NGUYEN VAN MAU
*BAi 'f121391. Cho tuttt gitic' ABC c'6 cdc'
dtdng c'ao AA', IIR', CCi' clong quy tai 11.
Chti'ng ninh riutg
llA llB llc - L. Lt b t
*
-
r-b"v -{ > () f-- +- +- .
IIA' TIB' IIC' LIA' I IB' IlC'
Ldi gitii. D6 c6c bi6u thuc trong hai v6 cria
BDT x6c dinh cdn c6 di6u kiQn tam gi6c ABC
kh6ng vu6ng. C6 hai trucrng hgp:
Trudng hW l.Tam gidc ABC nhpn (h.l).
86de.N6ubas6a)lz>0vix+l*z=ryethi vix+l*z=ryethi r- (x-lX y-l)(z -l ) < 6J3 -l 0. Chtmg minh.'Vr x, y, z> O n6nx<x+y+z=xyz.Do d6 yz> l, tuong \r zr> l; rv > l. Suv ra trons ba so x, !, \ kh6ng th6 c6 hon mQt s6 nho hon ho{c blng
L|inlt l
1.
N6u trong ba s5 x, !, z c6 dring mQt s5 nho hon hoflc b[ng 1 thi
(x*1Xy-1)(z-l) < 0 < 0..6-r O.limxn - limxn - N6u ba s6 x, .), z cLng l6n l.t=x-l;v=y-1 w=Z*7 vl; > 0). -1 hay t x,,*, -tK* tx,, - Lt<...<[il.' la -tt. * ('.i)'- '(..+)=#,
hon 1 thi dEt
(u>0;v>0;
TORN -6lirdi@HOC
l+ - -xx*7 x*7 $ +"lt'n + 54 18 g-Jw -t 24
Vi x+.)*z=.1ry2, nen
Lt + r) + w *3 = (a + 1Xv+ lXvy + 1).
. .i
Khai tri0n vi rrit gon dang thr?c trdn ta c6
Ltvw + vv) + wu + Ltv =2. Dil t = l/ttvw>O, ta c6 t3 +3t2 = ,rr*+3?,1ifi7 1 uvw*vw* wtt* uv =2 * 13 +3tz -2<0. Suy ra r <.6-l . Vay (x- l)(y-1)( z.-1) = tuw-r' < (..6 -t): =0..6-t O.
Dang thuc xAy ra khi vi chi khi x=y = r= Jl.
7-ro'lcri giai bdi todn.
Ti tanA,TiA+urnB.HE+anC,He =6, suy ra
I-IA ITB HC
-+-+---I.IA' HB' HC'
tanB+tanC tanC+tanA tanA+tanB
:---+---+-_--
tan A tan B tanC
tanB+tanC tanC+tanA tanA+tanB
:_+-+-
-tan(11+C) -tan(C+A) -tan(A+B)
: - I + tan 1l tan C - I + tan C tan A -1 + t an Atan B
= -3+(tanBtanC+tan Ctan A+tanAtan B) (1 )
^ ^
Ch( ), rdng RHC'=CHB'=BAC:
ffi' = 67, = {Ee ; ffii, =ffi, =fr*,
ta c6
abc
-+-+-FTA' HB' HC'
A,C+A,B B,A+B,C C,B+C,A
=_ +_
HA', HB' HC'
=2(tanA+tanB+tanC) (2)
Tu (l) vn (2) ta th6y BDT cAn chring minhtlrong duong v6i tlrong duong v6i
Z(tanA + tanB + tanC)
- (tanlltanC + tanCtanA + tanAtanB)
<ontl-q (3)
Til b6 dd, suy ra
(tanA - l)(tanB - lXtanC - l) <6\6-10.
Khai tri6n vh rtit ggn BDT tr6n, chti f ring
tanA + tanB + tanC = tanAtanBtanC, suy ra
BDT (3) dting. Ding thuc x6y ra khi vh chi
khi tam gi6c ABC d€u. Trad'ng tqp 2.Tam gidc
ABC ti (h.2) (theo banNgul,dn Vdn Quy, 11A, Ngul,dn Vdn Quy, 11A,
THPT chuy0n, DHKHTN, DHQG He Noi). Kh6ng m6t tinh t6ng qu6t, ^ gi6 su BAC>90'. I]A'C Hirit 2
Eil cotfii=m; cotfrEa=n; cotfidE= p
th\ m, n,p > 0 vh np + pm * mn = l.
