gi6tn lon nhat khi R = R*in (tai didm A).
v2
k^u* = -+- = 0,25; (& day ldy g = l0 m/s2) gft,in
Vay khi h0 s6 ma sdt k > 0,25 thi 6 t0 kh6ng
bi truot khi chuydn d6ng theo elip. D
(Nlan x6t. Ban NguyAn Manh Qudn,11 Li, THP|
chuy6n Nguy6n Hue, Hi TAy c6 ldi girii tot.
NGUYEN VAN THUAN
(4)
vd nh6 nhdt khi o to 6 B. R = R*o* tai B (hinh
vE), do d6 gia toc cria 6 tO sE lon
,,2
Tai A: a = a^a* =;; =2.5m1s2 (1)
-,2
TaiB: a=amin =
O: =l,Zlmlsz (2)
Luc ldm cho xe chuydn d6ng theo elip lh luc ma s6t f*, do mdt duong t6c dung lOn ldp xe; luc
ndy c6 phuong huong vio tAm O.
Luc ma s6t niy gAy
^ mv2
n€n J*,
PROBLEMS... Qiep ffang I\
FOR UPPER SECONDARY SCHOOLS
T61372. Do there exist two distinct positive
integers a, 6 such that bn+ n is a muJtiple of d+ n
for every postive integer n?
T71372. Let S be the set of all pairs of real numbers (a, F) such that the equation f -6* +M,- B=0
has three real roots (not necessarily disctint) and they are all greater than 1. Find the maximum
value of T =8a -38 for (a, p)e S.
T81372. Let ABC be an acute triangle and denote by p the center of its Euler's circle. The
circumcircle of ABC, which has radius R, meets AQ, BQ, and CQ respectively at M, N, and P.
Provetheineoualitv ,QMQNQPRI *', *' =3.
TOIVARDS MATHEMATICAL OLYMPIAD
T91372. Find the interger part of
x
+-
2008
where x is a real number in [-2008 ; 2008], and m, nare natural numbers, m>n>2.
Tl0/372. Let S be denote the set of all n-tuples
(n> 1) of real numbers (41, qz, ..., qn) such that
n
t.\Q =soz
a) Prove thal
,minnla, - a,lS
b) Give an example of an n-tuple (a7, a2, ..., an) such that condition (1) holds and for which there is an equality in (2).
T111372. A function f(x), whose domain is the
interval [1 ; +oo) I has the following two properties:
i) f(1)= I
;2008 2008
ii) f(x)+?,ffl(f 1x+t1)2 =f(x+l) for all x e [1 ; +co). Find rhe limit lim ( ttzl + f Q) *... * /(r*,)''l.
,,*-[/(l) f tz,t f@) )T121372. The angle-bisectors AAy, BB1, CCr of a T121372. The angle-bisectors AAy, BB1, CCr of a
triangle ABC with perimeterp meet B1C6 C141, and A$l respectively at 42, 82, and C2. The line through A2 and parallel to BC meets AB, AC at At, Aq.Constructthe points 83, Ba and C3. Ca in a similar way. Prove the inequality
ABo+ 3go+ CA++ BA.+ gUt+ ACs<P.
When does equality holds?
Translated bY LE MINH HA
(1)
(2)
n(n2 -1)
II I I I I I I t I I
phin sE brin lqi cho hgc sinh v6i gi6 re, mOt
phAn se t[ng cho c6c hoc sinh nghdo kh6ng
d0 tien mua s6ch vd tqng cho c6c Thu vi6n
trudng hQc. Tinh d6n ngdy 26-.5-2008,
NXBGD da trich 3 7.95 500 000 d6ng cho
c6c c6ng ti S6ch-Thi0t bi tru'ong hoc dC t6
chirc mua - b6n SGK cfr, c6c dia phuong dd
t6 chftc 1263 di€m mua - b6n SGk cff vi sO
luong S.GK dd mua vir quy€n goP duoc
trong tuAn 16 dAu ti6n ld 39 785 b6n. Tham muu-vdi So GD-DT chi dao c6c trudng ti6p
tuc c[rng cO t,i SCt< dirng chung cho hoc
sinh muon s6ch. Ngodi ra, NXBGD c6 c6ng
vdn sO 996AIXBGD ngiy 25-5-2008 ph6i
hop v6i S0 GD-DT chi d4o c6c c-Ong ti
S6ch- Thi6t b! trudng hoc hu6ng .dan c6c
trudng phd th6ng t4i dia phuong th6ng kd s0
lugng hgc sjnh con thuong binh. (c5 hAng),
con li6t sT dC cAp khdng SGK. Td chric t6ng
SGK cho c6c trudng kdt nghia. Ph6i hqp^vdi Qu! b.ao tro trd em TP. Hd NOi, Dd N[ng, TP. H0 Chi Minh tO chirc t[ng SGK cho hoc
sinh nghdo, vugt kh6. Hu6ng dAn c6c COng
ti S6ch - Thi€t b! truong hgc ph6t phi0u uu
ti6n giim gir{ SGK l0-- l'2% cho c6c d6i
tuong hgc sinh nghdo, hoc sinh hqc gioi gia
dinh chinh s6ch. DAm bAo vho ndm hoc m6i kh6ng dC mOt hgc sinh nio vi thii5u SGK md
b6 hoc.
