Đề thi và đáp án Toán vòng 1 Diễn Châu 2014-2015

pr{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 L- x-2 x'-5x+6 x-3 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. HUONG nAN CUAivr I\{ON rOAn Cffu. Dip 6n Bi€u *. I olem I a-2Ji + 7 -4Jj 3-2",6+1+ 1.A= a-all+3 1,0 : .6-r+2 6:1 / - r2 IVt - t,1 1,0 2. a)P 2x 4 1 2x A T l- ,dlk: x+2; x+3 x-3' :_ x-2 x'-5x+6 x-3 x-2 + (x- z)(x-:) 0,5 2x(x4)+a-(r-z) _ (x - z)(x -:) 2x2 -7 x +6 (x -2)(x 4) 0,5 (x -z)(zx -z) _2x-3 x-3 (x - z)(x -:) 0,5 )lr-7 ? b)P: -'" i : :2+ " ' x-3 x-3 0,5 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 ->X2014 -32ott:)X:3. 0,5 Vfly hq c6 nghiQm: x : y : z: 3. 0,5 2.Tac6 x2 -2y'=l (l) x - -l = /1)- _ _-r 0,25 e ("+1)(x -t)=2y' 0,25 :t (r+1)(x-1): 2 (x) 0,25 Md x + 1 vd x - I cins tinh chAt chdn, 16 do d6: 0,25 (*):>x+li2vitx-l:2 0,25 :>zyri 4 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 : (.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. +V6ix>0vdy:0,khid6A: *-2'[*y+y+2y-zJi+l :x - zJi + 1: .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 LK1BG IAC,tac6: AACE - AABG 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 0,25 'l'ua+ bl c:0 xiry ra ba trudng trop sau: l) 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 . 20rs 0,5 ->X2 014 -32ott:)X:3. 0,5 Vfly hq c6 nghiQm: x : y : z: 3. 0,5 2.Tac6 x2 -2y'=l (l) x - -l = /1) - _ _-r 0,25 e (" +1) (x -t)=2y' 0,25 :t (r +1) (x -1) : 2 (x) 0,25 Md. : : (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,. y+2v-zJi +1 : 3y + 1 :) A khdngt6n t4i cpc tri. +V6ix>0vdy:0,khid6A: *-2'[*y+y+2y-zJi+l :x - zJi + 1: .1 (.'6-r)' > 0v6imgi x> 0. A: Q e Jt-r=0 ex: 1 Z.Talu6nc6: