#€stu ffiffi#
nOr ,roAN Hoc
. yI0T NlM,
. r BAC/ qA
chi nhigm, ne uyfx C.&uH fo,A.n
Tru s,} : 70 I'rln llung Dao -. I{a l{d:
":-;*;i#*.;i$+** .ffi
rY
Thg kd rdir, so6.i , mOANG Cg{]nC
DAy.udi : 52825
tiri trong 2 bubi sl ng. m6i bubi
biii toar.r;
f.jAi rr'tAru{; vr}r
8 hoc sinh, bAi
4.ti6ng rd c6 3
j ri:t l:,,:i1: h :,''
,,-....i,,ta1 {r;liiuii,:i
ryqrygq=El'!al!sAc BAN rRi, ynu roAtf
KY 'r-HI T{}A,ru n}Ui}C hIAfvI 1980 p
"rLl
F LC ban bi6t ring nini i9g0, kh6:rg c6 tb chuc.
\, thi ToAn t1u6c t6. N6u kB tr,r, nlrn i95g, ki
thi To6n qu 6c t6 ctuoc tb chirc IAn driLr
(& fiumani) va ddu dr.rrr h:ng rrinr. ,ni Jly iu
lin dhu ti6n xiv ,ra su gian doa n. I in tlidirg
b6o v<ii cAc ban : ti'rang i rem nay (i9g2), kv
thi toAn qrrdc td Ijn ttrii 23 se riuoc t6,.huc ,i
Hung-ga-r:i, qud hrrong crSa nir:r t.tan hoc Bilva i.
llung-urr-ri, lA nrrr'rr. tl jrr tir.ir t1gr1 1ft.1 qio.i. Li.le
nEm tb chuc thi hoc sinh uii,i toiLr! [r.r]ir cguv
*ili, 3iil,,liii;'L-1li-l:11
^*,;,1 Yl;"i,1"il;
. lhti_7 l{ur:g-ga-ri ti6p uhAn t6. c}iirr tlii 1'o6n guilc
t6 (hai trHn'lrrri:c: vao cflc lim Igill va 1g?0,).
DA bb khuydt rli-.., s!r gr'6n <1r,nn. liLm ig,B0
m$t s6 nird.r: r"r chA'r iiu dI. tb chirc hrii ki. ilri
1'oArt qrrrlc td c& nlrir. n:ol l;i- thi ir piriir, i.:,:,
ng4)- I -2-7-1980, roi + .r.rir* ilram drr : Huiic-
ga-ii, llhAn Lan, Thuy $-ibn, Anh, vi r:it)t krl.
thi &. t,uc.xerii-bp;r. ngt -v l{r -- 1l-7-1gc0. vr.ri i
n,rt6c thanr du : Lric-xiLrn-buc, n?" Iii i.an, r.ain
. 'frr, Arrlr (tIOi Ihrr iruii. Cii t,ri L5 r;,i-r^r 1i,rr..
tb chwe theo tli6 thfc cfia ky thi T'oAn qu6c t6
th6ng Jhupng : m6i nudc cri mQt dQi tuybrr g6m
Xin gi,ri thi6Lr vdi c6c bqn 1'0u lc:dn oAc dE
to{n trong hai k.f thi dy. S
]{sdq- tltu nh?it:
1, 'f i'r.'r':g tam ciSc
cfia cflc canh -1 ti
irr X vd Y.
'3
,{BC, cic dusng trrrng lruc
rrA.4C cEt dircrng thilg BC
Chring iiiinh rr)ng dl BC = XF,
clir iA teB.tgC = li. Il:iv ch*rg t6
n{y kh(ing phii L\ c[n. vA hiv
r'iirt va dii rie dL'-XI .
mOt
ri.ng
tinr
diFrr hi0n
,liBu ki0n
diii:r hidn
(6 rti6rn )
clinh nhu
( tt{ n).
