cHeN Hec srNH GrOr euoc GrA Trrpr NAM.. Cho tam giitc ABC n$i titip du.[r]
Trang 1nQ irAo DUC vA DAo rao rY rm cHeN Hec sJNH cror eu6c GrA rHpr
, fiAhI ilHIhiffi Ngiy Thdi gian : thi thil 180 nh6t phrit (kh6ng : 06t01t2016kO thdt gian giao di)
I
Bni 1 6 aiAm1 GiAi he phuong trinh
I a*-y+22 =3 1*' -y' -22=-l (x,y,zelR.)
I o*' - 3y' - y -zzz =o Bii 2 (5 diam\ a) cho day so (o,,) xitcdinh boi a,,=rn(2nz +l)-ln(n2 +r+l), vdi
n=1,2, Chung minh chi c6 hffu hpn s5 n sao cho {o,}<].,tt 2
b) Cho ddy s6 (b,) xhc dinh bryi b, =ln(Zn'+ l) + ln(n2 + n +l), v 6i n = 1,2, Chfng minh tdn t4i vO h4n si5 n sao cho {4,}.# i
Trong d6 {x} ln ky hiQu phAn 16 cria sti thpc x:{x} = r-[r].
Bii 3 (5 diad Cho tam gi6c ABC c6 B, C co dinh, A thay d6i sao cho tam gi6c ABC
nhgn Gqi D li trung di6m cria BC vd E, F tuong ung li hinh chi6u vudng g6c cta D
l€nAB, AC
a) Ggi O li tdm cria duong trdn ngopi ti6p tam giirc ABC nf cht,qO vil BC lan tuqr
tqi Mvdll Chung minh dulng trdn ngoqi ti6p tam giirc AMN di qua mQt eli€m cO Ainn.
b) C6c ti6p tuy6n cria dubng trdn ngoai ti6p tam gi6c AEF t4i E, F cit nhau tai Z
Chring minh Tnthugc mQt dulng thing cd Ointr.
Bii 4 (5 diA@ Nguoi ta tr6ng hai loai c6y kh6c nhau tr€n mgt miting d6t hinh cht nhflt
kich thudc mxn 6 vudng lmdi o tr6ng mQt cdy) MQt cdch tr6ng c6y dugc gli ld iin
tuqng n6u nhu:
i) SO luqng c6y dugc trdng cria hai lopi cdy bing nhau;
ii) SO lugng ch6nh l$ch cira hai loai c6y trdn mdi trang khdng nhd hon mQt nua si5 O pfia
hing d6 vi si5 lugng chOnh lQch cria hai lopi cdy tr6n m6i c6t kh6ng nho hon mQt nta
sti 6 cria cQt d6.
a) HAy chi ra mQt c6ch tr6ng cdy 6n tugng l<hi m = n = 20t6
b) Chfng minh n6u c6 mQt crich tr6ng c6ry iintugng thi ci m vd n t16u lA bQi cria 4.
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- Cdn bp coi thi khdng gidi th{ch gi th€m.
Trang 2Be GrAo DUC vA EAo rAo
oe rru csixu rHUc
r}Ar{ ffiI[NFf,
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ri rm cHeN Hec srNH GrOr euoc GrA Trrpr
NAM 2016
Mdn : roAN Thoi gian : 180 phrit (kh6ng k€ thdi gian giao di) Ngiy thi thir hai:0710112016
Bni5 $diem) Tim t6t cri cric so thgc a d6 t6n tpi hdm s6 ,r:lR.+R th6a m6n:
i) /(t) = 20t6;
ii) f .(x+ y,+ f (y))= /(r) +ay v0i msi x,y e R .
Bni 6 Q di1m') Cho tam giitc ABC n$i titip du<rng trdn (O) (vdi tdm O) c6 cic g6c o dinh
B vit C dAu nhqn LAy diOm M trdn.rrng 6D khdng chira A sao cho AM\<hdngvuong g6c
voi BC, AM chttrung truc cua BC tai f Oucrng trdir ngo4i ti6p tam giac eOfZEt fqlit w
(N * A).
a) Chung minh EIit = CAN.
b) Gqi / li tdm dulng tron nQi titip vd G li chdn duong ph6n gi6c trong g6c A cula
tam gi6c ABC AI, MI, NI cht(o) 6n luqr tai D, E, F Ggi ! ia'gtuJng *g li.iiao di6m
cin DF voi AM vi DE vdi AN Eucrng tron di qua P vi ti6p xric vdi ,qO tqil
"hlOf @i H
(H * D), duong trdn di qua evd ti6p xric v6i eb tqi t cit oa @i K ( K * D)
chung minh duongtrdn ngoai tiiSp tam gi6tc GHK ti6p xtic v6i BC.
BAi 7 Q diam) 56 nguy6n ducrng n duoc ggi ld s6 hoan chinh n6u n bing t6ng c6c udc sd
duong cria n6 (khdng k6 chinh n6).
a) Chung minh ring ntiu n li s6 hoin chinh l€: thi n c6 dpng
n= ptmr,
trong tl6 p ld s6 nguy6n t5 c6 dqng 4k+1, s li s6 nguy6n duong c6 dpng 4h+tvit m
li sti nguy€n ducrng khdng chia h6t cho p.
b) Tim tdt cir cric s6 nguyen duong n>l sao cho n-l vd n(n-+l) dOu li c6c s6
2
hoan chinh .
Thi sinh khdng &rp'c s* dqng tdi tiQu vd mdy tinh cdm my.
Cdn bp coi thi khdng gidi thich gi th€m.