bt lt olympic 30-4-
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Maths OlympicTRANSCRIPT
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 1
BI TP LUYN THI OLYMPIC TON HC TON MIN NAM LN TH XVIII
Ch : PHNG TRNH, H PHNG TRNH,
BT PHNG TRNH, H BT PHNG TRNH ( VN PH QUC- GV. TRNG H QUNG NAM)
1. Gii PT: 3 2 3 2 3 33 2012 3 6 2013 5 2014 2013x x x x x .
HD: t 3 23 2012a x x ; 3 23 6 2013b x x ; 3 5 2014c x
Ta c h sau:
33
33 3 3
201320132013 3 2013
a b ca b ca b c a b c a b b c c a
Suy ra: a b hoc b c hoc c a .
2. Gii BPT:
2 3 2012... 11 1 2 1 1 2 1 3 1 1 2 1 ... 2012 1
x x x xx x x x x x x x x
HD: k ta c:
1 1 1 1
1 2 1 ... 1 1 2 1 ... 1 1 2 1 ... 11 2 1 ... 1 1kxkx
x x kx x x kx x x kxx x k x
p dng cho bi ton trn, ta thu c:
11 1 1 2 1 ... 2012 1 0
1 2 1 ... 2012 1x x x
x x x
.
Nghim ca BPT l: 1 1 1 1 11; ; ... ;2 3 4 2011 2012
x
.
3. Gii HPT:
1
2
3
4
2
3
4
1
2 2 os
2 2 os
2 2 os
2 2 os
x c
x c
x c
c
x
x
x
xx
; 1 2 3 4, , ,x x x x .
HD: Theo : 1 1, 4 os 1, 4022 2
iix i c ix
Nu 1 3x x th 2 4 2 4 3 1 3 1os os os osx c x x x xc c x c x x . Do : 1 3x x .
Chng minh tng t ta c c: 2 4x x .
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 2
HPT cho tng ng vi: 2
1 3
2 4
1
2 1
2 os42 os
4
x xx x
x c
x
x
xc
th ca hai hm s: 212 os
4x c x , 2 1
1 arccos 2 2x x
; 110;2
x
, 220;
4x
trong h
trc ta 1 2Ox x ct nhau ti mt im duy nht c ta 1 1;4 4
.
HPT cho c nghim duy nht l: 1 2 3 414
x x x x .
4. Gii HPT:
2
2
2
30 4 2012
30 4 2012
30 4 2012
y yxz zyx xz
; , ,x y z
HD: Ta c: 2 23030 4 2012 4 2012 0 0y y y y
x x
. Tng t , 0x z .
Khng mt tnh tng qut, gi s: ,x y x z .
Tr v theo v ca phng trnh th ba cho phng trnh th nht ta c:
3 2 2 22 230 4 0 30 4 0x y x y x yz x z x yz x
.
V 0, 0y xx z nn 0x y ; 3 2 0x yz .
Do : 3 2
3 2 2 230 4 0x yz
x yz x z x y x y zx y
.
5. Cho 2013 s dng: 1 2 2013, ,..., 0x x x tha mn:
2 21 2 2 12 22 3 3 2
2 22011 2012 2012 2011
2 22012 2013 2013 2012
..........................
..........................
x x x xx x x x
x x x xx x x x
.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 3
Chng minh rng trong 2013 s c hai s ,a b sao cho: 12012
a b .
HD:
T 2 21 2 2 1 2 1 1 20 0x x x x x x x x . Chng minh tng t c: 2 3 2013...x x x .
Mt khc: 2 2 2 22012 2013 2013 2012 2013 2013 2012 2012 20130 0 1x x x x x x x x x .
Khi : 1 2 20130 ...x x x .
Chia an 0;1 thnh 2012 on con bng nhau, di ca mi on con l 12012
.
Theo nguyn l Dirichlet: ,a b trong 2013 s cho thuc v cng mt on con.
Nh vy 1
2012a b .
6. Gii HPT: 2 2 22012 2012 2012
33
3
x y zx y zx y z
; , ,x y z .
HD: Xt cc vect: ; ; , 1;1;1u x y z v
. D thy . . 3u v u v
.
Suy ra: ,u v
cng phng 01 1 1x y z x y z .
Kt hp vi phng trnh cn li ta c: 1x y z .
7. Gii BPT: 2 22012 2014 2 4028 2014 2 4024x x x x ; x . iu kin: 2012x .
BPT cho tng ng vi: 22012 2014 2 2014 2 2012x x x x .
t: 2012 0u x ; 2014v x .
BPT thnh:
2 2
2 2 2
2 2 0 0
2 2
0uu v u v u v u v
u v u v
.
8. Gii HPT:
201220131 2 3
201220132 3 4
201220132012 1 2
1 2 2012
30 4
30 4.......................................
30 4, ,..., 0
x x x
x x x
x x xx x x
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 4
HD: Gi s: 1 2 2012, ,...,x x x l mt nghim ca HPT trn.
t: 1 2 2012, ,...,M Max x x x ; 1 2 2012, ,...,m Min x x x .
Suy ra: 0M m .
Ta c:
201220131 2 3
201220132 3 4
201220132012 1 2
34 30 4
34 30 4......................................................
34 30 4
M x x x
M x x x
M x x x
20132012 2012 2012 20122013 2013 20131 2 201234 ; ;...; 34 34M Max x x x M M 4026 2013 402434 M M 2011 4026 2011 402634 34M M .
Chng minh tng t, ta c: m 2011 402634 . Suy ra: 2011 402634M m .
Do : 2011 40261 2 2012... 34x x x . Th li thy ng.
Vy h phng trnh cho c nghim dng duy nht: 2011 40261 2 2012... 34x x x .
9. Gii HPT:
1 22
2 33
2012 11
1 20122
1 20122
..............................
1 20122
x xx
x xx
x xx
; 1 2 2012, ,...,x x x .
HD: Ta c: 21 1 1 11
1 2012 1 2012 02 2i i i i ii
x x x x xx
1,2011i
Cc ix cng du 1,2012i .
p dng BT Cauchy ta c: 11
2012 2 2012 2 2 2012 2012i i ii
x x xx
1,2012i .
Ly phng trnh u tin ln lt tr cho cc phng trnh s 2, s 3,..., s 2012 v theo v ta c:
1 2 2 32 3
1 201212
x x x xx x
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 5
2 3 3 43 4
1 201212
x x x xx x
................................................
2012 1 1 21 2
1 201212
x x x xx x
V 21 2 012...x xx nn 2 3 3 4 2011 2012
2012 2012 20121 0;1 0;...;1 0x x x x x x
v 1 2
20121 0x x
.
Suy ra:
2 3
2011 2012
1 2
2012..................
20122012
x x
x xx x
. Kt hp vi 2012ix 1,2012i suy ra 1 2 2012... 2012x x x .
10. Gii HPT:
2
2
2
2
2222
x x yy y zz z tt t x
; , , ,x y z t .
HD: t: 1 , 1 , 1 , 1X x Y y Z z T t .
Ta c h phng trnh sau:
2
216 8 4 2
2
2
X YY Z
X Y Z T XZ TT X
. Nh vy: 15 01 0 1X
X XX
* Vi 0 0 1X Y Z T x y z t .
* Vi 1 1 0X Y Z T x y z t .
11. Gii HPT:
221 1 2
222 2 3
222012 2012 1
12
12
....................................
12
kx kx x
kx kx x
kx kx x
; 1 2 2012, ,...,x x x , k l mt s cho trc.
HD: Cng v theo v ca cc PT cho ta c:
2 22012 2012
21 2 2012
1 1
1 1 10 1 ...2 2 2i i ii i
k k kx k x x x x x
.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 6
12. Cho s nguyn 3n . Gii h phng trnh:
1 2 3
2 3 4
1 2
2012 4025 2013 02012 4025 2013 0...............................................2012 4025 2013 0n
x x xx x x
x x x
; 1 2, ,..., nx x x .
HD: t 1 1 2 2 2 3 2; ; ... ; n ny x x y x y y x x .
H cho thnh:
1 2
2 31 2 1 2 1 2
1
2012 20132012 2013
2012 ... 2013 ... ... 0...........................2012 2013
n nn n n
n
y yy y
y y y y y y y y y
y y
.
Nh vy phi c mt ch s j sao cho 0jy . Nhng 12012 2013 ,j jy y ... nn 1 2 ... 0ny y y .
Suy ra: 1 2 ... nx x x a .
13. Gii HPT:
2013 2 2013
2012 2 2012
2013 2 2013
2 20122012
22 2 122 2 1
xyx x yx x
xyy y xx y
; ,x y .
HD:
Cng v theo v ca hai phng trnh trn ta c:
2 2
2012 2 2012 2 20125
1 122 2 1 2 2 1
xy x yx x y y
2 2
2 22012 20122012 2012
1 121 2 1 2
xy x yx y
(*)
Nhn xt: 2 2* 2 *VT xy x y VP .
0
* * 11
0
x yx y
VT VP x y x yx y
.
Vy tp hp nghim ca h phng trnh cho l: 0;0 , 1;1S .
14. Gii PT:2 2
2 2 2
3 2 2 2 3 1033 3 4 4 3
x x x x x x x x x xx x x x x x x x x x x x x x x x
HD: t 22, 1, 1, 1, 1a b x c x d x x e x . Phng trnh cho tr thnh:
103
a b b c c d d e e ac d e d e a e a b a b c b c d
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 7
101 1 1 1 1 53
a b b c c d d e e ac d e d e a e a b a b c b c d
1 1 1 1 1 253
a b c d ec d e d e a e a b a b c b c d
1 1 1 1 1
25 *c d e d e a e a b a b c b c d
c d e d e a e a b a b c b c d
p dng bt ng thc Cauchy cho v tri ca (*) th ta c * 25VT . Du = xy ra khi v ch khi 1a b c d e x . Vy x = 1 l nghim duy nht ca phng trnh cho.
15. Gii HPT: 2012 2012 2011 20112x y
x y x y
; ,x y .
HD: H phng trnh tng ng vi:
2012 2012 2011 2011 2012 2012 2011 20112 2
2 2 2
x y x y
x y x y x y x y x y
2012 2012 2012 2011 2011 2012 2012 2012 2011 2011
2 22 2x y x y
x y x xy x y y x y x y xy
2011 20112011 201122 2
10 00
x yx y x yx y
x y x y x yx x y y x y
16. Gii HPT: 2
2 2
5 4 4
5 4 16 8 16 0
y x x
y x xy x y
; ,x y .
HD: Ta c: 2 2 2 25 4 16 8 16 0 4 8 5 16 16 0y x xy x y y x y x x . Xem y l mt phng trnh bc hai theo n y ( tham s x ).
Ta c: 29x , t : 5 4
4y xy x
.
+ Vi 5 4y x , thay vo phng trnh th nht ca h ta c:
20 4
5 4 5 4 4 6 5 4 0 5 04
x yx x x x x
x y
.
+ Vi 4y x , thay vo phng trnh th nht ca h phng trnh ta c:
20 4
4 5 4 4 6 4 04 0
x yx x x x x
x y
.
Vy tp hp nghim ca h phng trnh cho l: 40;4 ; 4;0 ; ;05
S
.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 8
17. Gii HPT:
4 3 2 2
24 2 2 2
6 12 6
5 1 11 5
x x x y y x
x x y x
; ,x y
HD:
+ Xt 0x , h phng trnh cho thnh: 20 6
5y
( v l)
+ Chia v theo v ca tng phng trnh trong h cho 2 0x ta c: 2 2 2 2
2 2
2 22 2 2 2
2 2
1 6 1 16 12 6 12 0
1 5 1 15 11 5 11 0
x x y y x x y yx xx x
x x y x x yx xx x
.
t 2 221 1 2t x x tx x
.
H phng trnh thnh:
2 2 2 2
2 2 22 2 2
6 2 12 0 6 0
5 1 05 2 11 0
t ty y t ty y
t t yt t y
+ Xt 0t , h phng trnh thnh: 0
1 0y
( v l).
+ Chia v theo v ca tng phng trnh trong h cho 2 0t ta c: 2 2
2 2
22 2
2 2
1 66 0 6
1 1 15 0 5 2. 5
yy y y y yt tt tt t
yy y yt t t t
t: 1; ya y bt t
H phng trnh thnh:
2
2 2
5 66 22 5 5
2
aaab
a b ab
3
2
5 12 0
52
a a
ab
22
3 3 4 03
5 22
a a aa
a bb
Khi : 2
111 213 2 3 2 3 1 0 1
12222 221
tt
yy t t tt tty y t ty tt y t
y
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MATHVN.COM - Ton Hc Vit Nam
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 9
* Vi 1t ta c: 21 1 51 1 02
x x x xx
.
* Vi 12
t ta c: 21 1 1 172 2 02 4
x x x xx
.
18. Gii HPT:
2 4 2 2 2 4
2 2 4 2 2
3 2 1 2
1 1 2 2 1 0
x y x y x x y
x y x x x xy
; ,x y .
HD: H phng trnh cho tng ng vi:
22 4 2 2
2 2 4 2 2
4 1 1 2 2
1 1 2 2 1 2
x y y x x
x y x x x xy
.
Suy ra: 4 2 21 2y x x 2 4 2 22 2 1x x x xy 4 2 4 6 4 3 2 22 2 2y x x x x x y x 26 3 2 4 3 2 3 22 0 0x x y y x y x y (*)
T (*) v du = xy ra bt ng thc trn ta suy ra: 1x y .
