tham s )Tm m : 1 2 5x x+ =
3) Rt gn biu thc : P =1 1 2
( 0; 0)2 2 2 2 1
x xx x
x x x
+
+ Cu 3( 1 im)
Mt hnh ch nht c din tch 300 m2 . Nu gim chiu rng3 m , tng chiu
di thm 5m th ta c hnh ch nht mi c dinbng din tch bng din tch hnh ch
nht ban u . Tnh chu vhnh ch nht ban u .Cu 4 ( 3 im )
Cho im A ngoi ng trn tm O . K hai tip tuyn AB , Avi ng trn (B ,
C l tip im ) . M l im bt k trn cung nh( M B ; M C ) . Gi D , E , F
tng ng l hnh chiu vung gc ctrn cc ng thng AB , AC , BC ; H l giao
im ca MB v DF ; K
giao im ca MC v EF .1) Chng minh :a) MECF l t gic ni tip .b) MF
vung gc vi HK .
2) Tm v tr ca M trn cung nh BC tch MD . ME ln nh
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Cu 5 ( 1 im ) Trong mt phng to ( Oxy ) cho i-3 ; 0 ) v Parabol
(P) c phng trnh y = x2 . Hy tm to cim M thuc (P) cho di on thng AM
nh nht .
Dng 2 Mt s khc
S 1
Cu 1.1.Chng minh 9 4 2 2 2 1+ = + .
2.Rt gn php tnh A 4 9 4 2= + .
Cu 2. Cho phng trnh 2x2 + 3x + 2m 1 = 0
1.Gii phng trnh vi m = 1.2.Tm m phng trnh c hai nghim phn bit.Cu
3. Mt mnh vn hnh ch nht c din tch l 1200m2. Nay nta tu b bng cch
tng chiu rng ca vn thm 5m, ng thi rchiu di 4m th mnh vn c din tch
1260m2. Tnh kch thc vn sau khi tu b.Cu 4. Cho ng trn tm O ng knh
AB. Ngi ta vng trnA bn knh nh hn AB, n ct ng trn (O) ti C v D, ct
AB ti E.cung nh CE ca (A), ta ly im M. Tia BM ct tip (O) ti N.
a) Chng minh BC, BD l cc tip tuyn ca ng trn (A).b) Chng minh NB
l phn gic ca gc CND.c) Chng minh tam gic CNM ng dng vi tam gic
MND.d) Gi s CN = a; DN = b. Tnh MN theo a v b.
Cu 5. Tm gi tr nh nht ca biu thc P = 2x2 + 3x + 4.
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S 2
Cu 1. Tm hai s bit hiu ca chng bng 10 v tng ca 6 ln s l2 ln s b
l 116.
Cu 2. Cho phng trnh x2
7x + m = 0a) Gii phng trnh khi m = 1.b) Gi x1, x2 l cc nghim ca
phng trnh. Tnh S = x12 + x22.c) Tm m phng trnh c hai nghim tri
du.
Cu 3. Cho tam gic DEF c D = 600, cc gc E, F l gc nhn ntrong ng
trn tm O. Cc ng cao EI, FK, I thuc DF, K thuc DE
a) Tnh so cung EF khng cha im D.b) Chng minh EFIK ni tip c.c)
Chng minh tam gic DEF ng dng vi tam gic DIK v tm
ng dng.Cu 4. Cho a, b l 2 s dng, chng minh rng
( ) ( )2 2
2 2 2 2 a b a ba b a a b b2
+ ++ + =
S 3
Cu 1.Thc hin php tnh1
a) 2 6 4 3 5 2 8 .3 64
2 2b)3 5 3 5
+
++
Cu 2. Cho phng trnh x2 2x 3m2 = 0 (1).a) Gii phng trnh khi m =
0.b) Tm m phng trnh c hai nghim tri du.c) Chng minh phng trnh 3m2x2
+ 2x 1 = 0 (m 0) lun c
nghim phn bit v mi nghim ca n l nghch o ca mt nghimphng trnh
(1).Cu 3. Cho tam gic ABC vung cn ti A, AD l trung tuyn. Ly i
bt k trn on AD (M A; M D). Gi I, K ln lt l hnh chiu vgc ca M trn
AB, AC; H l hnh chiu vung gc ca I trn ng tDK.
a) T gic AIMK l hnh g?b) Chng minh 5 im A, I, M, H, K cng nm trn
mt ng
Xc nh tm ca ng trn .c) Chng minh ba im B, M, H thng hng.