^
Y\ 6Pq'=HCB n6n
HA AA' BA'AA'-=l---i--.--:l-iip.HA' HA' HA'BA' -=l---i--.--:l-iip.HA' HA' HA'BA'
^^ ^ Y\ BAB'=BHC'. HAB' = HBC n€n HB BB' BB' AB' - N la--lr----lr- HB' HB' AB'HB' M , ,HC, P I uons tu co --l+-.HC' M Y\ fu'=frEi;6d'=fr8 n€,n a CA, BA,
---+-=n* HA' HA' HA' P.
b CB' AB' IV HAB'=HBC n6n V HAB'=HBC n6n *= * *=;-n. ,CI I uong tu co ---- p. HC' tn
BDT cAn chung minh tuong duong v,3i
l-np+t+L+t+L+6J1mm mm _11 26+tt+ p+--tt+-- p tnm n+D ) e\-np+-!--J'+6..13-4>:- e) mm
sjlrru tu-rorol T?EI#ff
rathdy oJl-q>z; ,n=\!l'- vd rp<I, nonn+p n+p Fnp+!7 P +0.6-+>o *!ry* P)' *, t1x |-np _(n-1)2 +(p-l)2 +2(n+ p) ,2(n+ p) _2 1*np l*rp tn Di0u d6 c6 nghia lI BDT (4) dfng.
Ding thfc kh6ng xiiy ra. fl
(NIr4" x6t. t) Ddy li biri toiin kh6, c6 it b4n giai
dLroc bii todn trong trudng hgp tam giiic ABC ti.
2) Xin n€u tOn mQt vii ban c6 loi giai tuong diii ttit.
Vinh Ptrric: Phan Xrfin Tnrlng, l0A, TIIPT Nguy6n
Vi6t Xudn, Vinh Tuong; Hir NQi: LA Duy tlodng,
101'3. DI.lKtlTN, DI-IQC Hn NQi;Hii Duong: Ngul'€n
l\fin Dtittg, llT, TIIPT chuy0n Nguy0n Trdi; Ninh
Ilinh: Vr7 l-hanh'l-ittg. l2T. THPT chuy€n Luong Vdn
'l'uy: 'l'hanh llo6: li 'thdnh
COng, llBls, THPT Hqu
I.6c II; NghQ An: Ngrry,An Th€ Anh, llAr, THPT chuy0n l)l-l Vinh; CAn 'l'h<v.' LA Dqi Thdrth, THPT
chuy0n I-! Ty Trgng; Qunpg Nam: NSrr-ydrt Hing Srrn, l211,
-'l'llPl' chuyOn.Nguy6n Binh Khi0m; Binh Dinh:
Ng,tq'Art'l'h! Bqch'l'uyAt. lUl'. THPI chuy0n I.€ Quj Dfin. NGTJYEN MINH T{A
*Bei Lt/391.
Mot chiAt' bi t'6
khoi Lur.mg nl
chro'c' kio kh(tn.g
ttan t6c bun clau
tir khoang c'dclr
ct vdo bd' nhd'
nfit .so'i d/i1, vdt cluu ring roc o'
,',, ;, ,a .,:. . ...=----=::il:=7 . . ...=----=::il:=7
tr?n bd'. Cho do cao c'tirt bo'ld h. Ddy ilro'c kdo
t,6i ltrc' kh1ng doi tci F (hinh vd). Tim ,an dc
c't)tt bi khi nri cdn cdch bd ntot.khodng ld b
(b < u). Gia thiet bi khOng bi nlnc bn,lirc c'an
ct)a nu(t'c kh1ng ddi lit F,. Yol t;h
b2 +h2
Lii gitii. C6ng cua ngoai luc tdc dung len bE
g6m c6ng cua lgc cdn F,. cua nudc vir cdng cira luc k6o 1r. Ki hi6u A.lh cdng cta luc cin
/l . N6u tinh tu khi be bit dAu chuy6n d6ng cho ct5n khi n6 c6ch bcy m6t khodng D thi luc can dI thuc hiOn m6t cdng lh
A, =-F,.(a-b).
