2 e Ạ Suy ra 2 e B. T5m lai tac6 0 e C, ndu_l<x<lthl_l< 4x3_ 3x<l
leAvd2eB.
Tir d6 bang quy n4p de thay
Ac-13k+l,k e Nl, Bc-l3k+2,k eN) vi
C c- l3k,li e N).
Nhung vi moi s6 n e N c5 bidu diOn duy nhat
rt = 3k* r. (r. = A, 1,2)ndnA = l3k + \,t e N{
6= l3k+2,k e N) vi 6 = l3k, k e N).
ii) Ndu 1 e B thi li luAn nhu tren tthay đi vai
trd gifra A vit B) ta duoc
4= l3k + 2, t + N), 3- l3k + 1, k e N)
v)C=l3k,keN).fl
( Nna, x6t. B)i niy cluoc khri d6ng cdc ban tham gia
giii vi cric ban đu cd ldi giai dfng. C6c ban c6 ldi girii
tot la: Phri Tho: Kholng Ngoc Trong. llT, THm
chuyOn Hing Vuong, Vlct iri; tlar NOi: Dodn Tri Diing, llAl, DHSP Hn Noi, Pham Duy Ting, l0A
chuyOn, DHKHTN Hn N0i; Bic Ninh: NgLrltl-n Minh Hdng, l0A, THPT Yen Phong; Thanh H6a: Ddo Dirc
Hutin, l0T, THPT chuyOn Lam Sdn, TP Thanh H6a; Ngh6 An: N guyin Vdn Thonh, l0H, THFrf D6 Luong; Quring Binh: Dring Ngoc Tlnnlt, l1 THPI chuyOn Quang Binh; LAm Ddng: NguvinThi Nlur Ngrr.r'dt, llT
chuydn Thdng Long, TP Da Lat; Dak Lak: \'o \:dn Tudh,9A5, THCS Nguy6n Du, Krong Buk; Phri Ydn:
Huinh KintTrid'n, I I To6n, THPT chuyOn Ph[ Ycn. DANG HUNG THANG
*Bni T9l345. Cho circ sd /htrc rr, ,rr, ..., .r2oo;
thuQc do4tn [-l; l] vir ,'i ting tt'rt' ltip pltLrong
c'ito c'hting hcing 0. Cluing minl'r rdng ịi, f .r' ... .- .\',,,,,1 * 4.
l
Dilng fu1\'c xay ru khi ndỏ
Loi gitiị Nhdn x6t
4x3 -3x+ 1 = (4x3 + 4l) - (4i + 4x) + (x + 1)
(1)
(2)
Ap dung vdo blLi todn ta c6 -l < 4x! 4x, < I v6i moi i = 1,2,3, ...,2007.
COng theo tlng vd cdc BDT trOn ta nhAn duoc
- 2007 S -3(x, + x2 + .. .+ xzt)oj) < 2007 (do xf + *) +...+ x)xrz =0) .
Do đ lx, + x, + ... + .r:,u,, I s 2o!7 = eeg.
-1
Cfrng tir (1) ta c6 ,rr + -trr + ... + x2rxr7 = -669 khi
Irl
vlchi khịr, e {--: I I voi moi i = 1,2,...,2007.
t2 )
Kdt hop vdi xf +
") +...+ r)rxrz = 0 , suy ra
xt + x2 + ... + x2ttj = -669 khi vi chi khi trong
ciic sd x1, x2, ..., -rruu1 c6 223 s6 bang I vi cdn lai
I
I 784 so bans "2-1.
Tuong tụrr + xz * ... ! rzut = 669 khi vd chi
khi trong c6c so x1, x2, .,., xzrx, co 223 sd bang
- I vd cbn lai 1784 so Uang "2]. D
( Nnan x6t. Bat clirng thrlc (2) c6 ngudn g6c tir luong
giiic: neu cosa = x thi cos3a = 4x1 - 3x. Hdu het cric ban hoc sinh THPT đu girii theo ciich luong gi{c tr6n.
