• So 2 tu'dng iJng vdi
• So 3 tufdng iJng vdi
• So 4 tifdng iJng vdi
• So 5 tu'dng iJng vdi
134 Cdc phuang phdp gidi todn qua cdc ky thi Olympic
vh. 100.0%. (Mot so'phan mem giao dich hien nay nhif Metatrader ttr dong sur dung cdc mu'c Fibonacci 0.0%, 23.6%, 38.2%, 50%, 61.8%,
M o t khi nhi?ng mu'c nay dUdc nhan dien thi cac dUcfng nam ngang dUcJc ve ra va du'cJc suf dung de nhan dien cac muTc h6 trd va cac mu'c khdng cif c6 the c6.
Day so Fibonacci nhu" sau: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...
• T i le Fibonacci 61.8% du'cJc tao ra bang cdch chia mot so bat k y trong day so Fibonacci v d i so hen sau no. V i du, , ^ = 0.6153, 1 = 0.6179.
• T i le 38.2% dUdc tao ra bang each chia mot so bat ky trong day so' Fibonacci v d i so d vi tri thu" hai ve ben phai cua no.
V i du„ ^ = 0.3819.
• T i le 23.6% du'cJc tao ra bang each chia mot so bat ky trong day so' Fibonacci v d i so'd v i tri thiJ ba ve ben phai cua no.
V i du, ^ = 0.2352.
Cac nha dau tiT thiTdng SIJ: dung cac mu'c Fibonacci thoai lui nhU cac mu'c khang ciT va ho trd hieu qua. Ho sijf dung mu'c nay de vao lenh mua/ban hoac xac dinh mu'c chan lo.
s-NWDEx Dôr^laQ012Opôl^B^.72.Hl^MJ7.Le^ô.1ằ,cl<ằ•^ae,8e^2J%)
Xung quanh so Fibonacci 135
^JTSNDM - Mtr 9f 10O012 OtMn 395.72. HI 398.17 Lo 3 ô 18 ClOM 36*88 (-3 JHl
Fibonacci Extensions thtfcJug du'dc diang de du bao cdc mu'c h6 trd va khang c\i trong tUdng lai va dUdc ve vufdt khoi bien dp 100%.
Nhi?ng mu'c nay du'dc tinh bang each phan tich cdc mu'c hoan lai giiJa hai diem cao nhat vd thap nhat trong mot khung giao dich.
Van de dat ra la dieu gi se xay ra khi gia vu'dt qua hai diem ma ta sijf dung de tinh cac mu'c Fibonacci? Tai diem nao thi chung ta ky vpng thoat trang thai? Cau tra I d i chinh cho nhiJng van de nay la Fibonacci Extensions {Fibonacci md rong).
Fibonacci Extensions cho cac muc tieu gia vu'dt qua mu'c hoan lai 100% cua xu hu'dng tru'dc. Cac mu'c Fibonacci Extensions du'dc tinh bang cdch lay cdc mu'c Fibonacci chuan cpng them 100%. Do do, cdc mu'c Fibonacci Extensions chuan nhu' sau: 138.2%, 150%, 161.8%, 231.8%, 261.8%, 361.8% va 423.6%.
136 Cdc phuang phdp gidi todn qua cdc ky thi Olympic
Tai li$u tham khao
[1] Nguyen V a n M a u (chu bien), Chuyen de chon loc ddy so vd dp dung, Nha xuat ban Giao due, 2008.
[2] N g u y i n V a n Ngoc, Mot so bdi todn ve cap so vd tdng hiiu han, Cac chuyen de boi du'Sng giao v i e n chuyen Toan T H P T tai Quang Ninh, thang 10/2012.
[3] Nguyen V a n Ngoc, Cdc ddy so Fibonacci vd Lucas, Cac chuyen de b o i du'Sng hoc sinh gioi Toan tai Thai Nguyen, thang 11/2012.
[4] N i k o l a i Nikolaevich Vorobev, Fibonacci Numbers, Birkhauser, 2001.
