Clro hlnh vu6ng ABCD c6 c4nh bing a.. Ctro hinh binh hdnh ABCD vdi AC > DB.. Goi E vd F lAn lucrt ld chdn duong vu6ng goc lcd tu C d6n cdc dudrng thing AB vd AD... HUONG nAN CUAivr I\{ON
Trang 1pr{cNc GIAO DUC VA EAO TAO nieN crra.u
DE TT]I VONG I, CHQN HQC SINH GIOI HUYEN LOP 9
NAM IIQC 20r4_20t5 M6n Toin - ('Itroi gian lArn bAi 150 phut) Ciu 1 (6 di6rn)
I 'l'irrh gid tri cua bi0u thuc: A : * Jl _ qJl ;
2 (ltro bi0u tliu.c: I) = .2' -r =- 4
a) l'im diOu ki€n cua x clO t' co nghTa vd rut gon bi6u thric p
b) f im tAt cd cdc gi|tri cua x nguy0n d0 p nguyOn.
CAu 2 (4 di6m)
2 'fin nghie
rn nguyOn t6 oria phucrng trinh: x2 - 2y2 - 1
CAu 3 (a cliOrn)
I cho bi6u thrl'c A = x -zJi +3y-2Ji +1 Tim gi6 trf nho nhAt md A c6 th€ dat 0u'0c.
2 (''ho x, Y, Z ld ba s6 du'cmg vd x + y + z:4 Chri.ng rninl, ring x -r y > xyz. C'iiu 4 (4 di0n)
l Clro hlnh vu6ng ABCD c6 c4nh bing a Lay di€m M tiy ii tr0n ducrng chdo AC,
kd ME r AB, MF r BC X6c dinh vi tri cria M tr6n cluong chio AC d6 dien tich
A I)EF nho nhAt, tim gi6 trf d6.
2 Ctro hinh binh hdnh ABCD vdi AC > DB Goi E vd F lAn lucrt ld chdn duong
vu6ng goc lcd tu C d6n cdc dudrng thing AB vd AD
Chu'ng rninh ring: AB.All + AD.AF : AC'
C'iu 5 (2 diem)
Cho ba s6 a, b, c thoa mdn a + b + c : 0 vd m6i s6 cl6 ldn hon - 1 vd nho hcrn 1.
/ir I J
Chri'ng rninh rdng: a' + b2 + c' <2
Trang 2HUONG nAN CUAivr I\{ON rOAn
* I
olem
-4Jj 3-2",6+1+
: .6-r+2-.6:1
/ - r2
l- ,dlk: x+2; x+3
x-3'
:_
(x - z)(x -:)
2x2 -7 x +6
(x -z)(zx -z) _2x-3
x-3
DO P nguyOn thi x - 3 phdi ld u6c cua 3
1,0
U(:): {t;-t;:;-:}
Vdi x-3 : I :) x : 4; Vdi x- 3 : -l :> x: 2(loai);
V6i x - 3 : 3:) X: 6; v6ix-3 : - 3 :> x : 0. 1,0
Vqy cdc gi6 tri cdn tim ld x: 6; x: 4; x: 0.
2 1 x' +y' + 22 :xy + yz* zx e (*- y)' *(y-t)' +(r-")' :0 0,5
<> x: y : z Khi do: x2014 + y'o'o +220t4 - 3rzot+ - 3 20rs 0,5
2.Tac6 x2 -2y'=l (l)
x-
Md x + 1 vd x - I cins tinh chAt chdn, 16 do d6: 0,25
0,25 :>y2i2
Md2 nguyOn t6:> y i 2
0,25
y lai ld sd nguyOn t6 :) y :2
Tir (1):) x: 3
0,25
Vay: nghiQm nguy0n td cria (l) ld X: 3, y :2.
-J 1.A: r-2,[ry+3y-2Ji+1.Di0ukiOn x>0 vd y>0 d0 Ji vd Ji "6nghia.
+Vdix>0vdy>0,khid6:
A : x-2.,1i + y+2y-2J; +r : (J;-.D)' +t-2H; -^t)- 2^[i +2y 0.5
Trang 3: (.F - J, -r)' -r [-, +2v +: :
: (G - Ji -t)' .f,1tO -t)' -:
:)A ,-; vay A datgi6trinh6 nrrat wri zJv - 1:0 vd J;-Jr-1=0
t9
eV: - x=- ' 4' 4
1
Dod6MinA:-1
L
+ Vdi x: 0 vdy > 0, khi d6 A : r-zJi + y+2v-zJi+1 : 3y + 1 :) A khdngt6n
t4i cpc tri
.1
(.'6-r)' > 0v6imgi x> 0 A: Q e Jt-r=0 ex: 1
Z.Talu6nc6: (*-y)t )0 :) *t *y'+2xy>4xy:> (" *y)'2a*y (l)
Ap dung (1), ta co thO vi6t: <+ [('*y) *"f' >a(x+y)z
Vi x + y + z:4 n6n: (+)' > a(x+ y)z
:> 16 (**y)>a(**y)' , (vi x> 0, y> 0):t 16(x+ t)>rcty, (theo (1))
Vflyx*y>-ry"
Dfr"-_'"ay *khi x * y: zngYfia ld x: y: I vd z:2
a ndn BE.BF lcrn nhAt e BE : BF
L(
", -4-\:10,
2[ 4) 8
i-n€ thAy S MEM : S *, (cirng chiOu cao, cirng
canh d6y).
Tucrngtu: S-rr:So
+ S-rr:S-rt*S-rt*S**
: Sorrc - S*r, :
IV' - BE-BF)
S -u, dat gi|tri nho nh o Ut.Ut lcrn nhat (1)
e M ld trung di6m cua AC vd S -rr:
0,25
0,25 0,25 0,25
0,25 0,25
0,5
AC AB,
:)
- =- nav
AC.AG : AE.AB (1)
AE AG
-Vi AF//CB ndn 6CB = EAF :> LCBG - A ACF
AC BC
_>_
AF CG
Hay AC.CG: AF.BC (2)
0,25 0,5
0,25
0,5
0,25
025 c0ng (1) vo-i (2) theo tung v0, v6i chri y BC = AD vd AC > BD non diom G
^ L^ - .-^ ^n A E r AT\ AD - LC2
Trang 4l) NOu b - 0 thi a2+ b2+ c' : a' + c'< I + | =2.
-
| u I v _4 ru \I-.1_L_2.
2) NOu I < a< 0 <b < c < l, thi 0 <b*c=- a< 1.
n€naz+b2 *c2 < (-u)+(b+c)< 1+ l:2.
3)NOu I :;ab.O.c ( l,thi0 <-a< l, 0 <-b< 1 vd0 <c<-(a+b) < l,
- Ildi hinh, khong co hinh vo ho4c v0 hinh sai kh6ng chAm bdi hinh
u thi