19. Gii HPT:
2 2
3 3
214 2 2
92 2
xy y x y x y x y
x y x y
; ,x y .
HD:
iu kin 00
x yx y
.
t , 2 2
x y x yu v ; , 0u v . Suy ra: 2 2 2 2, x u v y u v .
H phng trnh cho thnh: 2 2 2 2 2 2 2 2
3 3
2 414
9
u v u v u v u vu v
u v
3 3 3 33 3 3 33 3
7 0 0 79 99
u v u v u v u vu v u vu v
.
H 3 3
09
u vu v
v nghim.
Gii h 3 3
3 3
79
u vu v
ta c
21
uv
.
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MATHVN.COM - Ton Hc Vit Nam
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 10
Do : 2 8 52
2 31
2
x yx y xx y yx y
.
Vy phng trnh cho c nghim l: 53
xy
20. Gii HPT: 4 4
2009 2013 2013 20092011
2 1
23
xy x y
x y x y
; ,x y .
HD:
Ta c: 2009 4 4 2009 2013 2013 2009 20112 0 0
3xy x y x y x y xy .
T phng trnh th nht ca h phng trnh ta c:
4 41 2 2 2xy x y xy xy ( Bt ng thc Cauchy)
13
xy v 2
4 4 12xy
x y
.
Li c: 2
2009 20092009 2013 2013 2009 4 4 1.2xy
x y x y xy x y xy
2008 2008 3 20111 1 1 22 . . . 2 .
2 2 3 3xy xyxy xy xy .
Du = xy ra
4 4
12
1 13 3
xyxy
xy x y
x y
.
Vy tp hp nghim ca h phng trnh cho l: 1 1 1 1; , ;3 3 3 3
S
.
21. Gii HPT: 2 2
2011 20132011 2013
1
2014
x y
x y y x x y xy
; ,x y .
HD: T phng trnh th nht ca h ta suy ra: 1 , 1x y .
Do : 2014 1 1 2013 0x y xy x y . + Nu x y th phng trnh th hai ca h c v tri dng, v phi m. iu ny v l. + Nu x y th phng trnh th hai ca h c v tri m, v phi dng. iu ny v l.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 11
+ Nu x y th phng trnh th hai ca h tha mn. Thay vo phng trnh th nht ta c:
2 12 12
x x .
Vy h phng trnh cho c hai nghim l: 1 1;2 2
, 1 1;2 2
.
22. Gii PT: 4
62
cos 23 1 tan 7cos
x xx
HD: t 22
cos 2 , tancos 1
xa b xx
.
Phng trnh cho thnh: 4 33 4 7a b . D thy a, b 0 v 2a b . p dng bt ng thc Cauchy ta c:
4 31 1 1 4 , b 1 1 3a a b . Suy ra: 4 3 4 33 3 12 , 4 2 12 3 4 7a a b b a b . Du bng xy ra 4 4 1 1a b a b hay tan 1
4x x k k
l nghim ca phng trnh cho.
23. Gii HPT:
3 33 3
1 1 9
1 1 1 11 1 18
x y
x y x y
( ,x y ).
HD:
iu kin , 0x y .t 3 3
1 1, u vx y
.
H phng trnh cho thnh:
33 3 9 3 91 1 18 1 18
u v u v uv u vu v u v u v u v uv
.
t ,S u v P uv . iu kin 2 4S P .
H phng trnh thnh:
33
2
3 9 13 91 18 18 2
S PSS PSS S P PS S S
Thay (2) vo (1) ta c: 33 23 3 63 0 1 64 3 2S S S S S P .
Vi 32
SP
ta suy ra: ,u v l nghim ca phng trnh: 21
3 2 02
XX X
X
.
Khi : 12
uv
hoc 21
uv
. Suy ra: 181
x
y
hoc 118
x
y
.
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MATHVN.COM - Ton Hc Vit Nam
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 12
24. Gii BPT: 2 4 26 3 1 1 0x x x x ; x HD: Bt phng trnh cho tng ng vi:
2 4 2 2 2 4 26 3 1 6 1 0 6 2 1 1 6 1 0x x x x x x x x x x
2 2 2 212 1 6 1 1 6 1 0x x x x x x x x
222 2
6 1112 6 01 1
x xx xx x x x
( v 2 1 0 x x x )
t 2
2
6 1 , 0
1x x
t tx x
.
Bt phng trnh thnh: 2 32 6 0 02
t t t .
Do : 2 2
2
6 1 9 11 21 11 215 11 5 01 4 10 10
x xx x x
x x
.
Vy tp hp nghim ca bt phng trnh cho l: 11 21 11 21;10 10
S
.
25. Gii HPT:
2
2 2
1 1 4 3
12 2 3 7 1 12 3 5
x y x y x y
x x y xy y x
, ,x y .
HD:
t 1 0u x y ; 3 0v x y .
H phng trnh cho thnh:
2 22 2
24 2 2 4
3 33 39 9 4 9 9 3 4 9
u vu vu v u u v v
2 22 2
3 2 2 34 2 2 4
3 33 39 9 3 3 09 9 9 6 3 0
u vu vu v u u v uv vu v u u v v
2 22 3 6
2u v
u vu v
Khi :
611 122 263
2
x yx y y x
x y
. Thay vo phng trnh th hai ca h phng
trnh cho ta tm c:
4 53 6
1 76 10
y x
y x
.
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MATHVN.COM - Ton Hc Vit Nam
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 13
26.Gii HPT:
4 4
2 2 2
1 1 22
1 1 3 32
y xx y
y x x yx y
, ,x y
HD: iu kin , 0x y . Vi iu kin trn, h phng trnh cho tng ng vi:
4 4 2 254 5 3 2
4 5 2 3 54 4 2 2
2 5 10 32 5 101 1 5 10 15 10
y x x y x yxy x x yxx y y x y x yx y x y
y
5
5
5
3 13 2
1 3 12
xx yx y
y
( tha iu kin).
Vy h phng trnh cho c nghim: 5 53 1 3 1;
2 2
.
27. Gii BPT: 3 3 2 24 6 7 12 6 2x x x x x ; x . HD:
iu kin: 32
x .
Vi iu kin trn, bt phng trnh cho tng ng vi:
2 32 2 232 2 2 2 2 2 2x x x x x x x
t 2 233 2 3 234 6 2 , 7 12 6 2 7 12 6 2A x x B x x x x x x x x (*)
Th th 32
x ta c: 0 , 0A B
Khi 2 2 3 32 2 22 2 2 2 2 2* 2x x x x x x x
A B
2 21 12 1 0 2 0 2 2x B xxA
( tha iu kin 3
2x )
Vy tp hp nghim ca bt phng trnh cho l: 2; 2S .
