THTT so 449 tháng 11 năm 2014

xuflr siu rUrgo+
2014
s6 44e
rep cni Ra HArue rHAruc - ruAu rx05{
oAruH cHo rRUNG xoc pH6 rnOruc vA rRuruc uoc co s6
Tru s6: 1B7B Gi6ng Vo, He NOi.
DT Bien tAp: (04) 35121607; DT - Fax Ph6t hdnh, Tri su: (04) 35121606
Email: toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre
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KHÔNG THAY THẾ CHO TC TH&TT TRUYỀN THỐNG !

ffi*ffi q&x&'2c aX ruF€& r*B&,&s% K&#&&ieY s w&&*ffi&ru e%eretr&&ffieeffi x
ffiWffi W&ffiY ru&ru&
tffiffij
& q s r"#
*F*4-4 *8 g#*rmk$
,p-^0 s6ch cAe cHUYErrr nE ndt nu0nic
"CfrlUJ<Hoc srNr-r Gioi vrcN TeANr LCIp e (Tap
Hilnot Dai so vd TAp hai Hinh hqc) dugc
Suiit ban nhan ki ni6m 40 ndm Viet Nam tham du
ki thi Olympic Todn hoc Quoc t€i (lMO 1974 - ZAM)
vmr mu* dieh cLrng eap tdri !ieu tl'ram khdo giup cdic
em hoe sinh d&o s&u, n&ng cao ki6n thue , ren luyen
pnLiCIng phap gidi Toan chudn bi cho eiic ki thi hoc
sinh gioi va thi vao c*c khdi ehuydn Trung hoc phd
thong. $;ich eon la tai lieu huu ich cho gi6o vi6n
vA cAn hQ quAn li Giao c{uc, nhdm phAt tridn tu duy
logic, sang tao, gop ph&n nang cao ch;it luong day
vA hoc tr*ng nhdi truong.
Sdeh vi6t duoi dqng ciie chuyOn d0, nr5i ehuv6n
dd gdrn ki6n thue cdn nho, vi du minh nofl vd h0
thdng hai tap phong phrl M0t so dinh li, c0ng th[rc
mo r6ng duoc biOn soan duoi dang c6c vi du hodc
bai tap.
Trong cudn Hinh hoc, 'oan doc sO thay nnidu biii to6n
rnoi qua ciic ki thi hoc sinh gioi cua c6c nuoc nhu
Canada, My, Nga, T:ung Quoc, Bulgari, Slovenia,
Rumani, Singapore,..,, dac bigt la ki thi IMO (V0
dich ToAn Qudc td) va APM0 (ChAu A -Thiii Binh
Duong).
l-li vong ring, b0 sach sO la tai lieu tham khAo thieit
thuc, huu ich ddi voi cdc em hoc sinh THCS, c6c
thdy c0 gido dqy Todn vd ban doc yeu ihich Toan.
TAPHAI : IIINH HOC
g

1FF3H(6H6 @&pNF @F3HS
.{, A
EFFEFFGSHUEN
vO u6Nc pHoNG (GV THqT Tian Du 1, Bdc Ninh)
Khi g{p nfit phurng n'inh c6 chia phdn nflNen chring ta kh6ng chi
thdy r:di hcr troslg thudt totin gitii phtrong trinh nfi cdn rhdy ri d6
nh{htg titth chdt thu vi t:ila ohdn nguyitt daot. sit dung. Hi t ong btii
viii nii.v sd dun lai nht*tg diiu moi la t,a hd frh c'ho cdt. ban.
A. Mot sd tinh chdt ciia phdn ngu.y€n Dang thrlc xiy ra 6 BDT v€ trdi, vd phii lAn
Tru6c ti0n xin nh6c lai m6t vdi tinh chdt (TC) luot ldL: [{xr} + {*} +... + {;r,}] = 6 ;
ciraphdnnguyOn: Y6ix,y,a ld.c6c sd thuc,m, z rro r-n
iu'ti * {*,} + "'+ {x,}l = n-l'
ld sd nguy6n, tDttaphqp sd rguycr, lt'lt
hieu [x] ld sd nguyen lon nhdt kh6ng wor qu6 ' H€ qud (cria tfnh chdt 8, 9, 10): Bidu thfc
x, doc ld phdn nguyOn ctta x, phAn 16 cira x li P=lxt+x2+...*x*-h-lz-...-!n7
{x} = x - lxl . Khi d6 ta c6 cdc tinh chdt sau
Ttnh chdi
l. x -l < [x] S r. H0 qui: 0 < {x} < 1.
2.lx)=non<x<n+l.
Dacbi0t[x]=0<>0<x<1.
3. fx+ nf:lxl+n.
4. Vdi x eZ th [-x]: -[x]
Vdi x eZ th l-xl = -[r] - 1 .
5. V6i n>l tac6
f,r+[,*1.]* .[, .+1=tnxt L n)
6. Vdi x >.y thi [x] > [y].
7.Yot n >1 thi 0<fr{x}l < n-t.
8. Vdi n>l tac6: nlxl<Lnxl<nl,x7+n-t.
+ nlxl:lnxf<>0<{r}.f.
n
+ [nx]= nlxl+ n -1 o n -l < tx] < l.
n
9. V6i m)1,n>l tac6:
. ndxl+ nlyl <lot* + nyl 3 mlxl + nly) + m + n - l.
. mfxl- nly) - n < ltnx - nyl < mlxl - nlyl + m - l.
10. Vdi x, e JR ta c6 :
- lxr ] - lxzl- ...-l*^l+ lyr I + lyzl+ ...+ly,l
nhdn c6c gi6 t4nguyOn -n;- n + l;...; m -1.
ll. a) Ndu x) 0,y > 0 thi lxy)>lxllyl
b) Ndu y, <0,/ < 0 thi lxyl<lxlly)
c)Ndu x<0< y thlxyl>[x][y]+[x].
12.. Ndu a> 0 vb alxl =b/l thi -1 < M -y < a.
.Ndu a< 0 vd a[x] = [y] thi a- I < m - y <0.
Tdng qudt:Ya a,;f ,;6 e IR ; a, , O;Fi ,0.
Ndu qfx,l + arlxrl+...+ a*lx*l
= frlyi+ Qzlyzl+ ...+ Fnly,l + d thi
a{r+ dzx2+ ...* d,,fi*- fiilr- frlz- . .- f,J,
€ eA- Br- ...- B"+ 6, %+ d2+ ...+ d*+ O.
Chirng mink
3. Gie sit [r] = m th theo TC2 ta c6:
ml x <m+7 > m+n1x*n <m+n+1
=[x+ n]=myn:lxf+n.
4. - Vdi x eZ th -x eZ nOn -x ld sd nguyOn
lon nhdt kh6ng wot qu6 -x vd x ld sd nguyOn
lon nhdt kh6ng wot qu6 x nOn
[r] = x ;[-x] = -.r, suy ru l-xl: -;e : -[x].
- V6i x #Zththeo TC1 ra c6:
x -7 <[x] < x + [x] < ;r < [x]+1
lxrl+lxrl+...+[x,] S[x, + xr+...+ x,] = -[x]-1 < -x<-[x] = [-x] =-[r]-1.
S[x,]+lxzl+...+lx,)+n-1. 6.TheoTC1 c6 x<[x]+1 vd lyl<y.
t.; nnr,r-rorn, t*ilrH$ |

Gii srl [x] < [y] suy ra
[x]+1 <[y] = x <[r]+1 <[Y] 3 Y + x < Y
mAu thuAn v6i gii thtdt x2 Y.
7. Do 0 < {r} < 1 n0n 0 < n{x} <n
=0<1"{*1<n-1.
8. Do n[x] eZ nAntheo TC3 c6
lnxl=ln(lxl+ {x})l = nlxf+[r{x}]. Mh theo
TC7 c6 0 < [m{x}] < n-1, suy ra
nlxT < lnxl < nlxl+ n - l. nlxl = lnx)
e[r{x}] = 0 <> 0 S n{x <1 e0 < {,}'-1
n
lnxl= nlxl+ n-l el"{x)l= n-l
e n -!< n{x} < n e n -l
={x} < 1.
n
9. lmx + ny7=lmlxl+ m{x) + nlyl+ n{y)l
= mlx)+ nlyl+lm{x} + n{v1
Unx - nyl=lmlx)+ m{x - nlYl- n{Y)l
= mlxT - nlyl + lm {x} - n {v}1.
Do 0< {*};{y} <1n6n
0 < m{x + n{y < m + n ; -n < m{x - nltt) < m'
Suy ra: 0 <lm{x) + n{y})< m + n -l ;
-n <lm{x} -n{Y}l< m-1.
Do vdy mlxf + nlyl <lmx + nY)
< mlxl+ nlyl+ m+ n-1.
mlxl- nlyl - n < lmx - ny) < mlxl - nly) + m - l.
10. fxr + x2 + ...+ xn)
= [[xr] + {x,} + lxzl+ {x} +...+ [x,] + {x,}l
= [x1] + lx2l+ ...+lxn)+ [{x1} + {x2} + ...+ {xr}] (l)
Do 0 < {x,} < 1 nOn 0< {xr}+{xr} +.-.+ {x, <n
suyra: 0<[{xr} +{xzl+...+{x,}] <n-l (2).
TU (1) vI (2) ta c6 didu phii chtmg minh.
Chtng minh hi qud:
Bidn ddi tucrng tgTCS,9, 10 duo.c:
p = [ {x, } + {xr + ... + {x* - {y} - {y z - ... - {J, "}]
eZ (1).
Ta c6: -n < {x} + {xz} + ...+ {x-}
-{Y} - {Yz -...- {Y"} < m ndn
-n <l{xr} + {xr} +... + {x,} - {y} - Uz} -.. - - U")
< m-l Q).
Tt (1) vh (2) suy ra P nhan c6c grd tri nguy0n
-n;-n +L;...;m-I.
11. Do [x][y] eZ n€n
lxyl= [([r] + {x}Xtyl + {Y})l
= [x][y] + [[x] {y} + [y] {x} + {x} {y}l
+tixl {y} + [Y] {x} + {:r} {Y}l
= [r][y] + [[x] {y} + {xX[Y] + {Y})l
: [x][y] + [[r] {y} + {xy|.
a) vdi x,y20 thi [x]>0 md {x};{v}>O
nen [x]{y} ) 0, {x}Y ) 0, suy ra
trl{y} +{xy > 0 = ttxl{Y}+{x}Yl> 0
do vay lxyl>[x][y].
b) V6i x,y<0 thi [r]<0 md {x};{Y}>0
n6n [x] {y} < 0, {*}y < 0, suY ra
trl{y} + {x}y < 0 = ttrl{Y} + {x}Yl < 0
do vdy lxy)<[x][y].
c)V6ix<0<ytlri [x] <0<ymi0< {x}; {y}< 1,
suy ra [x] {y} > [x] vd {*}Y > 0, suY ra
txl{y}+{x}Y>[x]
> [[x] {y) + {x}y1> [[x]l = [x] (theo TC6)
do vAy lxyl> [x][y] +[x].
lz.Yot alxf =[y] thi
dx - y : a(lx7+ {x}) - ([v] + {v}) = a{x - {v)
-h$ia >0 c6
-l < a{x- {y} < a > -l < dx - Y < d.
-khirl<0 c6
a -1 < a{x -{y} < 0 > a -l < dx - Y 30.
-Tac6
a{l + d2x2 + ...+ dmxm - Ah - fzyz -...- frnyn
= drlxtT+ arlxrf + ...+ a*lx*)
-giYi- FzLY)- "'- fr,lY)+
+ar{xr) + a.r{xr + ...+ a*{x*}
-fi{Y} - Fz{Y} - "'- F"{Y"}
TONN HOC 2 ' ;4"aEA s,;.* ,rr-rrro

= 6 + a1{xr + ar{rz +...+ a*{x*
-A{v} - Fr{vzl -...- F,{y") (r)
Do o< {*,};{t1}<t ',on
5 + ar{xr} + ar{*z + ...* d*{"*l - fi{y}
-Fz{y} -...- B,{y,} thuQc khoang
Cfi- 0z - - Fn + 6;q-t d2 * ...* d* + 6) (2)
Tt (1) vd (2) ta c6 dpcm.
(Ban doc tu chrlng minh c6c tinh chdt cdn lai).
B. Mot sd thi du
O Thi du l. Gicii phurtng trinh
Ir+ll Ir+3-l t1'*1s-T 22 l-l-l ', - +..._l l=a* ixj rt) Lr6 lLr6 l L16 I3
Loi girii. Theo TC5 ta c6:
vr( r) : [t1-] * [l:] * 1l *... * ltt * Z-l L16l L16 8l L16 8l
=L[r 416l= I [lLt- 2L ll
1) )) 25 Do 0 < {x} < 1 n€n !<J3V3P(l) :a*tx} <-.
Laic6 VT(l) eZ nAn VT(l): VP(l) = 8.
vrr) : g .= [4] =8 <] 8 < r+l <9 L2 ) 2
e 15 < x <17 e [x] = 15 ho4c [x] = 16.
')
VP(l)=$<> {r}:i. Ma ;s=[x]+{x} n6n
3
PT(l) c6 2 nghiem * =!3u3a, =1.
O Thi du 2. Gitii phurtng trinh
"Ittx-tz-] [sx-zl J.
^ -, - ,, l9lil LJL-J
Ldi gitii. Theo TC12, fU FrI(*) suy ra
-r.u.llx*9137 -5x-2 <3 <+ 2<x<4
=93.5 x-32 <6 = z.lsL*-32.1l= vp(*")<5.
Me VT(*) ld sd nguydn chia hdt cho 3 n6n
vT(*) = vP(*) = 3, suy ra
[[*#l=, [r.!!L.z PT(*)e{l ' J el ' l[t,-r.l_, Ir= 5x-2.0
lL 3 l- 3
ll_z<e Y<_3s Itt---.11 26 t4
lll_ t4 ll s
ts 5
V4y t4p nghiem cira Pr(*) td. r =L[4ll, +s))
O Thi du3.Gidi phuong rrinlt
l*'l*'.ll+ {ro - x2,, :g# (l)
Ldi giii. Theo TCl c6 lx2l> ,2 - 1 ncn
x2lx2l> x2 (x2 -l) = xa - x2 .
Theo TC6 suy ra l*'l*'|)2L*a - x2l.Tac6
VT(1) 2lra - *21+ 1*a - *2) : *4 - 12 (2)
vn( *z-1)'
st s) o c> x4 - *2 >osx-2 - (3) =
2s
Tt (2) vn (3) suy ra VT(l) > VP(l). Vay
W(D = VP(1) e dd.u"=" xiry ratu Q) vd (3).
PT(l) c6 2 nghiCm * = ^pr* =- vs'
O Thi du 4.Gitii phu:ong rrinh
4[I4-] -s[**r-l. J
rox - 3 | [3.] L 6.ll o )
25x2 *130x+229
(l)
t, nn, ,r-rorn,
T?EilrHff
3

Ldi gitii. Ap dqng TC9 ta c6
vro)
=
[o. t'*'
",,,-L3-.(- "s.(- *A *1)))l.- ,r -* 1{'o *6,*I'l
Itox+:l Itox+3.l
:l-lf''-tr L o lt 6 )-
l0x+3 - l0x+33
=::::---:-r(=- t'r
66
Mb (5x -114)2 > O
(4,-1<0x + 33
6
(3)
non tD (2) vd (3) suy ra VT(l) < VP(l). Vay
rI(1)=VP(l) e ddu"=" xhyratai(2) vd(3)
[4[rlt]l - s[., * ll = ['0, * rl *,
<+l L 3 .l L 6l L 6 l
I
[(sr- l4)2 =g
V4y Pf(1) c6 nghiOm x =2,8.
O Thi dq 5.Gidi phuong n'inlz
ry. ry "+ - E,'l = 6r.r,r + #(*)
Loi gitii. Theo TC8 ta c6
Ptl.2lx2_l+t 1ly. t3{'l .31*')*2 e) 2233
y4.4lx2l+3
1:y , [s{'] . t[*'_]*o (o) 4 4 5- 5
C0ng vd v6i vd (l),(2),(3),(4) ta duo. c
vr(*) < 4lr'l*ry (s)
60
Md x2 z o non lt'l>0 suy ra
4lx2)+# = 6lx2)+W
PT(*) e
!
= {*') .,
?
= {,, .,
lrr*rr<1<> 1=" =1x2Y<t
!
= {*'} .,
l*'1= o
1I121 5) l"xl+l1Oxl+ +_=_
tJil+[3Jr]+1
l;.;1.1;.;1.
3lx2 +0,81+[3x+0,
ll *')
25x2 -l3Ox+229
TU (5) vd (6) suy ra VT(*) < VP(*).
VT(*) = VP(*) <> ddu "-" xily rat4i (l), (2),
(3), (4), (5), (6).Do d6
| ], *.t
..' I {5 Viy t?p nghiem ctra PT(*) -l 1 ' l-1<x<-+ I t- LV5
rdr=t(.- t'J,s* ))' [-2,r). lJs' I
nAt rAP
Glhicdc phuong trinh sau
1) txtxll =41*,
2) lxll-xl+s{x2}+4=O
Jv)l lzL*^2 -r3* 13-]l-- Lfz x3+ tl]
o', L,l rtxt-2zt ll*L[orrl*tl =o
6)
7)
8)
[x] [0x] l0
zr*t * rg _ t - +rJilrz - Jit
J;
,[_
L'
llx * {3x2 +3x +0,2} -1 1,8
4x6 -13x4 +7x2 +25,25
[x2 +0,5][3x2 +0,5]+ {3x
+1
*1-] =,
2)
_I 1
t1'
x7
h
+
8l
)x2 -0,75y
x"'+ -6 ll.
7)
(6) e) 2t3{x}):ry#+(t*t+z)2
10) o+ - 1.
=[
4- TOrfciNr HuOaC@
11)
l*')+12x2)+l4x2l+ 2