Ciic ban hoc sinh THCS sau c6 ldi giAi tdt: Phf Tho:
Ta Dt)tc tl,ii, NguyAn Horirtg H(;i,-9A3, THCS Lam Thao; Bdc Ninh: Ngrr-vlrt Xdn Vrong, 9A, THCS huy6n ThuAn Thinh; HII TAy: Trdn Nlnt Tdn, 9A3' THCS Ngo Si Li6n, Chrrong M!; Vinh Phtc: LA Duv D{ing,9C, THCS Vinh Tudng; Nam Dinh: Pham Phi
DiAp, 6A, THCS Ycn Tho, Y Ydn; Thanh H6a: Nsuvdn Minh Anlt,98, THCS Trdn Mai Ninh, Irizlr
Quang Tlmnh, 9P., THCS Him Rcing; Ngh-Q An:
Nguyin Thriy Vy, Phan Si Quang, 9A, NgLodn Drlc COng, gD THCS Li NhQt Quang, Do Ludng; Hi finh:
Trdn Qudc Ludt,9P, THCS Son Hdng, Huong Son. NGUYEN MINH DUC
T94? \tpg yf'Tqor rgs .t2
* Bni rl0/3 45. Tim tit ca cuc hitnr ,;ri 7' 7L --> z
thoa mdn ctic' diitr kiAn sau;
1
i) flf(nt) - n) : .flnf 1 + .fln) * 2n..1\nt) v6'i nr<ti
nt,nez;
ii)/(r)>0
Ldi gitiị Trong didu ki0n i) thay m = l, ta c6
f(f(t) - n) = f(t) + fjr) - znJQ) (l)
Tiep theo, trong ( I ) ta thay n biti f(l) * rt, ta cd
f(n) = f(t) + f(f(r) - n) * z(f(t) - ti)f(t) (2)Cong theo tirng vd ( 1) vn (2), ta thu du-ọ c Cong theo tirng vd ( 1) vn (2), ta thu du-ọ c
2l'0) - 2V(t))' = 0 hay/(1) = I (dol(l) > 0).VQy Q) c6 dang VQy Q) c6 dang
f(l*rt)=fQr)-2n+l (3)
Til (3), v6i l = 1, suy ra,f(0) = 0. Thdvio didu
ki6n i), v6i nr= 0 ta thu duoc
f(-rt)=fQt),Yn e Z (4)Khi d5 (3) c5 dang Khi d5 (3) c5 dang fQt) -fQt- 1) = 2n - I,Vl e N. Suy ra fQt) - f(0) = - (2n * l) + (2(n - 1) * l) + ...+ (2.1 * l) hay fln) = 12, V2 e N.
Til (4) ta thu duoc/( n) = fi v6i moi n e Z. Thir lai, f(n) = n2 th6a mdn c6c didu ki6n
bar rạ LJ
( runan x6t. Nhi6u ban giai duoc vI cla so c6c ban đu
giai tucrng tu nhu cfch dir tlinh biry rt tr€n. NGUYEN VAN MAU
*BAi Tll/345. fam giric ABC cti ctit, dwing lrt.tng ttrl,in AAt, BBt. CC1 t'lúrng ltrinh rdng n1u hin kfnh t'dc' clru)'ng lrdn n6i ti1p ctic tttnr
girlc l)('81, (',4('1, AR,41 bcing rtltau tlti ttuu gitic
AllC lr'r tctttt giric: d1tr.
Ldi gitii /. Ta dua vlo c6c ki hi6u saú. BC= a,
CA = b, AB = c; AA, = m,, BB, = n'tt,, CCr = fit,'
I, r vd 1r, rr tudng rlng ld mm vi b6n kinh dudng
trdn n6i tidp c6c tam gidc ABC vit BCB,; 2p
vb,2p, tuong fng li chu vi citc tam gi6c ABC
vi BCB,.
^ A rofirt hoc v,fL[gg*JI]6
24.--i++-!q..
s49 (7-2006)
Goi Á, Á, l) tidp didm tr€n canh BC cria hai dudng trdn (1, r) vi (1,, r',); thd thi: IÁ= r, IrÁr=1-r, CÁ = p - c, CÁr - 1t, - nt,,vd ta c5 b CÁt =.r,Ó, hou ,Uo t ,-"''' =L (= ). CÁ lÁ ' 2(1t c) /'
Theo gii thidt vA chring minh tuong ru (bang
ciich hodn vi vdng quanh) ta c5
o*b -,n, b+L-ttt ,' ,o--,,, , .
