1. TRI'ONG TIIPT
DAo DUY TTI - IIA NoI
of rrn rrnlDAr Hgc r,Arv r (2010-.-2011)
nndx, roAN rn6r a
Thdi gian: 180 phtu (kh6ng *6 nm g*n giaa di)
CAu I. Cho him s5 y: x3 - 3x (l)-
/ U Xnaosft sg bi6n thi€n vi v€ eO tni narn sO (t;.
'.t
u
2/ Tl^m etii phuong trinh x3 -3x = + c6 3 nghiQm phdn biQt.
m" +l
3/ V6iO(0; 0) vd A(2;2) h hai ttiiAm nim tr6n AO tni hdm sd (1), rim di€rn lvi
nim trOn cung OA crha dd thihAm sO (t) sao cho khodng c6ch tu M cli5n OA ld ld.n nh6t,
Cffu II.
1/ Cho b6t phuong trinh: JT3x+2) 7n-rt7 1x+4l,
v al Giai bdt phuong trinh khi m :4.
b/ Tim tdt cir chc giittri cria m ee UAt phucrng trinh tr€n nghiQnr ding viii riiqi x >:i.
,,1 2/ Gi6i phucrng trinh: tanx+cosx-cos2x =sinx(l +tan!-tarrx).
Cffu III.
V ttTrCn m{t phdne tqa d0 Oxy, tim phuong trinh tludrng thingr di qua di6rn
M(l; 3) sao cho dudng theng d6 ctrng v6i2 dudrng thang (d1): 3x-{- 4y + 5 : {j vi
(d2): 4x + 3y - I : 0 t4o th'nnh mQt tam gitlc cdn c6 dinh ld giao di6m cria (d1) ,,,d (d2).(
2l Cho hinh ch6p SABC c6 dhy ABC li tam gi6c cdn, c6 AB : AC : ?,a; RC - 2a
c6c m{t bon ctrng tao vdi ttax,m0t g6c 600. Hinh chitiu H cia dinh s xu6ng (ABC)
nam O trong tam gi6cABC.
'/ at cntng minh t6ng H h tem dulng trdn nQi ti6p cria tam gi6c ABC.
b/ Tfnh thiS tich cria trl di0n SABC
Cffu IV.
Cho tflp hW A gdm n phAn tri (n > 4). nii5t ra"g sO tgp con gOm 4 pfran tri cria A
bing}} Hn sO tdp con g6m 2 phan tfr cria A. Tim k e {1,2,3,..,n} sao cho si5 tfp con
g6m k phAn trl cria A ln lcrn nhdt.
Cf,uY, Cho 3 sO kh6ng 6m a,b,c. Chung minh ring:
a3 +bt +c'> a'Jbc +b'Ji +t'^[on .
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2. _x
oAp An vA rHANc orru rrrr ruuD+r rrgc IAN r - ivr0N roAN
Caulf NQi dung cho tli6m oi6m
al Khio s6t str bi€n thi6n vi ve d6 thj hdm sd y = x3 - 3x 1.00
TXD: R
Gioih4n:
L im1x3-3x;= to
r+ t@
Strbitinthi6n: y'=3*-3. Tac6y'=0<=>x= t I
Bnng bi6n thi6n:
HAm sii tt6ng biiSn tr0n (-co; -l) vi (l; +o); Nghlch biiSn ffin Gl; t)
Hnm si5 d4t cyc d?r t?r (-l;2),cgc ti6u 6(l; -Z)
x -dr -1 I 1-@
v' +0 0 +
v
*oo
-2
Ei0m u6n: Y" = 6x - 0 <:> x:0 . Di6m u6n U(0; 0)
VE hinh: 2 Ditim u5n: y" = 6x = 0 <:> x = 0 . Ei6m u6nU(0; 0)
Ve hinh
a2
Tim m ttti phuong fiinh x3 -3x = + c6 3 nghiQm phdn biQt.
m" +l 1.00
Sưu tầm: Nguyễn Minh Hải

3. SO nghigm cua phuong trinh x3 -3,c =#bdng sd giao <ti6m cria 2 dO thf :
[y=r'-3t
lz^
l'= *u
0.25
DU,a vio dO thi ta th6y dO phuong trinh t€n c6 3 nghigm phdn bigt thi
+. ?* q)q=7-rn'-l<m<m2 +l
m'+l
0.50
(=) -mr -m- I < 0 < m' -m + I (Lu0n dung voi mgi m)
V{y voi mgi m phuong trinh ludn s[ ]nghiQm phan bipt. 0.25
43. 1.00
Gpi M(xo; yo) voi 0 < xo < zlb,mOt diem bat kj' n[m tr0n cung OA ciad6 thi.
X6t duong thdng (d) qua M vi song song voi OA. Ta c6:
d(M; oA) = d((d); oA);
Khoang c6ch nay lcrn nh6t khi (d) cdch xa OA ntrAt <=> (d) chinh b ti6p tuytin song sonl
OA tar ti6p di6m M.
Phuong tinh duong thdng OA li y = x
=> Tiiip tuy6n t4i M song song voi OA c6 hQ sd g6c f (x6): I
<=> 3xo2- 3 = I =) Xo =
# r**(hrfl
,!S 4*a2, a -^ti aa a (.r < -2;x 2 -l)
1.00
THt.Ntiu 4-Jf 4x+4=ol-=t 16<xz -3x+4
Biit phuong tinh lu6n th6a m5n.
f _, t*.F
<-> x, -ix-r2zo<=>l
^-,
:*lx(
LZ
rH2,Ni5u q-J*z*+q>g4=1
+.*.* 0
Khi d6 hai vii cua BPT di cho ludn duong, binh phuong hai vi6 ta c6:
BPT <=> x' -3x+2r-16-sJ* 1r+4 +x2 -3x+4
[*. u-s,
a=2 4"{y, -{J )-9 <=> x, -*-!. o .=r l
^ - o,
: (*uu mfln dk x6c ttir
16 I 6+J53
l_r>-
L+
Sưu tầm: Nguyễn Minh Hải

