HA NGHIA ANH - NGUYEN THUY Mtl - TRAN KY TRANH (Tuyen chgn vd gidi thi$u) DE THI TUYEN SINH VAO LdP 10 MON TOAN OE THI CUA CAC TRl/CfNG CHUYEN, CHQN TREN TOAN QUOC (Tdi ban Idn thiiC ttC, c6 8v£a chUa bo sung) Tm Vi£N Tff,'H EINH THUAN OWL f/iiy^e / A5 NHA XUAT BAN DAI HQC QU6'c GIA HA NQI NH^ XUfiT Bf^N DRI HQC QUOC Gifl NQI 16 H^ng Chu6'i - Hai Trang - Hk NQI Di6n thoai: Bien tap-Chg ban: (04) 39714896: H^nh chinh: (04^ 39714899: Tona bien tip: (04^ 39714897 Fax: (04^39714899 Chiu trdch nhifm xuat ban Gidm doc - Tong bi&n tdp: TS. PHAM THI TRAM Biin tdp Idn ddu: NGUYEN VAN TRONG Biin tdp tdi ban: NGUYEN THUY Chi ban: NHA SACH HONG AN Trinh bay bia: THAI VAN B6i tdc Mn ket xudt ban: NHA SACH HONG AN SACH LIEN KET DE THI TUYEN SINH VAO L6P 10 MON TOAN Ma s6': 1L-351DH2012 In 1.000 cuon, klio 16 x 24cm lai Cong ty Co phan VSn lioa VSn Lang. So xuat ban: 1557 - 2012/CXB/02 - 253/DHQGHN, ngay 26/12/2012. Quyet djnh xuat ban so: 355LK-TN/QD - NXBDHQGHN. In xong va nop lUu chieu quy I nam 2013. LCFl NOI DAU Cac em hpc sinh than men! HQC la qua trinh ren luyen vat va nhat cua ddi ngudi. Dieu gl vat va mdi c6 dupe thi do chi'nh la dieu minh quy nhat. Nhim giup cac em chuan bj thi vao Idp 10, giup cac em c6 them tu lieu va c6 mot cai nhin tong quat ve nhQng van de trong cac de thi tuyen sinh vao Idp 10 trong nhOng nam gan day, Chung toi bien sogn cuon: "DE THI TUYEN SINH VAO LClP 10 MON TOAN". Sach gom de thi cua cac trudng chuyen, chpn tren toan quoc tC/ nam hpc 2000 den nay. Chung toi hi vpng cuon sach se rat hufu ich cho cac em tren con dudng hpc tap. Trong qua trinh bien soan kho tranh khoi nhufng thieu sot, rat mong nhan dUpc S[jl gdp y cua bgn dpc gan xa. Chuc cac em thanh cong trong ki thi sap tdi. CAC TAC GlA DE 1 TRl/dNC PnH CHUYEN H6NG PHONG - TRXN DAI NGHTA L6P CHUY^N NGUYiN THl/ONG HIEN - L^P CHUYEN GIA DjNH De thi tuyen sinh vao I6p 10 PTTH chuyen tqi TP.HCIVI nam hoc 2006 - 2007 Cau 1. Thu gon cdc bilu thiJc sau : A = (2V4 + Ve - 2^)(VlO - V2), B = -y/a - 1 Va + 1 Va + 1 Va - 1 a + 1 vdi a > 0, a 7^ 1. Cau 2. Vdi gid tri nao ciia m thi dudrng thang (d) : y = — x + 2m cit 2 3 Parabol (P) : y = — tai hai di§m phan biet ? Cau 3. Giai cac phiTcfng trinh va he phuomg trinh : b) a) Vs - x^ = X - 1 X y X y c) V-x^ + 4x - 2 + V-2x2 + 8x - 5 = V2 + A/3. Cau 4. a) Cho hai so' dirong x, y thoa : x + y = 37xy. Tinh —. y b) Tim cac so nguyen dufong x, y thoa : — + i = i. X y 2 Cau 5, Cho tam gidc ABC c6 ba goc nhon (AB < AC), c6 dudng cao AH. Goi D, E Ian lucrt 1^ trung diem ciia AB va AC. a) Chijfng minh DE la tiep tuyen chung ciia hai du^ng tron ngoai tiep tam gidc DBH vk ECH. ) Goi F la giao diem thuT nhi cua hai dudng tron ngoai tigp tam gidc DBH va ECH. Chilng minh HF di qua trung diem cua DE. ' c) Chiifng minh rkng diibng tr6n ngoai tiep tam gidc ADE di qua diem F. BAIGIAI caul. A = V16 + 4V(V5 - 1?1 (Vio - A/2) = (VI6T4(X^(A^ - 5 A = (V2A/6 + 2V5)(A/10 - V2) = (A^VCVS + 1)^ (VlO - N^) = A/2(A/5 + - V2) = (VlO + V2)(VrO - V2) =8. B = Va-1 >/a + l -= + —= \ ( 2 ^ 2 1 = / I a + l> a - 2Va + 1 + a + 2^3 + 1 a-1 a-1 a + lj 2(a +1) a-1 ^a-l^' a + lj 2(a -1) a + 1 Cau 2. Phuang trinh ho^h dp giao dilm cua (d) (P) \k : o 3x^ + 6x + 8m = 0 3 o 3 2 — X + 2m = — X 2 4 (*) Dilu ki$n de (d) c^t (P) tai 2 diem phan bi|t 1^ (*) c6 2 nghiem phan 3 <=> m < —. 