SÔÛ GIAÙO DUÏC VAØ ÑAØO TAÏO BÌNH PHÖÔÙC
KÌ THI TUYEÅN SINH VAØO TRÖÔØNG THPT CHUYEÂN QUANG TRUNG
NAÊM HOÏC 2006 – 2007
MOÂN THI: TOAÙN (BAØI THI CHO LÔÙP CHUYEÂN TOAÙN)
Thôøi gian laøm baøi: 150 phuùt (khoâng keå thôøi gian giao ñeà)
------------------------------------------------------------------------------
Baøi 1
Cho phöông trình
( 2) ( 1)( 3) 0 (1) mx x x x
, (m laø tham soá).
a) Chöùng minh raèng phöông trình (1) coù nghieäm vôùi moïi m.
b) Tìm m ñeå phöông trình (1) coù hai nghieäm aâm.
Baøi 2
a) Giaûi phöông trình
22
3 1 2 6 1x x x x
b) Giaûi heä phöông trình
22
33
1
33
x y xy
x y x y
Baøi 3
a) Cho
n
laø soá töï nhieân lôùn hôn 1. Chöùng minh raèng soá
42
1nn
laø moät hôïp soá.
b) Tính toång
1 1 1
1.3 3.5 (2 1)(2 3)
Snn
Baøi 4
Cho ñöôøng troøn (O) vôùi daây cung BC coá ñònh(BC < 2R) vaø ñieåm A treân cung lôùn BC (A
khoâng truøng vôùi B, C vaø ñieåm chính giöõa cuûa cung). Goïi H laø hình chieáu cuûa A treân BC, E vaø
F laàn löôït laø hình chieáu cuûa B vaø C treân ñöôøng kính AA’ cuûa ñöôøng troøn (O).
a) Chöùng minh raèng HE vuoâng goùc vôùi AC.
b) Chöùng minh tam giaùc HEF ñoàng daïng vôùi tam giaùc ABC.
c) Khi A di chuyeån, chöùng minh taâm ñöôøng troøn ngoaïi tieáp tam giaùc HEF coá ñònh
Baøi 5
Cho a, b, c laø ñoä daøi ba caïnh cuûa moät tam giaùc vaø
2abc
. Chöùng minh raèng:
2 2 2 22a b c abc
.
HEÁT
!""#$!""%
&'()
*+,*-./0123/*
( 2) ( 1)( 3) 0 (1)+ + + + =mx x x x
456&'1*&78+9:;
&:*-</07(/*2&=/0,*-./0123/*5):>+</0*(?@7A.<(7+B(
(&C(
2
( 1) 2( 2) 3 0m x m x⇔ + + + + =
1 0 1m m+ = ⇔ = −
3
2 3 0 2
x x+ = ⇔ = −
1 0 1m m≠ = ⇔ ≠ −
!
"#
2
2 2
1 3
' ( 2) 3( 1) 1 0
2 4
m m m m m
∆ = + − + = − + = − + >
$# $ !
%&'# ( $# !
D:37E?F,*-./0123/*5):>+<*&(/0*(?@7&G7;
(&C(
)# $
0
' 0
0
0
a
s
P
≠
∆ ≥
⇔
<
<
1
1
2( 2) ( ; 2)
0( ; 2) ( 1; ) 0
1
3 1
0
1
m
m
m R
m R
mm
m
m
m
m
≠ −
≠ −
∈
∈
+
⇔ ⇔ ⇔ ∈ −∞ −
− < ∈ −∞ − ∪ − +∞ <
+
< −
<
+
%&# $ ( ; 2)m⇔ ∈ −∞ − !
&'(!
&:(&C(,*-./0123/*
2 2
3 1 2 6 1x x x x− − = − −
(&C(
*%
2
2 6 1 0x x− − ≥
*+
2
2 6 1, 0t x x t= − − ≥
!#
2
2 2 2 2 2
1
2 6 1 2 6 1 3 2
t
t x x x x t x x +
= − − ⇔ − = + ⇔ − =
% #
2
2
1 2 ( )
11 2 1 0
21 2 ( )
t N
tt t t
t L
= +
+− = ⇔ − − = ⇔
= −
'#
1 2t= +
#
( )
2
2 2
2 6 1 1 2 2 6 1 1 2x x x x− − = + ⇔ − − = +
2
3 17 4 2
2
3 2 2 0
3 17 4 2
2
x
x x
x
+ +
=
⇔ − − − = ⇔ − +
=
, -. !
%&#
3 17 4 2
2
x+ +
=
3 17 4 2
2
x− +
=
!
D:(&C(*?@,*-./0123/*
2 2
3 3
1
3 3
x y xy
x y x y
+ − =
+ = +
(&C(
*% ,x y R∈!
0x y x y+ = ⇔ = −
2
3
3 1
4 4
x
x x
=
=
2
2
2
2
3 1
10
4 ( 1) 0 1
x
xx
x
x x x
=
=
⇔ ⇔ ⇔ ∈∅
=
− =
=
/#0# 0x y+ = 0$ !
0x y x y+ ≠ ⇔ ≠ −
2 2
3 3
( )( )
3 3
x xy y x y x y
x y x y
− + + = +
⇔+ = +
3 3 3 3
3 3 3
(*)
3 3 2 2 (**)
x y x y x y x y
x y x y y y
+ = + + = +
⇔ ⇔
+ = + =
#,11
0
1
1
y
y
y
=
⇔ =
= −
'# "23"0,1#
3
0
1
1
x
x x x
x
=
= ⇔ =
= −
* 0# -.
0x y+ ≠
"## ,453,6453!
'# "24"0,1#
3
0
1
1
x
x x x
x
=
= ⇔ =
= −
* 0# -.
0x y+ ≠
"## ,354,454!