[5] Fredric T. Howard, Applications of Fibonacci Numbers.
[6] Alfred S. Posamentier, Ingmar Lehmann, The fabulous Fi- bonacci numbers, 2007.
[7] Thomas Koshy, Fibonacci and Lucas Numbers with Applica- tions, 2011.
[8] Constance Brown, Fibonacci Analysis.
[10] Robert Fischer, Fibonacci Applications and Strategies for Traders, 1993.
[11] Carolyn Boroden, Fibonacci Trading, M c G r a w - H i l l , 2008.
[12] h t t p : / / v i . w i k i p e d i a . o r g .
[13] h t t p : / / e n . w i k i p e d i a . o r g / w i k i / F i b o n a c c i - n u m b e r . [14] h t t p : / / p h o c h u n g k h o a n . v n .
[15] T a i l i e u phan tich ky thuat cua ngan hang A Chau A C B .
LCJl GIAI MOT BAI TOAN Md ^ .
1. Gim thieu
V 6 Quoc B a C a n ,
3 oO?'sV fiig 'jkl' ,[11
Trong ky ihi Olympic Toan Canada nam 1999 c6 b a i toan sau:
B a i toan 1. Chiing minh neu a. b. c Id cdc so thvtc khong dm c6
tdng bang 1 tin: ^^^-^^
a^b + h~c + c^a^—. .''-fv^ ^ • Sau do. ngirdi ta da t i m dufdc mot ket qua khac tong quat hOn nhiy sau:
B a i toan 2. Cho a. h, c la cdc sdthuc khong dm sao cho a+b+c = 1.
Chi'aig minh rdng vdi moi n ^ 2. ta luon c6 bat dang thilc:
a/'b + h"c + c"a^ "
{n + l ) " + i '
Vao khoang nam 2005. Vasile Cirtoaje da de xuat tren dien dan AoPS mot bai toan m d i v d i hinh thufc phat bieu tiTdng tiT:
B a i toan 3. Cho a, b, c la cdc sdthuc khong dm sao cho a + 6 + c = 3.
Chi'rng minh rdng
o'/^b + b'^'c + c'/'a^3.
Co the thay rhng ta't ca cac ket qua nay deu nham muc dich danh gia cac bieu thufc c6 dang a''b+b^c+c^'a khi da biet triTdc tong ciia ba so a. b. c (tdng nay c6 the la nhffng gia tri difdng tiay y ) . Nhu- vay, cau hoi dat ra la: Vdi mot gid tri duong bat ky cua k. lieu ta CO tun duac mot chdn tren that "-tot" cho bieu thvCc a''b + h^c + c^a?
V d i y tiTcing nhu" vay, Vasile Cirtoaje da de xuat bai toan mc!
ciia minh nhU sau:
137
138 Cdc phuung phdp gidi todn qua cdc ky thi Olympic
B a i t o a n 4. Cho a, b, c la cdc soducmg c6 tong bang 3. Chiing minh rang vdi moi k > 0, ta luon c6 bat dang thiic:
,/ ; of'h + h'^c + c'^a < m a x < 3,
T r o n g [1], tdc g i ^ V a s i l e Cirtoaje da g i d i t h i e u I d i g i a i cho cac trifdng h d p 0 < A; ^ ^, ^ I n 3 ^ " ^ I n 2 ^ ^ ^ ~ -^'^^^
n g h i e m cua phUdng t r i n h [k + = (3A;)^ ': ' T r o n g [ 2 ] , tac gia P h a m K i m H u n g c u n g da nhac l a i ke't qua n a y d i f d i d a n g b a i t o a n m d va de xuaft t h e m m o t so b a i toan l i e n q u a n k h a c .