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Cu 4. Tm nghim hu tca phng trnh 2 3 3 x 3 y 3 =
S 4
Cu 1. Cho biu thc( ) ( )
a 3 a 2 a a 1 1P :
a 1 a 1 a 1a 2 a 1
+ + + = + + + a) Rt gn P.
b) Tm a 1 a 1
1P 8
+
Cu 2. Mt ca n xui dng t A n B di 80km, sau li ngc n C cch B
72km, thi gian ca n xui dng t hn thi gian ngc
l 15 pht. Tnh vn tc ring ca ca n, bit vn tc ca dng n4km/h.Cu 3.
Tm ta giao im A v B ca hai th cc hm s y = 2xv y = x2. Gi D v C ln
lt l hnh chiu vung gc ca A v B lnhonh. Tnh din tch t gic ABCD.Cu 4.
Cho (O) ng knh AB = 2R, C l trung im ca OA v dvung gc vi OA ti C.
Gi K lim ty trn cung nh BM, H lim ca AK v MN.
a) Chng minh t gic BCHK ni tip c.b) Tnh tch AH.AK theo R.c) Xc
nh v tr ca K tng (KM + KN + KB) t gi tr ln nh
tnh gi tr ln nht .Cu 5. Cho hai s dng x, y tho mn iu kin x + y =
2.Chng minh x2y2(x2 + y2) 2
S 5
Cu 1. Cho biu thcx 1 2 x
P 1 : 1x 1 x 1 x x x x 1
= + + +
a) Tm iu kin P c ngha v rt gn P.b) Tm cc gi tr nguyn ca x biu
thc P x nhn gnguyn.Cu 2.
a) Gii phng trnh x4 4x3 2x2 + 4x + 1 = 0.
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b) Gii h2 2
2
x 3xy 2y 0
2x 3xy 5 0
+ =
+ =
Cu 3. Trong mt phng ta Oxy cho (P) c phng trnh2
xy
2
=
(d) lng thng i qua im I(0; - 2) v c h s gc k.a) Vit phng trnh
dng thng (d). Chng minh rng (d) lu
(P) ti hai im phn bit A v B khi k thay i.b) Gi H, K theo th t l
hnh chiu vung gc ca A, B ln
honh. Chng minh rng tam gic IHK vung ti I.Cu 4. Cho (O; R), AB
lng knh cnh.ng thng (d) l tip tca (O) ti B. MN lng knh thay i ca
(O) sao cho MN khng vgc vi AB v M A, M B. Cc ng thng AM, AN ct ng
thntng ng ti C v D. Gi I l trung im ca CD, H l giao im ca MN. Khi
MN thay i, chng minh rng:
a) Tch AM.AC khng i.b) Bn im C, M, N, D cng thuc mt ng trn.c)im
H lun thuc mt ng trn cnh.d) Tm J ca ng trn ngoi tip tam gic HIB lun
thuc
ng thng cnh.Cu 5. Cho hai s dng x, y tha mn iu kin x + y = 1. Hy
tm g
nh nht ca biu thc 2 21 1
Ax y xy
= ++
.
S 6
Cu 1.a) Gii phng trnh 5x2 + 6 = 7x 2.
b) Gii h phng trnh3x y 5
x 2y 4
= + =
c) Tnh 18 122 3
Cu 2. Cho (P) y = -2x2
a) Trong cc im sau im no thuc, khng thuc (P)? ti sao?
A(-1; -2); B(1 1
;2 2
); C( 2; 4 )
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b) Tm k ng thng (d): y = kx + 2 ct (P) ti hai im bit.
c) Chng minh im E(m; m2 + 1) khng thuc (P) vi mi gi tm.
Cu 3. Cho tam gic ABC vung ti A, gc B ln hn gc C. KnAH. Trn on
HC t HD = HB. T C k CE vung gc vi AD ti E.a) Chng minh cc tam gic
AHB v AHD bng nhau.b) Chng minh t gic AHCE ni tip v hai gc HCE v
HAE
nhau.c) Chng minh tam gic AHE cn ti H.d) Chng minh DE.CA =
DA.CEe) Tnh gc BCA nu HE//CA.