CUOC vAu BOUC
#w stnch nfi,
t/,i td,ilo 'adAB6e) &t 6dah 4,no ilDa d
ffnd oln ,tilti tttfuo
)rc\7ADAO4Oa
rong tinh hinh gi6 ch, citc ni6t hdne ti6u ding d€u c6 nhi6u bi6n dQng
ph[rc tap, inh huong d6n doi s6ng
cta ngudi dAn, d6c biet le c6c hd nghdo,
thuc hdnh chir tru<rng ti0t kiQm. ch6ng l6ng
ptLi theo tinh thAn chi duo cira Chinh phfi, ngdy 13-3-2008, B0 Gi5o dr;c vd Ddo tao d5
c6 c6ng vin sO }1341BGDET-VP chi d4o
c6c So Gi6o duc & Dio tao trong cA nu6c t6
chirc tuy6n truy6n v6n d6ng Hoc sinh stt
dwng lqi sdch gido khoa (SGK) cfi, phdt d6ng
c6c hinh thric quydn g6p, t{ng ban, xdy dlmg tt SGK dirng chung nhdm sri dung SGK
trong nhieu. ndm. Tiep d6. ngdy.19-3-2008.
cdng v5n sd 4313/BGDDT-VP vC cuQc v6n
dQng Hpc sinh tQng SGK cfi cho
"lp
trudng, BQ Gi6o duc vd Ddo t4o hu6ng dAn
c6c So Gi5o duc & EAo tao m6t s6 c6ng
,iC"
", the cdn thuc hi6n nnu: tO ctlii-' : ,': -' : ,':
Tuan l,i quyAn gop sdch gido khoa cfi vito
dip bC gi6ng n[m hec 2007-2008;. cdc
truong p.hAi sr}' dung kinh phi t6i thi6u tir
6-10%"t6ng ngdn s6ch chi cho GDPT hing
n6m d6 mua sim trang thi€t bi, SGK cho
thu viqn tlng cudng tri SGK dirng chung;
tdng cudng lugng s6ch cho hoc sinh mugn
ti 15-20yo. nhlm dAm bio moi hoc sinh, dflcbiQt ld hoc sinh nghdo c6 dir SGK dC hoc. biQt ld hoc sinh nghdo c6 dir SGK dC hoc.
Huong irng cuQc vAn dQng niy, Nhd xu6t bin Gi6o dqc (NXBGD) dA chi dao c6c.don
vi thdnh vi6n, c6c C0ng ti S6ch - Thiet bitrudng hoc o cin dia phuong triOn khaithgc trudng hoc o cin dia phuong triOn khaithgc hiOn. c6c hoqt d.Qng cu th€ nhY
'_ tuy6n
truy6n rdng rdi chri.truong ndy cria B0 Gi6o
duc vd Ddo t4o d0n c6c th6y c0
"gi6o, cdc
bic ohu huvnh. c6c em hoc sinh tO ch*c t6t
LJ
TuAn 16 tqng SGK cfr tai c6c trudng hgc.
Trong hai th6ng ph6t hdnh SSch gi6o duc
phuc vq ndm hoc 2008 - 2009, cdc c6ng ti S6ch - Thi6t bi trudng hgc. sE t6 chri'c c6c
cira hhng mua SGK cfi. 56 s6ch ndy m6t
Theo d6nh gi5 cfia du lupn b6o chi, cudc vAn d6ng Hoc sinh sh dryng lqi SGK cfi vd
tdng SGK cil cho til sdch dilng chung cua
nhd trudng cria BQ Gi6o dpc vd Ddo lao v1i
su d6ng g6p tich cpc cta NXBGD chic chln s€ d4t dugc hiQu qui cao kh6ng chi vC mgt gi6o dqc (hgc sinh c6 1i thirc b6o qu6n t6t
SGK d6 h.gc vd tqng lgi cho c6c b4n hqc sinh
nghdo), vC mflt kinh td md cdn g6p phAn vdo
vi6c thuc hi6n nhff'ng chfi truong chinh s6ch
ve kint t6, xd hQi, chinh tri c0a Ding vd
Chinh phir trong tinh hinh hi6n nay.
iPvi
II I T a I I I a I ! I I I t a