I{uong fdru girii
uE nu ilrYfu $ffi uAo m ro HffrEr{ tontt
TnU0ilG IHPT GHUYEil HA TiilH NAru Hec zol4-201s
@d thi ddng trdn TH&TT 5d 448, thdng 10 ndm 2014)
Ciu 1. Yl ac: -1 < 0 n6n PT 1u6n c6 hai Gid tri nhd nhiit
nghiQm ph6n biQt x1tx2.Ta c6 t - rr-1 = 0 f2 =6[a+b +JO+r *Jr*o)' =2(a+b+c)+
<> q +1 = xr2 suy ra x1>-1 vd
33x, + 25 = 9(xt +I) + 24xr+ I 6 = (3x, + 4)2
=P(x,) =3*r-JYrr*N =3xr -(3xr +4):4
(Do 3x, +4 > 0 vdi x, > -1).
Tuong at P@) :4 .Ydy 4x) = 4x) (dpcm).
Cffu 2. a) Ddp s6: x =
b) Di6u ki6n: ry 2 0.
t+Jt:
OC n9 PT c6 nghiOm thi x+ y >0 .
Tt PT thri nh6t ctra hQ ta c6
x2' + y2' - .l- -xy + 6,,!xy +9 (1)
Tt PT thri hai cira hQ suy ra
r--:-
64=1tlx2 +7 +11y2 +7)2 <2(x2 +7 +y2 +7)
rct hqp (1) ta c6 (r[i -3)' < 0 e ,[*y =3
Tnd6 x+ y - 6. Ddp s6: x= ! =3.
Cflu 3. a) Tti hQ dd cho ta c6
(*+Y)2 =22 +2(x+Y-z)
r-e
(x + y - z)(x + y + z -2) = o e l' = :* o'
lz=z_x_y
Thay vio PT ban dAu ta c6 k6t qui:
v =3,! = 4,2 =-5 ho{c x = 4,! =3,2 : -5 .
b) Gid tri lon nhiit
P = 1"{o * 6 + Jn * + r[, + o)' s 6(a+b + c) = g
1 ;J
z(lG. q@.
")
+,{@ +
"11"
+ o1 + I @ + rY,, + q)
Ta c6 (a + b)(b i c) = b2 + ab + bc + ca) b2 .
Ding thric xhy ral<hi ab + bc + ca = 0.
Tucnrg tu cho 2 BDT kJtdctac6: F >4>F>2.
Ding thirc xtry rakhi c6 mQt s6 bing 1, hai s6
b5ng 0. B
Cffu 4.
A M
a) . Tam gi6c ACE cdnt4i Cn6n
r
C4E =9ff -1
2
BAH .
7
+ BAE:i + AE ld phdn gi6c cua
2
Tucrng W AF li phdn gi6c cua CAH . Suy ra A,
Iy E thdnghdng (dpcm).
o Do Clph dn gi6c cir. IdE , LACE cdnt4i E
n€n CI ld trung tryc cua AE, do db IA : IE.
Tucrng W IA: IF.V$y IE: IF.
b) Ki hiQu (O ld duorrg trdn dudng kinh EF.
Tri cdu a) ta c6 / li t6m duong trdn ngopi ti6p
MEF suy ru EIF =2EAF = 90o , do d61 c (q.
Do CI ld trung truc cira AE nln1am gi6c I2AE -^
cdn tAi 12 > I,AE = AEIz = 45o suy ra
iD =90o hay Iz e (Q.Tucrng tulr e (O.
Do d6 (O h tlucrng trdn ngopi ti€p N{21.
(Xem ti€p trang 13)
t.nn, or-rorn, '?EI#S 5
+ F <G. Oi"g thirc c6 l<hi a = b = c -

#rur Tui- sr,$1 vlo r^& u rrofu flr[ rl(triT?,r+' rf MNH
NAM HOC 20T4-2015
VONG I
Cdu l. a) Gi6i phucrng trinh
(3 - x) /t: + x)(e + x2l = +/st: - ry
b) Tinh
x
v
bi6tx> 1,y<0vd
(x+/xx3 -rr,r[-V-{
T
(r - r'a;;1 *2 y2 + *y3 + ya)
Ciu2" a) Gi6i hQ phuong trinh
(r20 philt)
b) Tim zz cl€ phucrng trinh (l) c6 2 nghi6m phdn
biQt xr, x2 sao cho 2lx, + 7 m(2 + x, + $1 = 59.
Ciu 4. a) Goi *=#^tA, 2'
Y=Jab ldn luqt ld trung binh cQng vd trung binh nhdn cta
2s6
duong a vd b. Bi6t trung binh cQng cta x
vir y
bing 100. Tinh =Ji*Ju.
^s b) Gia st hai tlai lugng x, y ti lQ nghich (x, y lu6n
duong). N6u r tdng ao/o thi y giAm mo/o. Tinh m
theo a.
Ciu 5. Hinh r.u6ng ABCD co AB :2a, AC cit An
t4i 1. Gqi (6) la tluong tron ngoai tiiip tam gi6c
CID, BE ti6p xric vdi (6) tqi E (E ldtic Q, DE
cit A,a tqi r.
a) Chrmg minh LABE c6,n.TinhAF theo a.
b) BE cit AD tqiP. Chimg minh tludng trdn ngoai
tii5p tam gi6c ABPti6p, PxuDc voi CD.finh {.
Q EA c1t(e)@i M(MV,hircE"). Tinh AMtheo a.
(150 phtit) a..6a..6,a.-6
b) Chrmg minh ring n6u r > 1 thi a + c vd b + c
kh6ng the dOng thdi le sii nguy6n til.
Ciu 4. Cho diiim C thay dOi tr6n ntra duong tron
duong kinh AB : 2R (C * A, C +B). Ggi Hh hinh
,.i chieu r,u6ng g6c cua C lfu AB; I vit J 16n luqt ld
tAm dudng tron nQi tir5p c6c tam gi6c ACH vit BCH.
C5c ducrng thtng CI, CJ cit,qn lAn luqt tqi M, N.
a) Chimg minh ring AN: AC, BM: BC.
b) Chung minh 4 di6m M, N, J, I ctng nim tr6n
m6t duong trdn vd c5c ducrng thing MJ, NI, CH
tl6ng quy.
c) Tim gi6 tri lon nh6t ctra MN vir gi6 tri lon nh6t
cira diQn tich tam gi6c CMN theo R.
Ciu 5. Cho 5 s6 t1l nhi€n ph6n biQt sao cho t6ng
cua ba sO b6t ki trong chfng lcrn hcrn t6ng cria hai
s6 cdn l4i.
a) Chimg minh ring tdt ca S sO da cho tl6u kh6ng
nh6 hcrn 5. , -.( ). ' b) Tet cA citc b0 g6m 5 s6 tho6 mdn d6 bdi md
t6ng cria chring nh6 hon 40.
NGUYflN DIJC TAN (TP. H6 Chi Minh) gicti thiQu
= -6.
Il; - r. r)(il7. ex,. ?) - rs) = o
t_
[r/x'+9+rry+7 =8
b) Hinh thoi ABCD c6 diQn tich ld tar6 lmet
vu6ng), tam giitc ABD ddu. Tinh chu vi hinh thoi
vd b6n kinh <lucrng trdn ngo4i titip tam giilc ABC.
Cf,u 3. chophuongtrinh ni +(m-3)x+2m-l=0 (l)
x+3
a) Gi6i phuong trinh khi m: -1.
a".6a.-6a..6 VONG 2
CAu l. Chophuongtinhlz'z+ Sf -zmx*6m:0(l)
(rz ld tham s6).
a) Tim m sao cho phucrng trinh (1) c6 hai nghiQm
phdn bi6t. Chimg minh rdng khi d6 t6ng cria hai
nghi€m kh6ng thC ld s6 nguy6n.
b) Tim m sao cho phuong trinh (1) c6 hai nghiEm
xy, x2 thoirmin tli6u kiQn (x,x, - "!i3 a)o =rc.
Ciu 2.
lz(r* *,[i' =evJi
1) Giai hQ phucrng trinh ] ' "' ',
[z(t+yG)'=g*Ji
2) Cho tam gi6c ABC v$ng tai A vli c5c dudng
phdn gi6c trong BMvit CN. Chimg minh b6t ding
(MC+MA)(NB+NA) ,O, "
>Z+2Ji.
MA.NA
Cflu 3. Cho c6c s6 nguyOn duong e, b, c sao cho
1l I
abc
a) Chimg minh r6ng c + b kh6ng tfr6 n sO nguy6n t6.
TORN HOC 6 t.ruaLa sd aas trr-zorel

Chu6'n [i
cho lrithi
6t nshitp THPI
vi thi uio
Oai hoc
TNUG TAil IAIII GIC
NaOYEN TRUOITG sON
Oi tht nyAn sinh vdo Dsi hsc, Cao ddng hiQn ,: nay, theo cdu tnic cfia BQ GD&DT, cdc bdi
odn vd ea dA ffong mfit phdng thaong xuyAn
xudt hi€n. D€ gidi qrryiit cdc bdi todn nay cdc
)J thi sinh cdn ndm virng mQt ttnh chdt hinh hoc
phdng ndo d6, diiu dd ldm cho cdc thi sinh cdm
thay tilng tung. Bdi vt€t nay mong mudn ghip
mt chut kidn th*c nh6 cho cdc thi sinh sdp
bwdc vdo ki thi tuyAn sinh Dqi hpc, Cao dting.
I. KIEN THUC CAN NIIO
Cho tam gi6c ABCnQi ti6p duong trdn (.1),11ld
tr.uc tdm cira tam gi6c. Gqi E, F ldn luqt ld
chAn dudng cao hp tir B, C. Mld trung di6m
cira c4nh BC (h.1).
NhQn xet 1.78 =27il =ZT grong d6 -r H
trung di6m c;iula do4n AIt).
P
NhQn xit 2. IA L EF .
C6 nhi6u c6ch chimg minh nhan x6t ndy, c6
th6 sir dpng nhpn x6t 1. Sau tl6y li mQt c6ch
kh6c:
Ta c6 CFB = CEB =900 n6n th gi6c BCEF
nQi ti6p dunng trdn, do d6 frE =frE .
D.vngAt ld ti6p tuyi5n cua duone frdn (1). Khi d6
A -^
ACB = BAt . Ti d6 AFE = BAt ndn At ll EF.
Suyra IALEF.
NhQn xit 3. Gqi P ld giao diOm thri hai cria
duong thdng BH v6i dudng rdn (1). Khi d6, P
ld di6m eOi xtmg ciaH quadutrng thhngAC.
NhQn xit 4. Gqi Q ld ch0n ducrng cao h4 tu
dinh A cua A,ABC. Khi d6 H ld tdm nQi ti6p
cua LEFQ.
Chrmg minh c6c nhdn xdt 1,3,4 h kt6 dC Aang.
rr. rrri Du AP DuG
QThi dlr 1. Trong mqt phdng voi h€ truc tea
dQ Oxy, cho dudng trdn (C) ; r' + y' = 25 . .:
ngogi tiep tam giac nhon ABC cd chdn cac
dudng cao hq t* B, C lin laqt td M(-l; -3),
N(2; -3). Tim t7a d6 cdc dinh cita tam gidc
ABC biA ring di€m A co tung d0 dm.
Lli gif,i (h.2)
C{ch 1. Duong
tron (C) c6 t6m
O(0;0), b6n kinh
R=5. Ta c6:
ffi =(3;0).
Theo nhdn xdt 2,
tac6 OALMN.
Khi d6 ducrng
thtng OA qua O,
nhan MN = (3;0) ldm vecto ph6p tuy6n c6
phucrngtrinh: x=0.
Toa d0 di6mA h nghiQm cria hQ phuong tinh
[x=0
I . a _ _. VilcotwrgdQdmn0n,,4(0;-5).
lx' + Y- ='25
Hinh I
Hinh 2
ss *, or-roro
T?3I#?E 7

Ta thdy Vfr = (L;2,Vfi = (2;2) lin luqt ld
vecto chi phucrng cira rtudng thingAC, AB.
Phucrng trinh duong thing AC: 2x + y+ 5 = 0 .
Phuong hinh duong thtng AB: x - y -5 = 0.
Toa d0 tli6m C ld nghiCm cira h0 phuong trinh:
[2x+y*5:0 [x:0: v=-5 i el >C(-4:3).
lx- + y- =25 lx = -41y =3
Tqa d0 di6m B ld nghiQm cira hQ phuong trinh:
I x- v-5 = 0 [x:0: v =-5 i," el lx'+y':25 [x=5;y=Q =B(5;0).
Do 7E .Ve > o,Ee .Etr > o,cE.e) > o , n6n
tam gi6c ABC nhon. YQy A(0;-5), B(5;0),
c(a)) . a
Chch 2. Gi6 sir H(a;b) ld tr.uc t6,m LABC.
Goi P, Q l6,n luqt ld giao tli6m thri hai cira
cluong thing CH, BH voi ducrng tron (C).
Theo nhqn xdt 3, P, Q ldn luqt li di6m dOi
xr?ng cria H qua AB, AC. Vfly
P(4 - a;-6 * b),Qe2 - a;-6 - b) . Ta c6 hQ:
Y6i H(I;-2) th HM =(-2;-I),
Hfi = (1;-1) + ufr.nl: -l < o.
Yor H(I;-10) thi Hfr =G2;7),
Hfr=(1;7)= HM.HN:47>0.
Suy ra BAC lil g6c tu (mAu thu6n). Do d6
H(t;-2).
Tac6 AC: 2x + y+5 = 0 ; AB: x- y -5 :0.
Gqi 1ld trung di6m cua ,BC. Tt 7E =ZOi
su'y ra ,(l):',)1 ). rrr.rd: x+ 3y- 5:0. - -/)
Tri d6 suy ra c6c tli6m A(0;-5), B(5;0),
C(a))th6a mdn y€u cAu bdi to6n. D
QThi dqr 2. Trong mfit phdng voi h€ trc tga
d Oxy, cho tam gidc ABC cd A(1; -l), nryc
tdm H(2;1), BC =ZJi . Gpi E, F tdn lwqt ld
chdn d*dng cao hi t* B, C cila tam gidc ABC.
Lqp phuong trinh &rong thdng BC, bi6t trung
di*m M cua BC nim ffAn &rong thdng
d: x -2y - | : 0 vd M cd tung dp duong.
Ldi gi6i. Do M thuQc ducrng thing d n€n
M(2a+l;a)(a > 0). Ggi I ld tdm tlulng trdn
ngopi ti6p tarn gi6c ABC.
TacoZfr =(4;2),AH =2Ji -Afi
Yd
=27fr,
suy ra I(2a-l;a-l),IM:16. V M ld trung
di6mBCn6n IM LBC. Do tl6:
,o, =(T) + IM2 = lo = (2a+t)2 +a, =to 2 )
o5a2 +4a-9:0 <>a=l hod'5c o:-2.
Do a> 0 n6n a: I > M(3;l). Dudng thing
BC di qua M(3;t), nhfln 1fi =(a;l lim
vecto ph6p tuy6n c6 PT: 2x + y - 7 : 0.
@Thi dqt 3. Trong mfit phdng voi h€ trUc tqa d0
Oxy, cho tarn gidc ABC cdn tai A, tryc tdm
H(-3;2). Gpi D, E ldn lu.ot ld chdn daong cao hi
tu B, C cila tam gidc ABC. Bt€t diem A nim tAn
doons thdng d : x -3y-3 = 0, diem F(A))
thuQc &rdng thdng DE vd HD:2. Tim t7a dQ
diAm A.
Loi gini. Do A nim h6n ducrng thing
d: x-3y -3 = 0 n}n A(3t +3;t) va / e lR .
Ftr= Qt +5;t -3), H): (3t +6;t -2). Do
tamgi6c ABC cdntqiAn€n AH L DE .
Tac6 ADz = AHz *HD2. Khid6:
FA2-FH2=DA2-DH2
= FA2 *FH2 : AH2 -2HD2
> (3t + 5)'z + (t -3)' - 2 = (3t + 6)2 + (t - D'z - B ) t - 0. Ydy A (3;0). tr
Ldi binh: MOt tinh ctr6t ttli vi dugc sir dgng
trong thi dr,r 3, thudng gip d6 ld: Cho 4 di€m
{t+-rf +(6+ b)'=25=[, =t"b=-2
l{z+a)' +(6+b)'=25 la=l,b =-10
Ldi binh: R6 rdng khi ldm theo c6ch 2 thi diOu A, B, C, D, n€u AB L CD thi
ki€n tung d0 di6ml 6m ld kh6ng cdn thi6t. ACz - ADz = BC2 - BD2 .
^ TORN HQC U - ctrdifta s6 aas (n-2ou)

rpa iti Ox.r,, cho taw giric A{3C c:dn tqi A,
hai dvt)'ng ua BE, CF cdr nhau tqi Ilt2;21,
biet {tE,*3. T'irn taa d$ dtnh A c*cr taru giac
ABt' bi1{ riinh A thu6c du'tntg thiing
cl : x-r y + 12 =' 0 ru khodng tfick tir .A ct,in
.hrong tlwng Ef' nho nhat.
Ldi gi,fii Ta th6y ring H kh6ng thuQc ducrng
thing d . Do A nim tr6n duong thing
d: x+ y +12 = 0 n6n A(t;-t -1,2) vor / e IR..
u)= Q -2;-t -14) .
Vi tam gi6c ABC cdn t4i An€n AH L FE .
X6t tam giric vu6ng HAE ta c6:
AE2 = AHz -HE2 =Q-2)2 +1t+t+1, _e
=2t2 +24t +t9l
vit cl(.4.EF:nA::H. = Wtl2t' +24r +2oo
=Jrtut +2oo - Jrt%t +roo
9
2(t +6)2 +128 - Jra.* .nB
>r28-9 _tlsJ?
8J, t6
Ding thric xiry rakhi vd chi khi t = -6 .
Khoing c6ch tu A d€n EF nhb ntr6t Uang
t'ro^ ll
khi l(-6;-6).D
16
* fis{ dr1 5. }'rong wiit phareg viri hQ trUC iett
de {}}:.y, r:krs funt. giac Af}C cdn tqi dinh A. Goi
ht li t'r ung diem ctw dottn thang AB. {ioi /*t rr
E('i;l),,t| ;.:i llan lwpt ld r:hdn dwmg cao ii - 5J
ha tu-cac dinh 8, C ctia tarn gidt ,48C. tim
*la dp c{ta dir*t A hiil ring phuong trinh
&d"*g thiing C] /ri 2x + v * 13 = 0.
Ldi gidi Gqi G ld trgng t6m A,ABC. Do
LABC cdn tai A nln AG chirt';, li tlucrng trung
tr.uc cua tlopn thing EF. PT AG h
-3x+y+12=0.
Tqa dQ di6m G ld nghiQm cira hQ PT:
)lz x+'v -13: O ei lx:5^=G(5;3). f-fx+ y+12-0' |.y=:' --'-l
Ae AG > A(a;3a-12),C eCN = C(c;13-2c).
Do G li trqng t6m tam gi6c ABC n6n suy ra
B(15- a-c;8-3a+2c),
CAS - a-2c;-5 -3a+ 4c)
EEfg - a - c;7 - 3a + 2Q,Ee@ * 7 ;12 * 2c)
Ta c6 AG I BC;EB L EC n€n
ItS - a -2c +3(-5 - 3a + 4c) = Q l
[(S - " - c)(c - 7) + (7 -3a + 2c)(12 - 2c) = g
aa*c-7.
Khi d6 A(7;9),8(l;l),C(7;-l) . J
*Tki dqt 6. Trong nzst phdnS; v,ni h( truL' t{)a
d$ Oxy, *ut tarn giac AB{ n$i rii;t dwdng tr':)n
tum l.1;21, b(rn kfnh .R * 5. Cltttu &rdng cao
ha tir B, C r:fict iant p:itit: ,4tiC ltin lwr,tt l&
ff(3;3j, i.(0;-i ). tiiir phunng trinh du'rhtg tron , ..: ngoqti tidp t{r g;,;r' P{:UX, hi,6t ring tung do
diew A dwmg.
Ldi gidi Ta c6 Kfr = Q;4) .
Theo nhdn xdt 2, ta co IALHK. Do cl6
duong th6ng IA c6 phuong trinh ld:
YA'y A(-3;s).
Eucrng thing AB c6 phucrng trinh:
2x+y+1=0. Euong thhng AC c6 phuong
trinh: x+3y-12=0. Dudmg thing BH c6
phucrngtinh: 3x -y-6=0. Eudng thtngCK
c6 phucrng hinh: x *2y -2 = 0.
Khi d6 d6 dang suy ra B(l;-3),C(6;2).
Gqi -rh tAm duong tdn ngo4r tiCp tu grfuc BCHK
r.BC. Khi
" '(:,-:)
lx=l+4t
] ^ - ' '-
ft € IR). I thuQc duong thhng IA ly:2-3t'
n€n A(l+4t:2-3t), voi ,.?.
J
f. .l
Tac6 IA- 5 <> l6t2 +9t2 :25= I ' -'
[r=-1
i{:*r*?.,,-_W"HBE s

