MB NC a chring minh ring MBCN la tf gi6c ngoai ti6p dugc m6t dudng trdn.. b Tim gi6tri lcyn nh6t cria diQn tich tam gi6c AMN theo a... Chfing minh rdng vd,i mgi s6 nguyAn dwong n> 2 thi
Trang 1sO crAo DUC vA DAo TAo rAv Nrr\H
Ky rHr cHeN Hec srNH Gror Lo'p 12 THpr voNG rixu
NAtu Hoc zot3 -zot4
I\giy thi: 26 thftng 9 nf,m 2AI3
MOn thi: TOAN BtrOi ttri thir hai TIrd'i gian: 180 phut (khong ke tho,i gian giao dA)
or csixn rsrlc
@a thi gom co 0l trang, thi sinh khong phai clt6p di vdo giay thi)
Bai 1 (4 diem)
Giai phrrong trinh: (Z*- 3)
Bi i 2 (4 diLrn)
= x2 -2x+ 4
Giai hQ phucmg trinh:
vol rnoi n nguyOn clucrng.
Tim limu,,
Tim t6t cA cdc s6 tq nhi0n m sao cho v6i n ld mOt s5 tU nhiOn nAo rlri, ta cci
mn =1(modn) thi m=1(modn).
Biri 5 g aie4
Cho tam gi6c dOu ABC canh a M vi N ld hai cli6m di dQng lAn lucrt tren hai canh AB vd
AC sao .ho AM * AN = l.
MB NC
a) chring minh ring MBCN la tf gi6c ngoai ti6p dugc m6t dudng trdn
b) Tim gi6tri lcyn nh6t cria diQn tich tam gi6c AMN theo a.
I*' - 6x + y + 1 1
lv'-v-x-3
Bni 3 g arcm1
Cho ddy s6 (r,.,) xac dinh boi: ur=2 vi un*r =1[r, * t I
z\ un) Chimg minh ring v6i moi sO nguyOn ducrng n Z 2 thi ,n , ,6.
Bei 4 & aiam)
+2x+2
s inir:
IIeI
-He ve ten thi
Trang 2s0crAo DUC vAEAo rAo rAy NrNH
rY rnr cHeN Hec jsrNH cr6r lop 12 Trrpr voNG riNn
NAM HQC 2013 -2014
uudxc oAN cnAna rnr nnoN roaN (Bu6i thi rhrl hai)
CACH GIAI
Bei I
@ diem)
Gidi phwong trinh (z* - 3)W= x2 - 2x + 4
Di6ukien: xrl
2
EAt t- W.t>o thi x'=tz -2x-2.
Phucrng trinh dd cho tucrng ituong v6i t2 - (Z*-:)t - 4x + 2 = 0
Giai phucrng trinh tr6n tim dugc t - -2 (lopi) , t = 2x: l
Vdi t - 2x- I thi dugc phucrng trinh
Giai phuong trinh trOn tim dugc nghiQm
lopi
BAi 2
@ diem) Girti h0 pltwnng trinlt [*'
1,,
=6x+y+11
-y-x-3
He tuong ducnrg v6i I(":
-L(v'+
I ti;:;;i*r*;;.;r;
Lfv +2)(y2 -Zy+3) -3
Iyg$-ju cho trudrne hqp x
Suy
'a ne;;;hio* ilt;h
27)-6(x-3)-y+Z
8)-(y+2)=3-x
3>0 n6n x>3=+y> -2=)x<3.V6 lli
<3.
at (3;-2) .
Trang 3BAi 3
G diem)
i
Cho ttdysd (u" ) xdc illnh bdi: vr =2vri un*, = 1[r" * t ) vdi mgi n nguy\n
clwong Chfing minh rdng vd,i mgi s6 nguyAn dwong n> 2 thi un, Jj n*
limun
Gia sri b6t ddng thfc dring vdi
k+r :r["k +
",1
,,
fl-k,tac6 Uk ,.16.
- 6) *
*(6 - u
)
1(", - .6)
1/
Suy ra: 0 ( rn -,.,6 #(r, -.6),nr, > 3 0r5
rim#(,,-.,6) =o non lim(u, - 6) - o Suy ra limun - ,6
Bni 4
@ diem)
Tim tdt cd ctic sd 4r nhihn m sao cho vdri n ld mQt sa qt nhihn ndo itri, ta cd
mn = I (modn) thi m: I (modn)
N6u m =2,tac6 2" =1(modn) yoi -oi n >t.Z
Gi6 sti p ld u6c nguy0n t6 ntrO nhdt cria n, khi dO Z":1 (modp) (*) . 0,5
Tt (*) ta c6 p ld s0 16, suy ra n li sO le, do d6 (n, p-1) = 1 0,5 VO,y t6n tai cdc so nguy0n d,b sao cho an + b(p - l) - 1 0r5
Theo dinh lf Fermat nh6 thi: 2t 2nu.2@-r)b : I (modp).V6 lf. 0r5
N6u m > 2,ta"6 *(m-l)' =[1+(m-1;1('-rt2 = 1 (mod(m-t)r)
MAt khSc m/l (mod(m -t)') Bdi to6n kh6ng tho6 man.
0r5 - - 0r5
Trang 4Bni 5
ft diem)
Cho tsm gidc it1u ABC cgnh o M vd N ld hai iti6m di ctQng ldn lwgt trhn hui
cgnh AB vd AC sao
a) Ch*ng minh rdng MBCN ld tir gtdc ngogi ti6p itwgc mQt itwdng trbn
b) Ttm gid tr! td,n nhdt crta diQn tich tam gidc AMN theo a.
XA,
,A
B
a) Dat AM = x, AN = y Ggi E, F l6fl luqt ld trung rti6m cria AB, AC
ri, **g = 1 suy r" {Y:Y vay M, N ran ruqt thuQc c6c do4n
AE vd AF
Ttr gi6thi6t ta.o, u
= YB * u, IC = I =+ =L*j= = 3
:i1YP:XE3P_I9:_ f
0,5
I
>
MN2 = AM2 + A}-.{2 _ AI\4.ANI
= (a - MB)2 + (a - NC)' - (a - MB)(a - NC)
= a2 +MB2 + NC2 - 2a(MB + NC) + a(MB + NC) - MB.NC
0,5
= a2 +MB' +NC2
- a' +MB2 +NC2
- (MB+NC -a)'
-2a(MB + NC) + 3MB.NC - MB.NC
+ MN = MB +NC - a )MN +BC = MB +NC + MBCN ld trl gidc
ngoqi.ti6p dugc Dudng trdn nQi ti€p tti gi6c MBCN ctng ld <tucrn! trdn
nQi tiep tam gi6c ABC
0r5
lapl.AN.sin600
2
b) DiQn tich tam giac AMN la So*, = _ xy.\6
M4t khric a = x + y + MN = x + y+.ffiy' -t >Z,t"y+.ffi = 3r&y 0r5
Trang 5- a2
'u'Ji
=+ xy =;3 Sor* 36
v6y maxsor* :U=f ex-y=
36
a ANd ANI 1 /' A
\-3 AB AC \-3.
0,5
0,5
f
HOt o o .