2 " 2 ' ) " "t /i l .
= ' =zll=-i(r)
p-c p-o p-b [ , /
Mat kh6c, theo gia thiet cdc tam gidc BCB, CAC, vi ABA, c6 diOn tich bang nhau vd biin
kinh dudng trdn n6i tiep bang nhau n€n chLing
c5 chu vi bang nhaụ Bdi vAy, ta c6 cric đng
thfc sau
bca
a+-lntr, = OUr+ilt, = c+- | tnd = .\=2pt ) (ii)
Tir c6c hC dang thric (i) vd (ii) ta duoc
ft'+(p-ctA =2a+blv+(p-a))":2b+c lv+(p-a))":2b+c I-+1p-b))=2c+a (1) (2) (3)
Sau d6, tit c6c cap ding th1c (2), (3) vd (3), (1)
ta duoc h€ thrlc
c+a-2b
=u*b-2, (= ))
a-b b-c
Cudi cilng, til (iii) ta duoc
!ffo_,r f +lb-c)2+(c-a)21 =Q (iv)
2'
Suy ra ABC ld m6t tam gi6c d<iu, dpcm. fl
b
FF F
Ldi gitii 2. (DuatheoV6Thdnh Loi,llT, THPT
chuyOn Thoai Ngoc Hdu, TP Long Xuy€n, An
Giang). Sfi dung ki hi6u nhu dI duoc ding
trong ldi giAi 1. Tru6c het, ta chrlng minh bd dti sau dAỵ
Bd dẠ Trong mdt tam gi6,c, ftng v6i canh ldn
hon lh trung tuyen nh6 hon, vi nguoc laị
Cuthdlia>b >c<> nt,, 1tn1, 1il:,. (v)
Thlt vAy ta c6
4ntl=2grz +r'1*ó ; 4nf,:2gz +ó1- b'
Tn d6 4Qnj -rr,',1=31b' -ó) (vi) Tir (vi) suy ra: á> b' <> ,rl <,nf; (vii)
TD (vii) ta thu duoc ciic BDT (v) v) bd đ da
duoc chrlng minh.
Tro lai b)i todn, ta cfing thiet lAp duoc ciic ding
thric (ii) nhu 6 ldi giii 1, bidu thi chu vi ba tam giiLc BCB,, CAC, vi,ABA, bang nhaụ
Bay gid đ 6p dung bd đ ncu tren, ta vidt lai (ii)
dudi dang sau
Tir h0 ba ding thric (1'), (2') (3') ta chrlng minh rang a = b = c bang phuong phrip ph6n chring.
Thdt vdy, ndu m,,> nz, thi c > b (theo bd dC) vn
tt (1') ta duoc2c > b + c > 2a,hay c > a; suy ra
tn, a mu . Til (2') ta du-d. c: c + a < 2b vd do d6: 2a < c + a <2b. Y4y a < b; suy ra tnu > m1,hay
lb, m, - rno> 0. Do d5, tiI (3') suy ra a + b > 2c.
Do a < b nOn ta lai c6: 2b > a + b > 2c. Ydy
D ) c, n6n lai suy ra mt)< 2,. Nhu vAy li, xudt phrit tir gii thidt mu) nt, rdi tir (1'), (2'), (3') vi
nhd bd đ ta lai suy ra: fit, 1ry,.vd nguoc lai tfc li til c > b l4i suy ra b > c, vit ngđc laị Didu d6
chrlng t6 rlng chi c6 thd le b = c, đng thdi
ffih = ffi,. Vd do d6: b + c = 2b = 2c, thay vio
(1') ta duoc a = b = c vd ABC ld mdt tam gi6c
đụ dpcm. D
( Nnan x6t. I ) Dcii chieu hai Idi girii trtn dAy ra rhdy
rAng: Ve co btin, ldi giai I chi đi hoi vAn dung kien
thfc Hinh hoc 9 vi cho clnlrng ninh tr(c lllf đng thdi
Idi giai cflng gon gdng. Ldi giai 2 đi h6i su dung c6ng thrlc dudng trung tuydn (Hinh hoc l0) v) thi6t lap
m6t tinh chdt drroc phdt bidu trong bd đ n6i tren. Ldi
gi:ii cfrng gon ging, nhung su dung phuong phrip phAn chfng, m6t trong nhftng phuong phip chtlrng minh gidn tiip.
