=> XMK = 5CDK = DBM = DAM
= > M K / / A D
Do do MK // AD // BC. Ap dung he qua cua dinh li Ta-let, ta c6
IT _ MT ^ ZT ^ ZT ^ JT IC ~ MB " AB CD OC IC ~ MB " AB CD OC
AIJT AICO (c-g-c) => f u = CIO
Cty TNHH MTV D W H Khanp Vi^t
=> Hai tia IJ, lO trung nhaụ
=> I , J O th^ng hang. Vay XT, YZ, OI dong quy tai J. Gpi E, L la giao diem cua tia MK vdi AB, cung AB. Ta CO M I = IK, M I EL. .> .;. Do d6 Ta c6: IK M I MT IL JK ME IJ MB JZ IT IC JT jj_ lO => KJ // OL I OL 10 OD OC 5) .ill Ma OL = OD = OC (= R)
Do do JK = JZ = JT => J la tarn duTdng tron ngoai tiep tarn giac KZT. Vay XT, Y, OI cung di qua tam dufdng tron ngoai tiep tam giac KZT
Nhfin xet: Day la bai toan hinh hoc hay va kho, ve them cac diem, E, L de
c6 diTPc J la tam dUcJng tr6n ngoai tiep tam giac KZT that hpp lỵ
S6 4 3 >
O E THI T U Y I ' N S I N H V A O L(3P 1 0 C H U Y E N TRl/ClNG T H P T
C H U Y E N K H O A H Q C Tl/ N H I E N , DAI HQC QUÓC GIA H A NQI
N A M HQC 2 0 1 2 - 2 0 1 3 J
Cfiu Ị (3,5 diem) ^nXflsdxl Y« > 1) Giai phi/dng trinh: Vx + 9 + 2012Vx + 6 = 2012 + ^(x + 9)(x + 6)
2) Giai he phúdng trinh: x^ + y^ + 2y = 4 4* t + -Xi;. 2x + y + xy = 4 2x + y + xy = 4
Cfiu IỊ (2,5 diem)
1) Tim tat ca c^c cSp so nguyen (x;, y) thoa man ding thtfc f <() + x = u (x + y + l)(xy + X + y) = 5 + 2(x + y) (x + y + l)(xy + X + y) = 5 + 2(x + y)
2) Gia suf x, y la so thiTc diTcfng thoa man dieu kien ^>/x + l)(^/y + l) > 4.
J ,
x2
Tim gia tri nho nhat ciia bieu thiJc P = — + —
y X
Cfiu IIỊ (3 diem) Cho tam giac nhpn ABC npi tiep di/dng tron tam Ọ Gpi M la mot diem tren cung nho BC (M khac B, M khac C va A M khong di qua O). Gia sijT P la mot diem thupc doan thing A M sao cho diTdng tron diTcfng kinh
MP cil cung nho BC tai diem N khac M. <
1) Gpi D la diem dol xtfng vdi diem M qua Ọ Chỉng minh r^ng ba diem N, P,