trinh him vir d6y so. Thuc ra d6i vdi bii to6n dd cho chi cdn x6t diy so tr€n tAp circ s6 nguy0n duong lI du'
NGUYEN VAN MAU
*Bei TDl372. Cdc dudng phdn gidc trong
AA1, BB1, CCt cua tam gidc ABC c6 chu vi p
J , - ., .l
cdt citc doan thdng 81C1, C1A1, AtBr ttong *ng tqi Az, Bz,.C2. Drdng thdng qua A2 song song vdi BC citt AB, AC theo th{r tu tqi At Aq.
Dadng thdng qua 82 song song vdi CA cdt
AC, n,q thei thtir ta tai Bz, B+. Drdng thdng qua C2 song song voi AB cd1 CA, CB theo thtt
iu tai Cy Ca. Chtng minh rdng
AB4+ BC4+ CA4+ BA3+ CBr+ ACt< P.
Ddng thac xay ra khi ndo?
Ldi gitii.
b+c Za+b+c 2a+b+c
Gi6 srl BC = a, AC = b, AB = c, p = a A b + c.
Y\ AA4ll BC nan
BAt CA+ , AAz BAt + C,4q
\I,'
c b AAt b+c
Ap dung tinh chdt dtdng phdn gidc, ta c6
AC' AC IT'- bc
AB AC+BC a+b
Tuong tq ABt= o'
.
. a+c
Str dung cdng thftc dadng phdn gidcLABC vi MB,C,, ta thdy LABC vi MB,C,, ta thdy 2bc.cos4 2ABt.Act.or4 AA,= ' 2'. A-4.= / b+c - A$+ AC1 Do d6 AAz - b+c AAr 2a+b+c Tt (1) vd (2) ta c6 BAt+CAq b+c 2a 1 c+a+b BC4+AC3.a O (5) Tt (3), (4) vn (5) suy ra
AB4 + BC 4 +Cfu + B,$ +Cfi + AC3 < a+b + c = p
(dpcm). Ding thfc xhy rakhi vi chikhi a = b
= c,hay tam gi6c ABC ddu. D
(Nnan xCt. 1) Dd cli den bdt ding thric clin chfng
minh, didu cdn thiet li bidu dien duoc ,1 *6 441qua dQ
t\Al ddi c6c canh ciia tam gi'6c ABC.
Ngoiri c6ch su dung tinh chdt vd c6ng thrlc dudng phAn
gi6c trong tam giiic dd di ddn he thrlc (2), ta cbn c6 thd su dung m6t trong c6c ket qui sau:
cVdi M, N, P lA cdc didm cing ndm tr€n m6t dudng
,J ..N14 0
thang -MPad vd J = L (ct+ B+ 0), cdn O ld didm bdt ki
@ e athi otrt - a'ofr d+B+ 7-'oF
S,4, 8,,, ,
a -:
2abc
cho Sasc @+b)(b+c)(c+a)
tdp 5 ndm Tap chi Todn hoc vd
(xem cu6n Tuydn
Tudi trd, NXBGD, trang 120).
2) C'6c ban sau dAy c6 ldi giAi tot hon cai:
Hi Ndi: Trdn Th€' Khdi, 10Al Todn, Trcin Nhdt Tdn, 12A1 Tor{n, khdi THF f chuyen, DHKHTN - DHQG, Nguydn SonTing, I lA2, THP| NgO Quydn, ChAu Son, Ba Vi; Quing Ninh: DzTng Thu Httong,l 1 Torin, THPT chuyOn Ha Long; Bic Ninh: Trdn Anh Tudn, I0Al,
THPT Thuan Thlnh 1; Vinh Phrlc: Kim Dinh Son,
l0Al, Nguyin Hodng Hdi,llAl, Nguydn Khdnh Duy,
l2Al, THP| chuy€n Vinh Phric; Hh Nam: Hd Phtrong
Anh, l2Al, THPT chuyOn Hi Nam; Nam Dinh: NguydnVdn Quj,,9A, THCS Hrii Hau, TrdnThi Hdng Vdn, l1A, THPT chuy€n L€ Hdng Phong; Di Nfing: Ld VdnTd'n Quydn, 10A2, Nguydn Diic Tdm,l2A2, THPf
chuyOn LC Quf Don; Quing Ngai: Nguyirz Tdn Hmg
11T, THPI chuyen Lc Khidt; Phri Ydn: Pham Quang
Thinh,9I, THCS Hirng Vuong, TP. Tuy Hba; Ben Tre: L€ Phtic Lit,llT, THPT chuyOn Huj,nh MAn Dat, TP.