Phuong trinh dudng tron ngoai ti6p ru gi6c
/ n:,2 f 1' 25 BCHKId: It x-1?1l *t ' v+)-l | --.) --l
Ldi binh: C6 rdt nhiOu c6ch x6c dinh tga dQ
tdm ducrng trdn ngoai ti6p tu gr6c BCHK. Ta
c6 thO x6c dinh toa d0 t0m Zdudng trdn ngopi
ti6p tam gi6c AHK. Sau d6 suy ra toa d6 tdm J
bing c6ch su dpng nhQn x,!t l.
*Thi dy 7. Viet phaong trinh ba canh ctia
tctm gidc ABC bi€t E(-l;-z), F(2,2), Q(l,2) ldn
lrcctt ld chan ba danng cao ha tii A, B, C cua
tam giac ABC.
Ldi gidi. Theo nhdn xdt 4, truc tdm H cin tam
gi6c ABC chinh ld tdm dudng trdn nQi titip tam
gi6c EFQ. Do d6, ta tim tqa d0 di6m ll nhu
sau:
Ggi U ld giao tli6m cua AE voi QF.I{hi d6 ta
c6:
tlQ
=
!Q- :!= uo : -4 * - u( !,2)
UF EF s ' s (3,
Z HU FU I I
l_'_ HEFE33
='l: + -
HU = -: HE + H(0;l) . Phuong trinh duong thingABlit-x+y * 3 :0.
Phucnrg trinh duong thhng AC lit 2x + y- 6 : 0.
Phuong trinh dudng thing BC ld x + 3y + 7: O.
rrr. BAr TaP Tu LUYEN
1. Trong m[t phiiLng vdi hQ tr.uc tga d6 Ory, cho
tam gi6c ABC voi C(_3.0), ducrng thlng di qua
chdn ducmg cao h4 tir A, B c6 phucrng trinh ld
7x+y+5:0. Vi6t phuong trinh rlucmg trdn ngoai
d6p tam gi6c ABC bi6t ring M$;1) thuQc duong
trdn d6.
2. Trong m{t phing v6i h0 trgc tga dQ Oxy, cho
/ z r
tam giSc ABC can hi A. goi M vit Kl l;] lrA, [5' s ,
luqt ld chdn ducrng cao h4 tit A vd B cua tam giitc
ABC.Di€mE(-3;0) ld di6m d6i xrmg ciua M qua
trung di6m i/ crja carr,h AB. X6c dfnh tga clQ c5c
clinh cira L ABC bi}t U nim tr6n cluong thing
d:4x + y -2:0.
3. Trong mflt ph[ng v6i hQ tnrc tga d6 Oxy, cho
tam gi6c ABC cdn tai ,4, dudng thing BC co
phuong trnh 2x+y2:0, E, F lAn luqt ld ch6n
tlucrng cao kd ti B, C ctra tam gi6c ABC. BE c6
phu<rng trinh x+y+1:0, di)m MQ;I) thuQc
cluong thing CF. Tim toa <10 c6c tlinh cria tam
gi6c ABC.
4. Trong m[t phing v6i he trirc tga dQ Oxy, cho
tam gi6c ABC niri tiiip ducrng trdn c6 b6n kinh
/rr z
- .3 3'
C6c ditlm K(;4),H(3:1) 16n iuqt ld chdn ducrng
cao ha Li A, B c.6atam gi6c ABC. Tim toa tlQ c6c
clinh cira tam gi6c ABC.
5. Trong mpt phing v6i hq tqa d0 Oxy cho tam
gi|cABC c6 chdn cluong cao hp tuB, Cxudng
canh d6i diqn lAn luqt ld K(-2;2), E(2;2).
oiA", p([1s9'5;?l ] rir hinh chit5u vuong s6c
ctn[nt6ngBC. Tim toa c10 c6c clinh cira
AABC.
6. Cho tam gi6c nhon ABC vbi AK, CD ld hai
rludng cao vd H ld Wc tdm A ABC. BiCt PT
cludng trdn ngo4i ti6p tam gi6c DHK:
(* - 2)' + y' = 5 , kung di6m cta AC ld P(7 ;5).
Tim to4 d0 c5c di€m A, A, C biSt ring BC di qua
di6m Q[1;a) vd hodnh d0 diem D lon hon 3.
7. Trong mat phSng toa tlQ Oxy, cho IABC c6
A(2 ;3), chdn hai cludng cao kd tit AvdB lAn luqt
( -t _rt /-t 'r't ld Hl ' : " l. Kl j:11. coi 1 ld tam I13 13l Ir0'r0,
dudng trdn ngo4i ti6p LABC, E lir mot tlii5m
thu6c cung nh6 AB. Ke EMLBC, ENMC. Tm
toa d0 di6m E aC tnV cO dp dei lon nh6t.
8. Trong mpt phdng v6i hQ tr.uc tga dQ Oxy, cho
tamgi6c ABC. Gqi ,tr,rl,r(!'gltar luqr ld [5 5 )
chAn <luong cao h4 tu c5c dinh B, C cintam gi6c
ABC. Tlm tqa d0 cira dinh Abilt ring phucrng
trinh dudng thing BC lit 2x + y -73 =0 vd diOm
B c6 tung dQ ducmg.
9. Cho tam gi6c ABC c6 t4rc t6m 11, iluong trdn
ngoai titSp tam gi6c HBC c6 phucmg trinh:
*'+y'-x-5y+4=0. 11 thuQc ducmg thing
L:3x-y-4=0, M(2;3) ld trung di€m AB.
Tim tga tlQ c5c tlinh cira tam giSc ABC.
TOAN HOC 10'cludifi@

THUSUCTRUOCKiTHI
aEs6z
(Thdi gian ldm bdi:180 philt)
C0u 1 p aia4.Cho hdm sO y=; -512 +9x+m
(m ldtham s6) cO dO ttri 1C.;.
a) Kh6o s6t sg bi6n thi6n vd vE d6 thf cira hdm
sO (C) l<hi m:0.
b) Tim m d6 tdn t4i ti6p tuyOn v6i d6 thi
(C*) di qua di6m AQ;O) vd cit dudng tron
(S) c6 phuong trinh (.r+1)2 + (y-2)2 =25
theo m6t d6y cung MN c6 dO ddi nh6 nh6t.
Ciu} (I diA@.Gi6i phuorg trinh
cos4x -Jisin2x+2 =vtJ;'
sin4x - J3 cos2,
Cflu 3 (1 die@.Tinh tich phdn
r' - .ili' 2x' -4x+3
) r*-I'[-* +2*+3
I_ dr.
Cflu 4 (1 diAm).
a) Gi6i phuong trinh
bg^rr(t-4) = logro,, $ + 2)2+ 1togro,, (r-:)a
b) Cho s6 phric z thIamdn z+(l-Ap:2{l-21).
Tim phAn thuc vd phAn 6o cua s5 phric
o = z2 -32.
CAu 5 Q die@.Trong kh6ng gian v6i hC toa
dQ Oxyz, cho mflt phing (P) :2x + y - z = O
vd hai cluong thing
thtng L2 Vit5t phucrng iludng thing A di qua
M, vt6ng g6c voi A1 vd tpo voi mflt phing
(P) mQt g6c 30o.
Ciu 6 Q diA@. Cho hinh ch6p S.ABCD co
ddy ABCD le hinh ru6ng, SAL(ABCD),
SA=a. DiQn tich tam gi6c SBC bing
2t;
ry. Tinh the tich t<tr5r ctrop s.ABCD
2
theo a. Ggi 1, J ldn luqt ld trung di6m c6c
c4nh SB vd SD. Tinh khoAng cdch gifia hai
ducrng thing AI vd CJ.
Cffu 7 (1 diAfi.Trong m6t phing vcri hQ toa
d0 Oxy, cho hinh thoi ABCD c6 t6m I(2; l)
vir AC:28D. oi6m u(o'l] ,n O. clucrng ''3l
thing AB, N(0; 7) thuQc cluongthtng CD.
Tim toa d0 di6m ;' bi0t rang EF = 5Ei vit
di6m B c6 tung dQ duong.
Cfru 8 (1 die@.Giai he phuong trinh
lJr+3+Vr-2-"lyo+5=! 1t'^
lx2 +2x1y -2) + y2 -By + 4 = 0
CAu 9 (1 die@. Cho a, b, c ld c6c s6 thuc
ducrng th6a min obr=!- Tim giri tri nh6
6
nhAt cira bi6u thric
x-4 v A.,.1
1
P_
1) 7 )-)
22
di6m N tr6n dudng thing A1 sao
N ddi ximg vdi nhau qua duong
oa 7zb+ 1)(3c +t)' l6b4 (3c + 1)(a+ 1)
1
I-' 81ca(a+l)l2b+l)
PHAMTRO- NGTHU
(GV THPT chuyn Nguydn Quang Di€u, D6ng Thdp)
aa nn, or-rorr, t?EI#S 1 1
-J
x-6 aAr.".-1-
phing (P),
cho M vit

iruohro uAnr oAt.oE s6
Cflu l. b) Ta c6 d: y: -x + m (m *2) vit I(-t; D.
PT hoinh tl6 giao diiim ctra (H) vd d lit
x-l )
x+1
(do x = -1 kh6ng thoa mdn).
Ta c6 A = *2 +8 > O;Vm n6n (.F/) vd d lu6n
cit nhau tai2 di6m A,B vot A(xr;-xr+m);
B(x2;-x2 + m) trong d6 x1,x2 ld 2 nghiQm
cira PT (1) thoa mdn xr+xr=m-Z xfiz:-m-L
Tri ^s/rB =zJi >d(I;d).AB=4Ji
lml
,12
e*'1*' +8)=48e m=-2 (do m*2).
Cfiu 2. DK: cosx.cot{ * 0. PT dd cho tuong
2
. cos2x . sinx . r dUongvo'l
cos.r cos.r
-+-=Zslnx+[
<> cos 2x - sin 2x = cos.r - sin x
//n)
o
2x+L l= cosl x+- l. 4/ 4/
"o.[
Ddp s6: x = khn;x = -I6 + f!,3t' .2.
i/iT:r-(r+x)+"{-JiIx;
Cf,u3. Tac6 IL=lim x-+0
f+i
c6 1 c{p bdt tay vd m5i nguoi kh6ng bfit tay
vg/ch6ng minh). Ta co 2n(n -1) = 40 e n - 5 .
n 5n-11k
b) Ta c6 tr(x) : lc!1-t!*k zk * 2 . Theo
k=0
bdi ra zdeD*t *+Ce't =2n. Do 2" >o
vd acf;>2c, n€n n chin. Khi d6 n=2k
(k e N-). Thay vdo du-o'. 2c ry - 22k-4.
Suy ra k = 2 o n: 4. HQ si5 cira s6 hpng thri
4 c6n tim ld -32.
Cf,u 5. Ta c6 C(0;0;c) voi c > 0. Do
BC=CA=AB n€n c2 +9:18<>c=3. Ggi
G ld t6m LABC ta c6 G(l;l;l). PT duong
thAng A di qua G vd vu6ng g6c vtri (ABC) liL
*-l =Y-l :Z:). yi,S e A n6n .s(s;s;s). 111
I
Ta c6 lsc.stnc)=9<> sG=2J1<)s=3
3
hoflc s = -1. Do vQy S(3;3;3);5(-1;-1;-1).
Cflu 6. Ta c6 0<sd -5,4 =td -Irl <AE
nOn tam gi6c SAB vu6ng t1i S. DAt
HA=HO=x ta c6 OB=2x. Theo dinh li
c6sin ta c6 BH =*J1;BC=2xJ1. Ta c6
SA2 + SB2 = AB2 o o2 + o' - x2 +7 x2 =!2x2
or=far. Ir(,t1u. d+o, rY, t*=_1l .J8, o^7fl .nn, -=l-23 S-.
Gqi 1 li tAm duong trdn ngopi tii5p tam gi6c
SAC thl1 li trung di6m cua AC.Do HI ll OC
n€n Q1Xxx11=4r;fwD=IlL trong d6 K,L lAn
luqt ld hinh chi6u cua H tr6n c6c duong thing
l;
Co vir SK. Ta c6 HK=J+r!21 =o. 2'
HK.HS JN
,tHK'z + HS'z ll
l_-r
(x+lXl+Jt-rl_] 2'
Cf,u 4. a) Ggi s6 cflp vg ch6ng ld n(n>2). Ta
c6 si5 luqng cbi blt tay li C7, - n :2n(n -l)
(do m5i c6ch chgn 2 nguoi trotg 2n nguoi thi
. ^ TORN HOC 12 tcftdiU@
dtrttco)) -- HL:

CAU 7. Do tam g76c ABCvuitgtaiA c6 H e (Q
vit CA ld ti6p tuy6n cria (Q n6n B e (Q. Ta c6
Ac=2sA-cB = J7r ncn BH=L=J1. ,l,qB +,qC
='
- IBH =^,13
Gi6 su B(a;b)(b > o). Khi d6 {u'
.,lt -tf *B =, oo=!*=fi.ve" Bfl,f).
la-zf +8:3 2' 2 " 2' 2 )
Cflu 8. DK: y > l;*3 - *2 +1 > 0. PT thri nh6t
cira hQ hrong tluong voi
1x + lli)z + (y,[y *t)2 = 2(x * li1.y,[y a
<=' x+#: J(r-l)' * O:
o
o Vi = ,1-r., {',=
l
[x2 =1y-t;3.
PT thi hai ctra h9 trd thenh
*4 +# -f +l: x3 +l
e *4 -*3 +*2 -1'rt[; -; a1-x2 =o
/
e (xa -r3 +r' -,l[t-ffi;.] : o
el [(r,-_t )(r'+x+1):0< +[xl= 1^ (dox>0).
lrl*3 - *2 +7 :l- x2 [x = 0
Edp si5 : @; y) =(0;1); (x;y) = (I;2).
caue. ru,o (.+.+rY++
, 1 A
'*{a =l +3c2 +28=3a2 +2rt +5c2 >2(a+b)(a+c).
MIt 4a
kh6c
a2 +bc+7 2a2 +a2 +(b+c)2
' /- 8a42 /- 2a2 +-:
2a(b+c) a+b+c' t[o1U+4'
DovdvP<2 - 5
=*L- 3
a+b (a+b)' ,!a(b+c) a(b+c)
| .( r r')'.1_r( | _1)'.t =--l
I f--{l :--l < 5 -lo+b s) ' 3 -[ra1a*"y s) - rs
I{hi a=3;b=2;c=lttri F=!. Vfy maxP:4.
rpANOu6cmAr
(GV THPT chuy€n Hd Tinh)
HUCTNG nAN cTAI DE ...
(TiAp theo trang 5)
Taco LBAM = A,BFM (c.g.c)
ndn ffii[ =fu =%f suy ra EF LFM (dpcm).
Cflu 5. C6c s6 tlugc vi6t tr6n bing h 1, 5, 11,
23,47,71...
. Nh0n x6t ring c6c sO dugc viiSt tr6n bing (tni
sO t; cO tinh ch6t chia3 du2.
ThQt vQy, c6c s5 dAu ti6n tr6n bang (tru sO t;
c6 dang (3k+2) .
Ntiu str dung s6 1 d6 vi6t thi sO moi c6 dpng:
(3k + 2).1 + (3k + 2) + | : 6k + 5 chia 3 du 2.
Ni5u kh6ng sri dung sO t eC vitit thl sO moi c6
&ne Qk + 2)(3m + 2) + (3k + 2) + (3m + 2)
ckna3 du2.
Tathdy 20152014 : (3.672-17zor+ chia 3 du 1
n€n kh6ng thc vict dugc s,5 20152014.
o Do z : xy + x + y nln z t I : (x + L)(y+ 1) (1)
N6u cQng th6m 1 vdo c6c s5 duo. c vitit trCn bang
thi dugc di,y cdc s6 2, 6, 12, 24, 48, 72,...
C6c si5 dAu ti6n c6 dpng 2-.3' ndn tu (1) suy
ra cilc si5 dugc vitit th6m cQng v6i 1 cfrng c6
dnng d6.
Mat k1r6c 2015 + | = 2016 : 256.63 : 2a.32.7
n€n kh6ng ttr6 vii5t durv. c sO 2015.
Tr/ HoU SON (^Sd GD-DT Hd Tinh) gioi thiQu
ta *,,rr-rorn, t?lI*Hff
1 3