T r o n g [ 3 ] , c h u n g t o i c u n g da g i d i t h i e u I d i g i a i cua m i n h cho b a i t o a n 4 d t r e n . D a y qua thifc la m o t b a i toan k h o nhifng c u n g rat thii v i . T r o n g qua t r i n h t i m t o i Icfi g i a i cho no, c h u n g t o i da due ke't du'dc cho m i n h m o t k y thuat m d i g i u p difa m o t b a t d a n g thiJc ve x e t t a i m o t triTdng h d p dac b i e t : c6 m o t b i e n n h a n gia t r i b i e n .
V a d day, t r o n g b ^ i v i e t nay, c h i i n g t o i se g i d i t h i e u l a i I d i g i a i cua m i n h cho b a i t o a n m d t r e n c u n g nhif muo'n g i d i t h i e u c u n g b a n doc k y t h u a t xur l y cua m i n h . T u y n h i e n , do t h d i g i a n c h u a n b i c6 h a n n e n c h u n g t o i chu'a c6 d i e u k i e n t r i n h bay cu the phu'dng p h a p m d i nay, x i n diTdc h e n l a i trong m o t b a i v i e t k h d c .
Ta't n h i e n , do day la m o t b a i todn m d k h o n e n I d i g i a i cua no kha d a i d o n g va phiJc tap. H y v o n g r a n g , sau k h i doc b a i v i e t nay x o n g , se CO m o t b a n doc nao do t i m ra du'dc m o t I d i g i a i n g a n g o n va sang sua h d n .
2. Lcfi giai bai toan md
D e g i a i b a i todn nay, ta se x e t ba triTdng h d p nhtf sau:
Trifcfng hi/p 1. 0 < A; ^ 1. Suf d u n g bat dang thufc B e r n o u l l i k e t h d p v d i k e t qua c d b a n :
' ab + bc + ca ^
Ldi gidi mgt bai todn md 139
a^b + b'c + a'a = b[l + (a - l)f + c[l + {b - l)f + a[l + (c - 1 ) ] '
^ b[l + k{a - 1)] + c [ l + k{b - 1)] + a[l + k{c - 1)
= k{ab + bc + ca) + 3-3k ''
{a + b + cY ' '
< k- + 3 - 3 / c = 3.
Trift^ng h(?p 2. \ k < 2. T r o n g t n f d n g h d p nay, b a n g each sxi d u n g k e t qud ddn g i a n (ban doc c6 the tvT chiirng m i n h ) : Neu X ^ y > z ^ 0 vd m > I, thi , , ;
x'^y + y'^z + z'^x ^ xy"" + yz"^ + zx"^, • (*) de d a n g n h a n thay r a n g ta c h i can x e t hki t o d n t r o n g t r i f d n g h d p a ^ 6 ^ c ^ 0 la d u .
B a y g i d dat K = m a x <^ 3, ,a — c-\-svh.b — c + t {k + \f+\
(s ^ t ^ 0), b a t d a n g thiJc c a n chiJng m i n h c6 the v i e t l a i t h a n h
J 9{c) = K
T a c6
V 3
-{c + s)\c + t)-{c + tfc-c\c + s) ^ 0.
k
g\c) = K{k + l)(c+ '-^)' -k[{c + s)'-\c + t) + {c + t)'-'c + c ^ - ^ c + s)] - [{c + s f + (c + tf +
^ 3 ( f c + l )
= 3(A: + 1)
/ s + t\
V
-k[{c + s)''-'{c + t) + {c + t)'-'c + c'-\c + s)] - [{c + s)' + {c + t)' + c']
(^±^y -k{a'-'b + b'-'c + c'-'a) • -{a' + b' + c').