Cu 4.Cho hm s y = f(x) xc nh vi mi s thc x khc 0 v tha
( )21
f x 3f xx
+ = vi mi x khc 0. Tnh gi tr f(2).
S 7
Cu 1.
a) Tnh9 1
2 1 5 : 1616 16
b) Gii h3x y 2
x y 6
= + =c) Chng minh rng 3 2 l nghim ca phng trnh x2 6x +
0.
Cu 2. Cho (P): 21
y x3
= .
a) Cc im ( ) ( )1
A 1; ; B 0; 5 ; C 3;13
, im no thuc (P)? Gii t
b) Tm k (d) c phng trnh y = kx 3 tip xc vi (P).c) Chng t rng ng
thng x = 2 ct (P) ti mt im duy
Xc nh ta giao im .Cu 3. Cho (O;R), ng knh AB cnh, CD lng knh di
ng. l tip tuyn ca (O) ti B; cc ng thng AC, AD ct d ln lt tQ.
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a) Chng minh gc PAQ vung.b) Chng minh t gic CPQD ni tip c.c)
Chng minh trung tuyn AI ca tam gic APQ vung g
ng thng CD.
d) Xc nh v tr ca CD din tch t gic CPQD bng 3 lntch tam gic
ABC.Cu 4. Tm gi tr nh nht ca biu thc 2 2A 2x 2xy y 2x 2y 1= + + + +
.
S 8
Cu 1.
1.Choa a a a
P 1 1 ; a 0, a 1
a 1 1 a
+ = +
+ + a) Rt gn P.b) Tm a bit P > 2 .c) Tm a bit P = a .
2.Chng minh rng 13 30 2 9 4 2 5 3 2+ + + = +Cu 2. Cho phng trnh
mx2 2(m-1)x + m = 0 (1)
a) Gii phng trnh khi m = - 1.b) Tm m phng trnh (1) c 2 nghim phn
bit.
c) Gi hai nghim ca (1) l x1 , x2. Hy lp phng trnh nhnx
x
lm nghim.Cu 3.Cho tam gic nhn ABC (AB < AC) ni tip ng trn tm
O,ng knh AD.ng cao AH, ng phn gic AN ca tam gic ct tng ng ti cc im
Q v P.
a) Chng minh: DQ//BC v OP vung gc vi QD.b) Tnh din tch tam gic
AQD bit bn knh ng trn l R v
tgQAD =3
4 .Cu 4.
a)Gi s phng trnh ax2 + bx + c = 0 c nghim dng x1. Chminh rng
phng trnh cx2 + bx + a = 0 cng c nghim dng l xx1 + x2 0.
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b)Tm cp s (x, y) tha mn phng trnh x2y + 2xy 4x + y =sao cho y t
gi tr ln nht.
S 9
Cu 1.
1.Cho ( )2 2
2
1 2x 16x 1P ; x
1 4x 2
=
a) Chng minh2
P1 2x
=
b) Tnh P khi3
x2
=
2.Tnh2 5 24
Q 12
+ =
Cu 2. Cho hai phng trnh n x sau:( )2 2x x 2 0 (1); x 3b 2a x 6a
0 (2)+ = + =
a) Gii phng trnh (1).b) Tm a v b hai phng trnh tng ng.c) Vi b =
0. Tm a phng trnh (2) c nghim x1, x2 tha m
+ x22 = 7Cu 3. Cho tam gic ABC vung a v gc B ln hn gc C, AH
l
cao, AM l trung tuyn.ng trn tm H bn knh HA ct ng thAB D vng thng
AC E.a) Chng minh D, H, E thng hng.b) Chng minh MAE DAE; MA DE =
.c) Chng minh bn im B, C, D, E nm trn ng trn tm O. T
gic AMOH l hnh g?d) Cho gc ACB bng 300 v AH = a. Tnh din tch tam
gic HEC
Cu 4.Gii phng trnh2 2ax ax - a 4a 1
x 2a
+ = . Vi n x, tham s a
S 10
Cu 1.
1.Rt gn ( ) ( ) ( )2 3 2 2 3 2 3 2 3 2 2+ + .