28. Gii BPT: 35 3 2 2 21 1x x x x x x x ; x . HD: iu kin: 0x . + Nu 0x th BPT lun ng
+ Nu 0x th chia c 2 v ca BPT cho: 2 2 1 0x x
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
Vn Ph Quc, GV. Trng H Qung Nam - D: 0934 825 925 -Mail: [email protected] 14
Ta c:
32 2 2
2 2
5 3
2 2 2 2
1 1
1 11x x x x x
x x x xx x
x x
2
2 2 2
1 1 1 111 1 11 11x x xx x
x x x x x xxx
xx
11
11 1 11 x
x xx
xx x
x
.
t 1t xx
; 2t .
BPT trn thnh: 2
1 11 .. 11 . 0t tttt t
( lun ng 2t ).
Vy nghim ca BPT l: 0x .
29. Gii HPT:
22 2 2 2
22 2 2 2
22 2 2 2
3 1
4 1 ; , ,
5 1
x y z x x y z
y z x y y z x x y z
z x y z z x y
.
HD: + TH 1: 0xyz .
Nu 0x th h c nghim 0;0; , 0; ;0z y . Tng t cho trng hp 0y hoc 0z .
+ TH2: Chia c hai v ca cc PT trong h cho 2 2 2 0x y z ta c: 2
2
2
2
2
2
1 1 1 13
1 1 1 14
1 1 1 15
z y x x
x z y y
y x z z
t 1 1 1, ,a b cx y z
. HPT thnh:
2 2
2 2
2 2
3
4
5
b c a a
c a b b
a b c c
.
Cng v theo v cc PT trn ri rt gn ta c: 24
12 03
a b ca b c a b c
a b c
.
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30. Gii HPT:
2
2
2
222
x x y yy y z zz z x x
; , ,x y z .
HD:
HPT cho tng ng vi:
2
2
2
1 2
1 2
1 2
y x x
z y y
x z z
(I)
V mt trong cc gi tr , ,x y z bng 1 u khng tha h phng trnh ny nn: , , 1x y z .
Khi
2
2
2
21
21
21
xyxyI zyzxz
.
t tan ;2 2
, x t t
. Suy ra: tan 2y t ; tan 4 , tan8z t x t .
Do : tan tan 8 7
kt t t k . Nh vy: 2 4tan ; tan ; tan7 7 7
k k kx y z .
V ;2 2
t
nn 3; 2; 1;0;1;2;3k .
31. Gii HPT:
3 2 2
2 2
2 3 2 3
3 3 2
36
x x z z
y y x xy z zz
; , ,x y z .
HD:
HPT cho tng ng vi:
23 2 2
2 2
3 2 3 3 3 0
3 3 2
36
x z x z
y y x xy z zz
PT th nht c nghim 0
03x
zx
z
(1)
PT th ba c nghim 2 6 60 0y z z z . (2) T (1), (2) v kt hp vi 3z ta suy ra: 0z hoc 3z . p s: C 2 nghim: 1;0;0 , 2; 3;3 .
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32. Gii HPT:
3 2
3 2
3 2
3 3 1
3 3 1 ; , ,
3 3 1
x x y x
y y z y x y z
z z x z
.
HD:
Xt PT: 3 23 3 1x x y x . V 13
x khng tha PT nn 3
2
33 1x xy
x
.
t tanx t vi ;2
\3 6
t
.
D dng suy ra c: tan 3 , tan 9 , tan 27y t z t x t .
33. Gii HPT: 2012
1 1 1 1 13 2 3 2 3 2 2 2 2 2 2 2
x y z
x y y z z x x y z y z x z x y
; , ,x y z
HD:
iu kin: 2 2 2 2 2 2, , 0
; ; 0
x y z
Min x y y z z x
.
p dng BT Cau chy ta c:
22 2 2 2 2 21 1 2
3 2 2 2 3 2 2 2 2 22 2
x y z x y zx y x y z x y x y z x y zx y z
.
Tng t chng minh c: 1 1 2
3 2 2 2 2 2y z y x z x z y
; 1 1
3 2 2 22
2 2z y x zx z y x
.
Suy ra: 1 1 1 1 13 2 3 2 3 2 2 2 2 2 2 2x y y z z x x y z y z x z x y
.
Du "=" xy ra khi v ch khi x y z 20123 2012
9x x . Do : 2012
9x y z .
34. Tm mi cp s thc ;x y tha h: 22
6 3 2 2 2
3 3 2
2142
1 2
y y x xy x y
xy y x x y
.
HD: iu kin: 2 2 0 0 1xy xyx y .
Ta c: 2 22
22 1 12 44
1 12
xy x y x xyy x y
.
Khi : 6 3 2 2 2122
y y x xy x y .
Suy ra: 3 3 3 3 6 214 4 2 22
xy y xy y y x (1)
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M: 23 3 214 2 1 22
xy y x x y (2)
Cng v theo v (1) v (2) ta c:
223 2 3264 1 4 1 2 1 1 2 2xy y x x y y y xx 2
3
1 1 2 0
2 0
x y
y x
.
35. Gii HPT:
21 1 222 2 323 3 424 4 1
1 0
1 0
1 01 0
x x xx x xx x xx x x
; 1 2 3 4, , ,x x x x .
HD:
HPT cho c dng:
1 2
2 3
3 4
4 1
f x x
f x x
f x x
f x x
vi 2 1f t t t ng bin trong 1 ;2
, nghch bin
trong 1;2
v 1 52 4
f t f
. Suy ra: 5 1, 44k
x k .
* Trng hp: 1 41 1 11 52 2 6 4
x f x f
, m 454
x nn 41 ;2
x
.
Lp lun tng t: 3 21 1;2 2
x x .
Nu 21x x th 3 2 3 3 41 2 3 42 4 1x f x f x x x f xf x f x x xf x x . T : 1 2 3 4 1x x xx x nn 1 2 3 4x x x x .
Thay vo mt trong bn PT ca h ta c 1 2 3 4 1x x x x .
* Trng hp: 112
x , nu c 1k 12k
x th theo trn 1 2k
x k l mu thun.
Vy 1 2k
x k .
Nu 31x x th 3 2 4 2 4 3 1 1 31 f x x x f x f x x x x xf x . Tng t 2 4x x .
H tr thnh:
1 3 2 4
1 2 1 1 2 2
2 1
,x x x xf x x x f x x f x
f x x
.
t 2 2 1g x x f x x x . th ca hm s ny c trc i xng l ng 1x .
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T 1 1 2 2x f x x f x suy ra: 1 2g x g x 1x v 2x i xng qua 1x tc l: 1 1x m
v 1 1x m . Thay vo h
1 3 2 4
1 2
2 1
,x x x xf x x
f x x
ta tm c 0m . Suy ra: 1 2 3 4 1x x x x .
36. Gii PT: 2012 2011 22011 4023 2012x y
x y y x z
; , ,x y z .
HD:
* Ta c BT thc sau: 2
1 4ab a b
; , 0a b .
* VP 2 2 zz
. Du "=" xy ra khi 1z .