8 bi|t <=> A' = 9 - 24m > 0 Cfiu 3. x^ = X - 1 X - 1 > 0 5 - x^ = (x - if X>1 X = -1 hoac X = 2 X > 1 2x2 - 2x - 4 = 0 x = 2. b) Dieu ki^n : x ^ 0, y 0. Dat u = i; V = —. H? da cho c6 dang : X y Vdi u = 2, ta c6 : - = 2 => x = i. X 2 Vdi V = 1, ta c6 : — = 1 => y = 1. y So vdi dieu ki$n ta nhan x = —; y = 1. 2 3u - 4v = 2 4u - 5v = 3 <=> fu = 2 V = 1 V$y h§ da cho c6 nghi$m (x; y) la c) Khi cdc dieu ki$n xdc dinh thoa ta c6 : V-x^ + 4x - 2 = V2 - (x - 2)2 < V2 V-2x2 + 8x - 5 = V3 - 2(x - 2)2 < ^3 Do do : V-x2 + 4x - 2 + V-2x2 + 8x - 5 < + S Dau "=" xdy ra o x = 2. V^y phifcfng trinh c6 nghiem duy nha't x = 2. Cfiu 4. a) Ta c6 : Dat (1) x + y = at u = }-, u > 0. u^ - 3u + 1 = 0 <=> ^ + l = 3jii (1) u = 3± VS x _ (3 + V5)2 y 4 b) Vi vai tr6 cua x va y la nhu nhau n§n ta c6 the gia suf: x > y. x>0 o —<— => y>2. m . Ill Ta c6 : - + - = -; X y 2 x>y>0 => —<— => — = — + — <— => y<4. X y 2 X y y Do do : 2 < y < 4. Ma y la so' nguyen dUdng {gik thiet) nen y = 3 hoac y = 4. Vdi y = 3 X = 6. Do tinh doi xuEng ta cung c6 x = 3 va y = 6. Vdi y = 4 => X = 4. Cdch khdc : 1 1 — < — => y 2 i-i L<1 2 " X ^ X y 2 o 2(x + y) = xy o 2x - xy + 2y = 0 o x(2-y)-2(2-y) = -4 o (2 - x)(2 - y) = 4 Vi x, y > 0 nen 2 - x < 2, 2 - y < 2. Do d6 ta c6 cdc tru6ng hap sau • 2-x = -l va 2-y = -4 o x = 3 va y = 6 (nhan) • 2- x = -4va 2-y = -l<» x = 6vay = 3 (nhan) • 2-x = -2 va 2-y = -2 o x = 4 va y = 4 (nhan). Vay ta c6 ckc cap so nguyen dUdng (x; y) la : (3; 6), (6; 3), (4; 4). Cliu 5, (Hinh 1) a) Taco: DE//BC nen HDE = BHD. Tam giac vuong ABH c6 HD la trung tuyen nen DH = DB, suy ra BHD = DBH. Do d6 HDE = DBH. Suy ra DE tiep xiic diTdng trbn ngoai tiep tam gidc DBH tai D. Tucfng tir nhu tren, ta c6 DE tiep xiic difcfng tron ngoai tiep tam gidc ECH tai E. Vay DE la tiep tuyen chung cua hai difdng trbn ngoai tiep tam gidc DBH va ECH. b) Ta CO : IDF = IHD va IFD = IDH (= DBF). Do d6 hai tam gidc IDF va IHD dong dang. Suyra: — = — => ID^ = IF.IH (1) _ IH ID Tuong tu ta c6 : lE^ = IF.IH (2) Tif (1) va (2), suy ra ID = IE hay HF qua trung diem I cua DE. c) Vi cac tuf giac BDFH va CEFH npi tiep nen : DFH + DBH = 180°; EFH + ECH = 180°. Trong tam giac ABC c6 : BAC + DBH + ECH = 180°. ^ ~ ^ b) 2x2 ^ 2V3^ -3 = 0 c) 9x^ + 8x2 - 1 = 0. 5x + 3y = -4 Lai CO : DFH + EFH + DFE = 360°. Suy ra : BAC + DFE = 180''. Vay dirdng tron ngoai tiep tam gidc ADE qua F. DE 2 THI TUY^N SINH VAO l6P 10 TAI TP. HCM NAM HOC 2006 - 2007 Cau 1. Giai cac phifOng trinh va he phucfng trinh sau : a) Cau 2. Thu gon cac bilu thiJc sau : ^1-2 Va + 2^ r _ _ . Va - -V3' {yla+2 Va-2j (vdi a > 0 vk a 7t 4). Cau 3. Cho manh dat hinh chO nhat c6 dien tich SeOml Neu tang chieu rpng 2ra va giam chieu dai 6m thi dien tich manh dat khong