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thi vo THPT
2.Choa b
xb a
= + vi a < 0, b < 0.
a) Chng minh 2x 4 0 .b) Rt gn 2F x 4= .
Cu 2. Cho phng trnh ( ) ( )2 2
x 2 x 2mx 9 0 (*) + + = ; x ln, m l thas.
a) Gii (*) khi m = - 5.b) Tm m (*) c nghim kp.
Cu 3. Cho hm s y = - x2 c th l (P); hm s y = 2x 3 c th(d).
1.V th (P) v (d) trn cng mt h trc ta Oxy. Tm ta giao im ca (P) v
(d).
2.Cho im M(-1; -2), bng php tnh hy cho bit im M thucpha trn hay
pha di th (P), (d).3.Tm nhng gi tr ca x sao cho th (P) phi trn th
(d)
Cu 4. Cho tam gic nhn ABC ni tip (O), E l hnh chiu ca B trnng
thng qua E song song vi tip tuyn Ax ca (O) ct AB ti F.
1.Chng minh t gic BFEC ni tip.2.Gc DFE (D thuc cnh BC) nhn tia
FC lm phn gic trong v
giao im ca BE vi CF. Chng minh A, H, D thng hng.3.Tia DE ct tip
tuyn Ax ti K. Tam gic ABC l tam gic g th
gic AFEK l hnh bnh hnh, l hnh thoi? Gii thch.Cu 5. Hy tnh 1999
1999 1999F x y z = + + theo a. Trong x, y, z l nghimca phng
trnh:
( )x y z a xy yz zx a xyz 0; a 0+ + + + + = S 11
Cu 1.1.Gii bt phng trnh, h phng trnh, phng trnh
22x 3y 12
a) 2x 6 0 b) x x 6 0 c)3x y 7
+ = + = =
2.T kt qu ca phn 1. Suy ra nghim ca bt phng trnh,phng trnh, h
phng trnh sau:
2 p 3 q 12a) 2 y 6 0 b) t t 6 0 c)
3 p q 7
+ = + = =
Cu 2.
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thi vo THPT
1.Chng minh ( ) ( )2 21 2a 3 12a 2 2a + + = + .
2.Rt gn ( )2 3 2 3 3 2 3
2 24 8 63 2 4 2 2 3 2 3 2
++ + +
+ +
Cu 3. Cho tam gic ABC (AC > AB) c AM l trung tuyn, N lim btrn
on AM.ng trn (O) ng knh AN.1.ng trn (O) ct phn gic trong AD ca gc A
ti F, ct ph
gic ngoi gc A ti E. Chng minh FE lng knh ca (O).2.ng trn (O) ct
AB, AC ln lt ti K, H.on KH ct AD ti
Chng minh hai tam gic AKF v KIF ng dng.3.Chng minh FK2 =
FI.FA.4.Chng minh NH.CD = NK.BD.
Cu 4. Rt gn
2 2 2 2 2 2 2 21 1 1 1 1 1 1 1T 1 1 1 ... 12 3 3 4 4 5 1999
2000= + + + + + + + + + + + +
S 12Cu 1.Gii cc phng trnh sau
1) 4x 1 = 2x + 5 2) x2 8x + 15 = 0 3)2x 8x 1
2x 6
+
Cu 2.
1.Chng minh ( )23 2 2 1 2 = .
2.Rt gn 3 2 2 .
3.Chng minh ( ) ( )2 2
1 13 2 17 2 2 17
2 2 7 2 2 17
+ = + Cu 3. Cho ba im A, B, C thng hng (im B thuc on AC).ntrn
(O) i qua B v C, ng knh DE vung gc vi BC ti K. AD ct ti F, EF ct AC
ti I.
1.Chng minh t gic DFIK ni tip c.2.Gi H lim i xng vi I qua K.
Chng minh gc DHA v gDEA bng nhau.
3.Chng minh AI.KE.KD = KI.AB.AC.4.AT l tip tuyn (T l tip im) ca
(O).im T chy trn n
no khi (O) thay i nhng lun i qua hai im B, C.Cu 4.
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thi vo THPT
1.Cho tam gic ABC c BC = a, AC = b, AB = c, G l trng tm. Gy, z
ln lt l khong cch t G ti cc cnh a, b, c. Chng minh
x y z
bc ac ab= =
2.Gii phng trnh25 4 2025
x 1 y 3 z 24 104x 1 y 3 z 24
+ + + + = + + + +
S 13
Cu 1.Gii h phng trnh
2 2
2
x 2x y 0
x 2xy 1 0
+ = + =
Cu 2. Gii bt phng trnh (x 1)(x + 2) < x2 + 4.Cu 3.
1.Rt gn biu thc1
P 175 2 28 7
= + +
.
2.Vi gi tr no ca m th phng trnh 2x2 4x m + 3 = 0 (mtham s) v
nghim.
Cu 4. Cho tam gic ABC c ba gc nhn. V trung tuyn AM, phn gAD ca
gc BAC.ng trn ngoi tip tam gic ADM ct AB ti P vAC ti Q.
1.Chng minh BAM PQM; BPD BMA = = .2.Chng minh BD.AM = BA.DP.
3.Gi s BC = a; AC = b; BD = m. Tnh t sBP
BMtheo a, b, m.
4.Gi E lim chnh gia cung PAQ v K l trung im on PQ.Chng minh ba
im D, K, E thng hng.
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thi vo THPT
S 14Cu 1.
1.Gii bt phng trnh (x + 1)(x 4) < 0.2.Gii v bin lun bt phng
trnh 1 x mx m+ + vi m l tham s
Cu 2. Gii h phng trnh
3 61
2x y x y
1 10
2x y x y
= + =
Cu 3. Tm gi tr nh nht ca biu thc 2 2P x 26y 10xy 14x 76y = + +
+Khi x, y c gi tr bng bao nhiu?Cu 4. Cho hnh thoi ABCD c gc nhn BAD
= . V tam gic u Cv pha ngoi hnh thoi v tam gic u AKD sao cho nh K
thuc mphng cha nh B (na mt phng b AC).
1.Tm tm ca ng trn i qua 4 im A, K, C, M.
2.Chng minh rng nu AB = a, th BD = 2a.sin2
.
3.Tnh gc ABK theo .4.Chng minh 3 im K, L, M nm trn mt ng
thng.
Cu 5. Gii phng trnh ( ) ( )2
x x 2 1 1 x= +
S 15Cu 1.Tnh
( ) ( )22 2 4m 4m 1
a) 5 1 5 1 b)4m 2
++ +
Cu 2.
1.V th (P) ca hm s y =2x
2
.
2.Tm a, b ng thng y = ax + b i qua im (0; -1) v tip vi (P)
Cu 3. Cho h phng trnh( )mx my 3
1 m x y 0
+ = + =
a)Gii h vi m = 2.
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thi vo THPT
b) Tm m h c nghim m (x < 0; y < 0).Cu 4. Cho na ng trn ng
knh AB = 2r, C l trung im ca AB. Trn cung AC ly im F bt k. Trn dy
BF ly im E sao cho BEAF.
a) Hai tam gic AFC v BEC qua h vi nhau nh th no? Ti sab) Chng
minh tam gic EFC vung cn.c) Gi D l giao im ca AC vi tip tuyn ti B
ca na ng t
Chng minh t gic BECD ni tip c.d) Gi s F di ng trn cung AC. Chng
minh rng khi E di
chuyn trn mt cung trn. Hy xc nh cung trn v bn knh ca ctrn .
S 16Cu 1.
1.Tm bn s t nhin lin tip, bit rng tch ca chng bng 302.C th tm c
hay khng ba s a, b, c sao cho:
( ) ( ) ( )2 2 2a b c a b c
0a b b c c a a b b c c a
+ + = + + =
Cu 2.
1.Cho biu thcx 1 x 1 8 x x x 3 1
B :x 1 x 1x 1 x 1 x 1
+ =
+ a) Rt gn B.b) Tnh gi tr ca B khi x 3 2 2
= +.
c) Chng minh rng B 1 vi mi gi tr ca x tha mn x 0; x
2.Gii h phng trnh( ) ( )( ) ( )
2 2
2 2
x y x y 5
x y x y 9
+ =
+ =
Cu 3. Cho hm s: ( ) ( )2 2 2y x 1 2 x 2 3 7 x= + + + 1.Tm khong
xc nh ca hm s.2. Tnh gi tr ln nht ca hm s v cc gi tr tng ng ca
x
trong khong xc nh .Cu 4. Cho (O; r) v hai ng knh bt k AB v CD.
