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Duong tron ngoai tiSp tarn giac BHC clt tia phan giac cua g6c ABC tai diem thu hai M.. Gpi P la trpc tarn tarn giac BCM.[r]
(1)SOf GIAO DVC VA DAO T^O K I THI CHQN HQC SINH G l l hCfP CAP THANH PH6 HANOI NAMHOC 2016-2017
BE CHINH THtTC Mon: TOAN
Ngaythi:2£/J/2077 Then gian lam bai: 150 phut (De thi gom 01 trang) Bkil (5,0diim)
1 Chung minh «5 +5«3 -6n chia het cho 30, vai moi so nguyen duang n
2. Tim tat ca cac c5p so nguyen duang (x;y) cho x2 + 8y va y2 + 8x la cac s6 chinh phuang
B a i H (5,0 diim)
1 Giai phuang trinh 2 * - i + j£-2* = l + - L
x \x 2x
2. Giai h? phuang trinh
Ax
x
= ^x + y-yjx-y
— =Jx + y+<Jx-y
Bki H I (3,0 diem)
1
Vai cac so thuc khong am x, y, z thoa man x2 + y2 + z2 = Chung minh x + y + z<2 + xy
2. Tim gia tri Ion nhat va gia tri nho nhat ciia bieu thuc P = —— ' y
2 + yz + zx + xy BhiJV (6,0 diem)
Cho tarn giac nhpn^-BC (BC >CA> AB) noi tilp dubng tron (O) v2 co true tarn H
Duong tron ngoai tiSp tarn giac BHC clt tia phan giac cua g6c ABC tai diem thu hai M
Gpi P la trpc tarn tarn giac BCM
1 Chiing minh bon diem A,B,C,P cung thuoc mot duang tron
2 Bucmg thing qua H song song vai AO cit canh BC tai E. Goi F la diem tren canh BC cho CF = BE. Chung minh ba diem A,F,0 thing hang
3 Goi N la tarn duang tron ngoai tilp tarn giac ABM. Chung minh PN = PO
B a i V (1,0 diem)
Tren ban co 100 tarn the dugc danh so tu den 100 Hai ngudi A va B lln lugt moi nguai lay mot tlm the tren ban cho neu nguai A lay tim the danh so n thi dam bao nguai B chgn dugc tarn the danh so 2« + Hoi nguai A co the lly dugc nhieu nhat bao nhieu tim the tren ban thoa man yeu cau tren?
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-Lieu y: Can bp coi thi khong giai thich gi them