*VT 2012 20114023 2011 2012
y xx y y x
2012.4023 2012 2011 2011.4023 2011 2012
y x y x x yx y y x
2 2 2
2 22012.4023 2012 2011 20114 4
4023 4023xy y x x y
x y x y
2 2 2
2
4023 2012 20112 2
4023
x y y xx y
.
Du "=" xy ra khi 2011, 2012x y .
37. Tm m bt phng trnh sau v nghim:
2
2
1 12 s inx s inx 7s inx s inx 2.
1 13 s inx s inx 12s inx s inx
m
HD: Bt phng trnh cho tng ng vi:
2
2
1 12 sin sin 1sin sin 21 13 sin sin
sin sin
x xx x
x x mx x
.
t 1sin , tsin
t x x kx
.
Bi ton tng ng vi vic tm m bt phng trnh sau ng vi mi t: 2
2
2 1 23
t tt t m
.
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V mu thc xc nh t nn 11 12 0 .12
t m Khi 23 0 tt t m .
V 22 1 0 tt t nn 2
22
2 1 2 4 3 2 1 0 3
t t t t t m tt t m
359 16 2 1 0 .12
m m
Vy 3512
m l gi tr cn tm tha mn yu cu bi ton.
38. Gii phng trnh: 2 3 3 244 4 41 1 1 1x x x x x x x x ; x HD: iu kin: 0 1x
t 4
44 4
00
1 1
uu x
vv x u v
T phng trnh cho ta c :
2 2 3 2 2 2
2 2 0
u uv v v u u v
u v u v u uv v
1 0
1 0
u v u v u v
u v u v
( do u , v khng m v u , v khng ng thi bng 0 nn 0u v ).
Ta c cc h :
4
4 44
10 2)
112
uu va
u v v
. Do
112 .
1 212
xx
x
224 4 2 2
11)
1 2 2 1
u vu vb
u v u v uv u v
2 2
1 01
0 110
2 4 0 1 12
2 0
u v uu v
uv vu vuv
u v uv u v uuv
uv v
. Do
01 1 0
111 0
xx x
xxx
.
39. Gii phng trnh : 2012 20102 21 1 2x x x x ; x . HD:
iu kin: 2
2
1 01
1
xx
x x
.
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Khi : 2010 2010 20122 2 21 1 1x x x x x x ( do 1x ).
Suy ra: 2012 2010 2012 20122 2 2 21 1 1 1x x x x x x x x p dng bt ng thc Cauchy ta c: 2012 2012 2012 20122 2 2 21 1 2 1. 1 2x x x x x x x x
Do : 2012 20102 21 1 2x x x x . Vy phng trnh cho v nghim.
40. Gii PT: 338sin 1 162sin 27 0x x . HD: t 2sinu x K: 2 2u
PT cho thnh: 3 33 31 81 27 0 1 81 27u u u u . t 3 33 1 3 1v u u v . Do , ta c:
3 33
3 3 2 23
1 3 1 31 33 3 01 3
u v u vu vu v v u u v u uv vv u
3
332
2
1 31 3
3 13 3 02 4
u vu v
u uvu v u v u v
Lc :
3 3 16sin 8sin 1 3sin 4sin sin 3 sin2 6
x x x x x
23 26 18 35 5 23 26 18 3
x k x k
x k x k
.
41. Tm m PT: 2 21 1 2012x x x x m c nghim. HD:
2 22 22 2 1 3 1 31 1 2012
2 2 2 2x x x x m x x m
Trong mt phng vi h ta Oxy, xt: 1 3 1 3; ; B ;2 2 2 2
A
v nh ;0M x ta c: 1AB .
Vi mi im M th 1AM BM AB .
M 2 22 21 3 1 3 ; BM=
2 2 2 2AM x x
Suy ra: 1 12012 1 1 2012 12012 2012
m m m .
42. Gii HPT: 2
2
1 1
1 3
x y
y x
; ,x y .
HD: K: 1 , 1x y .
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t os ; y osx c c , , 0; .
H phng trnh thnh:
2 2
2 2
os 2cos .sin sin 1 1os sin 1
os sin 3 os 2sin . os +sin 3 2
cc
c c c
Cng v theo v ca (1) v (2) ta c: sin 12 2
(3)
Kt hp (3) v PT: os sin 1c ta gii c: 1os2
c hay 1 32 2
x y ( tha K)
Vy 1 3, ;2 2
x y
l nghim duy nht ca h.
43. Tm m phng trnh sau c nghim duy nht: 2012 2014 2011 20142012 201130. 2011 4. 2012 68378x x m .
HD: iu kin : Nu phng trnh c nghim 0x th 0x cng l nghim ca n. Do phng trnh c nghim duy nht th iu kin l: 0 0 0 0x x x . Thay vo phng trnh, ta c: m 1 iu kin cn: Vi 1m , phng trnh c dng:
2012 2014 2011 20142012 201130. 2011 4. 2012 68378x x
nh gi 2012 2014 2011 20142012 201130. 2011 4. 2012 68378x x . Do phng trnh c nghim khi v ch khi 0.x Vy phng trnh c nghim duy nht khi v ch khi 3.m 44. Gii PT: 2 2 211 14 9 11 2 3 17 2 3 2 2 2x x x x x x x ; x HD:
Ta c: 2 2 2 2 2 2* 3 1 2 2 3 1 2 1 3 1 2 2 1VT x x x x x x
2 2 22 2 2 1 2 2 1 2 2 1 2 1x x x x x x
2 2 1 2 1 2 2 2x x x x .
Vy
13 1 0 32 0 2 1* 2 2 2 .1 0 1 32 1 0 1
2
xxx x
VT x xx x
x x
45. Cho h phng trnh: 2 2 2
2 2
2 1 2 2 0
2 9 0
m m x m y m m
x y x
.
Chng minh rng h phng trnh trn lun c hai nghim phn bit 1 1,x y v 2 2,x y . Tm m biu thc 2 21 2 1 2P x x y y t gi tr nh nht.
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HD: Nghim ca h phng trnh ny l to giao im ca ng thng
2 2 2: 2 1 2 2 0m m x m y m m v ng trn 2 2: 1 10C x y tm 1;0I , bn knh 10R . Ta c: 2 2 22 1 2 2 0m m x m y m m
21 2 2 2 0x y m x m y . Phng trnh ny nghim ng vi mi m khi 1, y = 2x ngha l: lun i qua im c nh 1;2A v A nm trong C . Do lun ct C ti hai im phn bit 1 1 2 2; , N ;M x y x y . Vy h cho lun c hai nghim phn bit. Ta c: 2P MN nh nht khi v ch khi 2 2. 0 1 .2 2 .2 0IA IAu m m m
2 1 34 4 2 02
m m m . Vy 1 32
m .