Rach Gi6; TP. Hd Chi Minh: Bii Trdn Long, l0T,
THPT chuyOn Trdn Dai Nghia.
ud queNc uNu
*Bdi Lll372. Vi phia ddu cua mot con tiu
: ,^
dang chuyin dQng vdi gia t6c kh6ng ddi2,00 mls', ngtrdi ta ddt mot mdy phdng bdng 2,00 mls', ngtrdi ta ddt mot mdy phdng bdng
tennis. lvtdy ph6ng ra mot qua bong vdi vdn
(2)
TU d6, theo bdl ddng thac Cauchy cho hai sd
duong ta nhAn duo.c
(2a+(b+c1\2
8.4.+C.4,_2a(b+c). I r .,l
2a+b+c 2a+b+c
-2a+b+c (3)
4
Hobn todn tuong tu
")+a+cABa+Cg. ABa+Cg. - O \4) A1 . TgnN noc s6 176_{10-2008) SudiUA 25
t6c 25,0 mls d6i v6'i fiu vd hudng vi phia
du6i con tdu. Qua bring dat daqc dQ car,t 10,0m.
Bo qua sti'c can cttct khong khi. Hdi
a) D0 l6n g6c ndm (g6c tao boi phrco'ng ndnr
vd sdn tau) ld bao nhi€u ?
61 Di1m chant sdn titt cach mdy phong bao xu ?
Ldi gitii. a) Chon hq quy chidu gin v6i con
tdu sao cho truc Oy huong thing dung l6n tr6n
vd truc Ox nguo.c hudng chuydn dOng cua con
tlu. Qui b6ng chiu ti{c dung cria trong luc P = nE vd luc qudn tinh Fq = -mA.
Phuong trinh chuydn ddng cua quA b6ng ld
.x=vocoSd.t+!o1: (1)
2
y = vosina.t -!sP 2" e)
Phuong trinh (1) ldy dau + ndu tdu chuydn
d6ng nhanh ddn ddu vi ldy ddu - ndu tiu
chuydn d6ng chAm ddn ddu.
Tai thdi didm qu6 b6ng dat d6 cao cuc dai thi
vAn tdc theo phuong thing drmg bang kh6ng
nOn ta c6
0=vosina-gtt=0=/, = V6 SlIld
g
Thay bidu thric cfia /, vho phuong trinh (2) ta
rft ra
Thay so vdo bidu thrlc trdn, ta duo. c
. x x 66,3m, ndu thu chuydn d6ng nhanh ddn ddu.
. x = 50,3m, ndu thu chuydn d6ng chAm ddn
ddu. tr
(Nnen x6t. Trong s6 c6c ldi giai gui vd Toh soan chi
c6 ban Nguy€n Ngoc Duy, 1l Li, THPT chuy€n Le
Khidt, Quiing Ngii li x6t du hai trudng hop v) c6 ldi giai dring.
NGUYEN XUAN QUANG
*Bni L,21372. l'lor xt'lonh
ctat thiing dirng ctl tidt di|n
S= 40 cm' drroc'chio liun hai
ngdn B vit C' boi mot pittong (hinh vd) Trgng lrc lac clung
lAn pittong c6 dO l6'n l; :
60N. 1.ric rtdu hai ngdn B ttd
C chtba khi soo cho piltdng ndnt clilng ch[nh
giira xylanh va thi tich m6i ngdn la V: 2,41.
ip srat ctia khi o ngdn B ld p = 1,5. 104 Nim2.
