Suy ra uzott=22011 *52011 +1p'2011-13
: -10 (mod 20ll) : 2001 (mod 2011)
fet U4n: VQy sO du k*ri chta u2sllcho 201 I ld 2001. YNhQn xit Nhidu ban d5 thdnh thao $ thuyi5t
phuong trinh sai phdn n6n cho c6ng thic nghiQm
t6ng qu6t th6ng qua da thirc d4c tnmg vd x6c dinh
ngay du-o. c dang nghiCm u,=T +fi +l2n-13, Vn eN.
C6c b4n sau ildy c6 loi gi6i dring: B4c Li0trz Lit Hodng Thibn, llT, THPT chuy6n Bpc Li0u; Bic
Giang: NghiAm Vdn Nghla, llAl, THPT Lqng
Giang 3; Bic Ninh: NSuyA" Thanh Hiiiu, llT,
THPT chuy6n Bic Ninh; Binh Phufc: Biri CdngMinh, l2A, THPT chuy6n Quang Trung; Binh Minh, l2A, THPT chuy6n Quang Trung; Binh Thu$n: T6 Quiic Hung, l1T, THPT chuy6n TrAn
Hrmg Epo; Gia Lai: Nguydn Quang NhAL fiC3A,
THPT chuy6n Htng Vucrng; D6ng Nai: 0n TuQuiic Dqt,12T, THPT chuy€n Luorg ThiS Vinh; Hn Quiic Dqt,12T, THPT chuy€n Luorg ThiS Vinh; Hn
NQi: Hodng Le Nhil Tirng, 12T2, THPT chuy6n KHTN, Dfing Thanh Ting, Nguydn Vdn Cao, 10A1,
THPT Ung Hda A, Nguydn Vdn Dfing, Trin Thi
H6ng,10Al4, THPT Nggc T6o, Vil Bd Sang, L2Tl,
THPT chuy6n Nguy6n HuQ, Ninh Duc Cudng,llT,
Chu Vdn An, Phsm Ngpc Khdnh, Tq Khdnh Hd,
llT1, THPT chuy€n DHSP Hd NQi; Ilung YOn:
Nguydn Thi Thanh Lan, ll1,:8, Nguydn I/iCt Dthc,
Tri€u Ninh Ngdn, Nguydn Mqnh Hi€p, Duong Hing
Son, Nguydn Philc Hodng, 11A9, THPT Dunng
Quang Hirm, Vdn Giang; Long An: Phan NhQt Duy, l2Tl, Dfing Thdnh Trung, Nguydn Binh An, ILTZ,
Phqm Qu6c Thdng, Bili Thanh Long, llT1, THPT chuy6n Nguy6n Binh Khi6m; Nam Dinh: Trdn
Minh Hidu, 10T2, Dodn Thi Nhdi, llTl, Phqm
Hdng Trudng,llT2, THPT chuy6n L6 H6ng Phong; NghQ An: Cao Hieu Dqt,IlAl, THPT chuy6n Phan BQi Chdu; Phri Thg: Nguydn D*c ThuQn, llT,
THPT chuy6n Hirng Vuong; Phrri YOn: Nguydn
Hu)nh Huy Mdn,10Tl, Ngri L€ Phuong Trinh, H6
Minh Hodng, Ddng Bdo Wnh, LA Vil Minh Huy,
llT, THPT chuy6n Luong Vin Chd'nh; QuingNam: Br)i Nggc Giao,KO4-07, Nguydn Xudn Anh Nam: Br)i Nggc Giao,KO4-07, Nguydn Xudn Anh
Qudn, l1T, THPT chuy6n Nguy6n Binh Khi6m;
Quing Ngfli: Hu)mh Tidn Phdt,l2A7, THPT Duc Ph6
l, Vfi Vdn Tiiin, l0T, TIIPT chuy6n LO Khii5t; Thanh
H6a: Nguydn Ti6n Tdi, l2T , Dd Thily Anh, llT , TIIP.T chuydn Lam Son; V-mh Phfc: Hd H{bu Linh, Dd Vdn
QuyeL llAl, THPT chuy6n V"rnh Phirc.
NGUYEN VAN MAU BitiTlzl4sg. Cho LABC va drong thiing d btit tq) G1i A,.8,.C, ldn luot ld hinh chi€u cua A, B, C
n'An d. Khi do cac dudng thdng qua Arvd
t'ttortg g6c vrri BC, qua Brvd vu6ng goc vdi AC,
cltrct C, t,a vu6ng goc vcti AB d6ng qult tqi mQt
diAm t'ir diO:ru ndlt dtrqc g\i ld cAc trryc giao ctia drong rhiing d d6ivAi LABC. Chang minh rdng Irottg nt6t tarn gidc, dudng thdng Simson ri'ng
voi mot diAnt nim tr€n drdng trdn ngoai ti€p thi cltia cloi doan thang n6i di€m d6 vd cac trac giao
cila dtrdng thdng nay aAi vai tum giac dd cho.
Ldi gidi.
Gqi P ld di6m thuQc ctudng trdn ngopi ti6p tam
gi6c ABC; d ld cluong thing Simson cira P <l5i
v6i tam gi6c ABC; d' ld ctudng thing qua P
song song vor d; At, Br, Cr theo thir t.u ld giao dii5m cira d vd BC, CA, AB; Az, Bz, C2 theo thir
t.u ld hinh chil5u cua A, B, C fiAn d; A3, 83, C3
theo thri t.u ld hinh chi6:u ct.r- A, B, C tr€n d'. OC ttr6y P, B, 41, Bz, Ct cr)ng thuQc mQt dudng trdn (tluong kinh BP).
f6t hqrp vli ,4rCr=dlld': PB3, suy ra PAtCrB3 ld hinh thang cdn vdi hai d6y ld AtCtvit PBt. Tuong W PA$C3 h hinh thang cdn vdi hai d6y
ldABtvd PCt.
YQy Bp$3C3 H hinh binh hdnh.
Tri d6, chir y ring CzBzBzCz H hinh chfi nhft,
suy ra Brq=CrBr:Cfr.
Do tl6 B$zvir CtCz c6 cung trung <li6m.
Tucrng W C{zvitAtAzc6 cirng trung <li6m.
Gqi K li trung dii5m chung cua A1A2, 8182,
C{2, Q h di6m d5i xtmg cua P qoa K.
Ducrng nhi€n QAIA2,QBIB2,QCPCz ld c6c
hinh binh hdnh.
Tt al6, chir f ring PA; PB; PCl theo thri tg
vudng g6c vdi BC, CA, AB, suy ra QA2, QB2,
QC2theo thir t.u vu6ng g6c vcri BC, CA, AB.
N6i c6ch ldtbc Q ld cgc truc giao ctla d d6iv6i
tmn gi6c ABC.
f6t hqp vdi KthuQc d, suy ra dpcm. D
YNhQn xit. Biti to6n ndy kh6 kh6, sO ban tham gia
gi6i kh6ng nhidu. Tru bqrNg6 Quang Duong,l2M,
THPT chuy6n KHTN, DHQG Hi NOi, tAtcececbqn
kh5c tl6u kh6ng khic phlrc dugc tlnh trang phdp chung minh phu thuQc hinh v€. Xin n6u t6n mQt s6
b4n c6 lcri gi6i tucrng tl6i tOt: Hir NQi: Phqm Nggc Khdnh, Tq Khanh Hd,llT, THPT chuyEn DHSP Hd NQi; Phr[ Thg: Nguydn Ti€n Long, LA Ba Hodng,
l0T, Nguydn Drtc ThuQn,l lT, THPT chuy6n Htng
Vuong; Bic Ninh: Nguydn Thanh Hidu,l lT, THPT chuy6n BIc Ninh; Thanh H6az Dd Thu) Anh, Dd
Duy Mqnh, llT, Nguydn Ti6n Tdi, l2T, THPTchuyCn Lam Son; NghQ An: Trdn Quang Huy, chuyCn Lam Son; NghQ An: Trdn Quang Huy, I2A1,THPT chuy6n, DH Vinh; Long An: Vd Quiic Thinh, l1T, THPT chuy6n Long An.
NGTTYEN MINII HA
Bni L1l459. Ddt di€n dp u=Uncosro/ (vdi U;'
kh6ng ddi, a=3l4radls) vdo hai diu m6t dogn
,-l
mach gdm tu di€n C mdc n6i ti€p vcti bi€n tro R.
Bit;t .: . + ... .+. vo'i L* lir dir)n
U ,, U,-, U,,tr-C- R-
dp gifi'a hei diu R. Dtra vito di thi dmti d6.y hdy
tinh giti tri di)n clung C.