46. Tm gi tr ca m h phng trnh sau c ng hai nghim: 8
20128 8
256
30 . 24
x y
x y m
HD: Gi s 0 0;x y l mt nghim ca h phng trnh cho. Khi 0 0 0 0 0 0; , ; , ;y x x y y x cng l cc nghim ca h. V vy iu kin cn h c ng hai nghim l: 0 0 00 , xx y . Thay nghim ny vo h phng trnh trn ta c 0 1x v 0m .
Ngc li vi 0m , ta c h phng trnh: 8
8 8
256
2
x y
x y
.
p dng lin tip hu hn ln bt ng thc Bunhiacoski, ta c: 8 7 8 82 256.x y x y Du bng xy ra khi v ch khi x y . T y, ta d dng gii ra c nghim ca h l: 1;1 , 1; 1 .
47. Gii PT: 3 24 14 1 3 8 408
x x x x ; x .
HD: iu kin: 1x . Phng trnh cho tng ng vi: 3 2 43 8 40 8 4 4x x x x . Xt hm s: 3 2 43 8 40 ; g 8 4 4f x x x x x x trn 1; . p dng bt ng thc Cauchy cho bn s khng m, ta c:
4 4 4 4 4 44 12 .2 .2 . 4 4 2 2 2 4 4 13.4
g x x x x
I
AM
N
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Du = xy ra khi v ch khi 44 4 2 3x x . Mt khc 3 213 3 8 40 13f x x x x x x
23 2 23 9 27 0 3 9 0 3 3 0x x x x x x x . Du = xy ra khi v ch khi 3x . Ta c: 13g x x f x . C hai du = xy ra khi 3x . Vy f x g x xy ra khi 3x .
48. Gii HPT:
2 22 2
2 22 2
2 21 11 1 1 1
11 11 1
x yx yy x x y
x yx yx y
; ,x y .
HD: + iu kin: 1 , 1x y . + Vi iu kin , h phng trnh cho c vit thnh:
2 2
1 1 2 2
1 1 1
x y y x
x y y x
+ T y iu kin ca bi ton l: 2 2
1 , 1
1 1 0 *
1 1 1
x y
x y y x
x y y x
+ p dng bt ng thc Bunhiakovsky, ta c : 2 2 2 2 2 21 1 1 1 1x y y x x x y y
Do : 2 2 2
2
0 , 1
1 1 1 11
x y
x y y x x xyy
( do kt hp vi (*))
2 20 , 1
11
x yx y
+ Li p dng bt ng thc Bunhiakovsky hai ln ( c kt hp vi (1) ), ta uc :
2 21 1 2 2x y y x x y x y x y
2 2 1 1 2 2 2x y .
Do : 2 2
0 , 0
11 11 1 2 22
1
x yx y
y xx y y x x yx yx y
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+ T (1) v (2) ta c h cho tng ng vi: 2 20 , 1
1121
2
x yx y x y
x y
Vy 12
x y l nghim ca h phng trnh cho.
49. Tm tt c cc cp s thc ;x y sao cho: 22 5 41
0x
x xy x yy
.
HD: 2 22 5 4 0 2 5 4 1 0x xy x y x x y x .
Ta c: 1 1 1x x y xx y . Nh vy: 22 20 2 5 4 1 2 5 4 1 2 2x x y x x x x x x x y .
50. Gii PT: 7 6 5 2011 2012 20132011 2012 2013 7 6 5x x x x x x
; x .
HD: iu kin: 2013x PT cho tng ng vi:
7 2012 6 2011 5 2010 02012 7 2011 6 2010 5x x x x x x
(*)
Cho 0b a vi 2018a b . Ta c: x a x b
a b x a bx a x b b ab a x a x b x a x bab
b a b a
Nh vy: + Nu 2018x th * 0VT . + Nu 2018x th * 0VT ., + Nu 2018x th thy tha mn PT (*). Vy 2018x l nghim ca phng trnh cho. 51. Cho , ,a b c , 0a sao cho a v 4 3 2a b c cng du. Chng minh rng phng trnh:
2 0ax bx c khng th c c hai nghim cng thuc khong 1;2 . HD:
Ta c: 1 2 1 2 1 2 1 24 3 20 4 3 2 4 3 2 1 2 2 1a b c b c x x x x x x x x
a a a
(*)
Gi s PT: 2 0ax bx c c hai nghim 1 2, 1;2x x th 1 2 1 21 2 2 1 0x x x x . iu ny mu thun vi (*). Vy ta c iu phi chng minh.
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52. Gii HBPT:
1 2 3 4
1 2 3 4 1 2 3 4
1 2 3 4 3 4 1 2
1 2 3 4
00
00, 0, 0, 0
x x x xx x x x x x x x
x x x x x x x xx x x x
.
HD: t 1 2 3 4A x x x x 1 2 1 3 1 4 2 3 2 4 3 4B x x x x x x x x x x x x 1 2 3 1 2 4 1 3 4 2 3 4C x x x x x x x x x x x x 1 2 3 4D x x x x . T HBPT cho ta c: , , , 0A B C D . Xt 4 3 21 2 3 4 0P X X x X x X x X x X AX BX CX D (1) V tt c cc h s ca PT (1) u dng nn n khng c nghim dng. V th 21 0, 0x x . Nhng theo HBPT th 1 20, 0x x . Vy HBPT cho v nghim.
53. Gii HPT: 1 1 13 4 5
1
x y zx y z
xy yz zx
; , ,x y z .
HD: T PT th nht suy ra: , ,x y z cng du. Nu , ,x y z l nghim ca h th ; ;x y z cng l nghim ca h.
Gi s , , 0x y z . t tan , tan , tan2 2 2A B Cx y z ; 0, ;,A B C .
T PT th hai ta c: , ,A CB C BA l ba gc ca mt tam gic. Gi R l bn knh ng trn ngoi tip ABC . PT th nht tng ng vi: sin sin sin
3 4 5A B C ABC l tam gic vung ti 1C z .
Nghim ca HPT l: 1 1 1 1; ;1 , ; ; 13 2 3 2
.
54. Gii HPT:
2 4 7
2 4 7
1 1 1 1
1 1 1 1
x x x y
y y y x
; ,x y .
HD: Xt cc trng hp sau: TH: 0xy . HPT c nghim 0;0 TH: 0xy . Khng mt tnh tng qut, gi s 0x y . Khi : 7 2 41 1 1 1 1y x x x . HPT v nghim TH: , 0x y ; x y . Khng mt tnh tng qut, gi s 0x y . Khi : 2 4 7 71 1 1 1 1x x x x y . HPT v nghim TH: , 0x y ; x y . Khng mt tnh tng qut, gi s 0x y . HPT cho tng ng vi:
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2 4 7
2 4 7
1 1 1 1 1 1
1 1 1 1 1 1
x x x x x y
y y y y y x
8 7 7
8 7 7
1 11 1
x x y xyy y x x y
Suy ra: 8 8 7 7 6 6x y x y x y xy x y . T 0x y , 8 8 0x y , 0x y , 7 7 0x y , 0xy v 6 6 0x y . HPT v nghim
55. Cho a, b l hai s thc. Hy gii HPT:
2 2
2 2
2 2
2 2
1
1
x y x y ax y
y x x yb
x y
; ,x y .