( r flletn,l
2, L\:'rrr. ,(ao, a1, a?',.,.. arr) cIuc..c xilc
sixu:.rr'D -= 1f2,. otnt - ak;l- (l/n) ni (0
CIrung minh ling
-I/r (,i" 1 r'.
3. Xe:r ph.unng'tlinlr
'
'n+l=9o+1'
1 _ THTT
. ::-r5r1-.
igi.*t9j1il1,;;1;--; *,-.. -.1.-i: --. "-.:r::.:t{a'J::'r
trong tl6 n ld mdt sd tu nhi0n l6n hon l. Chr'rng
minh ring phuong trlnh khOng c6 nghi(m nguyBn
duong (r; g), vdi r vd n * 1 nguvdn t6 r:ring
trhau. {7 .li}m )
:,J:, J:-ul,o, ,,,-canh n6i ti6p ,.o'g hinh
trdn, co n - 1 chp c?nh cl6i song song. Vdi
.nh&ng giSL tr! niro cria n thi hrri canh d[5i cdn
Iai ciing la sr:ng son{ ? (6 diBm)
5. MOt drrcrng tbirng s'lrrg song r.'6i trr.rc hoinh
iluoc goi ld < co tinh tam giie I i15i vrii duSng
coil g
432
I/ = r -'j- Px -f i/l' I r'rr -r $
n6u n6 c&t ilrrcrng eong n$y tai 4 di6rrr khAc
r,hau A, B, C, D (theo thir tU {y !) vA tri cfc
iloan AB, lC, JD, c6 thb lep drroc mdt tam
giAc ?
Ch*ng rninh ring vdi clrrdng cor:g tr€n. thi
moi ctudng thing song song v6i truer hoAnh, clt
dudng cong tai 4 ilibrn kh6c nbau, hoic dbng
thoi c6 tinh tam giirc, holc d0rg thoi khOng c6
tinh latr gi{e. (i diEm)
0' Trong dang thAp phan cira so tVi-r- r/'+ ;tsao'
hEy tim chir c6 hAng rlcrn vi ld t:hit sd dhu
ti€n sau d{u phhy. (7 tlibm)
II. Ki thi i! tuc-xem'1,u4'
Nqdu fliri" nld, :
1. IIay
voi rnoi r
ihdi thira
tinr rlr0t hirri sd ,f (r) cltto. c xric ilinh
triru tf. va c6 gi5 tri trttu ti, ddng
rniln riiLr di6tr kien sru:
f,1):2. ii.r'y) = i (.r)/(g) -* /{ t 1" r;) '* I
i{l d i}rn)
dibm B. Tr'6n circ
drrdng kinh, dung
nr$t phla c16i I'ri
ti€p tu,t6n clrung
nd c61 nua drrirng
P li circ ti6P diiin
voi hai nria drtrng
1{ch cdc tam giic
cira ba nria iluong
3. Gia sit P ln m6t sd nguv€tt
sC., nguy 6n tlucrng. Cilirng minh
dB sau rlAY li trror.lg dluong;
,)
(7 11i'8m,
tli v{ n I{ mqt
riing hai nrQnh
b) 116 n cti tirO biEu tiiin ducrc
rl:!r"(-1, trong dd s, q Ii nhirng
a) Kh6ng c6 s6 C$ = n !/Jr l(n - &) ! niro chia
hdt cho p:
s }0 vn O{s{.p
dutri dang
sd nguv€n,
(7 di6nr)
l{qrig illf lai :
4. Hai drrong trdn tidp ruc ngodi hay tron$
vdi nhau tai dibm P. Ivl6t tlrrt-rng thrtng, tidp xrie
vtii mQt dLrdris tr0rr tai di6m .{. c6t iluirng trdn
kia tai cac diSm B vi L'. {.hring rrrinh rilng
dudng thring l'.'L ln phin gi6e cira Indi soc tao
br-r'i c{e during th8ng PB "td. PC (ti dibm}
5. Muoi uguui luc dlu cd s6 tiErr nhu nhau'
FIo choi trd choi sau di-t' : li rr lucrt nr6i rguoi
gieo 5 qufln xric xic, vi tri cho nr6i ngutri trong
9 ngudi kia 1ln s6 tiEo cua nguoi dtrqc trf,
hi0n c6, v$i n lA tbng s6 di6m ddrn tr€n 5 qu:ic
ric xic tlii gieo. D6n luqt di c&a ngirtii thri 10.