Satt do ngro'i ta c'ho thAnt khf vio ngtin 13 dA
tdng ap ,srLtit khi d ngdn B len g(ip Lt'oi
f iltong ttich chuy1n xuong chrdi (ntu stitkhong ctitng kA). T'inh c'ong mit khi d ngiin ti khong ctitng kA). T'inh c'ong mit khi d ngiin ti
dd thtrc hi€n de day piltong ruong ctu'6i. Nl'tilt
do cila khi o ngtin (' t'oi nhr ktrcng doi.
Ldi giii. Gqi p, , pc , p theo thrl tu lh 6p sudt
ban ddu cria ngan B, ngan C vd cr)a pittOng.
Theo bhi rata c6 pc = ps+ p v6iO=+, b
dAy F th trong luc cira pittong, cdn S fi tigt
diOn cria xylanh. Goi A1|i c6ng n6n ci.ra khi dng[n C,tac6
Ar =LP76!' = prvrlnLtt V', Pc tt V', Pc
trong d6 Vc , pc ld thd tich vh 6p sudt & ngan
C lfic ddu;V'g, p'gld, thd tich vd 6p sudt &
ngln C sau khi d5 tang 6p sudt 6 ngan B 10n
gdp doi. Ta cfrng c6 p'c = 2pa + p . Suy ra SlnA = Thuyg = l0m/s2, h =10m, vo = 25mls vho W \/;
bidu thrlc tr0n ta tim duoc
haya=34,450.
b) Thdi gian chuydn ddng cria quA b6ng li . ^. 2vssina
l - -.1 -
o 6
Thay vio phuctng trinh (1) ta c6
vlsin2a , Zavlsin2 aoo oo 66 , 2J' Sll1d = - 5 2gy = V6 TOftNI F{QC 26 -clUelUru
'"Pu* P\
Ar =(ps + p\vrlnl I r
\PB+P )=( o, *L\r"rn(zpss + r\ =( o, *L\r"rn(zpss + r\
- (" -T I c "'[ pBs] r .,l'
Goi Arld cOng cfra trong luc pittOng, ta c6 Az = F.Lh, 6 day Aft lh qudng duong dich chuydn ctra pitt6ng. Do nhiOt d6 cta khf A
ngan C kh6ng dtii ncn pcVc = p'cV'c. Do do v,. - Ptvc -(pt * p) r, . P'c 2Pa + P Suy ra LV =Vc -V'c = ^ O' Vr. Zps+p "
Til day ta tinh duoc nl, ={ =;;4 ,Vc .
s S\zp+ p)
min) cira ba sd d6. Do d6 dd ldi giii bhi
T4l31l duoc ddy dri, ta cdn chrlng minh th6m BDT (1) trong trudng hgp: a) c )- b > 0.
Trong trudng hgp ndy chung minh nhu sau:
Do a> c> b> 0 non a'- b2 ) O, c2 - b2> 0,
Ja+c>Ja+b>Jb*, > 0. Khi d6a2 _b2 b2 _c2 c2 _a2 a2 _b2 b2 _c2 c2 _a2 !- !- Jb.r ' Jr.r' Jr-b , o'-b' -b'-r' -r'-o' - '1;*6 ' Jr., ' Jr+b c2 _b2 b2 _c2 -_r - '[r.b ' .l-r+ , c2 _b2 b2 _c2 J a+c .,lc+a
Thinh thAt xin l6i ban doc.
B?n hiiu DUONG ... (riap trang tt)
Tdc ldng cd qu6'c tha httong
Dudng kia n6i no ngdn ngang bdi bdi;...)
11) Thuat ngir ra dudng, ra ngodi daong (ttdi
nghla v6i vd nhd), chi su tho6t li kh6i gia dinh
ho hlLng, mh quan hQ vdi x6 h6i bOn ngodi.
(Con ldnh con d cing bd
Vdng minh sdt mdy con ra ngodi dudng; Ra dadng bd no bd kia
V€ nhd khbng kh6i cdi nia cdi sdng;...)
12) Chi tinh chat cria m6t hoat d6ng cfia con
ngudi giong nhu tinh chdt ctra con dudng vE ra (Dudng to kd tbc; Dudng kim m6'i
ch{; Dtdng ngay l€ phdi;...)