'# "274"0,1#
3
0
1
1
x
x x x
x
=
= ⇔ =
= −
* 0# -.
0x y+ ≠
"## ,3574,74574!
%&8#9 ,453,6453,354,454,3574,74574!
&'(H
&:*+
n
6&'8+91-B/*(?G/6.</*./);*-</07(/*2&=/08+9
4 2
1
n n
+ +
6&'7+@1*.B,8+9;
(&C(
# 4 2 4 2 2 2 2 2 2 2
1 ( 2 1) ( 1) ( 1)( 1)n n n n n n n n n n n+ + = + + − = + − = − + + +
/#0#
n
:( $#4
4 2 1n n+ +
$(:,!
D:I/*1+F/0
1 1 1
1.3 3.5 (2 1)(2 3)
Sn n
= + + + + +
(&C(
#
2 2 2 1 1 1 1 1 1 1
2 ... 1
1.3 3.5 (2 1)(2 3) 1 3 3 5 2 1 2 3 2 3
Sn n n n n
= + + + = − + − + + − = −
+ + + + +
/#
1 1 2 2 1
1
2 2 3 2(2 3) 2 3
n n
Sn n n
+ +
= − = =
+ + +
&'(J
*+E-.'/012+'/5:A.<(K&GL>M/0>+9EN/*5O!:A&'E(?F712?G/>M/06.</5
P*+G/012M'/0A.<(4A&'E(?F7>*I/*0(-Q&>MC&>M/0:;+B(6&'*3/*>*(?9M>MC&12?G/4
A&'R6&S/6-.B16&'*3/*>*(?9M>MC&A&'12?G/E-.'/0PI/*T>MC&E-.'/012+'/5:;
&:*-</07(/*2&=/0AM+G/00+<>A.<(;
D:*-</07(/*1&70(&<>RE+S/0K&B/0A.<(1&70(&<>;
>:*(K(>*ML?F/4>*-</07(/*1&G7E-.'/012+'//0+&B(1(?9,1&70(&<>R>+9EN/*;
(&C(
&:*-</07(/*2&=/0AM+G/00+<>A.<(;
# ;< == >? , 0$ # 0# @@A
EBH FCH=
,:!,U
/B " @;8< # #
BAH BEH=
! C+ .# #
'BAH FCA=
0
'ABH CA F∆ ∆
,
0
' 90AHB CFA= =
'ABH FA C=
D@>!
/##
'BEH FCA=
!,UU
,10,11
'HBE BEH HCF FCA+ = +
C
HBE BEH ECH+ =
,# # 0
' 'HCF FCA HCA+ =
!
/#
' // 'ECH HCA HE CA=
!
C
'CA AC HE AC⊥
⊥
,!
D:*-</07(/*1&70(&<>RE+S/0K&B/0A.<(1&70(&<>;
$,#8<==>@A
'HEF FA C
=
,E6&6!C
'ABH FA C=
D
@>!/##
HEF ABH
=
"
HEF ABC
=
,111!
I
O
BC
A'
F
H
E
A
/B"# #@8?># #
HFE HCA=
,D@8!
/#
HFE BCA=
,1111!
,1110,1111#8<?-)(0# #@;>5E,>7:;
>:*(K(>*ML?F/4>*-</07(/*1&G7E-.'/012+'//0+&B(1(?9,1&70(&<>R>+9EN/*;
F( G H;>#
OI BC⊥
!
$,#
HEF ABC ABC HEF∆ ∼ ∆
=
!
C
OBI OEI=
,# #;<IG
ABO HEI
=
!
$,#
HEF ABC BAC EHF∆ ∼ ∆
=
!
C
CHF CAF=
,# #@8?>
EHI BAO
=
!
/#
OAB IHE∆ ∼ ∆
!C
OAB∆
$JI
IHE
∆
$JG
IH IE
=
,4!
& ( #
OAC IHF∆ ∼ ∆
C
OAC∆
$JI
IHF
∆
$JG
IH IF
=
,K!
,4 0 ,K :" G L $
( # 8<?!'G H ;>
$G HM$ #(# !
&'(V
*+&4D4>6&'E+@K&'(D&>&B/*>MC&7+@11&70(&<>A&'
2a b c+ + =
;*-</07(/*2&=/0W
2 2 2
2 2a b c abc+ + + <
;
(&C(
N#$
, , 0x y z∃ >
, ,a x y b y z c z x= + = + = +
!0" 0# HN"O
P$#
2
a c b
x+ −
=
2
a b c
y+ −
=
2
b c a
y+ −
=
!'P(
#
H (#((
, , 0x y z >
!
% #;*-#
2 2 2
( ) ( ) ( ) 2( )( )( ) 2x y y z z x x y y z z x⇔ + + + + + + + + + <
2 2 2
2( ) 2( ) 2( )( )( ) 2x y z xy yz zx x y y z z x⇔ + + + + + + + + + <
'
2 1a b c x y z+ + = + + =
/#;*
2
2 ( ) 2( ) 2( ) 2(1 )(1 )(1 ) 2x y z xy yz zx xy yz zx z x y
⇔ + + − + + + + + + − − − <
[ ] [ ]
2 1 2( ) 2( ) 2 1 ( ) ( ) 2xy yz zx xy yz zx x y z xy yz zx xyz⇔ − + + + + + + − + + + + + − <
[ ]
2 4( ) 2( ) 2 1 1 ( ) 2xy yz zx xy yz zx xy yz zx xyz⇔ − + + + + + + − + + + − <
2 0 0xyz xyz⇔ − < ⇔ − <
,$#0N"OQ3$7N"OR3!
'";*#,!
X
I
O
BC
A'
F
H
E
A
W*&B7&Y/ML<412-.'/0>*ML?G/M&/02M/0
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