nguoi ::Av tluqc 12 dii:in I sau khi trh ti€n xong'
thi s5 tiEn cfir rn6i ngu<ri. lai tr& thirnh nhu
luc trtrut khi chni'
Hdy xAc dlnh (n6u c6 thb) s6 di6m cia m6i
nguoi khi gieo 5 qutin xric x[c' (? tli6m)
ch<vi trd chcvi t'lua rrgua dDu bi6t qui'n x*c x*c
la mdt hinh lap phrrong c6 6 mlt duoc c[6nh
c6 tu I d6n 6 dibrn. Chilng ta ch:ing chap nhAn
mQt tro chcri in ti6n: nhung r:ic ban chirng nOn
lrg3c nhi6n vrri bAi toe'n tren 6 nutic tu b&n' O'
1\,10nac6,, ngrrtri vi6t bti nay de thdy vdi viii
qu,{n xirc t[. to-t Yei cA bii tfl1-, ngucri ta ch<ri
,,1,l lrng ( tt d choi >r iin tiFn vA'.' sat nghiQp I I !)
{i, 'l'im tf{t cir tiic aghitlm rrg'trr€n {'r: g) rira
phirrrnp tri*ir #
o f , :\ .. 2 2
s'-i- .t-ry i' r./- * s3: E(.t2 * 'rrg * 92 '.1- t)'
(? tli6m)
Chti thich. t-oi gili cAc bAi todl tlen sE ding
vio sd 0ll9S2. I{ong rlng do s0 li bAi girii' cring
lhirrg su-v nghi (nhAn xdt, li0n h€' mtr rQng'"')
ctia e {c barr 1'6u toAn vd 1-€u b5o Tl{r'T1"
{
lr
vdi moi ;':, 17 hiru 1r".
2. TrEn doarr 3C, IdY rrtQt
tloan AB, B{,, Ce coi Ithir
c{c rr&r-i dtrorlg trt)n vi r:tirig
rtu<ing ihirng nC. Qua B k6
cia hai nira duorrg tlt-ru iiriu,
trdn thu ba tai Ii' Cioi li yzl
cii* ti6p tuYdn chrrng thir hai
tron tlhu,'finh tf s6 di6n
J9PK vA EA(J qua bAn k{ah
tro ti .
ln
it
t
\
;
Ti{vTT
t--!
r/
Coile TIIfC sol BIflI{ TROHG TIHr TIen PIIfiN
t:{
e
TINH {,Or-t)e*'*2dr lanr6r cAu rrorrg ttE ihi
chinh thfc mOn to{n hoc tt ong ki thi tuv6n
sinh Dai hoc vir cao iLlng phia {am thang T/i081
thu6c kh6i B.
Co mdt s6 nguoi gi6i nhu sau: d[t g(il=r- x2,
vri 6p dung cOng- rhrrc d6i bi6n vdi ham g nAy,
ta cr: :
1
f
l(2.r -
o
nbu g(r) = c2 r,6j tich
v6i tich phAn
Cech gi&i ndy giy ra nhi6u tranh luin, r.i hirm
g kh6.ng don diQu tr6n kholng cln x6t fa [0, f].