C6c ban sau d5 dua ra nhi<iu thi du hay, du-o.c nhdn tang phdm:
NguydnThi Hidn,9A, THCS Vfl Kict, Thudn Thenh, Bic Ninh; Nguydn Hfru Trong, x6m NOi, x6 M6o Didn, ThuAn Thdnh, Bic Ninh;
SdmThi Hoa,bhn Vi6ng, Son A, Vrn Chdn,
Y6n Bdi'
vAN KHANH
YQy Az = F.Lh = Fpuvc
ZppS + F
Gqi A li cong mh khdi khi ng[n B dd thuchi0n, ta c6 hi0n, ta c6
A= At'Az
=( o, *L)r.rr(zpas + r)- Fpevc
('" Sr' Ip6S+rJ 2psS+F
Thay sd ta tim duoc: A x 17,2J. J
( Nnan xlt. CAc ban sau day c6 ldi girli tdt:
Hi NQi: Nguy€n Manh Qudn, THPT chuy€n Nguy6n Huc; Hii Duong: NguydnVdn Nguytn,l2Al, THP|
Ke Sat, Binh Giang; Vinh Phric: D6 Dii".Thrty,THPI
Ng6 Gia TU, L+p Th+ch; Nghe An: Ngulcir TrungTlhnh, l0A5 - K48, kh6i THPT chuyOn, Dd hoc Vinh.
NGUYEN VAN THUAN
DOC LAI CHO DUNG
Trong ldi gi6i cira bdi T4l3i\, c6 m6t lQp ir:An
chua duoc chfnh x6c lh: "Do vi tr6i cfia BDT (1) khong ddi trong ph6p ho6n vi v,iag quanh
a, b, c nOn c6 thd gii thidt a >- b >- c > 0".
Thuc ra thi m6t bidu thfc kh6ng ddi trong
ph6p ho6n vi vbng quanh a, b, c thi chi c5 thd
gii thiet duo. c mdt trong ba sd ld max (hoac
TONN HOC
FKWffi $ffi$ !ffiTiffi$is nK.&WimFOR LOWER SECONDARY SCIIOOL FOR LOWER SECONDARY SCIIOOL
T11376. (For 6th grade) Write the numbers 82008 and 1252008 consecutively. What is the
number of decimal digits of the resultingnumber ? number ?
T21376. (For 7th grade) Find a rational number
I such that the following three conditions are b
satisfied
ii) IIa+5b =26;
iii) 200< lal+lbl<230.
T31376. Find all non-zero natural numbers n
such that the numbe r A = l'3'5'7"'(2n-l) ',
n'
an integer, here the numerator of ,4 is the product of the first n odd numbers.
T41376. Prove that
abc3+ + > :-(a+b+c-l).bca2 bca2
where a, b, c are positive real numbers such
that abc: l. When does equality hold ?
T51376. In a right triangle IBC with right
angle at A, the altitude AH, the rnedian BM,
and the angle-bisector CD meet at a
common point. Determine the ratio *.AC
FOR UPPER SF]CONDARY SCHOOL
T61376. Solve for x
fi.6 +Ja-1=*z - 1.
T71376. Let S denote the area'of a giventriangle ABC, and denote BC : a, CA : b, triangle ABC, and denote BC : a, CA : b,
AB : c. Prove the inequality
a2b2 +b2c2 +c2a2>
fts2 +!a2 @ - c)2 +!u' G - a)z + !c2 @ - b)'z .
222
When does equality hold ?
T81376. Given a triangle ABC with three sides
BC : a, AC : b, AB : c such that s + s : 2b,
let ho, h, be the altitudes fi'om I and C
respectively,' and let ru, r, denote the l-exradius and C-exradius respectively. Prove that
1111
rd rc ho h,
TOWARDS NIATTIEMATICT\L OLYMPIAD
T91376. The positive real numbers a, b, c, x,
y, and z are such that
lcY+bz:a
I
t,az+cx=b
I
lbx+aY=s'
Find the smallest possible value ofthe expression
x2 v2 22D - t-L- r- D - t-L- r-
1+x 1+y 7+z
T101376. Let f be a continuous function on
R such that f(2010) : 2009 and f(x).fa@) :