Kf hiQu Ap lh.cOng cira lgc k6o F. Ndu tfnh ttt
khi be b6t ddu chuyOn dQng cho d6n khi n6 c6ch bo m6t khodng b thi c6ng cua luc k6o
bing lyc k6o nhAn vdi chi6u dhi cua ph6n dAy cli qua di6m C trdn bo, tuc ld
4 = F.s = F.(J"' + tt' -Jb\ h'z)Theo dinh lf dQng ndng, vi bd chuy6n dQng Theo dinh lf dQng ndng, vi bd chuy6n dQng
kh6ng vAn t6c ban cl6u, ta c6
AWa=Y\-o=,4k+4
2"
h^y
ry = F.(,[d +t' -'f1; . 1P)- F,,(a*b)
Tn ddy d€ dbng tim dugc
lz(r.(J*. n -Jb, +h,)- F,.(a-d)
'=I .
Tri di6u kiQn F, ' .$, Ju'+h2 a thAy
= 7,, (6 ddy T,,lh hinh chi0u c[ra luc cdng cua dAy theo phuong ngang). 7,,
d6ng vai trd nhu lgc ph6t clQng, lgc nhy ldn
hon h'rc cdn n6n bb chuyOn cl6ng vdi gia tdc
duong. Vi vQy bi6u thirc cfia v tim clugc otr€n lu6n th6a m6n. D tr€n lu6n th6a m6n. D
(Nn6n x6t. Cric ban sau dAy c6 loi gidi tiit:
YGn Brii: Nguydn Nan Minh, l0 Lt, THPT chuy€n
Nguy6n T6t Thinh; Vinh Phric: Phimg Vdn Duy, I lA3, TIIPT DOi CAn; Hi Nam: Dinh Ngec HAi,lO Li, THPT chuyOn Hlr Nam; Quiing Ng5.ir Nguy€n Tdn
DOng,12 Li, THPT chuy€n LC KhiCt.
NGUYEN VAN THUAN
*BAi L}l3gl. B6n di€n tich gi6ng nhatt Q
dtrqc dqt ,ii dint't o btin dinh hinh vtt\ng c6
cqnh bdng a trong chdn kh6ng. Tai fim O cua linh ,vu\ng tAn dQt mA,t di€n tich didm q
(cing. ddu vo'i Q) cd khdi lurrng m. Dich
chuy€n q m|t doqn nln theo phtrcrng ct?a m6t dtrr)'ng ch{o roi tha nh7. Chlmg to rdng di€n
tich q sd dao dong di€u lrcii. Tim chu ki dao d6ng. Bo qua tdc dung cua trong llrc.
TOAN HOC26'cftrdi@ 26'cftrdi@
Liti gitii. Chon tmc toa d0 nhu hinh vE.
Xdt khi diQn tfch q o
toa d0 .r rat nho. Ki
hi0u nua d0 dii cua duong ch6o ctra hinh
/\vu6ns " lb, b Ib=+l.\ "12) vu6ns " lb, b Ib=+l.\ "12) Hqp lgc t:ic dqng lOn cli6n tfch q lh F, = Ft +21;.cos(5cD)- F, kqQ ^kqQ x kqQ (b +.ry:'' 6z * *z' rf 61 *z (b - x)' kqQ .^kqQ x kqQ a=( t* { )' ' h' + r' Jb+ u,( t -!\' \ b) \ b) = kqQ(r _2r) *2kQQ r _@( r* zI) b,I h) b3 Dr\ b) - -"kqQ - ht
Theo dinh luat II Newton ta c6
, ^ kcte ..i b =_1 ta duoc lll-\ = -/ b. .l . VOr tq uuwv x' + z!'t? *= o <+ r" + 4Ji k'12 * = o nb; ma'
Phuong. trinh nhy chung t6 cliQn tich q dao d6ng di0u hoh voi chu ki:
T = fi.ct
(Nnln x6t. Ciic b4n sau ddy c6 loi girii trit:
Thrii Nguy6nl Vfi Quang Minlt, I I Li, THPT Chuy€n;
Hlr Nam: Ditilt Ngpc Hai, l0 Li, THPT Chuy0n; Nam
Dinh: l.? Qtrung inmg, I I Li, TI{PT Chuy6n L€ H6ng
Phong: Ilinh Dinh: Phqm Minh Thdi, I I Li, THPT
Chuy0n L0 Qriy DOn; Qu:ing. Ngii: Br)i l)t?c Khdnh, I I Li, 'I'HPT Chuy0n LO Khi6t; Long An: Trdn Tldy
Anh, l2A. THPT Chuy0n l-6 Quj DOn; Cin Thoz Trdn ThA Tdtt, 12A3, Ngrt1,2tt Lottg, Phuoc Dwlng, llA3,
1'lJP'l'ChLry0n Li Tq Trgng; Viing Tiu: Phing I'hanlt
Ilrtt', l2T7, 1'HPT Ving 'liru; Cir Mau:. Dtcrng Thdi Dtrtrttg, 12T2, TIJP'I-Chuy€n Phan Nggc Ili0n.