Cd-ngurii cho riing kh6ng rh6 dbi bidn voi g
khOng don diQu iluoe, r,A eho m6t v{i thi du
clu niy, n6u
I'a xdt thi dr: illu v6i 9 (.u) = 12 vlltich ph6.n
Di+ ly: Cho a, b, c,d tit b6n s6 rhu.c. a a( D
vn c(d.-Cho / lA mQt hdm s6 x6c djnl vn li6n
luc tr6n [c,d], vA g lA m6t hdm sd x6e clinh vd
c6 itao hAm li€n tuc tr6n [a,6i, ve g ila,bDe
[., ,t l. Ltc d6 :
CAch chung minh dfnh I;f nAy "C, cring giin
di. chi dua vAo kh6i ni6m lguyGn hdm vA tich
phAn,. chri ng ta c6 thB xern troi,g tli liQu ddn
b6n t.:_". Nhtt vay. v6i o : c --- 0, b - d = t,
/(.) = ex vir g(r) = ? - xl, hp durrg tlinh lf n6u
tren. cach giii n6i tr6n Ia ding.
Di nhi6n n€u ai chua bi6t dinh ljr tr€n, nh6t
lA^ c{c hoc sinh phb th6ng,,vl chi bi6t cAc c6ch
dbi bi6n s6 vdi cAc hdm g clon tliQu, mA gi6i
theo cich tr€n, ta ph&i xem lA ho gi&i sai vi &
dAy 9 kh6ng don di6u.
2.CLc thi drf n€n tron c6 ld e6e ngbith
ly toiin hqc kh6ng ?
b- e (b)
f1.
I / tgtt) )g' (t) or = | / (, ) ae.
JI
" o(u)
slt) o-
l) rr- x2 d.,. -- - | ,", I
J /rr:--)e",|u=u.
s(o) o
I
phirr | .r?dr', o(r) = sinr.
-1
JT
J sinrrle-. Vdi cdc thi
o
{p dung phrr,:ng phep d6i bi6n v6i circ him g
nhu th6, c{c tieh phAn cria chfng ta sC blng
kh6ng. vi sau khi tl6i bi6n c6c dAu nr:rt l{_v tich
phAn lai bdng rhau, trong khi do, lri6n nhi€n li
c6c tich phdr nay.khAc khdirg.
fi Aay chring tdi st trilh bdy Ihn ltrot cic
tdn dE sau : C6 lha d6i bi6n r.6i cic hdm g kh6ng
don tlif u hay kh6ng ? N6u c6, thi eAc thi"du tr€n
cfl ntrai li cdc rrghich 11. tuurr irrJo lih0ng l, \,d
sffir hdt t.lrring tfii xin dd nghi vB cdng thrirc: r16i
bi€n trong chrrorg trirrh phii th6ng.
1. Co-thE dbi tridn vtri cr{e h&rn g kh0ng
don diQu lray khSng ?
'I'rong quyhn ca s& giAi tich toin hoc, tAp J
cira G. M. Fichteng6rr, tjo nhA _rufrt b6n Dai hoc
vri Tr,ung hoc ehuydn nghiip xu{t bAn iai Ha
NOi nbm lS77, & tran{ 2g7, c6 chirrrg minh nr6t
dinh lli dbi bi6n, mi chring tOi xin ,i6t lui du6i
dq.ng sau diy :
1
f"
I r'dr. -\ay gii s,i ta
1r
1l
tl& cho ! ,,'a, : If
, -l
c6 hdm s6 / li€n tr,rc vA
x5.c ilinh tr€n khoing [c, d] chri.a q ll-1, tl) =
Q @)) s'(,") dr. Lric
to, r l,
c16 ta c6 f (J): 3 vOi.moi r thu6c [ - 1,1J,
,)
tl5c bi6t, vdi r = I, ta
I
*#
ff-
t;'
I
co / {i) : l, vA r.6i
2
r:*1ta c6l(t) : - l- Do d6l kh6ng x6c
2
dinh tai ,1, vQy kh6ng thb c6 f tuong ung cho
c0ng thue dbi bi6n rr6u trOn trong iuring hr.rp
niy. 1't d6, dinh lf trdn ktrdng th& 6p ilung cho
-t
!;,-;:tr
'!'
i
Lua.,
J
-t
D
vot q \x):r
VAy kh0ng c6 gi Li nghicir lf
sau cilng tunng tr-r,
3.Gdp y vb co*gthl?e dbi
rrutrng phb lhong.