PHEP CONG HAY PHEP NHAN
Nguy6n Dinh Huy (GV THPT chuyAn NguyAn Quang DiAu, Ddng Thap)
OD {rJdi todn T6 hgp ngdy cdng xuAt hiQn nhi6u
hon trong c5c <td thi HSG Qu6c gia cflng nhu
Qu6c tti, nhim giirp c6c em hgc sinh ti6p c6n bdi
to6n ndy m6t c6ch bdi bin vd chuyOrr sdu hon, t6i
xin gicri thiQu il6n c5c em mQt sO vAn tl6 li6n
quan. Mo ct6u ld mOt 1d ndng g6c cira bdi to6n
d6m: cQng hay nhdn?
Thi du 1. Xdc dinh s6 lon nhiit thu duoc khi x6a
di I00 chir s6 trong sd sau:
1234567 8910t I t2t3 .. . 99 I 00,
vdi sii tAn duqc tqo thdnh t* cac sd nguyAn trb I
d€n 100 xdp theo t(ilr try t* trai sang phdi
Lfr gidl Tabdt cldu voi mQt vdi ph6p tlOm. C6 9
s6 c6 i cht s6. Tt 10 d)n99, c6 99 - 10 + 1 : 90
si5 c6 hai cht sti. Do d6, con sO t€n c6 g + 2.90 + 3
:192 chtr s6. Sau khi x6a di 100 chft s6, ta c6
tlugc s6 g6m 92 cht s6. V.oi.b6t qi hai sd c6
ctng s6 chfi s6, sd c6 cht s6 tlAu l6n hcm se l6n
hon. Do tl6, s6 chirng ta cAn tim ph6i Ut Aiu
bing cdng nhi6u s6 9 cang t6t. Vi v6y, clAu ti6n ta
x6a 8 cht s6 ngodi cr)ng b6n tr5i. Sau d6, ta x6a
chu6i tOt1 12... 181 g6m tong cQng 9 x 2 + 7 : 19
cht s6. Tuong t.u, ta x6a cfurdi 202722...282,
303132. . .383 , 404142. . .484. Vay ta dd x6a 8 +
19 x 4: 84 cht sr5, hiqn ta thu duoc sO sa.,:
99999505t52s3...99t00 (*)
T.a. ;c.A n x6a: .1 6 cht sd nta. Kh6ng cAn ngtri , nhieu, chi cdn x6a chu6i 505152...57 g6m 16
cht s6 dC thu tlugc s5:
99999s8s96061...99100
Drmg qu5 nhanh, ban a. N6u chirng.ta cl6 5 cht
s9 e grmg clAu, gi6 fi l6n ntr6t c6 thO c6 cira chft
sO ti6p theo ld 7, thu dugc khi x6a chu6i
505152...565 g6m 15 cht s6. Ctrt sO cu6i ctng
c6n x6a li 5 trong 58. Do d6, c6u tr6l<yi ld:
999997 8596061...99100. tr
Thi dqt 2. Gido sa A, B, C vd D dang cho sinh
viAn E thi viin dap vi toan td hqp. BOn giao sw
dang ng6i thanh hang. Vi ld d6ng chu tich cila
tiy ban bi thi, giao sa A vd D phdi ng6i cqnh
nhau. W ld cii vdn cho sinh vi€n E, gido sa C cdn
ng1i cgnh.ddng cfu) tich cria W thi. Cdc giao sw
c6 th€ ng6i theo bao nhi6u cdch?
Ldi gi.-e. l ^5i6 vi tri md gi6o su C c6 th6 ngdi sE
thay ddi khi vi tri ngdi cira gi6o su I thay ddi.
Di6u ndy c6 th6 ldm chtrng ta b6i rOi ,ra di5*
kh6ng c6 phucmg ph6p. Mgo cua bdi ndy kh6ng
i. . , ).. phAi ld xdp vi tri ng6i .u t46 cho mQt gi6o su b6t
kj, tru6c tiOn, md ta ph6i x6p b6n giSo su viro c6c
vi tri ng0i c6 tu<rng quan v6i nhau r6i sau d6 mdi
x6p ch6 cho hg. Theo di6u ki6n d6 bdi, gi6o sul,
D vd C c6 th6 ng6i theo mQt trong cdc cSch sau:
(A, D, C), (C, A, D), @, A, C), (C, D, A). Yo1
x. .( . i mdi cdch x6p ch6 tr6^n, giSo su B c6 th6 ng6i o
gho dAu ho{c cutii. Do d6, c6u tri loi ld
2+2+2+2:8.4
Quy aic cQng. Niiu sw ki€n A c6 thiS xay ra theo a
cach vd sw ki€n B cd th€ xay ra theo b cdch thi sry
ki€n hodc A hofic B c6 th€ xay ra theo a + b cdch.
Cd the dd dang ap.dufs !, tudng trAn cho nhiiu
sry ki€n. Ta cd thA di6n dqt quy tdc c|ng bing
ng6n ngic tqp hW. Cho S ld mt fip hW. NAu
A.,,L,...,A, ld mOt phdn hoqch c{ta S thi
lsl=lal *l+l* . *lA,ltuong d6lxl m tu hiQu :. ,: so lwng phdn tt cia tQp hqp X
Thi dyt 3. Xdc dinh sd luqng hinh vu6ng.vd dtqc
sao cho mqi dinh cfia hinh vu6ng d€u ndm trong
mdng l0 x lD.tqo thdnh tir cac ddy di€m nhu
hinh 1. (Cdc didm cdch d€u nhau).
aaaaaaoaao
aaaaaaaaaa
aaaaaaoaao
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaoaaaaoaa
Hinh I
Ldi gidl Ta ggi 4 di6m Uat fy ta mQt b0 tft n x n
n6u chring ld cdc dinh cta mOt hinh rudng mi
cdc cqr.th hinh vudng song song v6i m6p cua
ming. Ta cflng gqi mQt hinh vudng voi c6c tlinh
TONN HOC 14 ' ;4i,}iEE ss.* ,,,rrq

hqp thdnh m6t bQ fli ld hinh wdng b0 tu. C6
92 =81 bdtu 1 x 1.
aaaaaaaaaa
Hinh2.
OC ttrAy ring c6 8 bO tu 2 x 2 trongming 3 x 10
nhuhinh 2. Khdng kh6 thay rdng cp 8 m6ng 3 x 10
nhu viy trong ming 10 x 10 d dC bdi. Do d6, c6
8'zb$ hi 2 x 2. Suy lufln tuong tu, ta c6 72 bO fi
3 x 3 vir cir nhu th6. V6i I a k aS, cO (tO_ tr)'
bg tu k x /r. Nhrmg di6m kh6 cria bdi ndy ld co c6c
hinh w6ng md cpnh cria chring kh6ng song song
vcri mdp cira m6ng. Tuy nhi0n, m6i trintr lu6ng
nhu vfy d6u nQi ti6p v6i mQt hinh vuOng bQ tu.
Hinh3. . . . . . . . . . .
Do d6, AO ACm dir thi phni dOm t6t cA hinh vu6ng
b0 tu vd mqi hinh vudng n6i ti6p. Khdng kh6
th6y r[ng trong m6t hinh w6ng bO tu /. x k, c6 k
hinh ru6ng nQi ti6p, bao g6m chinh hinh vu6ng
b0 tu. Vi dq, vcri k:4,ta c6 hinh 4. [m K...Y
Hinh4. N f,
T6ng hqp l4i,ta co dugc cl6p 5n bdi to6n:
9^9
I(ro -k)'.k =f(roor -20k'z+F)
quAn mdt s6 tha ba vd kh6ng bi€t thti' try cira cac
s6,ndy. 56 tlxil'ba nhQn mQt trong cdc gia tri tit 1
d2n 40. N€u m6i ldn thtr nhQp mdt l0 gidy thi
nhiiu nhdt mdt bao tdu rtd An tha net tit cd cdc
kha ndng?
Ldi gidi. Ta xem x6t 6 tflp hqp con. D[t:
A,={(*,n,24)It<x<a0}
4:{(*,2+,ti)fi<x<a0}
4 ={(n,*,24) n< x < a0}
,qo = {(24, x.17) I | <x < ao}
A, = {(tl,z+,x) I I <x < ao}
A6 = {(24,17, x) I t <x < ao}
Kh6ng kh6 de thiy tulem5l t6p con c6 40 phAn
4 Po d6, theo quy tic cQng, c6 40.6:240 ddy
sd tl6 thir vd cAn nhi0u nh6t ld 40 phrit. VQi qu6
r6i, bpn a! MQt tliiru quan trgng nh.mg dC bi b6
qua khi 6p dpng quy tdc cQng ld cdc.tQp hqp Ai
ph6i ld mQt phAn hopch thi quy tdc ndy mdi
tlirng, tuc lit Ai n Aj : A vot i * j. Nhtmg trong
bdi ndy, day s-6 {17, 1,7 ,24} thuQc vd ci l vd A3.
Tuong t.u, m6i d6y s6 {17, 24, l7), {24, 17 , I7),
{t7,24,24}, {24, 17,24}, {24,24, 17} ctng
thuQc vC hai tQp hqp n6n chirng dugc cl6m hai
hn. Oo cl6, chi c6 240 - 6:234 day dO thu, vd
cdutrhloi ching ld 39 phrit. D
Phdp c6ng vd ph6p nh0n c6 li6n qual mflt thi6t
v6i nhau. Phdp nhdn ld c6ch vi6t ngdn gon cho
ph6p cQng l[p nhiOu lAn. Vi ds, 3.5 :3 + 3 + 3 +
3 + 3 : 5 + 5 + 5. Dtng phdp nh6n mQt c6ch hiQu
qu6 c6 th6 girip hiOu ttr6u d5o dC giii cdc,biitobn
tl€m..C6 ngudi sE dO dang b6 qua ddy sd bi d€m
hai l6n trong bu6c cudi ctrng khi gi6i Tlli dq a.
C6 thO c6 nguoi sE t.u h6i li6u cdn ddy s6 ndo bi
dr5m nhi6u lAn khdng. Nghi s6u hcm mQt chft, ta
the;V rlnenhtng day s6 bt d6m nhi6u 6n chi c6
th6 ld ddy g6m {a, a, b} voi {o, b} : {17,24). a
vd b c6 th6 nhfln hai gi|trlld (a, 9)
: Ql,24) vir
(o, b) : (24,17). C6 3 c6ch sdp x€p c6c s6 a, a, b
lit (a, a, b), !o, b,,a) vd.(b, a, a). Do d6, c6 chinh
xhc 6 ddy s0 bi <16m 2 l0n.
Tq cflng c6 th6 gihi Thi fu 2binp phdp nh0n.
Diu ti6n, ta sdp x6p vi tri tuong ddi cho gi6o su
A vit D. C6 hai c6ch x6p lil (A, D) vd (D, A).
GiSo su C c6hai c6ch dC ng6i cpnh gi6o su.4 vd
D, d6ldngOi O b6n ph6i ho{c b6n tr6i.
(Xem ti6p trang 27)
trrn, or-rorn,
T?EI#S
1b
= 1oolfr -20>k'z +Zk3
k=t k=1 k=l
= roo.e'lo - 2o.s'to'rs *(4)' 2 6 2)
= 4500 - 5700 + 2025 = 825.
fh! dW 4.. [Tdi liQu Todn PEA, Richard ParrisJ
D€"c6 thA md filt A{ng d6 cila minh tai phdng.tqp
th€ hinh, An phdi,nho md s6. Ddy md s6 g6m 3
s6 vd hai trong s6 d6 ld 17 vd 24, nhtrng anh lgi

frt
lp*
'tr{
wN
cAc rcyr rl{cs
Bii 1'11449 (L6p 6). Cho 5 s6 nguy6n ph6n
biQt sao cho t6ng cira 3 s5 UAt t y trong chring
l6n hon t6ng cria hai s0 cdn lpi. Tim gi6 tri nho
nh6t ctra tich 5 s6 nguy6n t16.
NCWBN EIJC TAN
qr. ui chi Minh)
Bhi TZl449 {Lfp 7)" Cho tam grilc ABC vor
AB > AC, AB > BC. Tr6n cpnh AB cin tam
gi6c ABC l6,y c6c ei6m P vd E sao cho
BC : BD vir AC : AE. Qua D vir E ke DK
song song vfi BC vd EI song song voi CA
(K =CA,I eCB). Chimg minh rdng CK: CI.
W n0u cufN
1Cf ruCS uing Bdng, Q Hing Bdng, TP. Hdi Phdng)
Bdi'rc1449. Gi6i phucmg hinh
J;+3 J3r+1
-+-:
2
1+ Vx
NGUYENTATTHU
(GV THPT chuy€n Luong Thi! Vinh, BiAn Hda, Eing Nai)
BitiT4l449. Cho tam gi6c nhgn ABC vbi H lit
tr.uc tdm. M ld mQt di6m nim tuong tam gi6c
sao cho MBA=MCA. Ggi E, F lAn lugt ld
hinh chi6u vu6ng g6c cria M t:)n cdc cqrilt AB,
AC vd I, J tuong r1ng U trung tti6m .oha BC,
MA. Chtmg minh rlng c6c tluong thdng MH,
EF vdIJ dttng quy.
rE vnirAN
(SV lop Todn 48, DH Suphqm Hui!)
Bii T5/449. Tim t5t ctr citc cflp s6 nguy6n (x; y)
th6a mdn phucmg trinh xa * y3 : xy3 +1.
TneNvANHANH
(GV DH Phqm Vdn Ding, Qudng Ngdi)
CAC LO? TrTr,T'
Bei T6/449. Gi6i phucmg tdnh 8'-9lrl =2-T.
CAOMINHQUANG
(GY THPT chuyAn Nguydn Binh KhiAm, Wnh Long)
Bni T7l449. Cho tam gi6c ABC v6i ba canh ld
AB: c, BC: a, CA: b, b6n kinh dudng trdn
ngo4i tiiip ld R, b6n kinh duong tron nQi ti6p h
r.Chrmgminhring ;=ryH#
DINHVANTAM
(GV THPT Binh Minh, Kim Son, Ninh Binh)
Bei T8/449. Cho ba sO ttrUc ducrng x, y, z thba
mfln x > z .Tim gi|frnhO nhat cria bi6u thric
D_ xz , y2 ,x+22 I - ) -r-T-' y'+yz xz+YZ x+z
DIJONGVAN SON
(GY THPT Hd Huy Tqp, NghQ An)
TIEN TCTT OLYMFIC T$NN
Bili T9/449. Tim phAn nguydn cira bi6u thric -B
1592013 vot B =-3{ -+7- +1.1.. *-. 20t5^
NGO QUANGHUNG
(SV K54, lop KTD, DH N6ng Nghi€p Hd N|i)
Bni T10/449. Tim tdt cd cbc da thricfx) voi h$
L^ s6 nguy6n sao cho vcri mgi s6 nguy6n duong
n,/(n)ldu6c ctn3n -L.
NGUYENTUANNGQC
(GV THPT chuyAn Tiin Giang)
Bii T11/449. Cho dey s6 (x,) thoa mdn di€u
ki6n: [xs = 4,x1=34
lx,*z'x, = x1*r + l8' lo'*l' Vn e N
26
DAt Sn=Z*r*ktn€N..Chimg minh ring
k=0
vcri mgi sO t.u nhi6n 16 n,ta1u6n c6 5r 66 .
NGUYENVANTHANH
(GI/ THPT Chdu Thdnh A, Bdn Tre)
TORN HOC I6 *GIksiU@

di6m E,F dn luqt di chuytin tr6n c6c doqn CA,
AB sao cho B,E : CE. BE cht CF tqi D. Goi H,
K thri t.u li tr.uc tdm tam gi6c DEF, DBC.
Chrmg minh ring duong thing HK lu6n di qua
mQt diem c6 eirf, khi E, F di chuyen.
TRAN QUANGHUNG
(GV THPT chuyAn, DHKHTN - DHQG Hd NAi)
]i.r : : I 'ri j.',i ly{Qf vi6n d4n ktrOi lucr-ng M dugc
b6n 16n v6i vfln t6c f, hqp v6i phuong ngang
g6c u. Ddn di6m cao nh6t thi n6 n6, vd thdnh
hai m6nh. Manh nh6 c6 ttrOl luqng mvoivQn
-'l 'i,1 ,. l:
lror-t;iti,ltr;l 'l'11448
1ti*1" l.*'l' gi*d*). Find the
minimum value of the products of 5 different
integers among which the sum of any 3
arbitrary numbers is always greater than the
sum of the remains.
i':'t,l-il,i,ri;. 'r"i,',,i.i$
iXlor ?il' gn;ad*'i. Let ABCbe
a triangle with AB > AC and, AB > BC. On the
side AB choose D and E such that BC: BD
and AC : AE. Choose K on CA and I on CB
such that DK is parallel to BC and E1 is
parallel to CA. Prove that CK: CI.
Fr*hl*r* :[''314.i,9" Solve the following
.112
equatlon ---:*--F- G+3 .,6r+1 - t+Ji'
[i ;'r; i'r i r: ;-i;'i'..{i;i,{ 1}. Given an acute ttiangle AB C
with the orthocenter H.Let Mbe apoint inside
the triangle such that ffi)=frA.Let E and
F respectively be the orthogonal projections of
M on AB and AC. Let I and -I respectively be
the midpoints of BC and W. Prave that 3
lrnes MH, EF and IJ are concurrent.
tdc c6 mddun v,bQtra sau theo phuorg ngang
so vcri m6nh l6n. Hoi tAm xa cua m6nh l6n
ting th6m bao nhi6u so v6i trucyng hqp d4n
khdng n6?
W rueNu KHIET
(Hd NAi)
i-]ir,,:i t..r;,r'{.$rl. O6 do chu ki T cta mQt chAt
ph6ng xa ngudi ta dirng m6y d6m xung. Bi6t :, rdng trong h: 45 gio dAu tiOn m6y d6m dgc
n 1 xung ; trong t2 : 2t1gio ti6p theo m6y
.q
clOm duoc nz: i-nt xung. X5c ilinh chu ki
64
bhnrdT.
DINH THI THAI QUYNH
@d Nai)
llrcrhl*:i: 'I"51.{4,$" Find all pairs of integers
(x;y) satisfuing xa +y3 =xy3 +1.
.FilH r*I'{.;rA S{:*{{}q}L
F*"*hltl r-n "$'6l.i,i$" Solve the following
equation 8'-9lxl =2*3'
Fr*hlem T71449. Given a triangle ABC wrth
the AB : sides c, BC : a, CA: b. Assume that
the radius of the circumscribed circle is R and
the radius of the inscribed circle is r. Show
.. r.3(ab+bc+ca)
-'-*" fi - 2(a+b+c)2
.{&144{}.Let
Fnr+hlesit
x, y, zbe 3 positive real
numbers with x > z. Find the minimum value
ofthe expression
D_ xz , y2 x+22
-:2.T-T-. y'+yz xz+yz x+z
(Xem ti€p trang26)
Tffiffizu H
1i::thl i,;
,wffiffiffiffiffi#ffiffiffiffi
S<i aas lt-zotl) **:*ffii*#i e$ ?

