D B1 OB^ CH ± OBC HO =90 "
1. Phtfcfng trinh chifa can dang quen thuoc
2. He phtfcJug trinh hai an, quan sdt he phi/dng trinh bién doi roi dSt u = x + 1, V = y + 1 de dang tim den IcJi giaị 1, V = y + 1 de dang tim den IcJi giaị
Cfiu 2. 5x^ + 2xy + y^ - 4x - 40 = 0 o (4x^ - 4x + 1) + (x^ + 2xy + y^) = 41 « ( 2 x - l) H( x + y)^ = 41 2 x - 1 l e . D o do, taco (2x - 1)^ = 25 2x - 1 = ±5 X + y = ±4 [ ( X + y)^ =^ 16 X = 3; y = 1 J i ^ x = 3; y. - 7 \ X = - 2 ; y = 6 X - - 2 ; y = - 2
Vay c6 bon cSp so nguyen (x; y) can tim la (3; 1); (3; - 7 ) ; (-2; 6); (-2; - 2 ) N h § n xet: Day \h hhi loin ve phifdng trinh nghiem nguydn, giai diTdc b^ng phiTdngphap "Tong"
Cfli
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Cfiu 3. Ta c6 ON // AB (gt) A N = BD=> A C I = BCD
X6t ACAI v^ ACDB cd CAT = CDB (Hai g6c npi tiép cilng ch^n cung BC) va A C I = BCD . Do 66 ACAI ^ ACDB (g.g) ^ BC ~ B D • IC BC Vay l A BD
ChiJng minh tiTdng tif c6 IC ^ AC IB A D
(1) (2) (2)
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X6t AMBC va AMDB c6: BMC (chung), MBC bdi tia tiep tuyen va day cung)
BC M B ; Do do AMBC ^ A M D B (g.g) = Do do AMBC ^ A M D B (g.g) =
TiTdng tir AMAC <^ AMDA (g.g)
BD M D AC M A AC M A
= M D B (He qua g6c tao
(3) (4);-. • (4);-. • AD M D . ^ . . . ^ . - Ma M A = M B ' ' ' ' • j ' ^ ' ' " TCrd), (2), (3), (4) c6 I A = IB b) Ve OH 1 d tai H. Ta c6 H c6 dinh
Goi K giao diem ciia OH va A B .
MA, MB la cac tiep tuyen cua diTcfng tr6n (O) (gt) M A = M B , MO la tia phan giac ciia goc A M B => AMAB can tai M
Do d6 M , I , O ihing hang, MO 1 ab
Xet AOIK va AOHM c6: l O K (chung), OIK = O H M (= 90") 01 OK
OỊOM - OK.OH D o d o A O I R ' ^ AOHM (g.g) = D o d o A O I R ' ^ AOHM (g.g) =
^ ^ OH O M
AOAM vuong tai A . A I Ih diTdng cao => OỊOM = OÂ = Do do OK.OH = => OK = , khong doị Nen K co dinh.
OH ^
Ta CO OIK = 90", OK co dinh. Vay I di dong tren diTdng tr6n co djnh diTdng kinh OK.
NhSn xet: Day la bai todn hinh hoc khi quen thuoc doi vdi hoc sinh gioi toan. Phat hipn diem co djnh K (giao diem cua OH va A B ) la chia kh6a ciia bai loan.
Luven giai ai truac ki thi vjoiapjObamign BSc, Trung. Nammgn Tc^n rJoivJ™:[!!I
Cfiu 4. Ta c6 ra b . b £ ^ I bj > 1 + 3,
be + 1 — + 1 ca tab J (1)
Dat - = X , - - y , - - z thi X, ỵ z > 0; xyz = 1. b c a
y + 1
(1) o ^(xy + yz + zx)(x + y + z) > 1 + 3
M a x y z = l . ^
Suyra V(x + y)(y + z)(z + x) + 1 > 1 + ^(x + y)(y + z)(z + x)
£ . 1 (2)
(2) trd thanh Vf^ + 1 > 1 + t o + t > 1 + 2t + o t(t - 2)(t + 1) > 0
(BDT dung)
Dáu "=" xay ra khi va chi khi a = b = c.
k Nh§n xet: Day la bai loan bat d^ng thufc hay va kh6, cic ban dpc hay tim them cac each giai khac cho bai toan naỵ T'*' ^
CSu 5. Gpi A, B la hai diem thupc canh cua da gidc sac cho A, B chia bicn da giac thanh hai dir5ng gap khuc c6 dp d^i b^ng nhau va b^ng N ;
Gpi O la trung diem cua AB. Ve hinh tron ) tarn Ọbdn kinh R = - .
4
Gia sijf ton tai mot diem M thupc canh da giac
f 1 ^
' va M nim ngoai hinh tron O;— . Khi d6 MA + MB < - (do dai diTdng gap khiic chuTa diem M). Gpi N diem
2
doi xtfng vdi M qua Ọ Ta c6; MA + MB - MA + AN > MN > 2R = - (2) (1) va (2) mau thuan nhaụ Nhif vay hinh tr6n nay chiJa da gi^c da chọ
Nhan xet: Day la bai todn ve hinh hoc to hpp, mot dang toan kh6. CAc bai
loin cau 4, cau 5 nh^m giiip tim kiem hpc sinh thifc sir gioi loan.
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D £ S6 5 6
Di THI TUY^'N SINH VAO LOP 10 CHUY§N,
TRl/ClNG DAI HOC SI/ PHAM TP. H6 CHI MINH ' NAM HOC 2011 - 2012
Bai 1: (2 diem) Cho phU"dng trinh : mx^ - 2(m - 2)x + m - 3 = 0 (x la an so) a) Xac dinh m dc phúPng u-inh c6 nghipm k^p. Tim nghiem d6.
b) Xac dinh m de phiTPng trinh c6 hai nghỉm phan bi^t Xi, X2 thoa man : xf +x^= 1
Bai 2 : (2 diem) !, A
a) Ve do thi (P) cua ham so y = -x^ va diTcfng thing (D): y = 2x - 3 tren cilng mot he true tpa dp.
b) Tim tpa dp cac giao diem M, N cua (P) (D) d cau tren bKng phdp tinh.
c) Gpi Q va P Ian lifdt la hinh chieu vuong g6c cua M v^ N tren true hoanh. Tinh dien tich ciia tuf gidc MNPQ. ^
Bai 3: (2 diem) a) Chijfng minh r^ng:
3Vx 'V X Vx + 4x + 4>/x j N / X + 2 1 vdi X > 0, X ?t 1.