Troog cfLc cdng thtic d
b4y 6 c6c truirng Phb t
don di6u. Trong khi d6'
hoc si*rh thur.ing g[P l+i
crla dinh 1;!' n6u tr€n, vh g lai kh6ng don diQu'
M{t khic, clinh Ij- n6u trdn c6 th6 chirng minh
rdt d6 ddng bing cic khdi niQrn c6 sirr & chrrong
trirh phb thOng. Cho n0n'chirug tdi 1'ghi ndrr
dua clinh If niry vAo ciruong trinh ph'6 thong'
'lrong cAc truong clai hoc vd cao diag chul'6n
rrghiQp, chirng ta se g+p c6c tluong hop f phuc
trp fron, lirc d6 he1: trinh bAy cdng thitc dbi
bi6l vbi cfc hir:r g dr;n tli6u.
IO
h
ah
tol
h6
:{r
c6
on
trc
6,
b
6
tr
li, co* ilri du
bi$n sd rv cftc
rlugc trinh
g eiin phii
herp rnA eirc
6n
h
rd
c
DL'iC'\G \II\1I DITC
Can b0 gtdng dau khoa T-oan,
Dqi hpc'thng hqp 'l'.P- llb Cht Minh'
*
*
s6
im
ng
af,
t
t
i
i
bi6 n
t. h
trud
5t cr :6cc di6u kiQn
rfunhl
&$ts.*$*
b) ,t'= 1890' Ngutli
cu6i cirng. N6u lgrrni
sao cho
- /.
-
t
Oi -r ali+l - / tz - l, r.
II 1A nguoi tii-t cht s6
I vi€t o1' ngrroi '[I r-idt ci;l
5,..,, lSES). Khi ii L,
Bei 1/121. Hai ngedi cing uiit m|t sd cd k
chft sd md chi Qing cac chir sd 1,2,3,4, 5, 6,
M6i ng*di ldn ta$l uidtl ctrfr sd tb lrtii sarg ph&i)
Ir6i ngtrbi uitt chit sd cudi citq c6 thh rlitm
b&o cho sd dtrcrc oidt cllia hdt cho Lt l;hdnlJ tt4tr
a) k = 1930,
ir) lt : 1890.
' c) /c = 19,15,
Liti (ti&i (cria Ngrrgin 7-ltanh
truong Phan Chiu'l'rinh, EA
rri :945 X 7 :66 lli:.0.
Vdv nguoi II (nguoi vi6t chii sr5 cudi ctrng)
ludrr dim bi.o ilucYc cho ,4 .9.
e) /i: /-q45' Ngudi I lA ngLroi vi6t cl.rL! s6 cu6i
cung. l'rrcrng ir,r nhu trlrdng hqp b), i,du ngudi I
vi6t oi thi ngudi II vi6t airl sao cl.to o1 * a1+1:7
(i = 1, 3, 5,.... 1913). Iihi do
19,t5 lliil
I ri: I ai*arc4s:927x7*otg+snri
.-l ;: I
l-r
9)iX7 =6i8gl!i, ndi'r vdi circ chii sd 1.2.3'4'5'6
nguoi I kh6ng thb vi6t at945 et6 ci.ro A i9 dugc'
Edi 2i121- : I du pirjl srl l9s0 thdnlt titrtq cat s6
chinlt plitro':tq tlti ciin ph&i co tt nh(tt Dao nftiea
si lt,.t ttg'.' .
