AM c&ttrung truc cua BC tqi f.. b Gqi / ld t6m rludng trdn nQi titip vd G li chdn durrng ph6n gi6c trong g6c A oiua tam giric ABC.. Ggi r ia gtuong ftng la giao tli€m ctia D.lq'vli AM vi
Trang 1BQ GriO DUC vA DAO TAO
oE rui cuiNn rHUc
rt rm cHeN Hec srNH Gror euoc GrA THpr
NAU zoro
M6n : roAN
Thoi gian : 180 phft (khong kA ilrcti g;an giao di) Nedy thi thf/ hai:0710112016
Bni 5 $ die@ Tim tdt c6 c6c s,5 thyc a rtc tdn tai him s6 ,r : IR -+ R th6a m6n:
i) /(t) = 2016;
ii) f"(x+y,,+"f (y))=/(r) +ay voi msi x,yeiR.
Bii 6 (7 di€m) Cho tam gi6c ABC n$i titip duong trdn (O) (vdi tdm O) cb c6c g6c o dinh
B vd C ddu nhqn LAy di0m M f:dn.ung fr khdng chta A sao cho AMV,h}ngvu6ng g6c voi BC AM c&ttrung truc cua BC tqi f OuOng trdn ngopi titip tam giec AOT;futOital lf
(N + A).
a) Chung minh Efu = AN.
b) Gqi / ld t6m rludng trdn nQi titip vd G li chdn durrng ph6n gi6c trong g6c A oiua
tam giric ABC AI, MI, NI cht(o) len luqr tai D, E, F Ggi r ia gtuong ftng la giao tli€m
ctia D.lq'vli AM vi DE vdi AN Eucrng trdn di qua P vd titip xric v6i AD t+i t cit Of @i H
(H * D), dudng trdn di qua Qvd titip xric vdi AD tvi t cit ogtai K ( K * D)
Chtmg minh dubngtrdn ngopi ti6p tam giirc GHK ti6p xric vbi BC.
Bii 7 (7 dia@ 56 nguy6n duons n duoo gqi ln s6 hoen chinh n6u n bing tdng c6c udc sd
duong ctia n6 tf<fronif<'e chinh n6').
a) Chrmg minh ring ntiu n li s6 hoin chinh 16 thi n c6 d4ng
n= pstfl2,
trong <16 p ld sO nguy€n t5 c6 d4ng 4k +l , s li s6 nguy6n duong c6 dpng 4h+tvit m
li sti nguy6n ducrng khdng chia h6t cho p.
b) Tim tdt ctr c6c sti nguy6n duong n>l
hodLn chinh .
sao cho n-l vd n(n+l)
dou ld cac s6 2
HET -Thi sinh khong dwo'c ,ya dqng tdi lieu vd may tinh cam ta1,.
Cdn bo coi thi khong gidi thich gi th€m.