^tE;' - t o;i +iF Jrr' - r dr;17
--)
l(- TbL'Lt+5u
Ldi gi,fii. Cdch l. V6i c5c sd thr,rc duong x,y ta
c6 (x-y)' > o nen suy ra Laz*-y. Do d6
v(lo'+6ab+5b2)' (lo'+6ab+5b2)'
Chimg minh tucrng t.u, ta c6: 7bz +6bc+5c2 __L\2\LUtwr) 'l3b' +lObc +5c -) - - ) lc- +oca+)a- > JT(zc +a). ,[-zl +toca+sd vivfy F>3JT(a+b+c). MEt kh6c (a+b+c)'>3(ab+bc+ca) n6n tu
gi6 thi6t, ta co a+b+c > 3 . Suy ra F > 9A .
D6u bing xity rakhi a = b = c :1 .
Vfly minF =9J, h,hi a=b=c=1.
Cdch 2. Ta c6:
7 a2 + 6ab + 5u' =f,Fo + 4b)' *f;tr - a)') )
, tG, + 4b)'z -
). .) 5 .r
3a' +t\ab+sb' =i(a+sb)' -;(r-bf
- Ji'-+ toob +iF < {?, + su).Do d6 U :3s!i:t-2 J-z (sa++u)' Do d6 U :3s!i:t-2 J-z (sa++u)'
3' 4a+5b
=
>z(td +6ab+5b,)-
3a'+loab+5b' Jri;Toab +iF
-(a"' +l\ab + 5b2) = tto' + 2ab + 5b2 .
Tucrng tu, ta cflng c6:
7a2 +6ab+5b2
=,1@ .u*!rtY lif;*;i/ A:jur:)!_2- 36 (5b+ 4c)'z
Suy ra
Jtr' +toiiiif \ 3r' +toab+5b2
>,6:,r+2ai+sF.
Ta lai c6:
rrd +?nb + 5t = z(u + t)' $(a - b)' >z(u + u)'
Suy ra Jr*ir;i+iF >JT(za+u).
Dort6 7d +6ab+5b2 >JZLa+o\. (adsbygoogle = window.adsbygoogle || []).push({});
J3a'+loab+5b2 \ / _a--a lc' +6ca +5a' t; l*iroi+iF - t 4b+5c ' (sc +qa)' 4c +5a . - " - ,rJ2(Fr++u)' 3 j5b+4c)' *(s.+ar)') l aa+5b 4b+5c ac+5a )'
Theo b6t clSng thric Cauchy, ta c6
q##.@a+sb)>z(sa+ab).
sa n, rr-ror.r TgEI#t[
( sa+ 4b\2 Suy ra
->
6a+3b. Tucrng t.u ta c6 ' 4a+5b
t.
dusc: F' >\(oo *ob +9c) =3J7 (a + b + c).
MAt kh6c (a+b+c)'>Z(aU+bc+ca) n6n tu
gidthi€t,tac6 a+b+c23. Suy ra F>9$.
Diu bing xby rakhi a = b = c =l .
V4y minF =gO l<hi a=b=c=1.
Tr1n dq' lit m6t so ti drt minh hoa cho vi(c st)'
clung c'cich phtin tic'h biiu thit' clcing c'ip bac' hui
hui biOn thdnh dang t6ng hinh phmtttg hoac: hiQr binh phurtng. flt' vong ring hai vi€t sd gitilt cac cnt hoc .sinh t:6 fuAm tur liAtr d€ dn thi. Cuoi c'itttg
ld nt(t t,} bni t€tp ttr luvAn.
BAI TAP
l. Tim cilc cip sd thr,rc (x;y) thOa min di6uwQ", ,@li'y++f +@ a2olef 42 wQ", ,@li'y++f +@ a2olef 42
=z,[ry +ro(G*6).
2 (Bdi ddng trAn TH&TT thdng 9 ndm 2012). Giai hC phucrng trinh
3 (Trich dA thi tha EH ndm 2074,nguoithay.vn). Giiti hQ phuong trinh nguoithay.vn). Giiti hQ phuong trinh
4. Giii hQ phuong trinh
=87;68*y+W
,[i + Zy -Z + (zx + y +Z),$ x.+ y I
=r'+y' +6x+8y-3
5. Cho x,y,z ld c5c sd thuc kh6ng 6m. Chimg minh ring:
'ffiq'+f +
*,[ry +yr+ rx.
6. Cho a,b,c ld c5c s6 thlrc duong th6a mdna3 + b3 + c' =3 . Tim gi6 tri lcrn nh6t cira bi6u a3 + b3 + c' =3 . Tim gi6 tri lcrn nh6t cira bi6u thitc: M = "{rd; tuob +roz + ^fifi lgbc + 4c2 (adsbygoogle = window.adsbygoogle || []).push({});
+Ju +l2cal3a').
7 (Trich di thi tuydn sinhvdo l0 chuyAn Todn -
Hdi Phdng, ndm hoc 2015 -2016).
Cho x,y,z ld ba sd thpc ducrng. Chimg minh
2rang: rang: 2 Z > x+y+z ,lg/ l3;l +t+zx - 5 8 (Bdi ddng tAn TH&TT thang l0 ndm 2012).
Cho ba sO thgc ducrng a,b,c thba mdn cli6u
kien ' tsfl+f .1)= ro[a*a+f)+zorz.
\a' b' c') \ab bc ca)Tim gi6 tri lcrn nh5t cua bi,5u thfc: Tim gi6 tri lcrn nh5t cua bi,5u thfc:
11 !-r- r- J5a' +2ab+2b' ,f 5b2 +2bc +2c' 1 ) x- -f -l J8x'+ 3y' +14ry P- '[5i +zro.+zo'
9. Cho a,b,c ld d0 ddi ba cpnh cira m6t tamgi6c c6 chu vi bdng 6. Tim gi5 tri nh6 nhri.t cira gi6c c6 chu vi bdng 6. Tim gi5 tri nh6 nhri.t cira bitiu thric: ,b, _n- --,ffi tl2b' +Jbc+3c''l2c' +7ca+3a' ^3 L 22 y +yz+z 2 +l4yz JrF +7 ab +rF
wsffiffi ffip&ffiI PHEP oiil nflnc IRUG
TRONC TilUC HAN}I CI{I TOAN
{(lar toan sau xuat hiQn trong kj, thi Romani
T9Master in Mathematic nam 2016
Bni 6. Chr,t tam gidc: ABC va didm D niint trOn cenh BC. Dur)'ng trdn ngoai fiAp nm giac'
DAB.DAC lcin luqt cfu CA,AB tai E,F khac'
A. Gcti X h aiA* d6i ru'ng vo'i A qtta BC.DE,DF lin lu'o't cat KB,KC tai M, N. Ch{rng DE,DF lin lu'o't cat KB,KC tai M, N. Ch{rng rninh ring BN.CM vd AD ding quy.
Ldi gidi. Gii sir BN cit i
CMtai S. Ta thay EDC= BAC =FDB n}n DE,DF ddi xtmg
nhau qua BC. KB d6i
ximg v6i AB qtua BC,
do d6 KB cbt DE tai