T a se chiirng m i n h g'{c) ^ 0 b a n g each c h i i n g m i n h fa + b + cV
V 3 j
3(fc + 1) ^ kia'-'b + b'-'c + c'=-^a) + {a' + b' + c'
140 Cdc phucmg phdp gidi todn qua cdc ky thi Olympic
Sur d u n g (*) m o t I a n niJa v d i x = a'=-\ = 6*^-1, z = c'""^ va m = ^ > 1, ta dirtJc r':., ('(i ,:V jI,* j , ; . . ^ ^
' " ^ a' = - ^ ( 6' = - i ) ^ + 6^ - i ( c ' = - ^ ) ^ + c ^ - ^ ( a ' ^ - i ) ^ , hay
á^-^b + fé^-^c + c''-'a ^ ab''-^ + bé'-' + cấK ^'^'^'^^
N h i r v a y , b a t d a n g thtfc g'{e) ^ 0 se du'Oc chiJng m i n h n e u ta c 6
' Hk + 1) ( ^ ^ ^ ) ' ^ 2(a^ + t'^ + c'^-) + /c J2 o"-'{h + c).
D o tinh t h u a n n h a t n e n ta c 6 the g i a sur a + 6 + c = 3 (chii y r a n g k h i a = 6 = c = 0 t h i b a t d a n g thufc t r e n h i e n n h i e n d u n g ) . B a t d a n g thu-c t r e n t r d t h a n h
6(A; + 1) ^ 2(a'^ + 6'= + c'=) + fc^a'=-i(3 - a ) , hay
i: (2 - k){a!' + 6*= + c^^) + kia!"' + b'"' + c'"'') ^ 6{k + 1).
T a Viet dUdc b a t d a n g thiJc n a y dirdi d a n g h{a) + h{h) + h{e) ^ 0, t r o n g d o
h{a) = {2-k)a!'^Zko!'-'-k{2k-l){a-\)-~2{k + l).
L a n lirdt t i n h d a o h a m c a p m o t v a c a p h a i cua h{a), ta du'dc
I h\a) = k{2-k)o!'-'+ ?,k{k-l)a''-^-k{2k-l),
h"{a) = k{2 - k){k - l)a'=-2 + 3kik - l){k - 2)0!^'^
=k{k-l){2-k)a''-^{a-Z).
D o 1 < /c < 2 v a 0 ^ a ^ 3 n e n h"{a) ^ 0. D i e u n a y chiTng to h'{a) la h a m n g h i c h b i e n t r e n [0, 3], D o h'{l) = 0 n e n ta c 6 h'{a) > 0
Ldi gidi mot bat todn md 141
v d i a e [0, 1) v ^ h'{a) < 0 v d i a G ( 1 , 3]. Tuf d a y suy r a h{a) t a n g thirc sir t r e n [0, 1] v a g i a m thirc sir t r e n [ 1 , 3]. Q u a nhiTng l a p l u a n nay, ta de d a n g n h a n thay h{a) d a t circ d a i t a i a = 1, tuTc ta c6 h{a) < h{\) = 0 v d i m p i a € [0, 3 .
H o a n t o a n tufdng tir, ta c u n g c6 h{b) ^ 0 v a h{c) ^ 0, do do h{a) + h{b) + h{c) ^ 0. V a y ta da chuTng m i n h diroc g'{c) ^ 0. T i r do suy r a g{c) l a h ^ m t a n g v d i m o i c ^ 0, v a ta c d
g{c) ^ 5 ( 0 ) = K{-
k+l
^ K 5 + A fc+1
- sH - K
^k4+t\
V 3 )
fc+i
fc+i
-k'
\ k+i
3fc+i K - 3'^+i/c {k + l) k+l
T r t f c f n g hgfp 3. k ^ 2. K h o n g m a t t i n h tdn g quat, g i a suf a = m a x { a , b, c}. Ap d u n g b a t d a n g thuTc B e r n o u U i , ta du'dc
. b{a + c)''^a''b(l + ^ y ^ a ' ' b / kc\ 1 + —
^ a'^M 1 + — = + a'^-'bc + a'^'^abc
> + b^'-'bc + c'^-'ac^ + b^'c + c'^a.