Anh 2, THPT chuy6n Lucmg ThC Vinh, Long
Thdnh; Binh Phufc: Nguydn Vdn Hqp, BK12, THPT chuy6n Quang Trung; Hn NQi: Vil Bd Sang,
12 Todn 1, THPT chuy6n Nguy6n Hug, Hd D6ng;
Hnng YGn: Nguydn Vdn Qudn, 72 ToLn 1, THPT
chuyCn Hrmg YCn.
DINH THI THAI QUVNH
Bdi LZl459. Mdt vdt nho hdt diu trrutt
"u6ng
fi"An mdt cua n1m dang nim ),On tr1n mqt bdn
,im ngang nhu hinh vd dtroi ddy.
NAm ci g6c' nghiAng u, vqt c'6 khdi ltrqng m, nAm cd kh6i tu'ong M. Bo qua ma sdt gifi'a nOnt va mdt bdn. Hdy tim vdn t6c cila nAm d6i vbi
bdn tai thr)'i rJiAm vdt nt c6:
a') vdn nc i aAi vo'i nAm.
^ : -- -:-
bl vqn roc r,, doi v6'i hqn.
Ldi gi,rti. Ggi v4n t6c cira n6m tt6i vdi bdn li i, , ^ t " ^. -t
vin t6c cua vat doi v6i n€m vd v6i bdn l6n lucrt
ld i vd io.
a) Ta c6: io=i +i, = v. coscr =vox+vt (1).
86o todn clQng luqng theo phucrng ngang:ffivo, = Mv, (2). ffivo, = Mv, (2).
Tt (1) vir (2)suy ra: u, = #ffi
b) io =il+i, = ro, = (vo, +v,)tancr (3).
El6o todn itQng luqng theo phuong ngang:ffivo* = Mv, (4). MIt kh6c: rt =fr.+ro', 15;. ffivo* = Mv, (4). MIt kh6c: rt =fr.+ro', 15;.
Tt (3), (a) vd (5) suy ra:
vr=
DNhQn xdt. Cbc b4n c6 ldi gi6i dring: Hi NQi:
Nguydn Vdn Diing,10414, THPT Nggc T6o; Nam
Dinh: Phqm Ngoc Nam, Od fn*y Trang, 12 Ly, THPT chuy6n L6 Hdng Phong; Hi finh: Nguydn
ViAt Thlnh,l0 Lti, THPT chuy6n Hd finh.
NGUYEN XUAN QUANG 0,175 0,0135 0,0095 0,0055 0,0015 0,00 Ldigidi.Tac6: U*= =U* =2,00. Ta c6: ru=--L ",!n'+21 j--(om-2 ) R' RUn - +=u](t.\ : H * ulrl*:il=ry '*)= i*=&l'.#o) 1212 +
---f-t-t,o- ula,c, n,' Ui'
Dat r, =!: x =\: o=-- 2 . b
= 2-.
-7'r ui, " R2,* ula2gz, " uro
y c6 dqng: y: aX + b, d6 thf ctray ld iluongthing. Theo ddi thi 0 dC bdi ta c6: thing. Theo ddi thi 0 dC bdi ta c6:
r 10-6
Khi .:0.0015thi'"; =0.
Ui R,
Ta c6 : o,oo15 : o + a Khi +: o,oogs thi
U; U;10-6 10-6 N 0.0095 :0.0015 + --1 . .2)0-6 +0.0015 314'C' @ )UVfa: L=r-' ! 3 r4' .(0,009s -0.001s) =1,950.10{F =1,95pF .A
)NhQn xdt. Cic b4n c6 loi gi6i thing tl6 ra ki ndy:
Nam D!nh: Trin Trung D*c, Phqm Ngpc Nam, Vil
Minh Thdnh, Od fnui, Trang, 12 L9, THPT chuyen
LO H6ng Phong; Diing Nai: T6 Ddng Xudn Hinh,72
tanz a+7
t\. TOnN H9C
TIEIUE TNH QUA GAG BiT TOiN
sAr s6 aProblem: Prove that the sequence {A,,1 (n : 1, Problem: Prove that the sequence {A,,1 (n : 1,
2, 3 ..), where A, : ln
] * . +. tends /o
po.sitive in/inifi;.
Solution: Consider the following subsequence{Ar*l (m: 1,2,3.. .). We have {Ar*l (m: 1,2,3.. .). We have (t l) (t r I l\ 3 Ia*zJ*[ s * a * a * s.J = t*1, "'
By grouping the terms in this way, we caneasily see that A2^ > I + + Hence the easily see that A2^ > I + + Hence the subsequence {A2*} tends to positive infinity. But the given sequence is an increasing positive
sequence, it tends to positive infinity too.