4. riit rrq,p f) ta c6,.e.P'L#f y rS:ftYf,
r6t hq,p THl, ta c6 tl6p s5 x e,-*,tftrt*,**l
IUlb. 1.00
. BPT quy v0 drrr1e J f a* *;+ .fir'--3r * 4 > m
A
xet nam so
f (x) ='[x' 1x +, +'[77)c + 4
(**)
=> /(x) =
2x-3 2x-3
D6 th6y f '(x)>0 Vx > 3 n€n (x) <l6ng biiSn ffin (3; +o1
EO 1**1 c6 nghiQm Vx 2 3 <-> M! f (x)> m <+ f (3) =2+ ,[2 > m
un. 1.00
tanx+cosx-cost x =sinx(l*t*{t*r) DKcosx. cos x/2 *0'2
'7 sinl.sinr
PT <=> tanr+cosx-cos'a =pinr(l *-4-)'' cos-.cos.r
2
xxcosr.cos-+sm-.slnI
(=) tflox* cosx -cost x = sinx(4) x
cos-.cosr
(n)
,O'I
(=> tanx*cosr-cos'x = sinx 2
.orI..ort
2
<=>cosx-cos2x-0<->cosx= I <->x:k2 r $e Z)
Nghr€m thoa m6n diAu kiQn (*)
IIu1. 1.00
Phuong trinh duong thdng (d) qua M(l; 3) c6 d4ng A(x * 1) + B(y - 3) = 0
Sưu tầm: Nguyễn Minh Hải

5. <:)Ax+By-A-38=0
Tac6 i(A;n'1,112;+S,ir(a;3) Hn luqt h cric vdcto ph6p tuy6n cua (d), (dr), (dz).
E6 (q t4o voi (dr); (dz) mQt tam gi6c cdn tai dinh ld giao cua (dr); (dz) <=>
t(d ; d,) = t(d ; d ) .-, I 3 A
! 4 B
! - ftA
f n !
--11o,*
o.:= o
: :t
u^
^.=rl
n =
' ' s"[t +E y17;g -
L3A+48=4A-38 LA=
Chgn B : I ta c6 A: I ho{c -1.
Vfly phuong ttnh ttuong thlttg (d) c6 d4ng: x + y - 4= 0 ho{c x - y - 2 : 0-
Go. i MN, P lan luqt h hinh chi6u cria H l€n c6c c4nh BC, AC, AB
Tac6 ISMH = ISNH = ISPH=600 => HM : HN : HP = SH/tan600, H h diAm
trong dudmg trdn n6n H ld t6m ducmg trdn nQi ti6p cua tam gi6c ABC.
I
Tac6 V*u, =;SH.SABc
li ducrng cao, dudng trung tuy6n cria tam gi6c
Sesc= %ANI.vc=
|Jst -+A.za=a2Ji
Ta l+i c6 Sesc : pr: P. HN =) HN :
ry =
=>sH=HN.tan60o =$.a:+4 j4
=) vsnsc=:+tJ3=f"
<=> (n - 2).(n - 3) = 240 <:> n : 18
X6t ddy sO {Cl,hf e {0;1;2;...;18}.
Trlgiathi6t tae6 cl =20c: --n(n-r)(rL
-2)(n-3) =2gn(n,-l)
242
Sưu tầm: Nguyễn Minh Hải

6. Xdt day sO {Ci};lr e {0;1;2;...;18}.
Ta c6 T : -* = .*t=< I <=> /r < 8,5 1=) k= 0;l;2...;g
ciJ' 18-k 'v v'r''"''e
Tuongt.uT> | 4=)k=9; l0;...;18
Tt d6 ta nh{n duo. c c,! Ma,r k{ri k = 8 ho{c 9. so srinh t.uc titip nhfn dugc k = p
vl. 1.00
at +b3 +ct > a2Jbc +b'.[* +t'JoO j
Ta c6 theo BDT cdsi: 2vP : z1a2 J-bc + b' Ji + t' J ony < a2 (b + c) + b2 (a + c) + c, 1a +
< ab(a + b) + bc(b + c) + ca(a + c) (l)
Lai c6 dE dang chimg minh ttugc a3 +b3 >-ab(a+b)
cq(Do a3+ b3 -ab(a+b) = (a+b)(a-b)' > 0 voi mgi q b > 0)
Tuong t:ty b3 +c3 >bc(b+c);ct +at > ca(c+a)
=> 2VT ab(a + b) + bc(b + c) + ca(c + a) (2)
Tir (1) vn (2) ta nhAn dugc dpcm. Ddu ":'xiy ra khi a = b =c.
r7
Sưu tầm: Nguyễn Minh Hải