HD:
t ,u x y v x y . Khi 2 20 1x y uv ; ,2 2
u v u vx v .
HPT cho thnh:
2 22 112 1
2 21
u v u v uva u v u v uv a uvuv
u v u v u v u v uv b uvuvb
uv
Cng v theo v v tr v theo v ca hai PT trong h ny ta c:1
1
u u uv a uv
v v uv b uv
Nhn v theo v ca hai PT trong h ny ta c: 2 2 2 21 1uv uv a b uv uv a b 56. Cho cc s thc , ,m n p tha mn: 30 4 2012 0m n . Chng minh phng trnh
2 0mx nx p lun c nghim. HD:
* Vi 0m th 20124
n p . PT tr thnh: 20120 1 04
bx c p x
+ 0p suy ra PT c nghim x .
+ 0p suy ra PT c nghim 42012
x .
* Vi 0m , ta c tam thc bc hai: 2f x mx nx p
Xt 2 24 16 4 16 2012 4 2012
2012 2012 2012 2012mmf m m n p m n p
2 16 2012.30 02012m m m .
Vy PT lun c nghim.
57. Gii HPT:
3 2
3 2
3 2
6 12 8 06 12 8 06 12 8 0
y x xz y yx z z
; , ,x y z .
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HD: Cng v theo v ca ba PT ta c: 3 3 32 2 2 0x y z . Nu 2x th t PT th nht ca h suy ra 3 6 2 8 8 2y x x y Vi 2y kt hp vi PT th hai ta suy ra 2z .
Do : 3 3 32 2 2 0x y z . Nu 2x th t PT th ba ta suy ra: 36 2 8 0 0 2z z x z . D dng suy ra 0 2y . Do : 3 3 32 2 2 0x y z . R rng 2x y z l nghim ca HPT cho.
58. Cho 1a . Gii HPT sau:
13
13
x a y a z a aa
a x a y a z aa
; ,x y .
HD: p dng BT Bunhiacopski ta c c: 2 3 3A a x y z ; 2 3 3a y zB x .
Suy ra: 2 2 18A B a . Li c: 2 2 1 19 18A B a a aa a
.
Do :
11
1
x a y a z a aa x y z
aa x a y a z a
a
.
59. Gii PT: 1 1 14 30 30 30 304 4 4
x x ; x
HD: iu kin: 0x .
t 1 130 304 4
t x ta thu c h:
14 30 30414 30 304
x t
t x
.
Gi s x t suy ra: 13014 30 304
30 44
t t x xx t . Do t x .
T ta c PT: 14 30 304
x x .
t 1 304
z x ta thu c h: 4 30
4 30
x z
z x
.
Gi s 4 30 30 4z z x z xx z x . Do : z x .
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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60. Gii HPT:
2 2
2
22 8
4 3
x yz z x y
z y x
; , ,x y z .
HD: Cng v theo v ca PT th nht cho PT th hai ta c:
2 22 2 22 2 2 2 12 6
2 2z zx y z zx xy x y
.
p dng BT Bunhiacopski ta c:
2 2 2
2 22 4 42 2
2 22 2
z zx y x yz zzy zx y x x y
2 8.6 4 3zy zx z y x .
Du "=" xy ra
0
2 22 2
z y xz zx y
y x
.
61. Gii HPT:
2 2
2 2
2 2
2 1 9
2 1 9
2 1 9
x y y y
y z z z
z x x x
; , ,x y z .
HD:
HPT cho tng ng vi:
2
2
2
2
2
2
9
2 1
9
2 1
9
2 1
y yx
y
z zy
z
x xz
x
.
t
2
2
9
2 1
t tf t
t
; t .
Gi s: ax , ,x M x y z tc l: x y ; x z .
Ta c: 2 2... 3 0y f y f z y z yz y z y z f z f xx . Lp lun tng t ta c z x . Vy x y z .
62. Gii HPT:
2 2 2 3 132 3
9
x x y y z z
x y y z z x x y z
; , ,x y z .
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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HD: iu kin: , 10 ,x y z .
Gi s: ax , ,x
x M x y zzxy
, c 2 trng hp:
+ Nu x y z th 0x y y z z x x y z . HPT v nghim.
+ Nu x z y th ta s chng minh 2 39
P x y y z z x x y z .
Tht vy! 4 4 4P x y y z z x x y z z y x z x y z
= 2 3 1 3 1z y x z x y z
31 2 3 1 3 127
z y x z x y z
= 3 31 1 8 32 3 3 3 2 3
27 272
993x y x P
.
Du "=" xy ra 11; 0;3
x y z .
Tng t cho cc trng hp ax , ,y M x y z ; ax , ,z M x y z
63. Tm tt c cc b ba ; ;x y z nguyn dng sao cho 2 2 2
20122012
x yy zx y z
; : tp hp cc s
nguyn t. HD:
Theo ta c: 20122012
x y mny z
vi *,m n ; ; 1m n
Suy ra: 2012xn ym yn zm .
V 2012 \ nn 200
ny mz x y m xz ynx my y z n
.
Mt khc: 2 22 2 2 2 22x y z x z xz y x z y x z y x z y .
Do 1x y z v 2 2 2x y z nn 2 2 2
1x y z x y zz z y
.
V , ,x y z nn 2 2 2 2 2 2; ; x x y y z z x y z x y z . Du "=" xy ra 1x y z .
64. Gii HPT:
2 2 2
2 2 2
2 2 2
8
2 2
3 18
x x y z xyz
y x y z xyz
z x y z xyz
; , ,x y z .
HD: t 2 2 2a x y z ; b xyz .
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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HPT thnh:
2 2 2
2 2 2
2 2 2
2 882 2 4 8 43 18 9 108 324
x a b bxa bya b y a b bza b z a b b
.
Cng v theo v ca ba PT trong h ta c: 3 214 116 392a b b . Nhn v theo v ca ba PT trong h ta c: 3 8 2 2 3 18a b b b b . Suy ra: 214 116 392 8 2 2 3 18 ... 2 6b b b b b b b a .
65. Gii HPT:
2 2
2 2
2 2
5
13
40
x y x y
y z y z
x z x z
; , ,x y z
HD: Ta thy x y hoc y z hoc x z khng l nghim ca h cho. Nhn PT th nht, th hai, th ba ca HPT cho ln lt vi x y ; y z ; x z ta thu c:
4 4
4 4
4 4
5
13
40
x y y x
y z y z
z x x z
.
Cng v theo v ca tng PT ta c: 2 518 45 27 03
y xy x z z
Ta c:
2 2
2 2
540 8
x y x y
x z x z
.