doi. Tinh chu vi manh dat Iiic ban dau. Cau 4. a) Viet phuong trinh di/&ng thing (d) song song vdi difcmg thing y = 3x + 1 va cit true tung tai diem c6 tung dp bkng 4. b) Ve do thi hkm so y = 3x + 4 v^ y = tren cung mot he true tpa dp. 2 Tim tpa dp cac giac diem cua hai do thi ay bing phep tinh. Cau 5. Cho tam gidc ABC c6 ba g6c nhpn va AB < AC. Dudng tron tam O difdng kinh BC cit cdc canh AB, AC theo thil tu tai E va D. VIH-A/12 __J_. V5-2 2-V3' 8 a) Chu'ng minli AD.AC = AE.AB. b) Goi H la giao diem ciia BD va CE, goi K la giac diem cua AH va B( ChuTng minh AH vuong goc vdi BC. c) TLT A ke cac tiep tuyen AM, AN den dudng tron (O) vdi M, N la ca tiep diem. ChiJng minh ANM = AKN. d) Chu'ng minh ba diem M, H, N thSng hang. BAI GIAI Cau 1. a) = ^ c H-6y = -3 |x = -ll [5x + 3y = -4 [l0x + 6y = -8 [y = 17 Vay he c6 nghiem (x; y) la : (-11; 17). b) 2x-+ 2V3x-3 = 0 A'= 9 ^ >/A^ = 3. Phuong trinh c6 hai nghiem phan biet : x, = ^ ^ • x.> = " ^ 2 ' - 2 c) 9x' + 8x- - 1 =»0. Dat t = X-, (t > 0). Phuong trinh da cho c6 dang : 91" + 8t - 1 = 0 Vi a-b + c= l ti = -1; tv = 9 So vdi dieu kien t > 0 ta nhaii t = - 9 Vdi t=i,tac6x-=- => x = ±- 993 1 1 3' "3- V15-V12 1 V3(V5~2) 1.(2+ V3-) Vay phuong trinh da cho c6 nghiem la : Cau 2. A = V5-2 2-V3 V5-2 (2-V3)(2 + V3) = Vs - 2 - x/3 = -2. B = V^~2 V^ + 2^ Va+2 Va-2 -aVa a - 4 _ (V^-2)2 ^(4a+2f a -4 4a) (Va+2)(Va-2) Va a-4 • Cau 3. Goi chieu rong ciia manh dat luc dau la : x (m), (x > 0). Chieu dai manh dat luc dau la : 360 (m), Chilu rong manh dat sau khi tang 2m la : x + 2 (m). OCA Chilu dai manii dat sau khi giani 6m la : 6 (m). X Tir dieu kien dau bai ta c6 phu'cfng trinh : (x + 2) 360 -6 = 360. Giai phucfng trinh nay ta dMc x, = 10 (nhan) va Xv = -12 (loai). Vay : Chieu rong manh dat liic dAu la : 10 (m). Chi6u dai manh dat luc dau la : = 36 (m) 10 Chu vi manh dat luc dau la : (36 + 10) x 2 = 92 (m). Cau 4. a) Phaong trinh (d) song song vdi ducyng th^ng y = 3x + 1 nen c6 dang y = 3x + b, (dj cat true tung tai diem c6 tung dp bang 4 => b = 4. Vay (d) : y = 3x + 4. b) Bang gia tri : X -4 -2 0 2 4 -8 -2 0 -2 -8 X 0 -2 y / y = 3x + 4 4 -2 4 Phuong trinh hoanh do giao diem ciia 2 -4 -2 / 0 2 4 (P) : y = - — va (d) : y = 3x + 4 la : = 3x + 4 c:> x%6x + 8 = 0 <^ xi = -2 va X2 = -4 Vdi x, = -2 y, = -2 Vdi Xv = -4 =o y, = -8. Vay toa do giao diem ciia (P) va (d) la : (-2; 2) va (-4; 8). Cau 5. (Hinh 2) a) Ta CO : ABD = AEC (cung chdn cung DE). Suy ra hai tam giac vuong ABD va ACE dong dang. AB AD Do do AC AE AD.AC = AE.AB. LO b) Ta CO ; BEC = BDC = 90" (goc aoi tiep chan niifa dudng tron). Suy ra : BD 1 AC va CE 1 AB, Hay BD va CE la hai dudng cao ciia tam giac ABC. Ma H la giao diem ciia BD va CE nen H la trirc tam ciia tam giac ABC. Suy ra AH 1 BC. 1 c) Ta CO : ANM = - MON (goc tao bdi 2 Hinh 2 tia tiep tuyen va day vdi goc d tam ciing chan MN). Ma OA la tia phan giac ciia goc MON (tinh chat hai tiep tuyen cat 1 nhau) nen AON = - MON, suy ra : AMN = AON 2 (1) Ta lai c6 : ANO = AMO = 90°, nen tiif giac ANKO noi tiep (tuf