Remark: The idea of the above proof was due
to Nicole Oresme (French, about 1323 - 1382).
The interested readers can find his biography in the following link. http://www.nicole-oresme.com/ seiter/oresme-bio graphy. html T.,VUNG day s6 .i tlen tol ducrng vO cirng ddy con nh6m lai (d6ng tir) s5 hpng tdng : ld cua, do : ti6u su
NGUYEN PHU HoANG LAN
(Trudng DHKHTN, DHQG Hd NAi) sequence tend to positive infinity subsequence group term increasing due to biography
H4n gt'i blri dich: Muon nhdt la hai thang sau khi ddng bai.
Bdi todn. Cho A={t,2,...,n1 vd ddt p{A) td tqp hW gim tiit ca cac tdp concira A. Tim .s6 phin tir ctia ttip P(A)? cira A. Tim .s6 phin tir ctia ttip P(A)?
Ldi gidi. Goi ,S li t0p gdm cic ddy nhi ph6n c6 dQ ddi n. Ta xdy drmg mQt tuong irng I - 1 gifra
P(A) vd.S nhu sau. Vdi m5i A = {,,,...,,,} e p(A), lS4<...<4 <n" ta d[t tuong rmg B v6i d6y
nhi ph6n b$2..b, trong d6 4=l khi vd chi khi le B. Yi dp, trong trucrng hW n=4, ta c6
a <+0000, {t,2,+} <+ 1101,... Kh6ng kh6 de th6y ring tucrng ung (anh xp) nhu vfly ld m6t song5nh tu P(A) dCnS. Vay, s6 phin tu cria tQp P(A) blng sd c6c ddy nhi phdn nvd do d6bing 2'. 5nh tu P(A) dCnS. Vay, s6 phin tu cria tQp P(A) blng sd c6c ddy nhi phdn nvd do d6bing 2'. NhQn xit. C6c ban tham gia chuyCn muc n6i chung ddu hi6u vd dich 16 n6i dung cira bdi todn. Cfucbpn c6 bdi dich t5t hcnr cd li: Hi NQit Nguydn Vdn Dting,l0Al4, THPT Ngoc T6o, Phric Thg, Tq Duy phuong, 9A,
THCS My Hrmg, Thanh oai; Hii Duong: Nguydn Th! Tuy€n,l lE, THpr rhanh Hd; Nam Dinh: Ldm vfi Tulin, 12 Anhl, THPT chuy6n LC H6ng Phong; Ninh Thugn : Nguydn Thdnh Khdi,l I To6n, THPT chuy6n
L6 Quf D6n, TP. Phan Rang - Th6p Chdm; Gia Lai: Trin Eilrc Vinh,llAl, THPT Mpc Dinh Chi, Chu Pdh; Vinh Long: Chdu Minh Khdnh,l0Tl, THPT chuy6n Nguy6n Binh Khi6m.
TH&TT
rlr mN T},d.LII{I
i''
67 l* gi6 tr! lon nhat (GTLN), gi6 tn nho nhdt
at (GTNNI cua bi6u thric ba bi6n lu6n ld bdi to6n c6 mat <r hiu h6t trong c6c dC thi chgn hqc sinh
gi6i (HSG) c6p THPT vd tuyiin sinh vdo D4i hgc vd
-ao tling. Kh6ng nhfing thrl n6 con ld bdi to5n hay
vi kh6 nh6t trong c6c dA thi.
Trong chucrnC trinh qieng dqy vd.hgc tQp tim GTLN, GTNN lu6n ld chtr d6 h6p ddn aOi vOi ngucri d4y 15n
nguoi hgc. ViQc gidng day dl ldm sao hqc sinh hgc
tdt chu di: ndy lu6n ld mQt vAn de kh6. Chu tl6 ndy
thudng ddnh cho hgc sinh c6 hqc lgc kh6, gi6i v6
to6n n6n c6c bdi to6n dua ra thudng hay vd kh6.