Tip tuyn ti A (O) ct ng thng BC v BD ti hai im tng ng l E, F. Gi P
vln lt l trung im ca EA v AF.
1.Chng minh rng trc tm H ca tam gic BPQ l trung im con OA.
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thi vo THPT
2.Hai ng knh AB v Cd c v tr tng i nh th no th tamgic BPQ c din
tch nh nht? Hy tnh din tch theo r.
S 17
Cu 1. Cho a, b, c l ba s dng.
t1 1 1
x ; y ; z b c c a a b
= = =+ + +
Chng minh rng a + c = 2b x + y = 2z.Cu 2. Xc nh gi tr ca a tng
bnh phng cc nghim ca phtrnh:
x2
(2a 1)x + 2(a 1) = 0, t gi tr nh nht.
Cu 3. Gii h phng trnh:( )( )
2 2 2 2
2 2 2 2
x xy y x y 185
x xy y x y 65
+ + + =
+ + =Cu 4. Cho hai ng trn (O1) v (O2) ct nhau ti A v B. V dy
AE(O1) tip xc vi (O2) ti A; v dy AF ca (O2) tip xc vi (O1) ti
A.
1. Chng minh rng2
2
BE AE
BF AF= .
2.Gi C lim i xng vi A qua B. C nhn xt g v hai tam g
EBC v FBC.3.Chng minh t gic AECF ni tip c.
S 18Cu 1.
1.Gii cc phng trnh:
2
2
2 1 9 31
5 2 10 4a) b) 2x 1 5x 4x 1
22
+= =
2.Gii cc h phng trnh:x y 3 3x 2y 6z
a) b)xy 10 x y z 18
= = = = + + =
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thi vo THPT
Cu 2.
1.Rt gn( ) ( )
( )
5 3 50 5 24
75 5 2
+
2.Chng minh ( )a 2 a 1; a 0 .Cu 3. Cho tam gic ABC cn ti A ni
tip trong ng trn, P l mim trn cung nh AC ( P khc A v C). AP ko di
ct ng thng BM.
a) Chng minh ABP AMB = .b) Chng minh AB2 = AP.AM.c) Gi s hai
cung AP v CP bng nhau, Chng minh AM.MP =
AB.BM.d) Tm v tr ca M trn tia BC sao cho AP = MP.e) Gi MT l tip
tuyn ca ng trn ti T, chng minh AM, AB
l ba cnh ca mt tam gic vung.
Cu 4. Cho 1 2 19961 2 1996
a a a 27...
b b b 7= = = = . Tnh
( ) ( ) ( )( ) ( ) ( )
19971997 1997
1 2 1996
19971997 1997
1 2 1996
a 2 a ... 1996 a
b 2 b ... 1996 b
+ + +
+ + +
S 19Cu 1.
1.Gii h phng trnh sau:1 3
2
2x 3y 1 x 2 ya) b)x 3y 2 2 1
1x 2 y
=
= + = =
2.Tnh ( ) ( )6 2 5
a) 3 2 2 3 3 2 2 3 b)2 20
+
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thi vo THPT
Cu 2.1.Cho phng trnh x2 ax + a + 1 = 0.a) Gii phng trnh khi a =
- 1.b) Xc nh gi tr ca a, bit rng phng trnh c mt nghim l
1 3x2= . Vi gi tr tm c ca a, hy tnh nghim th hai ca phng
trnh.2.Chng minh rng nu a b 2+ th t nht mt trong hai phng
trnh sau y c nghim: x2 + 2ax + b = 0; x2 + 2bx + a = 0.Cu 3. Cho
tam gic ABC c AB = AC. Cc cnh AB, BC, CA tip xc v(O) ti cc im tng
ng D, E, F.
1.Chng minh DF//BC v ba im A, O, E thng hng.2.Gi giao im th hai
ca BF vi (O) l M v giao im ca DM
BC l N. Chng minh hai tam gic BFC v DNB ng dng; N l trungca
BE.
3.Gi (O) lng trn i qua ba im B, O, C. Chng minh AB, cc tip tuyn
ca (O).
