D thing hang.
2) ChuTng minh MN la tiép tuyen cua duTctng tr6n (I) c
HU(5NG D A N GlAl Cfiul. Ta CO x^ = (yle^^yjTTs - V2 + V2 + N/3 V2 - V2 + 73 - ^2 + V2 + %/3 = 3 2 - V2W3) + 2 + V2 +73 - 273^2^ - (2 + 73) = 6- 372 + 73 + 2 + yj2 + S - 27372 - yf3 = 8 - 2V2 + 73 - 27372 - 73 ''!<, ••••• ':' Do do (x^ - 8)^ = (-272 + yf3 - 27372 - 73 ) 177
Luygn giai JB truoc ki thi vao lop ID ba mien-Bac. Tmng; warn mon rean _ wfluyen ucfc lan = 4(2 + V3) + 12(2 - V s ) + 8N/^V22 - 3
= 8 + 4V3 + 24 - 1273 + 873 = 32 Nen x"* - 16x^ + 64 = 32 « x" - 16x^ + 32 = 0 Nen x"* - 16x^ + 64 = 32 « x" - 16x^ + 32 = 0
Vay X = V6-^3V2 + ^ - V2 + ^2 + 73 Ih mgl nghiem cua phifdng tnnh x ' - 1 6 x ^ + 32 = 0
NhSn x e t : K h i c6 diTdc x^ = %-2^2 + S - iS^l - S ta tiep tuc (x^ - 8)^ = (-2V2 + 73 - 273V2 - N ^ ) = . . . = 32 la mot "quyet dinh" sdng suot, chung ta mm diTdc dieu n^y v i d dau bai c6 "x"* - 16x^ + 32 = 0"!
2y(y + l)(x + 1) + yx = 6
2x(xy + X + y + 1) + xy = - 6 (1)
2y(xy + X + y + 1) + xy = 6 (2) Tfif (1) (2) ta c6 2x(xy + x + y + l ) + xy + 2y(xy + x + y + l ) + xy = 0
<=> x^y + x(x + y + 1) = xy + xy^ + y(x + y + 1) = 0
o xy(x + y + 1) + x(x + y + 1) + y(x + y + 1) = 0 X + y + 1 = 0 X + y + 1 = 0 xy + X + y = 0 ' o (x + y + l ) ( x y + X + y) = 0 o X 6 t x + y + l = 0 o y = - x - l y = - X - 1, ket hdp (1) c6 2\\-\ 1) + x ( - x - 1) = - 6 o 2 x ' + 3x^ + X - 6 = 0 o 2x' - 2x^ + 5x^ - 5x + 6x - 6 = 0 o (x - l ) ( 2 x ' + 5x + 6) = 0 o (X - 1) X + — 5^ 4j 23 8 = 0 o x = 1 X = 1 thi y = - X - 1 = - 2 • X e t x y + x + y = 0 ; xy + X + y = 0, ket hdp (1) c6 2x - X - y = - 6 o X = y - 6
X = y - 6, ket hdp (2) c6 2y[y(y - 6) + y - 6 + y + l ] + (y 6)y = 6 o 2y(y^ - 4y - 5) + y^ - 6y = 6 o 2y^ - 7y^ - 16y - 6 = 0
o 2y^ + y^ - 8ý - 4y - 12y - 6 = 0 o (2y + l ) ( y ' - 4y - 6) = 0
o ( 2 y + l ) [ ( y - 2 ) ^ - 10] = 0 o •2y + 1 = 0 (y - 2)2 - 10 = 0 1 " = -2 y = 7lO+ 2 y = -7lO + 2 1 TQ y
Cty TNHH MTV [)VVH Klvim Vif-t
I 2 2 V = — thi X = y - 6 = - 6 — ^ 2 2 y = 7l0 + 2 t h i x = y - 6 = 7 l 0- 4 y = -710 + 2 thi X = y - 6 = -7l0 - 4 / 1 I ^
Thur l a i : Cac cap so (x; y) la ( 1 ; - 2 ) ; - 6 - ; - - ; ( 7 1 0-4, 7 1 0 + 2 ) ;
V 2 2y )
(-7lO - 4;-710 + 2) deu la nghiem cua he phi/dng tnnh da cho Vay nghiem (x; y) ciia hp phiTdng tnnh da cho la: ( 1 ; - 2 ) ;
(710 - 4;7r0 + 2); (-7l0 - 4;-7lO + 2)
Nh0n x e t : Tir viec nhan ra (1) va (2) cho ta (x + y + l ) ( x y + x + y) = 0, giup chung ta c6 nhanh ch6ng Idi giai cija b^i todn.
Cfiu 3. Chia tam gidc deu M N P c6 canh bkng 2cm n^y th^nh 4 tam g i i c deu nho, canh c6 do d^i b^ng 1cm (Xem hinh ve ben)
Neu CO 2 diem trong n diem lay thupc cac canh hoac d phia trong tam gidc deu nho thi khoang each giiya hai diem nay be hdn hoSc bkng 1 (Khong thoa yeu cau dau bai).
Do vay m o i tam gidc deu nho chi c6 nhieu nhát la 1 diem trong so n diem laỵ Nhir vay n Idn nhat chi c6 the la 4.
Ta CO néu c6 3 diem n^m d cac dlnh cua tam gidc deu M N P va 1 diem n^m ci tam cua tam giac deu MP. 4 diem nay c6 khoang each giffa hai diem tiiy y l<3n hdn I cm.
Vay n Idn nhát thoa man dieu kipn da cho la 4.
Nhfin x e t : D a y la bai t o i n van dung nguyen t^c Di-rich-le tir de toan ch^c h^n cac ban cung nghl den chia tam giac deu M N P thanh 4 tam giac nho, canh 1cm. Giup c6 IcJi giai ciia bai toan.
Cfiu 4. Gpi 10 so nguyen dirdng lien tiep la n, n + 1, n -1- 2 n -1- 9 (n e N*). Gia sur ton tai hai so trong 10 so nay c6 iTdc chung Idn hcJn 9 la d (d e N , d > 9) va hai so do la n + a, n + b (a, b e N ; 9 > a > b)
Do do n + a i d va n- I -b : d (n + a) - (n + b)! d => a - b i d. V i 0 < a - b < 9. Nen d < 9
t u y g n giai SS trUOc kl IK vSo Bp 10 ba mien BaCTrorTgrNam mOn^ToSn _ Nguyen uuc lan
VSy trong 10 so nguyen dtfcJng lien tiep khong ton tai hai so c6 vCdc chung
Idnh0n9.
Nhfin xet: ban doc hay timldi giai khdc cho biitodnn^y! i,
Cfiu 5. 1. AE, AF m cAc tiep tuyen ciia A»
di/dng tr6n (I)
=> AE = AF, AI Ih tia phan gidc cua
g6c EAF " ' AAEF can tai c6 AI 1^ diTdng phan giic. Do d6 AI la dtfdng cao cua tarn giac AEF. AEAI vuong tai E, EK la diTcJng cao => AÊ = AK.AỊ
X6tAAEN va AADE c6: EAN (chung).
AEN = ADE (He quS goc tao bdi tia tiep tuyen day cung) Dod6AAEN'^ AADE (g.g)
Ta c6 AK.AI = AN.AD (= AÊ)
AE _ AN
AD ~ AE AÊ = AN.AD X6t AANK AAID c6 NAK (chung), AN AK AI AD
AKN = ADI
(vi AK.AI = AN.AD) DođAANK'^ AAID(c-g-g)
Vay ttf giic DNKI noi tiep
Do d6 cac diem I, D, N, K cilng thupc mot di/6ng tr6n. 2. MD m tiep tuyen cua (I) => M D 1 ID.
Tỉ giac MKD c6 MKI + MDI = 90" + 90" = 180°
Do d6 MKID noi tiep => M, K, I, D ciJng thupc mot diTcJng tr6n Vay M, K, I, D ciing thupc mot di/dng tr6n
Ta CO MN U N , N e (I)
Vay MN la tiep tuyen cua diT^ng tr6n (I)
Nh§n x4t: Day li b^i tôn hinh hoc quen thupc.
MNI = MKI = 90°
1 o n
Cty TNHH MTV DWH Khang Vijt
E>]6 S 6 4 7
D i THI TUVéN SINH VAO L f l P 10 THPT CHUYgN TOAN, T P . C A N THO N A M H g C 2 0 1 2 - 2 0 1 3
Cfiu 1. (2,0 diem) Cho bieu thiJc P = 2a + 4 V a - 6 ^
1) Rut gpn bieu thtfc P.
2) Chtfng minh rkng P^"'^ < 1.
Cfiu 2. (1,0 diem) Cho x, y, z M cdc so diTPng.
2 a - 2 (a>0,âl) Churng minh + y^ + > xy + yz + zx. Dau "=" xay ra khi niỏ ( Cfiu 3. (3,0 diem)
1) Giai he phÚPng trinh: xy +x^y + xy^ = 84 X + y = 19
i r X •
1
5 .
2) Tim m nguyen de phiTPng trinh sau c6 it nhat mot nghiem nguyen: x^ + 2mx + 3m^ - 8m + 6 = 0
Cfiu 4. (1,0 diem)
X + 7y = 50 Cho X, y, z, t khong am, thoa dieu ki$n: • x + z = 60
y + t = 15 Tim gia tri Idn nhat cua bieu thiJc A = 2x + y + z + t
Cfiu 5. (1,0 diem) Cho dirdng tr6n (O), day cung AB(AB < 2R), mot diem M chay
tren cung nho AB. Xic dinh vi tri cua M de chu vi AMAB dat gii tri Wn nhat.
Cfiu 6. (2,0 diem) Cho diTdng tr6n (O; R) ve day cung AB < 2R. Cdc tiep tuyen Ax, By cua diTdng tr6n (O) c^t nhau tai M. Gpi 11^ trung diem cua MA va K 1^ giao diem cua BI vdi (O).
1) Gpi H la giao diem cua MO va AB. Yih day cung KF di qua diem H. Churng
minh r^ng MO m tia phan gidc cua KMF.