ĐỀ THI CHỌN HSG QUỐC GIA THPT MÔN TOÁN 2016

BQ

GriO DUC vA DAO TAO
oE

rt

rm

rui cuiNn rHUc

cHeN Hec srNH Gror euoc GrA THpr
NAU zoro

M6n : roAN

frALJ ilffiIHFr
Bni 5

Thoi gian : 180 phft (khong

kA ilrcti

g;an giao

di)

Nedy thi thf/ hai:0710112016

$ 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
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
(N + A).

fr

lf

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 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
r

a) Chrmg minh ring ntiu n li s6 hoin chinh

16

thi n c6 d4ng

n= pstfl2,
trong

li

<16

p ld sO nguy€n t5 c6 d4ng 4k +l

,s

li s6 nguy6n duong c6 dpng 4h+tvit m

sti nguy6n ducrng khdng chia h6t cho p.

b) Tim tdt
hodLn

chinh.

ctr c6c

sti nguy6n duong n>l sao cho

n-l

.

HET-------Thi sinh khong dwo'c ,ya dqng tdi lieu vd may tinh cam
Cdn bo coi thi khong gidi thich gi th€m.

ta1,.

vd

n(n+l)
2

dou ld cac s6