g aiam cho cludng thing d cri clua trtrc tam H cria ta,r gi6o ABC.. dudng ttring dtii xung vdi d qua BC, CA, AB.. DE CHfI\H THTIC trang, thi ,sinh khong phai chep di vao giay thi Gei ;,.
Trang 1SO GIAO DUC VA DAO TAO TAY NII\H
Ky rrrr cHoN Hec srNH Gror Lop 12 THpr voNG riNn
NAvl Hec 2ot3 - zot4
Ngay thi: 25 thing 9 nlnr Z0l3 Mdn thi: ToAx - Btroi ttri thri,ntrdt Tho'i gian: 180 phut (khong kA thdi gian giao dA)
@A thi gont co 0 t
Bii 1 (4 dient)
Cho ba s6 ducvnga,b, c tho6 mdn cli6u kiqn a+b+c = 3.
Chirngminhr6ng: -L * l+bc l+ca l+ab b * t ,32
Ri i 2 (4 client)
vy> o Chu'ng mi
Ilii 3, (4 diem)
Bni 4 g aiam)
cho cludng thing d cri clua trtrc tam H cria ta,r gi6o ABC
dudng ttring dtii xung vdi d qua BC, CA, AB Chring
dr, d2, d-, d6ng quy.
s6 f(x) xac clinh vcyi rnQi gia tri
Irfx + y) = f(x).f(y), Vx > 0.
Lrf28) - Q
nh r6ng: f(x) = 0, Vx > 0
DE CHfI\H THTIC trang, thi ,sinh khong phai chep di vao giay thi)
Gei ;,.
rninh
cho lrinh ch0'nh$t ABCD Tr0n c6c cluong thang BC va
cD, l6y lAn lugt c6c di6m di
fl:il*"x,,il;i?r:n" ffir = e00 Gqi H rd hinh chi6u vu6ng g6c cria A tr6n MN rirn qu!
dz,
rang
d3 lan lucyt la oac
ba dtrong th[ng
Bei s G aiiim)
tno't minh rf,ng trong I I s5 thuc khirc nhau thu6c croan
[l; 10001 , o6 the chon cluoc hai
so x vdy sao cho: 0 < x _y <l +:{.y
Her I-.1q,
So
va t6n thf
Trang 2-socrAo DUC vAoAo TAo rAy NINH
xi, rnr cHeN Hec srNH crOr Lop 12 THpr voNc riNn
NApr Hec zots -2014 nucrrvii ;iiii'&iM ffi rilbil idfi{''i;;il il ;il nn6t)
CACH GIAI
BAi 1
ft diem)
cho ba so dwong o, b, c thod mdn rtiiu kipn a + b + c = 3 chftng minh ring
abc3
1+bc 1+ca 1+ab 2
l+bc l+bc
)4 -i(ub+ac)
Bii 2
ft diem)
clto hdm saf@) xdc itlnh vdi mryi gid tri ki€n:ltf* + y) - f(x).f(y), vx 2 o, vy > o
'
Lf Q8): o
x ) 0 vd thod mdn itiiu Ch*ng minh rdng f(x) = 0, Vx > 0
X6t xo >28 thi f(*o)=f(xo -28+28):f(xo-28).f(28)=0.
t(28) = f(l 4 +14): f(l4).f(14) = 0 =+ f(l4) = 0 .
Lim tuong tU nhu trOn thi f(x) - 0, Vx > 14
LAp lupn tucrng tp thi co: f(x) - 0, vx >7; f(x) = 0, vx a!
Trang 3rlei 3
g diem)
Cho hinh chfr nhQt ABCD TrAn cdc itudng thhng BC vd CD, ldy tdn lw,qt cdc ifiAm di itQng M, N sao c/ro ffi = 900 Ggi H ld hinh chi6u vuAng gdc ctia A tr€n MN Tiim qu! tich ctic iti6m H
M(a; ma)
B(a; 0)
DUng hQ trpc toa d0 Oxy nhu hinh ve v6i A = O(0;0).
Gi6 sri B(a;0), D(0; b), & ) 0, b > 0 vd phuong trinh duong th[ng AM ld
y = mx thi M(a; ma)
l)Ni5u m*0:
Phuong trinh cludrng thdng AN h y = -l* Suy ra N(-mb; b).
m
Phucmg trinh dudrng thing MN lA (b - ma)x + (a + mb)y - ab(l + m2) : g
(1)
Phuong trinh dulne th6ng OH le (a + mb)x - (b - ma)y = 0 (2)
GillhA;il""s;'i"h (1, ;irfiil d;;;;
TU d6 thu dugc *
* I = l Phusng trinh ndy phuong trinh cfia dudng
ao
thing BD
2) NOu rD = 0 : Khi d6 M = B, N - D Suy ra H eBD.
Vfly quy tich cdc di6m H la dudrng thang BD
Gtti chfi: Ndu thl sinh s* dung ki6n ththc itudng thiing Simson diSi vdi tam
gidc CMN vd di€m A dd gidi thi:
- Phdn thudn: 2il
- PhAn ddo: 2d
trang 2
Trang 4Bni 4
G dieln)
Cho itudng *dng d iti,qua trgc tdm H cfia tam gidc ABC Ggi d1 ,
ld gdc itudng thdng ddi x*ng: vt6,i d qua BC, CA; AB Ch*ng minh
thdng & , dz, dj cl6ng quy
dz, dj lhn lwgt rdng ba dwd'ng
Ggi A', B', I' lAn luqt ld c6c <li6m ctOi xrtng cria II qua BC, CA AB thi
A', B', C' nim trdn du&ng trdn (O) ngoai ti-6p tu- gia AnC.
vi d cft it nh6t mQt tr"ong cic dudng theng AB, BC, cA nOn gi& sir d c[t
BC t4i E Gqi I ld gi,ro eti0m cira d1 vd d3.
N6uI=A'thi Ie(tr).
Xit I I'hdp dOi xftng tqrc IIC hi6n Id thanh iJi, phip dOi xring
trpc AI3 bi6n tr ilrdnh IrCt
DAt (BC,BA)_ er O6c tam gi6c A'IJC', IBI2
-ts-.+
-G-A'BC' - IBIz =Zrx II#n ARA'I: ABC'fr.
Suy ra
* BI,C' vA BA'I
cfin tai B vil
0,5
t'l6u I nim trong g6e ^{B' thi
A'IC': A'IB + RIC': BIzc'+ BIC'= lB00 -"2cr, cdn n6u I nim ngoii g6c
^
A'BC' thi A'IC'=2a.
Suy ra b6n ditim A', B, C', I dAu thuQc dulng trdn (O)
vi B'HI, - BI-IF = BC'I = BB'I n6n duimg thlng c16i xrrng v6i d qua AC
,lTLll P'I.VAv {f: it,d, cliing quy t4i I
l0sl
lo,ri
tl
t_: l
Trang 5Bni 5
ft diem)
Chftng minh rdng trong 1I sd thgc khtic nhuu thuQc ttogn [1;1000], cd thA chgn itugc hoi s6 x vd y sao cho: 0 < x - y < 1+ 3{/xy
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trang 4