Thay 2 53
y xz vo PT ny v bin i ta thu c 2y x ; 3z x .
66. Tm b ba s thc ; ;x y z phn bit tha mn:
1 1 1
4 3 10
x y zy z x
x y zxyz
; , ,x y z .
HD: iu kin: , , 0x y z .
t 1 1 1x y z ky z x
. Ta c: 1yz kz ; 1zx kx .
T suy ra: 2 21 1 1k x k zx kzx k yz x k xyz x k k x xyz k . Tng t: 2 1k y xyz k . Suy ra: 2 1 0k x y . Do x y nn 1k . Mt khc 0xyz k nn 1k . Khi : 1 11 1 1x y z x
y z x ; 1 1, 1
1y z
x x
.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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67. Gii HPT: 2 21 1 1 2 1
1 1 21 1 1
x x y
x y xy
; ,x y .
HD: iu kin: 1 , 1x y v 0xy . T PT th nht suy ra: 0 1x . Do : 0 1y .
Ta c: 2
1 1 1 121 11 1 x yx y
.
D dng chng minh c: 1 1 21 1 1x y xy
.
Khi : 1 1 2
1 1 1x y xy
.
Du "=" xy ra x y . Thay y x vo h u tin ta c: 2 21 1 1 2 1x x x . t sinx t ; 0;
2t
.
68. Cho cc PT: 2 1 0x ax (1) ; 2 1 0x bx (2) ; 2 1 0x cx (3). Bit rng tch mt nghim ca (1) vi mt nghim no ca (2) l nghim ca (3). Chng minh rng: 2 2 2 4a b c abc . HD: Gi 1 2,x x ln lt l cc nghim ca (1) v (2).
Ta c: 11
1x ax
; 22
1x bx
; 1 21 2
1x x cx x
.
Suy ra: 2 21 21
1 2x ax
; 2 22 22
1 2x bx
; 1 22 1
x x ab cx x
.
69. Gii PT: 2 2 22 3 2 1 4 2 3 2x y xy x y x x x ; ,x y . HD: PT cho 2 22 1 2 3 2 2 2x y x y x x x x 2x
21 2 2 1 2 2x y x x x x
21 2 2 1 2 2x y x x x x
2 22
1 2 03
xx y x
y
.
70. Gii HPT: 20121 2
1 2 2012
20121 2 ...2012 2011 1
... 2012.2013
xx x
x x x
; 1 2 2012, ,...,x x x .
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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HD:
Ta c: 1 2 201220121 2 ... 1 2 ... 201220121 2 ... 1
2012 2011 1 2012 2011 ... 1x x xxx x
Suy ra: 1 2 2012... 1x x x .
Tng qut ha ton trn Gii HPT:2012 20121 1 2 2
1 2 2012
1 2 2012
...
...
x ax a x ab b b
x x x a
; 1 2 2012, ,...,x x x
71. Gii HPT:
1 2 3
2 3 4
98 99 100
99 100 1
100 2 1
00
........................0
00
x x xx x x
x x xx x xx x x
; 1 2 100, ,...,x x x .
HD: Cng v theo v ca cc PT trn ta thu c: 1 2 100... 0x x x . V tri ca PT ny gm 100 s hng nn ta c th vit li PT ny nh sau: 1 2 3 4 5 6 97 98 99 100... 0x x x x x x x x x x . Theo mi tng trong ngoc n bng 0 nn 100 0x . Lm tng t nh trn, cui cng ta thu c: 1 2 100... 0x x x .
72. Tm cc s nguyn ,x y tha mn PT: 2012
...x x x y .
HD: Nu ,x y l cc s nguyn tha mn iu kin bi th x x m v x k l cc s nguyn. Suy ra: 21k k m . Nu 0k th 22 2 1 1k m k k m k v do m khng phi l s nguyn, tc 0k . Do 0x y .
73. Gi s 1 2 3 4 5, , , ,x x x x x l cc s nguyn dng tha mn h sau:
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
100000
00
0
x x x x xx x x x xx x x x x
x x x x xx x x x x
x x x x x
Tm gi tr ln nht ca 2 41 3x xx x .
HD: Nu k hiu 1 2 3 4 5, , , ,y y y y y theo th t l v tri ca cc PT th hai n th su trong h trn th
, 1,5i iy x i l cc s nguyn dng. Ngoi ra 1 2 3 4 5 1 2 3 4 5 1000y y y y y x x x x x v 1 2 1 2 3 2 3 4 3 4 5 4 5 6 52 ; 2 ; 2 ; 2 ; 2y y x y y x y y x y y x y y x . T suy ra cc iy cng tnh chn l.
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Bi tp luyn thi Olympic Ton hc ton min Nam ln th XVIII - Dnh cho HS lp 10 chuyn Ton
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rng: 1 3 1 2 3 4 51 1 10002 2
x x y y y y y ; 2 4 11 10002
x x y .
Nhng 1 5;y y l cc s nguyn dng v l s chn nn 1 2y ; 5 2y suy ra: 1 3 499x x ;
2 4 499x x . Do : 2 4 41 993 499x xx x .
74. Gii HPT:
1 2
1 3
1
1
1 01 0..................1 01 0
n n
n
x xx x
x xx x
; 1 2, ,..., nx x x .
HD: D thy cc n 1 2, ,.. , 0. nx x x . T PT th nht v th hai ta c: 1 3x x . T PT th ba v th t ta c: 3 5x x .... By gi nu n l th theo cch suy lun ny ta c: 1 3 5 2... n nx x x x x . Tng t ta cng c: 1 4 1... nx x x . T PT cui cng ta c:
21 11 1x x . Thay gi tr ny
vo PT th nht ta c: 2 1x . i vi n l ta c: 1 2 ... 1nx x x . Th li thy ng. i vi n chn, tng t nh trn ta c: 1 3 1... nx x x ; 2 4 ... nx x x .
T PT u tin ta c: 21
1xx
. Gi s 1 3 1 2 41... ...n nx x x a x x x a
0a .
75. Gii HPT:
1 2 1 3 2 3 4
1 2 1 4 2 4 3
1 3 1 4 3 4 2
2 3 2 4 3 4 1
2222
x x x x x x xx x x x x x xx x x x x x xx x x x x x x
; 1 2 3 4, , ,x x x x .
HD:
Tr theo tng v tng i mt cc PT ca h cho, ta c:
1 2 3 4
1 3 2 4
1 4 2 3
2 3 1 4
2 4 1 3
3 4 1 2
1 0
1 0
1 0
1 0
1 0
1 0
x x x x
x x x x
x x x x
x x x x
x x x x
x x x x
T xt 4 trng hp: TH1: 1 2 3 4x x x x TH2: 1 2 43x xx x TH3: 31 2x xx v 41x x TH4: jix x vi i j .
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