2) Hdu het cdc ban đu su dung c6ng thric dudng trung
tuyen holc su dung bd de dudi dang mOt nhAn x6t, khong chring minh. Tuy nhiOn, lAp IuAn cua da s6 cdc ban vd chung minh phan chring hliu hdt lai c6 tinh chdt cAm tinh, thieu chat che vi khong đy dt.
3) Ciic ban sau dAy c5 ldi girii tuong đi tdt:
Hung Y€n: Plnn Ti€ir Diing, 1l Toiin, THPT chuydn Hung YOn; Hii Duong: Ttiltg TltíNglria, l0Tl,
THPT Nguy0n Trii, TP Hrii Duong; Ngh€ An: Dau Li
ThLiy, I lA, THPI Qulnh Luu IV, Qu!'nh Luu; Hd
finh: Ngl,vlrr Tlri Hanh Drtng, I t Toiin, THPT chuydn
Hi Tinh; Quing Tri: Vd Thi Clumg. I I Torin, THPT chuyOn Lc Quf Don; Thia Thi6n - Hr6; NgụvitrTi1ir
Ctirtlr, I I Todn, THPT Quoc hoc FIue; Dh Ning:
Nguyin Nhr Dttrc Trung, 10A1, THP'f chuy€n Le Quf Don. TP Dir Nring; Long Xuy6n: N guytn Qu6'c Khdnh, 10A3, THFrf Long Xuy0n.
NGUYEN DANG PHAT
*Bii T121345. L'ho ntat ttiir tam O brin kinh
]1. ,.[6t hinh c'litp S.AUC' cltttf itr [lin{ scro cho ('(iL' L'onh Sr1, ,98, SC luon fidp .rtic' t,6'i ntdt c'iiu
lrAn lheo th{r tútoi ,1, l}, C, lttttt nfu, .RB-.=90",
,.\r-' = 60"" anl : 120". T'itu ttip h,rp ,funlt S.
Loi gitiị Pldn thudn' GiA su S th6a m6n didu kiSn đ bdị Ta c5, A,ASB vu6ng cAn tai S; ASBC
đu; ACSA cAn tai S ve 6G = 120". Suy ra
eB = Jise; BC = sA; AC = .6sa
) AB2 + BC2 = AC2::> LABC vu6ng tai B (l)
Goi / li trung didm cia AC.
Tn (1) suy ra 1 ld tdm dudng trdn ngoai tidp
MBC (2)
Mdt khrlc dC thdy:
[Sa=Sf:SC lcirng tiep xric v6i ducirrg trdn (O; R)
J
IOA=OA=OC lcirng bang R).
Til d6 vd (2) suy ra SI vd OI đu vu6ng g6c vdi
mp (ABC) n6n S, 1, O thing hdng. ToflD ngq vn rq6t rwe 121,,,,,-n,,)=b+c-2a (l') fZlr,, *,n,\=c+a-2b l2') l2(m,-tn,,\:a+b-2c (3') 349 (7-2006) 25
---*-
. Btt
3sin i{o = AS AI -AO -
SO
RJ32R- 2 2R- 2
J3
t20 (5)
di clua trung didm SO
Vi SO di qua 1 n0n: ASO : ASI = 60" (do AdSA can tai 5 v,i .frb = 120")
.40 2R
sin60" J:Iznj Iznj
-) J thuoc mịl[ cau I Ú.--- t .
i J:,t
Plrdtt dciọ GiA su S thLroc mat cdr-r iOr#\,\ vJ/
Goi SA ld tiep tuy0n ke t[ S t6i mat cdu (O, R)
(A ln tiep didm). Goi (P) Ii mit phing clua A, vu0n8 g6c vdi SO vI giao vdi SO tai 1. Giao cira
(P) vdi m[t c[iu (O, R) lir duirng trdn (rr.r) r,i\ l lir
tArn cia (o:). Goi C li diem doi xring cua A clua
1