giac c6 hai dinh ke cung nhin mot canh dudi hai goc bang nhau), suy ra : AKN = AON (2) Tif (1) va (2), suy ra ; ANM = AKN. d) Ta CO : ANE = ABN (goc tao bdi tia tiep tuyen va day vdi goc npi tiep ciing chan cung NE). Va NAB la goc chung ciia hai tam giac ANB va ANE. AN AE Suy ra AANE ^ AABN (g-g) Lai CO : AEH = AKB = 90^ Suy ra AAEH ^r, AAKB (g-g) Tif(3)va(4) => AB AN AE AH AK AB AN- = AH.AK. AN AH AN- = AE.AB AE.AB = AH.AK 3) (4) Xet AANH va AANK c6 : NAK chung va ^ ^ . AK AN Suy ra AANH or, AAKN (c-g-c) => ANH = AKN. Tren ciing mot niia mat phang c6 bd chufa tia NA, ta c6 : ANH = ANM- Suy ra hai tia NH va NM trung nhau. Vay 3 diem M, H, N thang hang. 1] DE 3 TRUdNG PTTH CHUYEN LE HONG PHONG, TPHCM De thi tuyen sinh vdo I6p 10 chuyen Todn nam hoc 2005 - 2006 Cau 1. a) Dinh m de hai phifo'ng trinh : x' + x + in = 0 va x" + mx +1 = 0 CO it nhat mot nghiem chung. b) Cho a, b, c la do dai ba canh ciia mot tam giac. ChiJng minh r&ng phirong trinh : b'"x" + (b'^ + c" - a")x + c" = 0 v6 nghiem. Cau 2. Giai phifong trinh va he phuong trinh : -y^ =3{x^y) 13x a) \) — + = 6. i X + y = -1 3x- - 5x + 2 3x^ + x + 2 Cau 3. a) Chiing niinh 2(a' + b"*) > ab' + a"b + 2a'b" vdi mpi a, b. ^ b) ChiJng minh Va^ - b" + V2ab - b^ > a, vai a > b > 0. Cau 4. Tim cac so' nguyen diTOng c6 2 chuf so', biet so' do la boi ciia tich 2 chCf so' ciia chinh so do. Cau 5. Cho hinh binh hanh ABCD c6 goc A nhon, AB < AD. Tia phan giac ciia goc BAD cat BC tai M va cat DC tai N. Goi K la tam ciia dudng tron ngoai tiep tam giac MCN. a) Chufng minh DN = BC va CK 1 MN. b) ChuTng minh rang BKCD la mot tii giac noi tiep. 2au 6. Cho tam giac ABC c6 A = 2B. Chiifng minh rang : BC- = AC- + AB.AC. BAI GIAI • :au 1. a) Goi Xii la nghiem chung ciia hai phuong trinh ta c6 : xl + x,, + m = 0 (*); XQ + mx,, + 1 = 0. Tirdo (xg+Xo+m)-(x5 + mx„ + 1) = 0 ^ (1 - m)(x„ - 1) = 0. Voi m = 1, ca hai phUong trinh deu c6 dang : x" + x + 1 = 0 (vo nghiem). Vd-i x„ = 1 tCr (*) CO m = -2, khi do phuong trinh x" + x + m = 0 c6 hai nghiem la 1 va -2. PhUong trinh x" + mx + 1 = 0 c6 nghiem kep la 1. Vay m = -2 thi hai phuong trinh da cho c6 it nhat mgt nghiem chung. b) b'-x- + (b- + c'-^ - a^)x + c" = 0 Ja CO : A = ih- + - a'f - Ah'c' = (b" + c'' - a")" - (2bc)" = [(b + c)- - a-ll(b - c)- - a-| = (b + c + a)(b + c - a)(b - c + a)(b - c - a). Do a, b, c la do dai ba canh cua tam giac nen A < 0. Vay phaong trinh da cho v6 nghiem. 'Cau 2. x^-y^=3(x-y) ^ |(x - yKx^ + xy + y^ - 3) = 0 x + y = -i .[x + y = -i x-y = 0 a) <=> Giai he (I) ta c6 nghiem (x; y) = X + y = -1 x^ + xy + y^ - 3 = 0 X + y = -1 r 1 _i) 2'"2j- (I) (II) Xet he (II) tir phirang trinh cuoi x + y = -1 => y = -1 - x. Thay vao phirong trinh dau ta difoc x" + x - 2 = 0. Suy ra he (II) c6 hai nghiem (x; y) = (1;-2), (-2; 1). ( 1 1' Vay he da cho c6 3 nghiem la - - ; - - V 2 2, b) Vi X = 0 khong la nghiem ciia phuong trinh nen ta chia ve trai ciia phuang trinh cho x ta difcfc : 2 13 2 13 , (1; -2), (-2; 1). 