Biti T21445 (Lop 7). Cho tam gidc ABC co
6tri ,90n vit dO ddi ba cqnh ld ba sii chdn
hAn ti€p. Tinh dp diti ba canh cila tam giac do.
Ldi gidi
Ye BH L AC tqi
H. vi 6Zd > eo'
n€n BC h c?nh
lon nhdt ci,a iam B
gi6c ABC vd A ndm girffa H vit C.
Tam gi6c HAB vlulngtqi H
+ AB2 : B# + AF? (dinh ti Pythagore)
Tarn grdc HBC vu6ngt4i H
> BC : Br? + cfr (dinh li Pyrhagore).
ra c6 BC : ar? + cr? : ar? + (AH + Aq2
BitiTll441 (Lop 6).Ch{mg minh rdng;
r 1 1... r 1 1222...222 - 333...333 ------vJL-v- ------vJ
201,1 chirso I 2014 chIsd2 2014 oh[isd3
ld m6t so chinh phmtng.
Ldi gidl Df;t a : 1 1 -Jl ld s6 vitit trong hQ thap
ph6n c6 20L4 chir sO t. tic d6 s6 ttugc vi6t Uoi
2014 chir sO tr ta bb*bb : lJ 1 : b. 1 b.a vir
102014- l: gg-gg :9a.Tac6
c : trr...ttt222...222 * 333...333
: 111-111 .lo2or4 + zzz-zzz
JJJ...aaa - JJJ
aaa
: : a.I02o1a + 2a - 3a a.l02o1a
-a
: a (lO2ola - 1): a.9a: (3o)r:1Zt-511, .
VQy sd C ld s6 chinh phucrng. tr
Y NhQn xdt. }/r(lt sii bpn bitin d6i ddi. C5c bpn c6 ldi
gi6i dirng, gqn ld: Phri Thg: Phqm Thu Thily, 6A,
THCS Thi Tr6n II, YCn LAp; Vinh Phrfic: Nguydn
Nhdt Loan, Ddo Ngpc Hdi Ddng,Trin Minh Huy,
Trdn Dan Trudng, Tq Thi Thu Hodi, Bili Thu Hiin,
Nguydn L€ Hoa, 6A, THCS Ly TU Trgng, Binh
Xuy6n; Ta Kim Thanh Hi€n, 6A1 Nguydn DiQu
Linh,LA D*c Thdi, Nguydn Thi Haong, Bili Tutin
Anh, Nguydn iinh Linh, 6A2 ; B&c Ninh: Tq Vi€t
Hodn,6C, THCS Nguy6n Cao, Qui5 VO; Hii Phdng:
Mai Quang Vinh, 6At, THCS H6ng Bdng; Hi Nam:
Nhtr Thi Thuong, 68, THCS Dinh C6ng Trdng,
Thanh Li6m; NghQ An Trdn Ngec Khdnh, 68,
THCS HO Xudn Huong, Quj'nh Luu; Nguydn Dinh
Tuiin, Thdi Bd Bdo,6C, THCS Li Nhat Quang, D6
Luong; Tdng Trung Ngha,6A, THCS Hda Hi6u II,
TX. Th6i Hda; Quing Ngii: Zd Tudn KiQt,58, TH
sO l, Hdnh Phu6c, Nguydn Dilrc Hdn,5B, TH Hdnh
Trung, Nghia Hdnh.
VIET HAI
> BI? + Afr + AC hay BC > AB2 + AC ()
Gqi d0 ddi ba canh cira tarn gi6c ld n - 2, n,
n + 2 (n chin, n > 2). Vi BC ld canh lon nh6t
n€n BC: n-t 2. .)')'))
Tir (*) ta co (n + 2)'> (n -2)' * n' + 8n> n'
= n < 8. Md (n - 2) * n ) n + 2 (BDT tam
gi6c) n6n n > 4.Tt 4 < n I 8, n ch1n = n : 6.
Vfly d0 ddi ba canh cira tam gi6cld 4; 6;8. A
Y NhQn xit
1) Bdi to6n tuy dcrn gi6n nhmg kh6 hay. Tet ca cilc
bdi eti d6u cho d6p s6 ihing. Nhi6u bpn sri dpng k6t
quh BC > AB2 + AC nhrmg kh6ng chimg minh.
2) Neu ta thay giA thi6t "ba c4nh ld ba s6 chin li6n
ti6p" bing gi6thi6t "ba cpnh ld ba s6 t.u nhi6n li6n
tir5p" hodc "ba c4nh ld ba s6 16 li6n ti6p" ta cfrng
dugc nhirng k6t qui thri vil
3) Cdc b4n sau c6 loi gi6i ti5t: Vintr Phic: Hodng
Minh Duc,7A3, THCS LAm Thao; Tg Kim Thanh
Hiin,6A1, THCS Y6n L4c; Thanh Ho6: Phimg Hit
NguyAn, TD, THCS TrAn Mai Ninh, TP. Thanh Ho6;
NghQ An: Nguydn Thu Giang, Trd,n LA HiQp,
Nguydn Thi Nhu Qu)nh A, Nguydn Nhu Qu)nh B,
7A; Hodng Trdn D*c,7D; Nguydn Thdi HiQp,7B,
THCS L), Nhft Quang, E6 Luong; Nguydn Trpng
Bing, 7A2, THCS T.T. QuSn Hdnh, Nghi L6c;
Quing Ngdi: Truong Quiic Binh,7C, THCS Hujmh
Thric Kh5ng, D6 Thi Mi Lan, Truong Thi Mai Trdm,
Nguydn LA Hodng Duy€n, Vd Quang'Phil Thdi,7A,
THCS Pham Vdn Q6ng, Nghia Hdnh; Binh D!nh:
Nguydn Bdo Trdn,7A, THCS Tdy Vinh, Tdy Son.
NGUYEN XUAN BiNH
TOAN HOC 18'clildiff@

Bni T3/445. Cho hai sd thUrc du.ong a, b thoa
mdn a + b vd ab ld cdc sii nguyAn drctng vd
lo'+on)+lb'+otl ta so c.hinh phroag, o
d6 ki hiQu lx) ld s6 nguyAn ktn nhiit kh6ng
vwrt qud x. Chang minh ring a, b ld cdc s6
nguy€n duong.
Ldi gidi.. Dox* 1 < Drl (x n6n
7l + abl + lb2 + abl < I * ab + * + ab : (a + b)2
vd ll + abl + yb2 + abl > (a + b)' * 2.Ta c6:
(a + b)2 - 2.ld + ab)+ lbz + ab)3 (a + b)2.
. N6u a + b : lthi 0 < a < | vdO < b <1, suy
ra ab < 1 trfiivoi gi|thiet.
. N6u a -t b > 2 th gifia hai sO 1, + b)z vit
(a + b)2 - 2 kh6ng t6n tai mQt s6 chinh
phucrng ndo. Do d6
7d + ab1 + lbz + abl : (a + b)' : d + b2 + 2ab.
MIt kh6c, do ab ngryln duong n6n
;d + abl + lb2 + abl: ldl + yb21+ 2ab.
Suy ra yd1+ ;n'z1: d + b2.
Ta c6 ldl < d; lb'l < b2 > yd1+ yt21< d + b2,
dlng thirc xity rakhi vd chi khi lil : i; t#l : #
> d vdb2 ldcilc s6 nguy6n duong. (*)
. Mdt kh6c, a + b ngty€n duong vd
, a'-b2 , a - b: -
-
hiru ti, suy ra a, bh*vfi. (**)
a+b
Tti (*) ve (**) suy ra a, b ngty€nducrng. I
Y NhQn xit
l).Ta c6 tni5 a6 Oang chimg minh cbc tinh ch6t sau:
NCua+ bvda- 6hiruti thia,b hiruti;Ni5uahiruti
duong vd d ngtydnduong thi a cingnguyCn duong.
2) Cdc ban c6 lcri gi6i dring ld: Binh Dinh: Nguydn
Bdo Trdn, 7A, THCS TAy Ninh, Tdy Son; Vinh
Phric: Nguydn Minh Hi€u, Nguydn Hibu Tilng,
Nguydn Kim D*c, 8A5, Nguydn Hing Anh, 8Al,
THCS Y€n Lac; Ngh6 An: Nguydn Hing Quiic
Khdnh,9C, THCS Ddng Thai Mai, TP. Yinh, Nguydn
Trong Bdng,7A2, THCS Thi Tr6n Qu6n Hdnh, Nghi
LQc, Tdng Vdn Minh Himg, Nguydn Vdn Manh,7A,
Hodng Trdn Dthc,7D, THCS L), Nhat Quang, D6
Luong; Qu6ng Ngni: Nguydn Dai Dwrng,SB, THCS
Nguy6n Kim Vang, Nghia Hdnh; Hi NQi: Ddng
Thanh Tilng, 88, Nguydn Thdnh Long,gB, THCS
Nguy6n Thugng HiAn, tlng Hda, LA Duy Anh,9A,
THCS Nguy6n Huy Tu&ng, D6ng Anh.
NGTIYEN ANH QUAN
BdiT41445. Cho tam gidc nhon ABC voi cac
dudng cao AD, BE, CF. Tr€n tia d6i cila carc
tia DA, EB, FC lin laqt liiy cac di€m M, N, P
sao cho BMC = CNA = APB :90o.
Chung minh riing cdc &rdng thdng ch*'a cdc
cqnh cila luc giac APBMCN citng ti€p xuc vdi
m6t &rdng trdn.
Ldi gi,rti
V BE, CF ld c6c ducrng cao trong tam gi6c
ABC n0nta co AE.AC: AF.AB (1)
Ap dung h6 thric hong c6c tam giilcvu}ngANC
vitAPB ta c6 AE.AC : dlf; AF.AB : Af Q)
Tt (1) vit (2) suy ra AN : AP. Tuong tu ta
nhfln dugc BP: BMvit CM: CN.
Gqi O ld giao dii.lm cta c6c ducrng trung truc
claa MN, NP, PM. Do c6c tam gi6c PAN,
PBM, MCN c6n n6n AO, BO, CO tuongimg
ld cdc duong phdn gi6c cin cdc go" Fffi ,
PBM , MCN. Mil kh6c, theo tinh ch6t d6i
ximg ta c6
OPA=ONA; ONC:OMC; OPB=OMB (3)
Laic6, m=6N) - 5FE=6fie .
r6t hqp v6i (3), suy ra 6fu:6ila .
Tucrng tu ta c6 OPB:OPA; ONA=ONC .
YQy cdc ducrng phdn gi6c cua cdc go" Eile , ---:.---_ ^ MCN , CNA, NAP, APB , PBM tl6ng quy
t4.i O. Do d6 cdc cqnh cira luc gi6c APBMCN ,.,( cung tiep xric v6i m6t tluong trdn. tr
ta *, or-rorn,
T?SHrHES
I g

Y Nh$n xet C6c bqn duoi tl6y c6 lcri gi6i t6t UA NQi:
LA Duy Anh,9A, THCS Nguy6n Huy Tudng, D6ng
Anh; Phf Thg: Trin fu6c LQp, Trdn Mqnh Cttdng,
8A3, Ddo Thanh Phuc,9A3, THCS Ldm Thao.
NGIryEN THANH HONG
Bii T5/445. Tim sd nguyhn m rtd phwong trinh
*3 + 1* + # - (2m- l)r- (2mz + m + 4) : o
c(t nghiQm nguy2n.
Ldi girtL Cdch t. ni6n d6i PT (1) nhu sau
x3 +(m+l)x2 -(2*-l)r- (m+l)(2m-1) = 5
o *2 1* + m +l) - (2*- 1)(, + m +l) = J
<> (r+ m+l)(xz -2m+l = 5 (2)
Do m vd x ld chc sd nguyOn n6n x + m * I vit
* -Z*+ 1 lA c5c sd nguyCn vd ld u6c ctra 5.
Ta c6 5 : 1.5 : (-1).(-5). NhQn thSy x * m * I
vd * - 2m + lU sO le n6nx vd lz ld sti chin.
Suy ra *' -2*+ 1 chia 4 du 1. Do t16 ,.
x' - 2m + I blng t holc 5. XAy ra hai khi ndng
lx+m*l:l lm=-x l)i " o{ " lx" -2m +l =5 lx" +2x-4=0 (*)
PT (*) c6 nghiQm x : -1tr6 t<trOng nguyCn
n6n loai.
2)l1x+. m*1:5 lm=-x+4 <+l " lx'-2m *l = I lx'+2x-8:61**;
PT (**) c6 nghiQm x:2 vd x: 4 d6u ld s6
nguy6n. Suy ra m:2vdm:8.
Cdch 2. gien d6i PT (1) thenh
2m2 -(x2 -2x-I)m-(x3 +x2 +x-4)=0 (3)
Coi (3) ld PT bfc hai An mvbi
L=(x2 -2x-1)2 +8(x3 +*2 +x-4)
=(x2 +2x+3)2 -40.
oe pr (1) c6 nghiQm nguy6n thi PT (3) phei
c6 nghiQm nguy6n, suy ra A phii ld sd chinh
phucnrg. D$d +2r+212 -+o: 121r e x;
€ d + b + 3+ D@2 + h + 3 - k1 : 49.
Dox e Z,k eN,
)1
(x' + 2x+ 3 + D - @' + 2x + 3 - k) : 2k,
i+x+3+k>on6n
_ _ TO6N HOC 2A -clfudi@
d*x+3+k)rd*2x+3_ k),
* + x + 3 + k ve i + 2x + 3- kctng ld s6 t.u
nhi6n chin. Ta c6 40 :20.2: l0.4.Xity rahai
khi n[ng sau:
l)l lx2+2x+3+k:20 lk:9 ^ <>{ " lx' +2x+3-k=2 lx" +2x-8=0
(k=9
e i Tim dugc ffi:2, m:8.
lx=2;x=-4
l12+2*+3+k=10 lk:3 2)1 ^ <>{ " lx' +2x+3-k=4 lx- +2x-4=0
e {lk =3 -, kh6ng th6a mdnx nguy6n.
[x=-1rJ5'
VQy k'hi m : 2 hoic m: 8 thi PT (1) c6
nghiQm nguy6n. E
Y NhQn xet. C6 nhidu ban tham gia gi6i bdi ndy vi
ldm theo hai c6ch tr6n. MQt s5 ban ldm c6ch I do
kh6ng dua ra nh{n x6t vA cdc nhen tu 6 v6 trili cria PT
(2) n6n ph6i xdt di5n b6n khi ning; mQt s6 ban ldm
c6ch2 cho keZnQnphdi xdt nhi6u ktri ndng hon dhn
dtSn bdi gini ddi ddng. Tuy6n duorg c6c bpn sau d6y
c6 loi gi6i tOt phrri Thg: Nguydn Thin Chi, Trdn
Mqnh Cudng, Trdn QuiSc LQp, 8A3, THCS Lam
Thao; Ngh$ An: Nguydn Xudn Todn,7A, THCS Llli
Nhft Quang, D6 Luong; Quing Binh: Phan Trdn
Hubng, 9A, THCS Qu6ch Xu6n Kj,, B6 Trech;
Quing Ngfli: Nguydn Dqi Duong, 98, THCS
Nguy6n Kim Vang, Nghia Hdnh; Kon Tum: LA Vi€t
Lmt Thanh,gA, THPT chuy6n Nguy6n f6t fnann.
PHAM THI BACH NGQC
Biii T61445. Chung minh riing vdi moi s6
thqrc a, b, c l6n hon I ta lu6n c6:
(logu a+ log. a - 1) x (log" b+ logo b - l) x
x (logo c +logbc - 1) < 1.
Inigi,fr. (Thm s6 dingcd,c bw Sfibdivi tdasoqn)
Do 1og, b.log6 c.log" a = I vir a, b, c l6n hon 1
n€n tdn tAi cdc si5 tfurc ducrng x, y, z th6a mdn :
1(V_Z
logob : -:; !-; yzx
loguc = log" a = 7-. e6t ding thric cAn chimg minh tunng duong / / /
voi: I Z +1-rl[ 1*!-r ][ (.r I*Z-r l< r x )ly y ), z )