pC tim GTLN, GTNN cira bi6u thirc ba bii5n c6
nhi6u phuong ph6p, tuy nhi6n kh6ng c6 phuong ph6p nio ld van nang tt6 giai dugc mgi bdi toSn v6
tim GTLN, GTNN mi chi c6 nhirng phucrng ph6p
gi6i dugc mQt nh6m c6c bdi to6n md th6i. MQt trong nhirng phuonC ph69 kh6 hiQu qui ld "VQn &trng
phuong phap hdm s6 vdo viQc tim GTLN,.GTNN c{ta -biau
ilic t) Uen thtfing grtp".Eudngl5i chung tl6
ei6i bdi torln tr6n bing phucrrg ph6p hdm s6 h dua
ii6u tflic chira ba bi,i; s6 uC U#" thric theo mOt bi6n
s6 m6i r sau d6 sir dqng c6ng cp tlpo him, thi6t hp bing bii5n thi6n cira hdm s6 y = f (t) tr6n tQp x6c
ttinh cira n6, tu d6 suy ra GTLN vd GTNN cta bi6u thirc cAn tim. Bdi vil5t nay nhim girip c6c b4n hgc
sinh c6 c6i nhin rQng hon v6 phuong ph6p hdm sl5
trong.c6c bdi to6n tim GTLN, GTNN cira bi6u thric
ba bi6n.
r. DINH NGHIA VA MQT SO LUU.'l QUaN
TRQNG
l. Dlnh nghia. Cho hdm s6 y= f(x)xdc tlinh tr6n
tpp hqrpD c IR.
a) Nr5u tdn tpi mQt diiSm -r" e D sao cho /(r) 3 f (x")v6i mgi r eD thi sA U =/(x.) dugc gqi ld GTLN v6i mgi r eD thi sA U =/(x.) dugc gqi ld GTLN cira hdm s5 7x6c <linh tr6n tpp hqpD,ki hiQu ld
M =max f (x\. xeD" -
b) NOu tdn tqi mQt di6m xo e D sao cho
f(x)>/(x.) v6imgi xeD thi s6 m=/(x") du-o.c
gqi ld GTNN cria hdm s5 I xec ilinh tr6n tQp hqp D, ki hiOu ' ld m = xeD'min f(x).
m
) r3'
" i uaN D|JNG pHUoNG pilAP HAt I0 ,
\ rr \ t-- -;. - '^ - t ? ^- ^- vAO \Irtc IIM GrA IRI t0l{ NHAI GIA TRI NHO NHAT
cie nrfra rnfuc iu1e nt nrfrN rnrtorte eip
.1 . \ PHAMTRONGTHLI
,:
(GVTHPT chuyAn Nguydn Quang DiAu,Ding Thdp)
2. MOt sO hru lt quan trgng
. Cho hdm sii y=f(x)x5c dinh tr6n tQpK(voi K ld mQt khoing hoic mQt do4n ho[c nua khoing). Hdm s6 dugc gqi ld:
- D6ng bi6n (t6ng) tr6n K, n6u:
Yu,veK, u<v = f(u)< f(v);- Nghich bi6n (gi6m) tr6n K n6u: - Nghich bi6n (gi6m) tr6n K n6u:
Yu,veK, u<v = f(u)> f(v).
Hdm s6 ddng bi6n ho{c nghich biiln tr6n K dugc goi chung ld him.s6 d<m ttiQu tr6n K.
. Khi dd cQp d6n GTLN hay GTNN ctra hdm sd
/ md kh6ng n6i " tr6n t6p hqp D " thi ta hi6u d6 li
gi6 tri lcm nh6t hay nh6 nh6t cria/ tr6n tpp x6c ilinh cria n6.
. Khi gdp bdi to6n tim GTLN vd GTNN cria hdm s6
/li6n fi,rc tr6n mQt doAn[a; b],c6 d4o hdm trdn
khoang (a; b) c6 thC trt mQt s6 hiru han di6m
vir f '(x)= 0 chi tai mOt sO ttiru han di6m cria
khoing (a; b) thi ta kh6ng cdn hp b6ng bii5n thien
ctra hdm / tr6n ilo4n d6.
. Ei6m m6u ch6t trong c6ch gi6i bdi to6n tr6n bing
pbuong phSp hdm sO h xay dlmg cho
"iluqc hdm s6/(0 vor t eD.Trudng hqp ta kh6ng th6 x6y fimg tr.uc ti6p him f (t),t eD th6a m6n P = f (t), ta di tim him s6 f(t)th6amdnP>-f(t)(d6i v6i bdi to6n tim
GTNN) ho{c P < /(/) (d6i vdi bdi to6n tim GTLN).