Ldt gt&i- Gii sii ttci i98C : nz +02 tn, f ff'v)
Do its0 jl ra
(24 + l)? -= i (nroil4), (2li)2 ='0 (rrrod4), (1)
rr6n ia i1:ri,r' o r'J. b dDLr Jihrii chirrr,
I)it o =2c1 vh 5:2b1, ta t:6 1930:+rri i 4f i
h, ]'
,195 : rr? + e!(ar, Dr {i })
, Do v€ trai t:t'ra clirrrg thtic I6 nen tlong o1 r'd'
br ph6i co nt6t s6 li, r,hung nhu r'&I thi r0 rdng
1 890
x
i =i
Du'ong, .t6p 1111
Nirng):
Eet A : ct ez.. a6 nhu v6.y db ,1 i I thl ciin
c6 (at t az +... .t uL) i g
a) k:1930 : Ta sd ci'rrilg minh rigtriri vi6t ilh&
s6 cu6i cung khOng th?: hfro dd rn cho ,4 1 9.
Thdt v{y: l6u ngtrdi I i'i6t sd thu nhAt at:2'
Ngudi /1 (ngrroi vi6t chir s6 cLItii ct)ng) r'i6t sfi
az rrAo c16y thi ngucri thr'r I vi6t o3 sao cho
tr2 * a3: ?. f iCp lurr, lli ngrrdi thri // vi6t a1
thi ngrrCri 1 lai vi6i (ri+l sir() cl-to ni f a;a1 = 7
(i = 4. 0, 8.-... iglx).
1 930
^\.
r\ntr vav ui* 2 '1 !lri4 .' I I ()1930
i-,
. :6Ti0 + rrtg3o
IUd 6;50 iI ,,lu rrig3g li nt(lt tt or)g ca<' chir' sd
1, 2, 3, 4, 5.6 n6n rrguoi II khdng thb chgu otgso
nhu th6 nao dB ,1:9.
4
J*
r
I
i
Bei 4/121. Gtai ha
,
a - i:1:
b
=a*bqe
f
&
t
,2, + U! .: I (mod,tr) cdn {Oi -= J (mocl,i), do vdv
khdng thb c6 d.ipg rhuc 495 : ,? + r,l t,ic ti
1980 =# a" * t,'
- G'i su ta r.o lcsrt : 12 .- bt j_ cz iu, 1.i, c r:.t.)
'lheo nh$n xct il.in su}. ra Li,- : rtz +b2*cg:tg80:4
thi phfr.i c6 u, D, tr rldu ch.iu
I)iit a=.2n1. b=2bt, c'=2c1. Ta c6
rt)s0=ao? + ii,? + 4c? h:,r.
495 = a?'+ rr? + ,?i (ur, irr, ct e .\ )
1 l:? = 49;r':= 5 (;nod 4) rr6n stir
ra at, bt. cr di!rr tri,.
D5t at = 2az] i, tti=,)g2* i, rt .'=2tz* t (.a2. b,t,
ce € N). Ta e6
a95 : aa! * 4az * t + 1bL + 4b2 + t* tcf,*tcz + l
hav
phuongl trinh:
,
a - .Ttr
:---'' h,b,c*0)
c
t2J: ai * a2 -t bi - hz- .i * .2
: u2(a2+ l) + bz\bz + t) + (ztcz t. t). , ",uiJr1'-r,l';j
Ta thdy cac s6 azkz*l), bz$zir ) vi cz(rz.l- l) ta di.tdi -bi6u
d€u chrirr mA ^t23 l4i Id sd 16. y;.. "o irr.-o"" ,li. sau (A =t 0):
1S80 =i= ,z + bz * cz voi a. b, c € N. 1)
I .oi qi di
i)rit XiI I:i gi:i tri chur:g cua bn ii sci trong
phrrong trinh ciiiu (rhri yi rang 1 .# A ,i thco gi6
thidt a. b, c dEu =t0), rbi tinir o, D, c theo / vA
r., it,, z.'fa aiur_yc (trong trrrdng htfp t,.';t - hay
t/t : o):
a : f(.r? - U.), b = t(gz - :r), c : tez -- ru\
1'rr do suy ra :
oa -- bc: /2.. (r3 + u, + za - a*yz)
vd h:ri he tlrri, rrila Lurrrril ir.r (tldi voi bZ * ca
YA C - ob).