e (r,+ z - x)(z + x- y)(x+ y - z) 3 xtz (1).
N6u c6 hai fong ba thria si5 tong v6 tr6i cria
(1) 6m, ching han y + z - x 10, z + x- y <0
) 2z = (y * r-r) + (z + x - y)< 0 . Di0u ndy
kh6ng xtry ravi z > 0.
N6u c6 mQt trong ba thria s6 y + z - x, z'r x - /,
x * ! - z dm vd hai thria sO con 14i duong
(ho[c bing 0), thi bdt dingthfc (1) <hing.
Ntiu ci ba thria s6 y + z * x, z * x - !, x * ! - z
duong (ho[c bing 0), 5p dune b6t tling thric
Cauchy, ta c6: {(.r, + z - x)(z + x - y) < z;
(x + y - z)(y + z - x) < y,l(z + x - y)(x + y - r) < *.
Nhdn theo vlaaahtding thric tr6n, ta duqc (1).
n6t ding thric trong dAu bdi dugc chimg minh.
Ding thric xhy rakhi vd chi khi x: y - z
-
Q a=b=c.J
F Nh$n x6t. E6y ld bii tor{n kh6 co ban n6n c6 nhi6u
ban gur bdi gi6i vC tda so4n. M6t s6 ban <lat
log, b = x;logu c = y;log" a = z * x,!,2 ) 0;xyz = |
viQc hinh bdy lcri gi6i phirc t4p hon.
Trong c6ch dflt log" 6 = L;bgu s = Z;log" o = 1, tu yzx
c6 th6 chqnx h s5 thuc duong b6t ki;
y = xlogu a;z = ylogcb + log. a =lo;, b.logu a =
z
x
C5c bpn sau ddy c6 bdi gi6i t6t: nic Ninh: L€ Huy
Cu.ong,1l To6n, TIIPT chuy6n B6c Ninh NghQ An:
H6 Xudn Hilng, l0Tl, TIIPT D6 Lufirg I, Dh Son,
DO Luong; Hi NQi: Vfi Bd Sang,10 Torin l, Trdn
Mgnh Hirng,l1 Toan A, TIIPT chuy6n NguySn HuQ,
Kim Vdn Hilng, l2Al, THPT M! Dric B, Trdn
Phwong Nam, 12A3, THPT Ngqc T6o, Phric Thg;
Tidn Giang: Ne"ye" Minh Th6ng,11 Toan, THPT
chuyCn Ti6n Giang, M! Tho; Long An: Chdu Hda
Nhdn,l2T),; THPT chuy6n Long An; Viing T}u: LA
Hodng Tudn, l2M, THPT Dinh Ti6n Hoang, TP
vflng Tdu'
NGUTENANHDLTNG
BhiT7l445. Cho tam gidc nhpn ABC (AB < Aq ./
nQi ridp dadng trdn {O}. Cdc dudng cao AD,
BE, CF
"dt
,ho, tqi H. Gpi K td trung didm
cila BC. Cdc ti€p tuydn v6i du'dng trdn (A) Ui
B vd C ciit nhau tqi J. Chilrng minh ring HK,
JD, EF d6ng quy.
Gii str EF r: BC: G; HK a EF: I;
GA a(O) : R (R + A); OA n EF : M. Ta c6
GB.GC=GR.GA=GF.GE, suy ra R n5m tr6n
dudng trdn dulng kinhAH,hay HR LAG.
Ap dpng dinh li Brocard cho tu gi6c nQi titip
BFEC voi BF a CE: A; EF ^ BC: G vit
chri f ring K ld tdm ducrng trdn ngopi tiiip tu
gi6c BFEC ta dugc HK L AG. Ti d6 ba ttitim
H, K, R thdng hdng. X6t clrc vd dOi cUc dOi
v6i tlulng trdn (O). OC ttr6y GDBq - -1,
n6n dudng d6i clrc ctla D di qua G (1)
M[t kh6c, tath6'y duong d6i cyc oia D di qua
-r (do tlucrng d6i cgc cta J ld BC tli qua tli6m
D) (theo dlnh li La Hire) (2)
Tri (1) vd (2) suy ra G/ ld tludng ddi cgc cira
D ddiv6i tlucrng trdn (O). Theo tinh ch6t cira
cpc - d6i cgc ta thdy OD I GJ. Ket hqp v6i
GK L OJ suy ra D ld tryc t6m tam gi6c GOJ,
dod6JDLGO (3)
Tiiip theo ta sE chimg minh DI L GO. ThAt
vfly, gei N: DI a GO, OE ttrAy OA L EF t4i
Mn€ntb giitc AUMnQitirip. Tir d6
GI .G 14 = G R.GA= G B.GC = GD.G K
(do (GDBQ - -1, K ld trung tti6m BC n6n
theo h€ thac Maclaurin GB.GC =CO.CX 1.
$ TOAN HOC - 44e (11-2014) & sTudiUA 21

Suy ra ta gi6c IMKD nQi ti6p >BDN=IMK'x,e t__n!Lc
md tu gi6c GMOK nQi ti6p n)n, IMK:NOK.
Vi vQy BDN=NOK, suy ra tfi gi6c DNOK
nQi ti6p. Do d6 DNO = 90o, hay DI L GO (4)
Tri (3) vd (a) suy ra ba di6m D, I, J thing
hdng, hay ba du<rng thing HK, JD, EF d,6ng
quy tpi 1(dpcm). tr
F Nh{n xlt.Tdtci c6c loi gi6i gui v6 Toa soan d6u
dring theo c6c hu6ng: Sir dpng tinh ch6t cua Tri giSc
diAu hod, Hing ditlm - chirm di6u hod, CUc - d6i
cuc, Phuong tich cua mQt tlii,lm O5i vOl mQt dudng
trdn... Cdc b4n sau c6 ldi gi6i t6t: tti NQi: Hodng LA
NhQt Tilng, l2A2 Tohr, THPT chuy6n KHTN,
DHQG HA NQi, LA Duy Anh,gA, THCS Nguy6n Huy
Tu&ng, D6ng Anh, Nguydn ViQt Anh, Trdn Mqnh
Hilng, llTo5nl, THPT chuy6n Nguy6n HuQ; YGn
B6i: Vil Hing Qudn,11To5n, THPT chuy6n Nguy6n
f6t fnann; Hir Nam: Hodng Duc Manh, 11To6n,
THPT chuy6n Bi6n Hod; NghQ An: H6 Xudn Hi.mg,
11T1, THPT E6 Luong l, Trdn Quang Huy, l0Al,
THPT chuy6n DH Vinh, Phan Vdn Khdi, l0Al,
TIIPT Cira Ld, TX Cria Ld; Hi finh: Nguydn Vdn
The, LA Vdn Trwdng Nhdt, Nguydn Nhu Hodng,
11To5n1, THPT chuydn Hd Tinfu Binh D!nh:
Nguydn Trpng Khi€m, 10A1, THPT Quang Trung,
Tdy Son.
N* md -El-. ,. ta nhdn duoc
tra :!lr(;)=
2n
i@.1) -, khi ru + tm
Tt d6 suy ra g(x) : 0 Vx e IR, tric U 71*1 = 1.
3
C6c bi6n d6i trCn ld tucrng duong, do d6 ta
kh6ng phdi thu lai. Vfly c6 duy nh6t mqt ham
si5 thoa m6n bdi toSn ldr f (x) =I Vx e lR. tr
3
F Nhfln x6t. Ddy ld bdi to6n tim hdm s6 gini bing
phuong ph6p dAy s6, 1o4i bdi toSn dd xu6t hiQn nhi6u
trong c6c ki thi hoc sinh gi6i to6n qu6c gia, thi hgc
sinh gi6i to6n cira c6c nu6c kh6c, thi IMO. C6c b4n
hgc sinh sau c6 ldi gi6i tOt: tti, NQi : Izfi Bd Sang,
l0Tl; Nguydn Vi€t Anh, Trd:n Mqnh Hilng,llTl,
THPT chuy6n Nguy6n HuQ; Hodng LA Nhdt Timg,
llT-42, THPT chuy6n KHTN DHQG Hd NQi;
Nam Dinh: 1ng Titng Daong,11T1, THPT chuy€n
L6 H6ng Phong; Hir finh ; Vd Duy Khdnh, Nguydn
Vdn Th€, llTl, Trdn Hdu Manh Cudng, 12T1,
THPT chuv.n Hd rTnh'
NGU'EN MrNH DIrc
BitiT9l445. Cho da thac:
/(x): *t - 3x' + 9x + 1964.
Chrimg minh riing tdn tqi s6 nguyAn a sao cho
fla) chia het cho 32ota.
Ldrt gidi. (Theo bqn Trin HQu Mqnh Cudng,
l2Tl, THPT chuy€n Hd Tinh)
Tac6lx):(x- l)'+6(x- I)+ l97l
2 >fl9* + 1): (9x)" + 6.9x + l97l
- -)1 ,,y1u- ,,'-t- + 2x + 73).
Xdt rla thfc g(x) : 27x3 -t 2x * 73 vd tqp
.q= {s}}!r. Ta chimg minh A ld mQt hQ dAy
clir mod 3".ThdtvQy, gih str tr5i lpi,4 kh6ng ld
hQ day dtr mod 3".t<hid6 t6n tai 1 ( i <j < 3"
sao cho S(,) = g(/) (mod 3')
> 27 i3 + 2i + 73 =27j3 + 2j + 73 lmod 3';
H6 QUANG VINH
Bdi T8/445. Tim hdm sa7: m. -+ R. bi chdn
ftAn m6t khoang chaa di€m 0 vd th6a mdn
2fl2x) : x + J(x), vdi moix e lR.
Ldi gidi. Gi6 sufix) ld hdm s5 thoi m6n bdi
)t
to6n. Chri i x = 2.4-{. oo AO
a JJ
2fl2x): x+f(x) oz(trz.t +): f@-;.
D[t g(x) : .f (x)*!. msuy ra
I (x r /x)_ _l-(r)_ 8(x) = ;rl, )= 7 rli ):... = 7t[7 )-
Tri gin thi}ttac6la € R*, 3M e IR.* sao cho > (t_ illZl1i, + j2 +iil+21i 3".
lru>l <MYx e IR, l*l .o.BoivpyVx e IR, v Z7Q2+ j2+ij)+2/: n6n j-ii3" (v6li).
zz'?!l#E!.*

Va1y A ld hQ dAy du (mod 3';. Do d6 tOn tpi
I lkn( 3'sao cho g(k,) i 3".
Dlt a,: 9k, + 1 ta c6 -f (,a,):27 g(k,) :. 3n*3.
Yoi n:2}ll ta co f (arorr) '. 3'o'0. 1
) Nhfln x6t. C6 kh6ng nhi6u b4n tham gia gi6i bdi
to6n ndy vfi cdc cilch giiri khSc nhau. C6c b4n sau
rtdy c6 ldi gi6i t5t: tth NQiz Trdn Mqnh Hi.mg, 11
To6n A, THPT chuy6n Nguy6n HuQ; Di Nfrng:
Nguydn Hiru Hodng Hd| IIAI, THPT chuy6n L6
Quy E6n; Quing Tri Trdn Trpng Ti€n, t2Toin,
THPT chuy6n Ld Qu;i D6n; Binh Dinh: Mai Ti6n
Ludt, l2Todn THPT chuy6n L6 Qu:f D6n; Nam
Dinh: )ng Timg Daong, 11To6n, THPT chuyCn Ld
H6ng Phong.
DANG HDNG THANG
Bni T10/445. Tin tqi hay kh6ng hdm sii li€n tuc
./: R -+ lR. sao cho v6'i moi x eN., trong cdc s6
flx),flx + 1),./(x + 2) luon c6 hai td hfr, fi ro
mot so v6 ti.
Ldi gidi. Nhfln x6t: Kh6ng th€ tin tqi hdm
h2n fuc /:1R -+ lR sao cho vcti mpi x, trong
hai sd f @),f(*+l), c6 mt sii v6 tjt vd mQt :,- so nwu ry.
Chrmg minh nhQn xdt: Gii st tdn tpi hdm
f thbamdn nhpn xdt. X6t c6c hdm s6
h(x) = f (r) + "f (, +1), s(r) : "f (r) - f (x +L) .
NCu ft(x) vd g(x) d6u ld hdm hing thi
f (x) =@#@ cflng ld him hing. Trudng
hqp ndy bi loai vi kh6ng th6a mdn di6u kiQn
cira nhdn xdt.
N6u ft(x) vd g(x) kh6ng d6ng thdi ld hdm
hing thi kh6ng mAt tinh t6ng qu6t gib sir h(x)
kh6ng ld hdm hing. Suy ra tdin t?i ,.x2 sao
cho: h(xr) < h(xr) = t6n tqi s6 hiru ty
q e lh(a);h(xr)l vd vi h(x) ld hdm li0n tuc
n6n theo dinh l)i gi6 tri trung gian, t6n tpi
n eln;x): h(n) : q . Do d6 f(Q+ f(a+t) :q .
Nhmg vi q hiru ty n6n .f (r), f @o+1) d6ng
thdi ld s5 t tu fj. hoac d6ng thcri Id s6 v6 fj,.
Di6u ndy trdi v1i gi6 thi6t. NhQn xdt dugc
chimg minh.
Quay lpi bdi to6n dd cho, vi trong c6c s5
f(r),.f(*+l),f(x+2) lu6n c6 hai s6 triro t'1'
a vd mdt so v6^ .t,y n6n c6 3 trucrng hqp xdy ra:
.f(x) la sO hiru ty,.f(*+l),f(x+2) ldhai
a ^., so vo ry.
. .f(x+l) ld sO tiro ty, f(x) vd f(x+2) lit
a ^., nal so vo ty.
o f(x+2) ld s6 hiru ty, f(x) vd f(x+l) lit
a ^., nar so vo ty.
Tt nhQn xdt tr6n ta thl,y trong c6 3 truong
hqp deu kh6ng t6n tpi himf. A
F Nhfln x6t
1) Bing chimg minh phin chring vd sri dpng dinh 1),
gi6 tri trung gian c6 th6 chimg minh nhQn xdt sau (tu
d6 gi6i tlugc bdi to6n dd cho).
N€u f :lR -+ IR ld hdm hAn fuc vd chi nhQn cdc gid
tri v6 fi ffan R thi f (x) = c , voi C td hiing s6 v6 t!,
ndo d6.
2) CLc b4n tham gia ddu giai dring biri niy, t6n cria
cdc bun ld: Y6n Biiz Vfi Hing Qudn, 10 To6n,
THPT chuy6n Nguy6n T6t Thdnh. Binh E!nh: Mal
Ti€n Luqt,12T, THPT chuy6n L6 Quy D6n, TP. Quy
Nhon. Long An: Chdu Hda Nhdn, 12T2, THPT
chuydn Long An.
TRAN H TU NAM
Biti Tttl445. Cho ddy sii {r,l? daoc xdc
dinh bcti c6ng th*c: q : l, az : 2014,
2013a,. (. 2013)
fttl | | I n-l lz n-l)
(t I l)
n : 2.3, ... T'im lim I -+-+...+- l. r+-( a, a2 a, )
Ldi gidi. (Theo da s6 cdc bqn)
cdch t.Ta c6 an+t =2ot3ra n *(r *4n!)-rl),-,
/
.nr"l an , ar-t 1 ,
/r n-r./
//
= 2or3[ %*tu+]+zor:[ tu4*!4]i*o,_,
I r n-l) n-2 n-3)
/
an *on-t +an-2 +on-3 l*o- , I r n-l n-2 n-3)
=2013[
te nn, or-rorn,
T?EI#S
28

=...=zo,(:tl*,,
"'=t i )
ann
(201 3 + 1)(20 t3 + 2)...(2013 + n - r)
) Nh$n x6t. Cbc b4n sau tl6y_c6 loi gi6i dung: Hi
Tinht Vd Duy Khanh, Nguy4n Vdn Th€, LA Vdn
Tudng NhQt, Nguydn Dinh Nhdt Nam, llTl; Trdn
Hdu Manh Cudng, 12T1, THPT chuy6n Hd Tinh;
Tidn Giang: Nguydn Minh Th6ng, 11T, THPT
chuy6n Ti6n Giang; Ilung YGn: Nguydn Thi Huong,
12A1, THPT chuy6n Hrmg Y€n; YGn Bdi: Yfi Hdng
Qudn, l0T, THPT chuy6n NTT; Hn NQi: Nguydn
Vi€t Anh, llTl; Trdn Mqnh Hilng, llTA, THPT
chuy6n Nguy5n HuQ; Quing Ngiii: Zd Thi Bich
Nga, Bqch Thi ThiAn Ngdn,llT2, THPT L6 Khi6U
NghQ An: Phan Nhu Trlnh,llAl, THPT DiSn ChAu
3; Binh Einh: Mai Tiiin Ludt,12T, THPT chuy6n
LC Quf D6n; Vinh Longz Trin Cao NhiQm,llTl,
THPT chuy6n Nguy6n Binh Khi6m.
NGUYEN VAN MAU
Bii 'f1?/445" Cho tu gitir Atit} nyrLai ilAp
dtr*ng trdn ([]. Cac csnh AB, BC tiilt .uii' :'(ti
tll li.n {uot tai M, N. Gpi {i la ;4iao r{iim .'ilrt
A{i') v'i.A{N: F' id giuo didm cia 8C t'd DE.
f-)tul cfi Ul t(ti di6yn 7- khdc fr,I. CiiLl'ng tninh
rdng FT ti ti1p tu.t,in ctiu |.fi.
Ldi girti (Theo bqn Philng Ddc Vil Anh, 1271,
THPT chuyAn Amsterdam, Hd Nfli).
Suy ra
an+2 =,oB(i+).", _2013an*, t , u^n+l' n+l
Vpy n6n an*t=20139 * % = r,ff *1,, =rr,.
Do d6 an+1 - n+20I3
,n =1,2,... vd
a"n = ,n=2r3r,,. (n-t)l
Suyra l*a+...+a
al a2 an
=1*l*i 2014 f'- (2013 +l)(2013 + 2)...(2013 + k -l)
:u 1 (r--ZL)
2012( 2or4)
.--Lir 2012- (=J
(k-1)!
(k -t)t
2012 74Q0 I 3 + l)(20 1 3 + 2)...(20t3 + k - 2)
kt
(2013 + t)(20t3 + 2)...(201 3 + fr - 1)
201 2 x 20 1 4 x 201 5 x (201 3 + n - l)
_1 - l+__
20t2
O6 y, rang
nl
lim -0
n-+a )Ql) x 2014 x 2015 x (201 3 + n - 1)
(t nen ,l-r-mf la_1+ _a+l 2. ..+_ r l:_. zorg o, ) Z0l2
Cdch2. OE ddng chimg minh an =Co*?rrr, tu
d6 suy ra
1 _- 2oBt( 1
%
20i2[(,r+r0t 1X/,+ r010] (,r+ l)n
1 t Gqi P, Q theo thu t.u h ti6p tlitim ctra CD, DA
l. ,U (4;.S le giao diiSm cira TNvd Pp (hinh vc).
/ Cdc kdt qud sau ld quen thu6c:
* P, Q,E thing hing.
(n + 2012)(n + 2011)...(n + 1)
Tt d6 ta c6 tlpcm. fl
TONN FIGC 24 ' *frrdiikA sii eag tu-zorer