3x^ - 5x + 2 3x^ + X + 2 = 6 2 2 3x + 5 3x + - + l x X = 6 (*i Dat y = 3x + - - 5 (*) dugc viet lai : x - + -1^ = 6 o 2y''* + 7y - 4 = 0 y y + 6 Giai phiicfng trinh nay ta diTOc : yi = - ; ya = -4- Vdi yi = - c=. 3x + - - 5 = i C5. 6x' - llx + 4 = 0 => x, = ^ , - - 2 X 2 3 Vdi y2 = -4 3x + - - 5 = -4 c:> 3x- - x + 2 = 0 (v6 nghiem). X 4 1 Vay phifcfng trinh da cho c6 hai nghiem x, = — , Xv = — . 3 ' 2 Cau 3. a) Ta CO <::> 2(a' + > ab-' + a''b + 2a-b 4(a' + b') > 2ab'' + 2a'b + 4a"b' (b' - 2ab'^ + a'b-) + (a'' - 2a-'b + a-b") + (Sa" + 3b'' - 6a-b^) > 0 (b- - ab)- + (a^ - ab)^ + 3(a^ - b^)' > 0 (diing) Vay bat dang thufc da cho diing. b) Vdi a > b > 0 thi Va" - b^ + V2ab - b^ > a ta' - b-) + (2ab - b") + 2yl(a^ -b^)(2ab-b^) > <^ 2b(a - b) + 2V(a^ -b^)(2ab-b-) > 0 (dung) Vay bat dang thiJc da cho dung. Cau 4. Gpi so can tim la ab (a, b khac 0). TCf gia thiet c6 ab = m .ab. Suy ra : 10a + b = mab, hay b = a(mb - 10), suy ra b chia het cho a. Dat b = na (n < 9). Tir na = a(mna - 10) c6 n = mna - 10 o n(ma - 1) = 10 nen chia het cho n => n e |1, 2, 5). - Vdi n = 1 thi ma - 1 = 10 ma = 11, suy ra a = b = 1. - Vdi n = 2 thi ma - 1 = 5 r=> ma = 6 => a = 1, 2, 3, tUcfng iifng c6 b = 2, 4, 6 - Vdi n = 5 thi naa - 1 = 2 <=> ma = 3 => a = 1, tu'Ong iifng c6 b = 5. Thii lai ta c6 cac so can tim la : 11, 12, 15, 24, 36. Cau 5. (Hinh 3) a) Ta CO BAN = DNA, ma BAN = DAN (gia thiet), suy ra DAN = DNA hay ADNA can tai D DN = AD = BC. Nhan tha'y ACMN can tai C nen CM = CN, ket hop vdi KM = KN ==> CK 1 MN. b) Xet hai tam giac KBC va KDN c6 : Hirili 3 BC = DN, KC = KN, KCB = KND (= KMC) AKBC = AKDN, suy ra KBC = KDC hay tu^ac BKCD la tiJ giac noi tiep (dpcm). 14 C&u 6. (Hinh 4) Tren tia do'i ciia tia AC chon Bi sao cho ABi = AB luc do BKC = 2B'B7C, ma BAC = 2ABC B, TCf do : BBjC = ABC, suy ra ABB,C oo AABC. BjC BC BC AC Hay BC- = AC(AB + AC). DE 4 Hinh 4 TRl/CfNG PTTH CHUYEN LE HONG PHONG, TPHCM De thi tuyen sinh vdo I6p 10 (Ban A, B) nam hoc 2005 - 2006 Cau 1. Cho phirong trinh (c6 an so la x) : 4x- + 2(3 - 2m)x + m" - 3m +2 = 0. a) ChuTng to r&ng phirong trinh tren luon c6 nghiem vdi moi gia tri ciia tham so m. b) Tim m de c6 tich ciia hai nghiem dat gia tri nho nhat. Cau 2. Giai cac phuong trinh va he phuong trinh : a) x^ + y^ = 2(xy + 2) X + y = 6 b) x' + 25x^ (X + bf = 11. Cau 3. a) Cho a > c, b > c, c > 0. Chiirng minh : Vc(a - c) + Vc(b - c) < 4ah. b) Cho a > 0, b > 0. Chiirng minh : 2Vab < VVab. Va + Vb Cau 4. Tim so chinh phuong c6 4 chC so biet rang khi tang them m6i chijf so mot don vi thi so mdi dugc tao thanh cung la mot so chinh phuong. Cau 5. Cho tam giac ABC c6 ba goc nhon noi tiep trong dudng tron (O; R), so do goc C b^ng 45". Dudng tron dudng kinh AB c^t cac canh AC va BC Ian \mt tai M va N. AB a) Chiifng minh MN vuong goc vdi OC. b) ChiJng minh MN = -j=r. Cau 6. Cho tam giac ABC cd ba goc nhon noi tiep dudng tron (0; R). Diem M di dong tren cung'nho BC. Tii M ke cac difdng th^ng MH, MK Ian lu'ot vuong goc vdi AB, AC (H thuoc dudng thang AB, K thuoc dirdng thang AC). a) Chirng minh hai tam giac MBC va MHK dong dang vdi nhau. b) Tim vi tri ciia M de do dai doan HK Idn nhat. 15 BAI GIAI Cau 1. a) 4x- + 2(3 - 2m)x + m'' - 3in + 2 = 0 Ta CO : A' = (3 - 2m)'' - 4(m- - 3m + 2) = 1 > 0. Suy ra phuong trinh da cho c6 hai nghiem phan biet Xj, X2 voi moi gia tri ciia tham so m. b) Theo dinh li Viet : x,Xv = m^ - 3m + 2 1 3^ m — 2 2 — > - — 16 " ~16 Dau "=" xay ra <:5> m = 2 Vay gia tri nho nhat ciia X|.xv la Cau 2. a) . 