ĐỀ THI OLYMPIC 30-4 NĂM 2007

KY
̀
THI OLYMPIC TRUYÊ
̀
N THÔ
́
NG 30/4
LÂ
̀
N THƯ
́
XIII TA
̣
I THA
̀
NH PHÔ
́
HUÊ
́
ĐÊ
̀
THI MÔN TOÁN LỚP 10







Chu
́

y
́
:





 



! 
"

Câu 1#$%&'(
)*+,-





−=+
=
+
++
yxyx
yx
xy
yx

.
..
/

Câu 2#$%&'(
0123a, b, x, y*4%56+
7
=−
byax
(-189:;&
<
aybxyxbaF
+++++=
....
(
Câu 3#$%&'(
01ABC=1=A, B9%56+
.
.
.
7

.
7

BABA
−
=+
(
0<1ABC>1%5(

Câu 4#$%&'(
0<1?ABCD.@AM>%&B!C()DP, Q, R, S>1%&
MPMDMCMB $
=++
E
MQMAMDMC $
=++
E
MRMBMAMD $
=++
E
MSMCMBMA $
=++
(
-F8G;%&MPA = QB = RC = SD.
Câu 5#$%&'(
HID%JJK1?=1%L>M%&=D%J
! (0<N H OK1=G:J%&=D%J
! (
PPPPPPPPPPPPPPPPPPPQRPPPPPPPPPPPPPPPPPPPPP
Ghi chúCán bộ coi thi không giải thích gì thêm
S111
TUVWXT) SVY
Câu 1: )*+,-





−=+

=
+
++
'.#!Z!Z
'#/
!Z
Z!
!Z
.
..
[S56+Z\!] ^_
[#' ⇔#Z
.
\!
.
'#Z\!'\Z!`/#Z\!'
⇔a#Z\!'
.
b.Z!c#Z\!'b/#Z\!'\Z!`
⇔#Z\!'
7
b/#Z\!'b.Z!#Z\!'\Z!`
⇔#Z\!'a#Z\!'
.
b/cb.Z!#Z\!b$'`
⇔#Z\!b$'a#Z\!'#Z\!\$'b.Z!c`

⇔
. .
Z ! $  #7'

Z ! $#Z !'  #$'
+ − =


+ + + =

^_
d#7'⇒Z\!`$^eF>#.'%,f
Z
.
\Zb$`.⇔Z
.
\Zb/`⇔
Z 7 ! g
Z . ! .
= − ⇒ =


= ⇒ =

(

#$'F+F-Z
.
\!
.
hF>Z\!]( ^_
ij!+=+>#b7Eg'E#.E.' ^_
TUVWXT) SVY
Câu 4: 0<1?ABCD.@AM>%&B!C()DP, Q, R, S>1

%&
MPMDMCMB $
=++
E
MQMAMDMC $
=++
MRMBMAMD $
=++
E
MSMCMBMA $
=++
-F8G;%&MPA = QB = RC = SD.
)*k=%&M9>1()DG>%&
MDMCMBMAMG
+++=
_
(
^_
d
MPMDMCMB $
=++
^=
GAPA _$
=
(
,3
GBQB _$
=
^
GCRC _$

=
^
GDSD _$
=
(

W%=PA = QB = RC = SD
⇔
GA = GB = GC = GD. 
TeABCD><1Je%,f%,lmO-G
B O F> M >  %&  n!  :  Z1  %8  o
( )
ODOCOBOAOM
+++−=
(p&O:!9PA = QB = RC =
SD(

TeABCD6*><1Je%,f%,lm-
6?O%&M(
^_
S111
TUVWXT) SVY
Câu 2: 01 23
a
^
b
^
x
^
y

94%56+
7
=−
byax
(
-189:;&<
aybxyxbaF
+++++=
....
(
ieO
( )
..
..
$
7
..
ba
a
y
b
xF ++






++







+=
(
^_
SH
( )
y;xM
=
^






−−=
..
a
;
b
A
^
( )
7
=−∆
byax:

(=
..
.
..






++






+=
a
y
b
xMA
(>
( )
∆∈
M
 
( )
[ ]
..

.
.
7
ba
;AdMA
+
=∆≥
(
SI<Z*!6
M
>-e;
A
 
( )
∆
(
^_
q!
( ) ( )
7
$
77
.
$
77
..
..
..
..
=+

+
≥++
+
≥
ba.
ba
ba
ba
F
(

ij! 
7
=
Fmin
 %O %,f  I  O  6
( )








−=
.
.
.
/

. ;;;y;x;b;a
(

S111
TUVWXT) SVY
Câu 3: 01ABC=1=A^B9%56+







.
7A
\






.
7B
`.







−
.
BA
(
0<1ABC>1%5(
=#
.
7A
'\#
.
7B
'`.#
$
'#7 BA
+
'#
$
'#7 BA
−
'(

≥
#
$
'#7 BA
+
']E#
.
BA

−
']

≤
.
BA
−

≤
$
7 BA
−
r
π

⇒
#
.
BA
−
'
≥
#
$
7 BA
−
'

⇒
#

.
BA
−
'
≥
#
$
7 )BA(
−
'

d#
.
7A
'\#
.
7B
'`.#
.
BA
−
'F>#
.
BA
−
']
q!.#
$
'#7 BA
+

'#
$
'#7 BA
−
']
Q!#
$
7 )BA(
−
'](

pefFo#
$
'#7 BA
+
'
≤
^=#
$
'#7 BA
+
'#
$
'#7 BA
−
'
≤
#
$
'#7 BA

−
'
W%=.#
$
'#7 BA
+
'#
$
'#7 BA
−
'
≤
.#
$
'#7 BA
−
'
≤
.#
.
BA
−
'

i-Fj!e#
.
7A
'\#
.
7B

'`.#
.
BA
−
'-*=







=
+
−
=
−
'
$
'#7
#
$
7
.
BA
BABA

⇔
A`B`
7

π
(
ij!1ABC>1%5(

S111
TUVWXT) SVY
Câu 4: 0<1?ABCD.@AM>%&B!C()DP, Q, R, S>1
%&
MPMDMCMB $
=++
E
MQMAMDMC $
=++
MRMBMAMD $
=++
E
MSMCMBMA $
=++
-F8G;%&MPA = QB = RC = SD.
)*k=%&M9>1()DG>%&
MDMCMBMAMG
+++=
_
(
^_
d
MPMDMCMB $
=++
^=
GAPA _$

=
(
,3
GBQB _$
=
^
GCRC _$
=
^
GDSD _$
=
(

W%=PA = QB = RC = SD
⇔
GA = GB = GC = GD. 
TeABCD><1Je%,f%,lmO-G
B O F> M >  %&  n!  :  Z1  %8  o
( )
ODOCOBOAOM
+++−=
(p&O:!9PA = QB = RC =
SD(

TeABCD6*><1Je%,f%,lm-
6?O%&M(
^_