Cu 4. Cho ( ) ( )2 2x x 1999 y y 1999 1999+ + + + = . Tnh S = x
+ y.
S 20
Cu 1.
1.Cho2
1 1M 1 a : 1
1 a 1 a
= + + + a) Tm tp xc nh ca M.b) Rt gn biu thc M.
c) Tnh gi tr ca M ti3
a2 3
=+
.
2.Tnh 40 2 57 40 2 57 +
Cu 2.1.Cho phng trnh (m + 2)x2 2(m 1) + 1 = 0 (1)a) Gii phng
trnh khi m = 1.b) Tm m phng trnh (1) c nghim kp.c) Tm m (1) c hai
nghim phn bit, tm h thc lin h gia
nghim khng ph thuc vo m.
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thi vo THPT
2.Cho ba s a, b, c tha mn a > 0; a2 = bc; a + b + c = abc.
Chminh:
2 2 2a) a 3, b 0, c 0. b) b c 2a > > +
Cu 3. Cho (O) v mt dy ABM ty trn cung ln AB.
1.Nu cch dng (O1) qua M v tip xc vi AB ti A; ng trnqua M v tip
xc vi AB ti B.2.Gi N l giao im th hai ca hai ng trn (O1) v (O2).
Ch
minh 0AMB ANB 180 + = . C nhn xt g v ln ca gc ANB khi Mng.
3.Tia MN ct (O) ti S. T gic ANBS l hnh g?4.Xc nh v tr ca M t gic
ANBS c din tch ln nht.
Cu 4. Gi s h
ax+by=c
bx+cy=a
cx+ay=b
c nghim. Chng minh rng: a3 + b3 + c3
3abc.
S 21cu 1:(3 im)Rt gn cc biu thc sau:
( )( )
.7
1;
3
1
4 91
1694
223312
22
3
323
2
1 51 2 0
4
156
2
1
2
2
2
+=
++
++
=
+=
xxx
xxxC
B
A
cu 2:(2,5 im)
Cho hm s )(21 2 Pxy =
a. V th ca hm s (P)b. Vi gi tr no ca m th ng thng y=2x+m ct th 2
im phn bit A v B. Khi hy tm to hai im A v
cu 3: (3 im)
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thi vo THPT
Cho ng trn tm (O), ng knh AC. Trn on OC ly B (BC) v v ng trn tm
(O) ng knh BC. Gi M l tim ca on AB. Qua M k mt dy cung DE vung gc
vCD ct ng trn (O) ti im I.
a. T gic ADBE l hnh g? Ti sao?b. Chng minh 3 im I, B, E thng
hng.c. Chng minh rng MI l tip tuyn ca ng trn (OMI2=MB.MC.
cu 4: (1,5im)Gi s x v y l 2 s tho mn x>y v xy=1.
Tm gi tr nh nht ca biu thc .22
yx
yx
+
.
S 22
cu 1:(3 im)Cho hm s xy = .a.Tm tp xc nh ca hm s.b.Tnh y bit: a)
x=9 ; b) x= ( )221c. Cc im: A(16;4) v B(16;-4) im no thuc th cas,
im no khng thuc th ca hm s? Ti sao?Khng v th, hy tm honh giao im ca
th h cho v th hm s y=x-6.
cu 2:(1 im)Xt phng trnh: x2-12x+m = 0 (x l n).
Tm m phng trnh c 2 nghim x1, x2 tho mn iu ki=x12.
cu 3:(5 im)Cho ng trn tm B bn knh R v ng trn tm C bn k
ct nhau ti A v D. K cc ng knh ABE v ACF.
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thi vo THPT
a.Tnh cc gc ADE v ADF. T chng minh 3 im E,thng hng.b.Gi M l
trung im ca on thng BC v N l giao imcc ng thng AM v EF. Chng minh t
gic ABNC l hnh
hnh.c.Trn cc na ng trn ng knh ABE v ACF khng cha D ta ln lt ly
cc im I v K sao cho gc ABI bng gc(im I khng thuc ng thng NB;K khng
thuc thngNC)
Chng minh tam gic BNI bng tam gic CKN v tam gic Ntam gic cn.d.Gi
s rng R
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thi vo THPT
b. Chng minh rng Parabol v ng thng (d) lun ct nhauim phn bit. Tm
to giao im ca chng. Vi gi trca m th tng cc tung ca chng t gi tr nh
nht?
cu 3: (4 im)Cho ABC c 3 gc nhn. Cc ng cao AA, BB, CC ct nhH; M l
trung im ca cnh BC.1. Chng minh t gic ABHC ni tip c trong ng trn.2.