16 dat duoc khi m 2 x^ + y'^ = 2(xy + 2) x^ + y^ = 4xy + 4 X + y = 6 <=> X + y = 6 Giai ra ta dUdc he c6 hai nghiem (x; y) la : (4; 2), (2; 4). b) Dieu kien x ^ -5. Phirong trinh da cho tuong dirong vdi : X + y = 6 x.y = 8 X + = 11 <=> (x + 5f X - 5x X + 5 \ 10 X + 5 = 11 X + 5 , 10 X + 5 = 11. Dat y = (1). Phuong trinh tren trcf thanh : y" + lOy - 11 = 0 X + 5 Giai ra ta dugc y, = 1, y;, = -11. - Vdi yv = -11, thay vao (1) thay phiTcfng trinh an x v6 nghiem. - Vdi yi = 1, tijf (1) ta c6 phi/dng trinh - x + 5 = 0. Giai ra ta duoc : X, = Thiif lai ta thay x, = 1 + V2I 2 1 + V2I X^; = 1 - V2T 2 1-V2I la hai nghiem ciia phiTcfng triiih da cho. Cau 3. a) Ap dung bat d^ng thiifc Cosi cho hai so duang, ta c6 : 16 c(a - c) ab fc(b - c) _ 17 \b ' V b c ''b-c^ [ a J a V b J 1 ^c a-c^ 1 (c b-c] < — — + + — — + 2 Kb a ; 2 la b ; = 1 => dpcm. Dau "=" xay ra co be b-c = a. b) Ap dung bat dSng thiire Cosi cho hai so duong, ta c6 : 2vVa-Vb < Va + Vb => Suy ra ^Vab ^ va + Vb < (Va + ^Ib}. VV^.Vb Va + Vb Cau 4. Goi so chinh phuong c6 bon chuT so can tim 1^ abed (a ^ 0). Dat n- = abed (n E N'). Ta eo n" < 10000 nen n < 100. Khi tang m6i chiJ so ciia so' abed len mot don vi thi dirac so : (a + l)(b + l)(c + l)(d + l). Theo gia thiet (a + l)(b + IKc + l)(d + 1) = m" (m E N"). Tirong tir nhtf tren c6 m < 100. Suy ra 2 < m + n < 200. Xet : m- - ir = (a + l)(b + l)(c + l)(d + 1) - abed = 1111. Hay (m - n)(m + n) = 1111. VI nil = 1.1111 = 11.101, nen chi xay ra m - n = 11 va m + n = 101. suy ra m = 56, n = 45. Vay abed = 45- = 2025. ThiJr lai ta thay so can tim la 2025. Cau 5. (Hiiih 5) a) Ke tiep tuyen Cx voi dudng tron (O), khi do BAG = BCx (cung ehan cung BmC); Mat khac BAG = MNC (cung bij vdi MNB) Suy ra MNC = BCx, tiT do MN // Cx Mat khac Cx 1 OC nen MN 1 OC. b) Ta thay ACMN c/^ ACBA. MN CN 1 AB AC V2 (vi AACN can tai N) MN = AB V2' THLH/IENTI.\!HBINHTHUAN 17 Cau 6. (Hlnh 6) a) Bon diem A, H, M, K cung nkm tren mot ducfng tron dudng kinh AM. Ta CO : MBC = MAC = MHK, MCB = MAB = MKH. Suy ra tam giac MBC dong dang vdi tam giac MHK. b) Theo tren AMBC oo AMHK, suy ra BC MB HK MH ma MB > MH BC HK > 1 HK < BC. Dang thilc xay ra <=> H = B, luc do ABM = 90'' <=> AM la dudng kinh ciia dirdng tron (O). Do do khi M la diem dol xufng ciia A qua O thi do dai HK Idn nhat. DE 5 TRI/ONG PTTH CHIYEN LE HONG PHONG, TPHCM De thi tuyen sinh vao I6p 10 chuyen Toan nam hoc 2004 - 2005 Cau 1. Giai he : 2x - y X + y 1 1 2x - y X + y = -1 = 0 Cau 2. Cho X > 0 thoa : x" + A- = 7. Tinh x"' + — Cau 3, Giai phiforng trinh : 3x = A/3X + 1 - 1. V3x + 10 Cau 4. a) Tim gia tri nho nhat ciia P = Sx" + 9y- - 12xy + 24x - 48y + 82. X + y + z = 3 b) Tim cac so nguyen x, y, z thoa he : x^ + y^ + z^ = 3 Cau 5. Cho tam giac ABC c6 3 goc nhon npi tiep trong dudng tron tam 0 (AB < AC). Ve dirdng tron tam I qua 2 diem A, C cat doan AB, BC Ian lirot tai M, N. Ve dirdng tron tam J qua 3 diem B, M, N cit dirdng tron (O) tai dii'm H (khac B). a) Chdng minh : OB vuong goc vdi MN. 18 b) ChiJng minh : tuT giac lOBJ la hhih binh hanh. c) Chijrng minh : BH vuong goc vdi IH. Cau 6. Cho hinh binh hanh ABCD. Qua mot dii'm S trong hinh binh hanh ABCD ke dudng thing song song vdi AB Ian lirpt cat AD, BC tai M, P va cung qua S ke dirdng thang song song vdi AD Mn lirpt cat AB, CD tai N, Q. ChiJng minh 3 dirdng thing AS, BQ, DP ddng quy. - BAI GIAI 3 6 Cau 1. 2x - y X + y 1 1 Dat a = 2x - y X + y 1 2x - y = -1 = 0 b = Dieu kien : 2x - y ^ 0; x + y ^0 1 X + y , , . f3a-6b = -l H? da cho dirac viet lai : <^ , „ o ^ • • a-b = 0 1 a = - 3 C5 a = b a = b = Giai he Cau 2. Ta c6 : 2x - y 1 ta dugc nghiem (ji; y) = (2; 1). Suy ra : X + y 3 X + — X 1 ^ X + — + — + X X 1^ X + — X 4 1 X + — X ( 1 + X De dang tlnh dirorc cac tong sau : 1 X + — X 1 X + — X = X- + 2 + = x^ + 1 = 9 Vay : x^+. 1 ( 1 + 3 X + - x^ / 1 ^ V / X + - x^^ V v X + - = 3 (vi X > 0) X 3 1 ( 1^ 3 ' 1^ X + — - 3 X + — I x; ^ x; = 18 4 1 ^ X + — ( 1 3 V X = 3.47-18 = 123. Cau 3. Dieu kien : x > - 3 i Phuong trinh da cho diroc viet dudi dang : 19 [...]... 2003^ - 4xy nen xy tang (giam) k h i I x - y | giam (tang) Theo gia t h i e t x, y la so nguyen duong, t a n h a n tha'y : - Gia t r i nho n h a t ciia P la 2003'' - 6007 .100 1 .100 2 dat duqrc k h i (x; y) bang (100 1; 100 2) hoac (100 2; 100 1) Gia t r i Idn n h a t cua P la 2003'* - 6007.1.2002 dat difoc k h i (x; y) bang (1; 2002) hoac (2002; 1) ^ C a u 4 (Hinh 30) 1 Do tuf giac A B H E n o i tiep nen E... AMSTERDAM D e thi t u y e n sinh v d o Idp 10 c h u y e n T o d n , Tin n a m h o c 2 0 0 5 - 2 0 0 6 (X 1 „ ^, , , , Cho he phu-olig t r i n h • \ + y)'' +13 = 6xV^ + m , xylx" + y") = m P = X CO IVdi v = -2 1 2 3 - + — + - = 6 Xet bieu thufc x y z , thay vao (1) t a diTcfc : 2v'' + I T v " - 100 = 0 Chufng m i n h r a n g t r o n g 2005 dUong t h a n g do c6 i t nhat 502 dudng dong quy V = ±2 thi u= 1... Cau ^ c+a " 2 OA = OB (= (gia thiet) Hinh 19 AOAB can tai O R) ID = IE (= r) DN 1 ON AOAB ^ AIDE Do do => ND la tie'p tuye'n cua dudng tron (0) AIDE can tai I ID" " DE ^ OA _ AB R ^ 3DE r ^ DE R = 3r.i b) N la trying diem cung BC, E la trung diem day BC (gia thiet) DE Hinh 18 T R U O N G PTTH CHUYEN LE HONG PHONG, TPHCM => O, N, E thang hang De thi tuyen sinh v a o Idp 10 c h u y e n Toan n a m h o... phan so co tuf so' la t i c h ciia ca n a m so' t r e n ( m l u so' la 1) 1 + 5 + 10 + 10 + 5 + 1 = 32 Cac p h a n so t r e n dugc chia t h a n h ttfng cap nghich dao ciia nhau va 32 khac 1 n e n so' phan so Ion h o n 1 la — = 16 2 D E 20 DAI HOC OUdc GIA HA NOI - TRlTCfNC DAI HOC KHOA HOC TlJ NHIEN De thi tuyen sinh vao I6p 10 chuyen Khoa hpc TL/ nhien nam