M a t khac, theo b a t d a n g thiJc A M - G M ,
b{a + cf = k'
\ / • 6 ^ /c'=
/, a + c
^ m a x < 3,
k + l
(A: + l)*=+i — ( / c +1)^+1 j •
Ke't h d p h a i b a t d a n g thiJc n a y l a i v d i n h a u , ta d e d a n g suy r a 3/0+1;,/=
a% + b'^c + d'a ^ b{a + c)^ ^ m a x { 3, B a i t o a n difdc chiJug m i n h x o n g .
{k + l) k+l
142 Cdc phuang phdp gidi todn qua cdc ky thi Olympic
3. Bai toan mof mdi ' ^
Ta thay rang b^i todn md cua Vasile Cirtoaje thifc cha't chinh la mot trirdng hop rieng cua bai toan sau (da dUdc Pham Kim Hung de xuat trong [2]):
Bai toan 5. Cho cdc so khong dm a, b, c co tong bang 3. Dat:
, . TA- V.
Hay tim gid tri Idn nhat vd gid tri nho nha't (neu co) cua P{r, s) theo r, ,s 6 M.
Ket qua cua Vasile Cirtoaje chinh la iJng vdi triTdng hop r > 0, s = 1 (va r = 1, 5 > 0). Ngoai ra, chung ta cung co lc(i giai cua no trong mot so trUdng hdp dac biet khac nhiT r = s; r = - \k s = -
1 1 2 2
(va ngUc(c lai); r = - va s = - - (va ngifdc lai); ... Nhiftig lieu
o 0
chung ta co tim difdc Icfi giai tong quat cho no? Day thiTc sir la mot van de rat kho, va chuTng toi rat mong nhan du'Oc nhffng y kien trao ddi cua ban doc gan xa.
4. Mot so' bai tap ttf luyen
Trong muc nay, chung toi xin de xua't mot so bai tap co hinh thffc phat bieu giong (hoac gan giong) vdi cac bai toan da dffcJc gidi thieu d tren. Cac bai tap nay deu da co Idi giai va ban doc co the thu" sffc vdi chiing de cung nang cao ky nang.
Bai tap 1. Cho a. b, c la cdc sothuc khong dm thoa man a+b+c = 3.
Chiing minh rang
' • ' ^" a^b + b '^c + c'^a + abc ^A.
Bai tap 2. Cho a, fc, c Id cdc so thuc khong dm thoa man + 6^ +
= 3. Chiing minh rdng
a'^b + b'^c + c'^a^2 + abc. i
iJUcfng doi trung vd m^t so bdi todn dp dung 143
Bai tap 3. Cho xi, X2, . • •, Xn {n ^ 2) Id cdc so thuc khong dm thda man xi + X2 H h a;„ = 1. Chvlng minh rang f?,/;/
4
X\X2 + xlxz + • • • + xlxi ^ —. _
Bai tap 4 (Olympic Todn Trung Au, 2007). Cho a, 6, c, d la cdc so
thuc khong dm co tong bang 4. Chiing minh rang a%c + b'^cd + c^da + d'^ab ^ 4 .
Bai tap 5 (Duf tuyen IMO, 2007). Cho a i , 02, . . . , Oioo la cdc so thuc CO tong binh phUOng bang 1. Chiing minh rang ; .
a\a2 + ala^-\ alf^ai <-.
Bai tap 6 (Anh, 1986). Cho x, y, z la cdc so thUc thoa mdnx + y + z = 0 vd x'^ + y'^ + = 6. Tim gid tri Idn nhdt vd gid tri nho nhat cHa bieu thiic:
P = x^y + y^z + z^x.
Bai tap 7. Cho xi, 0:2, . . . , x„ (n ^ 3) Id cdc so thuc khong dm thoa man Xi + X2 H 1- x„ = 1. Tim gid tri Idn nhat cua bieu thiic:
P(Xi, X2, . . . , Xn) = xlxl + xlxl + ••• + X^X? + U^^''-'^X^xl • • • X^
Tai lieu tham khao
[1] Vasile Cirtoaje, Algebraic Inequalities: Old and New Methods,