+ AC, BD, MP, NQ d6ng quy (tAi r).
YAy, 6p dt g dinh l{ Pascal cho s6u tlitim
MNO
^ _: , chri f rdng QQ^ MT = D:
QPT '
QP aNT = S;MP ^NQ:K, suy ra D, S, K
thing hdng.
X6t cgc vd d6i cgc d6i v6i (1). Ta c6 B ld cgc
cin MN, D ld cgc cin PQ. Do d6 E li clrc cira
BD (v E = MN
^PQ ). Sry ra E, S li0n hqp
(vi S e DK =BD). Di6u d6 c6 nghia ld S ld
qrc cira DE (vl D, ,S 1i6n hqp). VQy S, F 1i6n
hqp (vi F e DE ). Do tl6 F ld cgc cira SM (vi
ll, F li6n hqp). Suy ra F, T li6n hqp (vi
7 e SN). N6i c5ch ldthc FTti€pxic v6i (4. D
) Nh$n x6t
1) Ngodi bryt Vfi Anh, c6 8 b4n tham gia gi6i. Tuy
nhi6n vi kh6ng bitit sir dpng clrc vd tl6i cUc n6n loi
gi6i cua 8 bpn d6u ddi.
2) Xin n6u t6n c6 8 ban: Kon Tum: Nguydn Hodng
Lan,71A1, THPT Nguy6n T6t Thdnh, TP Kon Tum;
NghQ An: tti Xuan Hitng,l0Tl, THPT D6 Luong I,
D6 Lucrng; Thanh IJo{: Ddng Quang .,lnh, 7A,
THCS Nguy6n Chich, D6ng Sor; Hi NQi: Trdz
Manh Hitng,l1 Toan A, TIIPT chuy6n Nguy6n HuQ,
TX He D6ng; Hi Tinh: LA Vdn Trudng Nhdt,
Nguydn Nhu Hodng, Nguydn Vdn Th€, llTl, Trdn
HAU Mqnh Cudng,12Tl, THPT chuyCn He Tinh, TP
Hd rinh'
NG.TYEN MINH HA
Bliti Lll445. Mt thanh cilmg ding chdt, ti€t
di€n diu, chiiu ddi L ilwqc treo ndm ngang
boi hai sqi ddy mdnh, kh6ng gidn cilng chiiu
ddi I nhu hinh vd. Kich thich cho thanh c*ng
dao dQng nh6 trong mrtt phdng hai ddy.
Xdc dinh chiiu ddi I theo L d€ chu ki dao ilQng
ct)a thanh td nh6 nhdt vd tfnh chu ki il6.
Ldi gidi. Xdt khi thanh lQch khoi phuong
ngang m6t g6c nho q (duong cao OG l6ch
khoi phucmg thing dimg g6c <p). Phucmg trinh
quay quanh O: mgOGsinrp = -Ioq"
Nhu vpy thanh dao dQng di6u hda vdi chu lcj.:
2tt 2x 612 - L
., 3
'[4P 4'
Di5 chu lcj,dao dQng nh6 nhilttac6 th6 su d*ng
clao hdm hoic b6t ding thfc Cauchy ta sE tim
r
clug' c: / = i l; vitchu kj, nh6 nhAt trrhi Ay bing: VJ
.r _2n L
'mrn J|'s'
) Nh$n x6t. Cic bpn c6 ldi gi6i dfng: Nam Dinh:
Phqm Nggc Nam, 10 Lir, THPT chuy6n LC H6ng
Phong; NghQ An: Phqm Quiic Vwong,1241, THPT
DiSn Ch6u 3; Binh Phufc: Ngqtdn Vdn Hilng,7lB,
THPT chuyCn Quang Trung.
NGU}'EN XUAN QUANG
BitiL2l445. Mqch ilien v6 hqn ld mqch di€n
tao thdnh t* vd s6 mdt mqch gi(ing nhau, ndi
hAn fi6p theo mQt quy tuQt nhtit dlnh, sao cho
khi th€m vdo (hay bdt di) mAt m& mach thi
di€n trd cila cd doqn mqch vdn kh6ng thay d6i.
Cho mqch diQn vd hqn bi1u di6n tAn cdc so tl6
(a) vd (b).
Mqch (a) tqo thdnh tir vd s6 cdc mdt nhw nkau
gim c6 ba di€n tr?r,2r,3r; Mqch (b) tqo thdnh
tii v6 sd cdc hinh w6ng, cdu qo tir cdc day ddn
ding chdt, nAt nAi fidp ffong hinh w6ng khdc,
ma di€n trd cit"ax .m)i cqnh hinh vu6ng ld r. Xic
dinh diQn trf cfia mdi doqn mqch.
ta*, T?EI#S
or-rorn,
**F+=-*|f,.,'
Vcri g6c q nh6, bi6n d6i ta duoc:
,, Itl+t2 - t] r0"*---- _ <p=Q
6[' - ]:
L, il
4 )'
25

Ldi gidi. Trong s6 c6c bdi gi6i, ban Chu Minh
Th6ng c6 lcvi gi6i cdu b) hay, s6ng tgo. Xin
gi6i thiQu loi gi6i cira bpn Chu Minh Th6ng.
a) Ggi iliQn tro cira dopn mach ld R. Vi mach
v6 hpn n6n khi th6m hay bort mQt mit m4ch thi
cliQn tro ci:r- ch tlo4n m4ch kh6ng thay d6i n6n
ta c6 scv d6 tlopn m4ch nhu sau:
FB
Di€n tro cin cb tlo4n mpch:
R= 2'R +4r e R2 -4rR-Br2 =0. R+2r
Giii phucrng trinh ta thu duoc: R = 2r(l* ",5).
b) Do tinh d6i xtmg n6n nhirng cli6m c6 cung
) NhSn x6t. C6c b4n sau c6 loi gi6i thing: Nam
Dinh: Pham Ngoc Nam, 10 Li, THPT chuy€n LC
Hdng Phong; NghQ An: Chu Minh Th6ng, A3-K41,
THPT chuy6n Phan BQi Ch6u.
DANG THANH HAI
PROBLEMS...
(Ti€p theo trang 17)
TOWARDS MATHEMATICAL
OLYMPIAD
Prolrlem Tgl449. Find the integral part of the
l5 9 20t3
exD' resslon B - -3+7-t+t -+. . .+ 20t5
Froblem T101449. Find all polynomials ftr)
with integral coefficients such that fln) is a
divisor of 3n - 1 for every positive integer n.
Problem Tlll44g. Let (x,) be a sequence
satis$ring:
,VneN
26
Let S, =Z*n** , /l € N*. Prove that, for
k=0
every odd natural number n, Sni66 .
Problem T121449. Given a triangle ABC. The
points E and F respectively vary on the sides
CA and AB such that BF : CE. Let D be the
intersection of BE and CF. Let H and K
respectively be the orthocenters of DEF and
DBC. Prove that, when E and .F change, the
line FIKalways passes through a fixed point.
EQC LAI CHO DUNG
TrAn Tqp ch{ sd 448, trang 16, xin daqc dgc
bi di bai T5/448 nha sau;
Cho a, b, c ld c6c s5 thr;c dwtng thoa min
a' +b' + c' = 1. Chtmg minh ring
a2- +b12- b12- +c: - c2- +a2- - I^
-T-abla+
b)3 bc(b+c)t ca(c+a)'- 4'
Thdnh thdt xin l6i ban doc.
diqn thti c6 thO ch4p l4i v6i
cliQn tucrng ttuong nhu sau:
, Ta c6 mach
Tt hinh vE tr6n ta c6 th6 vE lai
hrong tlucrng nhu sau:
B'
Tuong tg m4ch a) ta c6 phucrng ffinh:
Ri,r,- t'Rt's,- r;=O
NghiQm cira phucmg trinh: R,u, :'t';t',
Tt d6 tinh duoc:
r d1+t
42
r (Ji+t)
-4+2r
lxs = 4,x1=34
l*,*2.x, = x|*r +l 8. 1 0'*r
mach cli6n
n* =l* ='J'
2
- TONN HOC 26 - cl'uOiga so as (11-2014)

PHEP CQlrG... (Ti6p theo trang t5)
CuOi cing, giSo su B c6haic6ch ng6i, d6 ld ng6i d
b6n ph6i ho{c b6n tr6i. Do d6, ddp i.nliL2.2.2:8.
Nht'ng suy luQn ndy d6n ta cttin mQt quy t6c <ttim
quan tlong kh6c.
Quy tdc nhfrn. lVdu ,str ki€n At co th€ .ra1t,ra theo
at ciich khac nhau vd su ki€n A2 c6 th€ xay ra,
theo a2 cach khdc nhau,... vd str ki€n A, c6 th1
xay ra theo a,, cdch khdc .nhau thi t6ng s6 cach
d€ str,ki6n Alxat- ra r6i d€n su'kiAn A2 xa1; ra,...,
r6i d€n str ki€n A,, xay ra ld ap2.. .a,,.
Ta c'ilng co th€ di€n ta quy tiic nhdn bdng ng6n
ngir tQp hep, ttrc ld n€u
S = {(r,,.!2,...,"!,) / s, e S,,1 < I ( z}
rhi lsl=ls,lls,l ls.l
f( dy 5. MOt bien s6 re c6 3 lq, ru' ddu ld dAy
g6m 3 chii cai trong bang chir cai vd 3 lqt ta sctu
ld ddy .gdm 3 con s6. C.6 th€ ldm ra bao nhi€u
bien s6 xe khcic nhau nAu kh.6ng,dactc dirng s6 0
va chti O trong cilng m6t hi€n s6?
Ldi gi,fii GSi S, ld t{p hqp c6c bii5n sd xe kh6ng
c6 sii 0 vdsrld tap hqp c6c bitin s6 xe khdng c6
cht O. N6u aBy - eh// ld mQt bi€n sd xe thuQc
^S1 thi P, 0, W # 0. Ti6p theo, kh6ng c6 y6u cAu gi
O5i vol a, f , y n6n m6i e, f, I c6 th€ nhQ-n26 gi6
trj, trong khi m5i 0, d, V nhfn clu-o. c 9 gi5 fi. Do
d6, lql = zo'.e' . Suy lufn tuong t.u,
ls, | = zs'. t o3 (vi vai trd cria cht vd so ctu-o. c doi
voi nhau). Dudng nhu <l6p 6n cira bdi to6n lA
ls,l*lsrl=263.93 +253.103. Tuy nhien, day
khdng phii ld ddp Sn,chinh x6c. Nhrmg m6i
bu6c ldm 4r*g nhu rdt hqp ly. Vqy sai o dAu?
CAu h6i m6u chdt hcrn ld: Lirm sao ta bi6t c6 sai
hay khdng?
Ta tra loi c6u h6i thir hai trudc. Ggi S $ taq hOp
moi bi6n s6 xe t4o du-o. c theo nhu y6u cau at Uai.
M6i cht trong d6y 3 cht c6i c6 26 lgachon vd m6i
con s6 trong ddy 3 con sd c6 10 lga chgn. Theo quy
t'ic nhAn, lSl=zO'.tOt. Khdng kh6 dC kitlm tra
du-oc: lS, | * lS, I = 263 .93 + 253 .lO3 > 26'. 10' = lsl.
R6 rdng lS,l*ls,l khdng ph6i h cdu tri loi ta
mu6n. Gid ta phii sria l6i sai. Luu y r[ng c6 vdi
ch6 tring nhau gita,S, ve E, d6 ld nhirng bi6n
si5 kh6ng c6 ci sti 0 ho{c cht O. cgi S, ld tfp
hqp cdc ,b .i;.6n sd nhu vay. Suy ta
S, = S, n S, v6i m6i cht c6i trong mOt bi6n sd
thuQc S, , co 251ga chgn vd voi mdi con s5, c6 9
lpa chsn. Do 116 lql=zs'.e'. Vi m6i bi6n s6
trong,S3 clugc d6m 2 lAn trong S,
"dS,
ndn cdu
tri lcri cu6i cr)ng cira bdi toan ld:
ls,l * ls,l - lq I = 263 .s3 + 2s3 .to3 - 2s3 .93
=17047279.
K! thuft bao hdm nhfrng tAp hqp ch6ng chdo l6n
nhau vd loai trt nhirng phAn dugc d6m hai 16n
goi li Quy tic Bao hdm - Lo4i trir.
Thi dy 6. [AIME_ 1996J Trong mdt gi(.ti tidtt co 5
dQi tham gia, m6i doi ddu mQt trQy v6'i,timgdi
cdn lai. M6i d)i crj 50'%, ca' h6i chiAn thdng bdt ki
trdn ndo ntd n6 tham gia (khdng co tran hda). Tinh
xdc sudt giai ddu khdng cd hoac m)t d)i kh6ng
thua trdn ndo hodc m6t d6i kh6ng thdng trdn ndo.
Ldi gi,rtL M6i doi phii choi a @1. Do d6, c6 5.4
trQn n€u m6i tr4n dugc il6m hai lAn. Vay 5 c10i sE
, .l 5,4
choi tdns cons - ' ' :10 k6n. Vi m6i hdn c6 th6
2
c6 hai k6t qu6 n6n c6 210 k6tqu6 cho gihi dfu.
C6 5 c6ch tl6 chgn ryQt dQi kh0ng thua kfln ndo.
Gi6 su ilQi A thdng tdt ch 4 tr{n md n6 tham gia.
Vfly m6i trQn trong 6 trAn cdn lai c6 thd c6 2 k€t
qua. trong t6ng s6 210-4 - 2u k6t qure. Vi chi c6
nhi6u nh6t mQt il6i kh6ng thua trAn ndo n6n c6
5.26 gihid6u cho ra mQt dQi khdng thua trAn ndo.
Suy iufn hrcrng t.u cho ta 5.26 tring}to glii dd.u
cho ra mQt dQi khOng thing tr6n ndo.
Tuy nhi€n, hai xhc su6t ndy kh6ng lopi tni l6n
nhau. C6 th6 c6 chinh x6c mQt dQi kh6ng thua
trfn ndo vd chinh x6c mQt dQi khdng thdng trQn
ndo trong ctng mQt gi6i d6u. C6 ,4 =20 ho6n
vi hai dQi nhu v{y. Gi6 str dQi I kh6ng thua trAn
ndo vd <tQi B kh6ng thing trQn ndo. C6 biy (cht
khdng phbi tim, v A vit B cl6u vdi nhau!) trfn
trong d6 hoac clQi Ahoic dQi B hoflc ci hai dQi
tham gia. KOt qui cua 7 trQn ndy tl6 du-o. c x6c
dinh. MOi trfn trong 3 trQn cdn lai c6 hai k6t qu6
trong tdng sd 2'0-7 =23 giaid6u. N6i c6ch khiic,
20.f : i2' t ong zto giai <l5Lu.c6 cn dQi kh6ng
thua trfln ndo vd dQi khdng thdng trfln ndo. Do
d6, theo quy tdc Bao hdm - Lopi bri, c6:
2'o - 2.5.2u + 5.2s = 2t (2t - 5.22 + 5) = 2t .17
gi6i d6u kh6ng cho k6t qui ho{c mQt dQi kh6ng
thSng lrfln ndo ho[c mQt dQj kh6ng thua tr0n ndo.
Moi k6t qui c6 x6c suAt gi6ng nhau n6n x6c sudt
). . 77.2s fi
Can tlm l8 ."- 2'u = -3.2
* T?3ilr58E
nn ,rr-rorn,
27

Th{ d1t 7. Hoa c6 cdc h6p son gdm 8 mdu khdc
nhau. C6 *udn ton m|t b0 biSn hinh vu6ng cila
rnQt tiim bdng 2 x 2 sao cho cdc hinh vudng canh
nhau dug'c son mdu khdc nhau. Tim sii phuong
dn son mdu khdc nhau md Hoa cd thA 4o ra.
Hai phwong dn son mdu &rqc xem ld gidng nhau
ndu co thd thu duqc phtrong cin nay bing cdch
xoay phwong an kia.
Ldi gidi Hoa cdn it nht,t 2 vd nhiAu nh6t + meu.
C6 3 trucrng hqp nhu Hinh 5.
Hinh 5. (i) (ii) (iii)
Trong trudng hqp (i), co -,( cilchdti chqn mdrA,
B, C vit D l&6c nhau. M6i cdch scrn mdu trong
trucyng hgp ndy c6 th6 dugc xoay 90 d0 nguqc
chi6u kim ddng hO 3 Dn dC c6 3 c6ch scrn mdu
kh6c nhau nhu trong hinh 6. N6i c6ch kh6c, m6i
c6ch son mhu trong truong hqp ndy bi cltfm 4 lAn,
tinh d6n cA trudng hgp xoay trdn. VAy c6
1- = Orlc6ch son miu kh6c nhau.
4ffiffiffiffi
Itinh 6
Trong trucrng hgrp (ii), c6 4 c6ch chon mdu
kh6c nhau. trzt6i cach son mdu trong trucmg hqp
ndy c6 th6 iluqc xoay 90 dQ ngugc chidu kim
d6ng hd 3 lin dC c6 3 c6ch son mdu kh6c nhau
nhu trong hinh 7. N6i cSch kh6c, m6i c6ch scrn
mdu trong trucrng hqp ndy bi d6m 4 lAn, tinh d6n
13
cd trudng hqp xoay tron. Vdy c6 5=84 cbch
4
scrn ffiffiffiw mdu kh6c nhau. HinhT
Trong trucrng hqp (iii), c6 I c6ch chon mdu
kh6c nhau A vit B. tvtdi c6ch scrn mdu trong
trucrng hqrp ndy c6 th6 dugc xoay 90 itQ nguoc
chi0u kim d6ng hO 1 6n dC thu dugc m6t cSch
son maru khSc nhu trong hinh 8. M6i c6ch s<yn
mdu trong trucrng hqrp ndy bi d6m 2 ldn, tinh dtfn
t2
ci trudng hqp xoay tron. Vdy c6 + =28 circh
2
sol mhu khSc nhau.
. Hinh I
Cu0i cing, ta co 420 + 84 + 28 : 532 c6ch son
mdu khSc nhau.
Ta dd xong chua? Chua i16u bpn 4! Nguoi dgc c6
th6 da tim ra m6t cdu tra ldi kh6c. Nhrmg tru6c
khi chi ra 16i sai cira minh, chring t6i mudn h6i
xem ldm c6ch ndo phdt hiQn ra 16i sai c6 thri c6
trong khi d6m. Vdng, m6t c6ch hiQu qud ld 6p
dgng phuong phfip tuong t.u cho c5c gi5 tri ban
dAu khSc nhau. Trong thi dU ndy, sO luqng mdu
dd cho khdng tl6ng vai trd quan trgng trong bdi
gihi cua chirng t6i. N6u ban dAu chring t6i clu-o.c
cho 7 mdu thi sao? V6ng, vQy ta sE c6 I los
---!- =
42
c6ch scyn mdu kh6c nhau trong trucmg -
hqp (ii). ThQt ra chirng ta kh6ng c6 4 c6ch scrn
mdu kh5c nhau trong hinh 9. C6ch s<yn thri ba
tinh tu tr6i sang gi6ng v6i c6ch scm diu ti6n vi
cdc cdch ph6n b6 mdu B vd C dugc dt5m khi
chon mdu c6 thir W (4) . fucr"g t.u, c5ch son
mdu thr? ba vd tu cflng gi6ng nhau khi chon mdu
13
c6 thu tu. Vav., 2c6 3 = 168 c6ch son miu kh6c
nhau trong trucmg hgp (ii). Vpy : d6p 6n chinh x6c
cho Th[ du 7 liL 420 + i 68 + 28 616.
BAI TAP
1. Tim s6 lugrg s5 nguydn duong c6 2 chir s6 chia
hi5t cho c6 hai cht si5 cta n6.
2. IAIME 2000] C6 2 hQp,m6i trqp chua c6 bi den
. i . :. /,. va trdng, va t6ng s6 bi trong hai hQp ld 25. L6,y
ngiu nhi6n mQt bi tu m6i hdp. Xric su6t Ce cd hai bi
4.7
tl6u ld bi den ld 1 . X6c su6t d0 c6 hai bi d6u ld bi
50
tr6ng h bao nhi6u?
3. C6 10 nt vd 4 nam trong lcrp t6 hqp cira thAy
Dfrng. 9,6 bao nhi6u c6ch d6 xt5p nhirng hgc sinh
ndy ng6i quanh pQt bdn trdn sao cho kh6ng c6 hgc
sinh nam ndo ng6i canh nhau?
4. Cho r ld mQt sd nglry0n l6n hcm 4, vh cho PrP, .1
ld clc da gi6c l6i n calrth. Binh m.u6nvE n - 3 tluong
ch6o phdn vung khdng gian b6n ffong.da giircthitthn
- 2 tam gi6c vd c6c dudng ch6o chi giao nhau t4i
tlinh cua.tla gi6c. Ngodi ra, anhmu6n m5i tam gi6c
c6 it nh6t 1 c4nh chung vcri da gilc. Binh c6 th)
chia nhu vfy theo bao nhi6u cSch?
zst?[H,H@