X 2005 - 2006 + y + xy = 3 => P = 1 V6-i S =... i n h R(b + c) > aVbc D a n g thiJc xay r a k h i va chi k h i t a m giac A B C vuong can t a i A TCr do cac goc ciia h i n h t h a n g A B C D R(b + c) = aVbc t h i t a m giac ABC vuong can t a i A BAI GIAI C a u 1 DE 7 a) A = T R U O N G PTTH C H U Y E N L E H O N G P H O N G , T P H C M 3V2 - 2V3 V2-V3 1 V2-V3 V3V2 + 2V3 ^ 76(73 + 72 De thi tuyen sinh v d o I6p 10 c h u y e n Todn n a m h o c... 5 ° = 1 0 5 ° -X x'-^ - X x^ - 3x + m = 0 Mm TRl/CfNG PTTH C H L Y E N LE HONG PHONG, TPHCM De thi t u y e n sinh v a o Idp 10 c h u y e n Todn n a m h o c 2002 - 2003 C a u 1, T i m gia t r i cua m de phifang t r i n h sau c6 n g h i ^ m va t i n h n g h i e m ay theo m : C a u 3 cac 1 x^ 48 ^Jx — + — = 10 A\ xy- - 2y + 3x^ = c) y^ + x'^y + 2x = C a u 4 A = x'" + x'' + 1 C a u 2 2 X • = - 4m 3 -... t r u n g d i e m O F c:> A la = DE 17 TRUONG PTTH C H U VAN AN VA TRUCJNG PTTH HA NOI - AMSTERDAM De thi tuyen sinh vdo Idp 10 chuyen Khoa hoc Tii nhien nam 2003 - 2004 4 4 + - > - Vk 5 5 Dau "=" xay r a = 60" AB =^ OA = AC = A T = — 2 Vsk^ - 2k + 1 CAT = k = i 5 - 1 a u 1 o u U ' Cho bieu thiJc : D P = x^-V^ = x + vx + 1 1 Rut gpn P V a y khoang each tir O den dudng t h a n g (d) Idn n h a... X = 3 Vay gia t r i nho n h a t ciia dien tich tani giac A M N la 2 x^ + 3x'* + 9x3 + 4 3 x 2 + 129x - 5 = 0 Ta thay x = 0 k h o n g la nghiem ciia (*) v i : - 5 D E 10 TRl/dNG PTTH CHUYEN LE HONG PHONG, TPHCM De thi tuyen sinh vdo I6p 10 c h u y e n Todn n a m h o c 2001 - 2002 C a u 1 T i m t a t ca cac so' nguyen x thoa : x"* + 8 = VVSx + 1 C a u 2 Cho n la so nguyen duong Chufng m i n h ta luon... - 4 (loai), tv = 2 x = ±2 Thiir l a i ta t h a y x = - 2 va x = 2 la hai nghiem ciia phiTcfng t r i n h Vay phuong t r i n h da cho c6 hai n g h i e m x = - 2 va x = 2 x^ + y^ - xy^ = 1 DAI HOC OUOC CIA HA N O ' - TRUCfNG DAI HOC KHOA HOC TlJ NHIEN (1) 4x'* + y-^ = 4x + y C a u 2 D E 19 (2) (2) CO dang : 4x'' + y"* = (4x + y)(x'' + y'' - xy") De thi tuyen sinh v d o I6p 10 c h u y e n Todn, Tin n... ( B a t d a n g thiJc B C S ) Dau,"="xayra Chufng m i n h tUo'ng t i f t r e n t a c u n g c6 : d ' a = b ' m + c ' n - a m n 2 B a i toan p h u : ChiJng m i n h r a n g Cdc/i 3 : d'a = b ' m + c ' n - a m n A ' B D =- A D E AD AB AE AD A D ' = AB.AE D o do A D ' < AB.AC 43 PhifOng t r i n h c6 n g h i e m TRl/dNG PTTH CHUYEN LE H 6 N G PHONG, T P H C M De thi tuyen sinh vdo Idp 10 (Ban A, B) n . van de trong cac de thi tuyen sinh vao Idp 10 trong nhOng nam gan day, Chung toi bien sogn cuon: "DE THI TUYEN SINH VAO LClP 10 MON TOAN". Sach gom de thi cua cac trudng chuyen,. TRAN KY TRANH (Tuyen chgn vd gidi thi$ u) DE THI TUYEN SINH VAO LdP 10 MON TOAN OE THI CUA CAC TRl/CfNG CHUYEN, CHQN TREN TOAN QUOC (Tdi ban Idn thiiC ttC, c6 8v£a chUa bo sung). 0). TCf gia thiet c6 ab = m .ab. Suy ra : 10a + b = mab, hay b = a(mb - 10) , suy ra b chia het cho a. Dat b = na (n < 9). Tir na = a(mna - 10) c6 n = mna - 10 o n(ma - 1) = 10 nen chia