P l im i xng ca H qua M. Chng minh rng:
a. T gic BHCP l hnh bnh hnh.b. P thuc ng trn ngoi tip ABC.
3. Chng minh: AB.AC = AA.AH.
4. Chng minh:8
1''' HC
HC
HB
HB
HA
HA
S 24cu 1: (1,5 im)
Cho biu thc:
x
xx
A 24
442
+
=1. Vi gi tr no ca x th biu thc A c ngha?2. Tnh gi tr ca biu thc
A khi x=1,999
cu 2: (1,5 im)Gii h phng trnh:
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thi vo THPT
=+
=
52
34
12
11
yx
yx
cu 3: (2 im)Tm gi tr ca a phng trnh:
(a2-a-3)x2 +(a+2)x-3a2 = 0nhn x=2 l nghim. Tm nghim cn li ca
phng trnh?
cu 4: (4 im)Cho ABC vung nh A. Trn cnh AB ly im D khng
vi nh A v nh B. ng trn ng knh BD ct cnh BC ng thng AE ct ng trn
ng knh BD ti im th haing thng CD ct ng trn ng knh BD ti im th F. Gi
S l giao im ca cc ng thng AC v BF. Chng mi1. ng thng AC// FG.2.
SA.SC=SB.SF3. Tia ES l phn gic ca AEF .
cu 5: (1 im)Gii phng trnh:
361122 =+++ xxx
S 24cu 1: (2 im)
Cho biu thc:
1,0;11
11
+
++
= aaa
aa
a
aaA .
1. Rt gn biu thc A.2. Tm a 0 v a1 tho mn ng thc: A= -a2
cu 2: (2 im)
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thi vo THPT
Trn h trc to Oxy cho cc im M(2;1), N(5;-1/2) v thng (d) c phng
trnh y=ax+b1. Tm a v b ng thng (d) i qua cc im M v N?2. Xc nh to
giao im ca ng thng MN vi cc tr
v Oy.cu 3: (2 dim)Cho s nguyn dng gm 2 ch s. Tm s , bit rng
ca 2 ch s bng 1/8 s cho; nu thm 13 vo tch ca 2s s c mt s vit
theo th t ngc li s cho.
cu 4: (3 im)Cho PBC nhn. Gi A l chn ng cao k t nh P xung
BC. ng trn ng khinh BC ct cnh PB v PC ln lt M v NN vi A ct ng
trn ng knh BC ti im th 2 l E.1. Chng minh 4 im A, B, N, P cng nm
trn mt ng trn
nh tm ca ng trn y?2. Chng minh EM vung gc vi BC.3. Gi F l im i
xng ca N qua BC. Chng minh AM.AF=AN.AE
cu 5: (1 im)Gi s n l s t nhin. Chng minh bt ng thc:
( )2
1
1
23
1
2
1 >
+
+= ,0,0;
2: .
1. Rt gn biu thc trn.
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thi vo THPT
2. Tm gi tr ca x v y S=1.cu 2: (2 im)
Trn parabol 22
1xy = ly hai im A v B. Bit honh ca
A l xA=-2 v tung ca im B l yB=8. Vit phng trnh thng AB.
cu 3: (1 im)Xc nh gi tr ca m trong phng trnh bc hai:
x2-8x+m = 0 34+ l nghim ca phng trnh. Vi m va tm c, ptrnh cho cn
mt nghim na. Tm nghim cn li y?
cu 4: (4 im)Cho hnh thang cn ABCD (AB//CD v AB>CD) ni tip
tron
ng trn (O).Tip tuyn vi ng trn (O) ti A v ti D ct nh
E. Gi I l giao im ca cc ng cho AC v BD.1. Chng minh t gic AEDI
ni tip c trong mt ng trn.2. Chng minh EI//AB.3. ng thng EI ct cc
cnh bn AD v BC ca hnh thang ng R v S. Chng minh rng:
a. I l trung im ca on RS.
b.RSCDAB
211=
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