Kd d4,4 ouattl; mw&m
.Kffieru
rur,ue.
qd-&Y
ffi-w lt*
:,.]:irri: siicii Li? i:,ti, ',1o.rl;ii.ii 'i"1,;-'"i; r,rli ii'; l::.ll iii:li t:iti !'l;i :.t ttrli-'i i-:r,'r: ,;{'ii ll.14. :
ry -*' CiAi ]iu t siir-: {3 girii}
t. od tlguydn Vinh Huy,10 Todn, PTNK - DHQG
fP. nO ChiMinh. l
2.Ittguydn Trung Hi€u, 12 Todn 1, THPT chuyCn
Hrng Y6n.
"e GiAi NhAr t.: gi,li:
l. L€ Phudc Dlnh, 9ll, THCS Kim Ddng, HQi An,
Quflng Nam.
2. Ng.rydnDthc ThtAn,gA3,THCS LdmThao, Phri Thg.
3. NgLryen VdnThd,l0 Tor4n 1, TFIPT chuy6n Hn Tinh.
'1 {Jiei Shi { 11} fii*ii
l. Trin LA Hi€p,7A, THCS Ly Nhat Quang, E6
Luong, Ngh$ An.
2. Nguydn Bdo Trdn,7A, THCS Tdy Vinh, Tdy Scm,
Binh Dinh.
3. Nguydn Thi Hq Vy,7A, THCS Henh Phuoc, Nghla
Hdnh, Quing Ngni.
4. V{i Thi Thi,8A, THCS Hanh Phudc, Nghia Hdnh,
Quing Ngni.
5. Li D4t Anh,gA,THCS Nguy6n Huy Tuong. E6ng
Anh, Hi NOi.
,
6. LA fuang Dilng, 9D; THCS Nhfi Be Sy, Ho5ng
7. Ngttydn HieuHuy,gAl,THCS YCnL4c, Vftrh Phric.
8. Phqm Quang Todn.gC.THCS Dqng Thai Mai, TP.
Vinh, NghQ An.
9. Hd Xudn Hilng,l0Tl, THPT Do Lucrng I, NghQ An.
10.Trdn Hdu MqnhCudng,llTl, TIIPT chuydn Hir Tinh,
11. Nguydn Long Duy, ll To6n 1, THPT chuydn
I{trng Y€n.
2. TrAn Bd Trung, 1l Tofu: 1, TIIPT chuydn Hmg YGn.
13. L€ Anh Tudn, ll To6n, THPT chuyen Bi6n Hod,
TP. HeNam. Hir Nam.
14. Vil Tudn Anh,12Todn2, THPT chuy6n LC H6ng
Phong. Nam Dinh.
15. Chu Thi Thu Hiin,l2T THPT chuyen Long An.
$. LA Minh Phaong, 12 Toin, THPT chuy6n Phan
Ngqc Hi6n, Cir Mau.
17.L€Th€ SnI,LLAS,THPTBim Son, Thanh H6a.
18. Trdn Nguy€n Try, l2C3A, THPT chuy6n Himg
Vucrng, TP. Pleiku, Gia Lai.
D. LA Eilrc VieL 12 To6n, TFIPT chuyen Hodng Vdn
Thq, Hda Binh.
; ilri, iB* r23 g.rii!
l. Nguydn Dinh Tuiin,6c, THCS Li Nhat Quang, D6
Luong, NghQ An.
2. Dqng Quang Anh,7A, THCS Nguy6n Chich, EOng
Son, Thanh H6a.
3.'Nguydn Daong Hodng Anh,7C, THCS V[n Lang,
TP. ViCt Tri, Phri Thg.
a. NguyAn Dqi Dwtng,7B, THCS Nguy6n Kim Vang,
Nghia Hdnh, Quing Ngli.
5. Nguydn L€ Hodng Duydn.7A. THCS Ph4m Van
D6ng. Nghia Hdnh, Quing Ngfli.
6. Nguydn Phuong DuyAn,7C, THCS Li6n Huong,
Vfr Quang, Hi finh.
7. Phqm Thiin Trang,7A, THCS Hanh Phuoc, Nghia
Hdnh, Quing Ngfli.
8. Phqm Thi Vy Vy,TA,THCS Nghia M!, Tu Nghia,
Quing Ngdi.
Sti aas Ol-'2ot4)
"r#$qPd ,H#{-
--.- -ffir,ex*iffi,r 1#W

9. Nguydn Thi Hing,88, THCS Li Nhat Quang, E6
Lucrng, NghQ An.
10. Nguydn Hiru Hodn,gB, THCS TrAn Phri, TT.
N6ng C5ng, Thanh H6a.
ll. Ngty)dn Thi Th€m,9A1, THCS YCn [ac, Vinh Phric.
2. LA Vdn Tructng NhQt,10T1, TIIPT chuydn Hi Titth.
B. LA Himg Ctdng,11A7, THPT Lucrng Eic Bing,
Hoing H6a, Thanh H6a.
14. Vd ThA Dry. 1 iA1, THPT SO t rr. Phi My. Binh Einh.
15. Bqch Xudn Dso, 11 To6n, THPT chuydn Bi6n
Hda, tlir Nam.
16. Trin Manh Hi.mg,11TA, THPT chuy6n Nguy6n
HuQ, Hi NQi.
17. Dfing Quang Huy, 11 To6n, THPT chuy6n Bi6n
Hda, Hir Nam.
18. Mai Tiiln Luqt, l1T, THPT chuyCn Ld Quf D6n,
TP. QuyNhcrn, Binh Dinh.
19. Trdn Duy Qudn,11T1, THPT chuy6n Nguy6n
Binh Khi6m, Vinh Long.
20. Eodn Phu Thi€n,11A1, THPT L6 Hdng Phong,
Tdy Hoa, Phri '6n.
21. Nguydn Minh Tri, llTl,TlIPT chuy6n Long An.
22. Trlnh Ngpc Til,11 Toan, THPT chuydn Bi6n Hda,
Hi Nam.
n, Vrt Vdn Quy,1241, THPT Nguy6n Chi Thanh,
TP. Pleiku, Gia Lai.
.lr t,lirii i-"im;,r,!:r i.lticn i65 gi*i3
l. Ng6 Ngqc Hudn,6A, THCS Phpm Vdn D6ng,
Nghia Hdnh, Quing NgIi.
2. Ngrydn Th! Qu)nh Trang, 6A, THCS Ho Xudn
Huong, Qujnh Luu, NghQ An.
3. Ngd Thi l{g7c iinh, 7A, THCS Cao Xudn Huy,
Di6n Chdu, NghQ An.
4. I,{yydn Cao Bdch,7B1, THCS Nguy6n Nghi6m,
TP. Qudng Ngii, Quing Ngii.
5. Kiiu Xudn Bdch,7A, THCS Le Htu Lflp, Hflu LQc,
Thanh H6a.
6. Trin Cd Bdo,7Al, THCS Phu6c LQc, Tuy Phu6c,
Binh Dinh.
7. Nguydn Thity Dung,7B, THCS Li Nhat Quang, D6
Lucrng, Ngh$ An.
8. Trin Minh Hi€u,7C, THCS Vdn Lang, TP. Viet
Tri, Phti Thg.
9. Nguydn Khdi Hcmg,7D, THCS Nhir BA S!, Hoing
Hoa, Thanh H6a.
lO. I/d Thj H6ng Kiiu,7A, THCS Ngtria M!, Tu
Nghia, Quing Ngfli.
1I. D6 fhi W Lan,7A, THCS Hanh Phuoc, Nghia
Hdnh. Quing Ngni.
12. Nguydn Vdn Msnh,7A, THCS Li Nhat Quang,
D6 Luong. Ngh$ An.
B. VA Phaong Tdm, 78, THCS H6 Xudn Huong,
Quynh Luu, NghQ An.
14. tlguydn Ydn Todn,7A, THCS Li Nhat Quang, D6
Luong, NghQ An.
15. Ng4)dn Thdnh Vinh,7A1, THCS vd THPT Hai
Bd Tnmg, TX. Phric Y6n, Vinh Phric.
16. Nguydn Eqi Daong, 8B, THCS Nguy6n Kim
Vang, Nghia Hdnh, Quing Ng6i.
U.NgLryAnTii*tlong,8A1, THCS Ldm Thao, Phri Thq.
18. Daong Xudn Long,8B, THCS Li Nhat Quang,
D6 Luong, NghQ An.
19. Chu Mai Anh,gAl, THCS Y6n Lpc, Vinh Phtfc.
20. Hodng Th! Minh Anh,9A7,THCS Y6n Lpc, tnh Phric.
21. LA Phuc Anh,gA, THCS Nguy6n Huy Tuong,
D6ng Anh, Hn NOi.
22. Cao Hibu Dqt,9C, THCS Dqng Thai Mai, TP.
Vinh, Nghp An.
23. Nguydn Thi Thanh Hwrng,gA, THCS Y6n Phong
Bic Ninh.
24. Vil Thu) Linh,9A3, THCS Ldm Thao, Phti Thg.
25. Ng6 t{hQt Long,9A2, THCS Tran Phf, Pht Li,
IId Nam.
26. Hodng Dac Mqnh,gA, THCS Dinh C6ng Tritg,
Thanh Li6m, Hi Nam.
27. Td Minh Ngpc,9At, THCS Ydn Lpc, Vinh Phfc.
28. Nguydn Thu!, Qu)nh,9A2, THCS Gi6y Phong
Ch6u, PhirNinh, Phti Thg.
29. Hodng Huy Th6ng,9c, THCS Phan Chu Trinh,
TP. Bu6n Ma ThuQt, D[k LIk.
30. Trdn Thanh Binh,10 ToriLn, THPT chuy6n Quing
Binh, Quing Binh.
31. Nguydn H6ng Ddng,10 To6n 1, THPT chuydn L6
H6ng Phong, TP. Nam Dinh, Nam Dinh.
32. Nguyin Dodn Hidu,10T1, THPT D6 Lunng I, E6
Luong. NghQ An.
33. Ldm Bt?u Hang,10A1T, THPT chuyEn Nguy6n
Thi Minh Khai, S6c Tring.
34. Nguydn Tudn Htrng,10 To6n l, THPT chuy6n L6
Hdng Phong. TP. Nam Dinh, Nam Dinh.
35. Nguydn Trdn LA Minh,l0 To6n, THPT chuyCn L€
Quf D6n Ninh Thu$n.
36. Ng.qt€n H6ngNgpc, 10A1, THPT chuy6n tnh Phtfic.
37. Nguydn Minh Ng7c, 10 To6n, THPT chuy6n
Qu6ng Binh, Quing Binh.
38. Trurmg Minh Nhqt Quang,10T, THPT chuy6n L6
Quf Ddn, TP. Quy Nhon, Binh Dinh.

53. Nguydn Hibu Khoi, 11 To6n 2, THPT chuy€n
Nguy6n HuQ, QuQn Hd D6ng, Hn NQi.
54. Nguydn Duy Linh, 11 To6n, THPT chuy6n B6n
Tre, B6n Tre.
55. Dinh Chung Mung, ll To6n, THPT chuy6n
Hodng Van Thr,r. TP.Hoa Binh, Hda Binh.
56. T*Nhfu Quang,11 To6n, THPT chuy6n B6n Tre,
B6n Tre.
57. Ngo Hodng Thanh Quang, I I To6n, THPT
chuy6n Quang Binh. QGng Binh.
58. DAu Hing Qudn,1lAl, THPT chuy6n Phan BQi
Chdu. TP. Vinh, Ngh$ An.
59. Nguydn Minh Thdnh. l1 Todn, THPT chuy6n
Ti6n Giang, TP. My Tho, Tiijn Giang.
60. Trin Trpng Ti€n,1 I To5n, THPT chuydn L6 Quy
D6n, Quing Tr!.
61. Trdn Eac Anh, 12 Toin, TIIPT chuyCn LC Quf
D6n, Quing Tri.
62. Phqm Tudn Huy, 12 ToLn, PTNK - DHQG TP.
HO Chi Minh, TP. ttO Ctri Vfinn.
63. Luu Giang Nam, 12 Tobn 1, THPT chuy6n Phan
Nggc Hi6n, TP. Cd Mau, Ch Mau.
64. Nguydn Nhu ThiQp, 12A1, THPT fran quOc
Toan. Eakar, DIk LIk.
65. Nguydn Vdn TuyOn,l2AtlK25, THPT D6ng Hj',
TP. ThAi Nguy6n, Th6i Nguy0n.
ry * Giei her (l gini)
I'tguym Mqnh Ddn, 10 A3 Ljz, TF{PT chuy6n rmh Phfic.
* Gi:ii hi (6 gidi)
1. ttguy€n Manh Dfing, 10 A3 Li, THPT chuy6n
Vinh Phrfic.
2. Phan Quiic Yaong,
NghQ An.
3. Biti Vil Hodn, ll
Quing Ngii.
4. Vrt Ydn Dilng,ll Toim2, THPT chuy6n Th6i Binh.
5. Nguydn Vdn Hirng, l1B, THPT chuy6n Quang
Trung, Binh Phufc.
6. LA Xudn Bdo,12A3, THPT chuy6n Phan BQi ChAu,
TP. Vinh. Ngh$ An.
* (;iri Ba (9 si,ii)
l. Vfi Dlc Thhrg,l0 A3 Li, THPT chuy6n V-rnh Phric.
39. Vil Hing Qudn,10 To5n, THPT chuydn Nguy6n
Tdt Thdnh, YGn B6i.
40. Vatrng Hodi Thanh, 10A2T, THPT chuy€n
Nguy6n ThiMinh Khai, S6c Tring.
41. Nguydn Thi Trang,10 To5n, THPT chuy6n Bic
Giang, TP. Bic Giang, B6c Giang.
42. Nguydn Vdn An,11 To6n, THPT chuy6n Bic
Ninh, TP. Bic Ninh, B6c Ninh.
43. Nguydn Ydn Cao,11Al, THPT S6ng Son, S6ng
L6, Vinh Phric.
44. Truong Hadng Duy,l1T, THPT chuy6n Nguy6n
Einh Chi6u, D6ng Th6p.
45. Phqm Trung Dilng, 11A1, THPT chuydn DH
Vinh, TP. Vinh, NghQ An.
46. Nguydn Tidn Dqt,11T, THPT chuyCn Lam S<rn,
Thanh H6a.
41 . Nguydn Thi ViCt Hd, I I To6n l, THPT chuydn Hd
TInh, Hi finh.
48. Le Vdn Hdi, 1147, THPT Luong Dfc Bing,
Hodng H6a, Thanh H6a.
a9. Ngtyln Vdn Hdi,118, TF{PT Tdy Son, Binh Einh.
50. Phqm Minh HQu,11 Toiin 1. THPT chuydn Long
An, TP. Long An, Long An.
51. Tdng Trung Hidu,1141, THPT Thrii Hoi, TX.
Th6i Hod. Ngh$ An.
52. Nguydn Th! Phuong Hodi, l1 To6n, THPT
chuydn L6 Quf Ddn, Quing Tri.
11A1, THPT DiSn Ch6u 3,
Li, THPT chuy6n t6 Khi6t,
2. Nguydn Vidt Sang,10 Li, THPT chuy€n Nguy6n
Du. Ddk Ldk.
3. Tdng Trung Hidu, l1Al, THPT ThSi Hda, TX.
Th6i Hda. NghQ An.
4. Chu Minh Th6ng, 11A3, THPT chuy6n Phan BQi
Ch6u, TP. Vinh, NghQ An.
5. Nguydn Thi Oanh,11C1, THPT Hoing H6a fV,
Thanh H6a.
6. NguyAn Vi€t Tudn,1245, THPT chuy6n EH Vinh,
Ngh$ An.
7. Phqm Neqc Bdch,12A4, TFIPT Tirrh Gia 2, Thanh Hda.
8. NguyAn Hodi Nam, 12A1, TTIPT Ducrng Qu6ng
Hirm, Vdn Giang, Hung YOn.
9. Phqm Thanh Binh,12A1, THPT Luong Phir, Phi
Binh, Thii Nguy6n.
ilic l-ran doai giii nho gir'i gdp d!;r chi n:di c*a rnlnh vd T'drr soa;i cid nh.ln Gidy Chr'rng nh{n v;i ling 1:hdm ciia 'Iap c}ri.
eg11lsl:qq_*W#ffi[st

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