NGUYEN XIJAN QUANG
.B
,na
rrnffz
PROI}I,EMS... Qiep tang l7)
T71395.Let ABC (BC = a, AC = b, AB = c) be
a triangle where A, B, C satisfying thecondition condition
cosA+cos^B = ZcosC. Prove the inequality
8
c 2:maxla.b).
9
When does the equality occur? T8/395. Solve the equation:
,logrl I ,-3loPqx =2y.
TOWARDS MATHEMATICAL OLYMPIAD
T91395. Solve the equation
thx+4=-tr3+3x2+x-2.
T10/395. Let X be the subset of the set { l, 2,
3, ..., 2010) satisfying the following twoconditions: conditions:
l) lXl = 62.
2) For every x e X there exista,be X u{0;2011 }
a+b
(a and b differ from x) such that x= Z .
Prove that there exist two elements x, y in X
such that I *- yl>l I and Il/ tt not in x.
2
T11/395. Make a torus-shaped chessboard by first identifying a pair of opposite edges of an n x n chessboard to get a cylinder and then identifying the opposite bases of the resulting cylinder. Prove that it is possible to place n queens on this torus chessboard so
that none of them are able to capture anyother using the standard chess queen's moves other using the standard chess queen's moves
if and only if (n,6) = 1. (A queen can capture another if they share the same row, column or diagonal.)
T121395. Let ABCDEF be an inscribed hexagon, AC is parallel to DF and BE is the circumdiameter. AB cuts EF at M and BC cuts
DE at N; / is the intersection point of AN and
CM.Prove that EI is perpendicular to AC. Translated by LE MINH HA
\[hiOu crlctr giai cho mOt bii tbdn
VE MOT BAI IOAN XAC DII{H DUONG VUONG GOC CHUI{G
cira hai duomg cheo nhau
rong chuong trinh 6n thi tuy6n sinh vho c6c truong D4i hgc, bhi to6n "x6c
dinh vi tfnh d0 ddi duong vu6ng g6c
chung cita hai duong thing ch6o nhau"thuo'ng le kh6. Sau ddy li bdn c6ch gi6i cta thuo'ng le kh6. Sau ddy li bdn c6ch gi6i cta
cirng m6t bhi to6n vO vAn d€ ndy. Hi vgng bhi
.:
viOt sd gifp fch cho c6c em hgc sinh trong vi0c tim loi giAi c6c bii tuong tU.
IIAI TOAN. Clut linh ltip pluxtttg AIICD.EI'.GIL curtlt u. Iltil' rtic clinh vd tinh do ddi &rong
vtrOtt,q .git' t:htut,g c't)cr AH vti DB.
o llhu'rrng phirp tdng hqrp
Ctich I
I linlt I
'l'r0n hinh l: M ffAn AH; N trln DB; MN lit
dudng vu6ng chung g6c cira AH vd DB.Ti M
kA MP L AD th\ MP L (ABCD) vh PN L DB
(theo clinh li ba duong vuOng g6c). Tuong tg, ke N0 L AD thi NQ L (ADHE) vd QM L AH.
[]ai tam gi6c AMQ vb, DNP vu6ng cAn nOn
DQ=QN=QP=PM-PA-
PHAM eAO Ha NQi)
C6ch x6c dinh vi tri cdc didm M vd N suy ra
tu hai di6m P vh Q chia clo4n DA thhnh ba
phAn bing nhau. O
Cdch 2. 6.2)
D
Hinlt 2
Ta c6 HFIIDB vh tam gi6c AHF ddu. MEt
phing @Hn qua AH vh song song vdi DB.
Gqi 1, O, P theo thu t.u li tAm cira c6c hinh
vu6ng EFGH, ABCD, AEHD. CE cit
mp (aafi tai K li giao ctra AI vd CE. O6 ttr6y
K llr trgng tim cira tam gi6c AHF. FI L EG
nOn F/ L (EGCA) do d6 FI L CK. Tuong tU
HP L (CDE) n€n HP L CK. Suy ra CK L
@Hn.Tu O ke OJ//CK. Tu / ke JWIHL Ti