Ddi chi6u ba clins thrrc dd, ta cluoc:
o2-bc ir-ro cz*ob
chung cria'ba ti sd nAy bflng l/tr
phunng trinh cuiri cring cia hQ,
thuc nghidm cfia h6 cl[ cho nhrr
i.ri.g ri6: i----.- 2(q 1 a+ tl
i)
tr; ,1 + o?
' lldt khAc ta th{y 19ti0 = +2 + lr1 f lu2 _r 382.
t Ua-" mu6:r biOu rt ii.n sd lgg0 thCrrh i6ng cria
csc sd chinh phuclo thi ciir, it nhdt la.tr sij haira..
Bai 3/121 ()Jlin!l^ ntirt!t rdngs uo,i nioi sl tt
ahien n thi hiQu (2n)r,_1 khonj thi 1d 1Ar. ,n,;;l
, :i:o t-n|t sd la nhiin uoi sd ntfi"nguy|n du,ong ttd
' ldn hon l. - r-.r"'
Loi girii. T'a cht!.rq nrinh bang nhiri chu,iis
: , Cii srr._cd nrdt so trl r:iri0ir o iC rrrot s6 guyen
guulg A')2 s.act cho r .- "
.f (2n)a'- I^= ok h,,1 {(2n)o * j I l(2,,), - r l= rrk {I)
i: V"i (2n)'* I v,1 (2r,)o - 1 ld hai s6 16 iien ti6p.
#r:iro "h.ing irsuy6n 16 ct)ng nhau. Nhuilg tich
*.ii chring ii Iir.r.thric blrc A. cda rr:ot si, ttr
nhi€1. l)o do rnriq si, (tn)o+ r "e iir,,i ll " ,ln ,*
ld lf,1' thr}a bAc .k e iia rnOt s6 to ul.,;en. .I,rrc ia
(2n)o11=Dk, (:r)'- I =ck. h)clt
Do do bk *-,,k - [{:,,)n + 1l - [(lrr)" - i j= 2
Vi .k > ?, , > c ) j. n6n rling rhirc bk _ .k: &
kh0rrg th6 xfi_r ra.
VQy kh6ng c6 sd a nAo th6a miin diug thuc (1)
(b-c)7*(c- o)2+(o-il?
(\6tr i, = 0 tlri x:: lJ = z.= 0 ctrrg th6a nrfrn
eAc bi6u ttnic nghi6m ring ubi truon! hop l1=0).
2) Trrrong hep 1- e ha]- 1lt:0, la ituryc:
t=g=z=(qt,b+c)l:j.
. N,E P.
Bai 5,LZl. Cid si, () ld td.m dubn{t tr6n nli
titp ud ll ki trvc tdm cia tam gid.c ABC. Chnng
minh
t) o,q2lGB.Ao + oBz l(BC.BA) | oczl(ca.ca)=t
2) ( H B. IlC)I (A B, AC) + ( H C, I] A\ I I BC.
B A)
+ (HA. HB)iCA.CB):t n1u A, B, C <90a. far
qud tne ndo ntu..4>90o.
_L& gitti. 1) GQi M, rr/, p liin Irrot lA ti6p dibrir
cfia elnong tron n6i ti6p tam giitc AllC,idi .6"
canh AB, BC, CA. 1'a co
M = o.,{ l(AB.,tc) + oa2l(sc.na) + oczlrce.ca,
= O.{? sin Al2{tt(Ats} + OBBsinB/2dt t,tACl +
-f }Cz sin
C I Zd t(..1 AC).
( ,:7, tn3 - hc)
I v=1. (b2 - cu)
(.=L(t2-ab)
"a: ::: ,
\':r : i"i i .. .
L. T. H.