gi6i mQt bdi to6n nhu th6 ndo?
sai ,iCt ndy d6 cQp t6i m6t.s6 phucr.ng thrlc dinhhu6ng dring c6ch huy dQng ki6n thfc dC gi6i mQt bdi hu6ng dring c6ch huy dQng ki6n thfc dC gi6i mQt bdi to6n vd tu d6 ldm s6ng to k! thu4t thUc hiQn loi giii.
Phmmg thric l. T4o nhii:u co hQi tI6 hgc sinh
nhfn thri'c vai trd cria mdi li6n hO nhAn quri
trong ho4t tlQng tim kitSm Ioi gini.
'i'nghia tritit hgc sdu xa ndy thti hi6n trong ho4t
ri
dQng giai to6n o cho: S6ng t6 diOucdn tim, c6n chimg minh trong bdi to6n c6 ngudn gdc tu tri thuc ndo da biCt. Ching hpn, n6u didu c6n tim
li6n quan tdi dO ddi do4n thing, tich cdc dq ddi,
binh phucrng dQ ddi, d0 lon g6c thi ta thucrng
dtrng tich v6 hu6ng hay dinh lf d6n xu6t tu tich v0 hu6ng ld: clinh lf c6sin vd dinh lf sin trong
tam gilc. ThAt vQy, tich c5c d0 ddi a; b chinhld tich v6 hu6ng ctra hai vecto i; i .r'rrg hu6ng ld tich v6 hu6ng ctra hai vecto i; i .r'rrg hu6ng
r-lt-l-ll'l
vd lal= a:lrl:h. Khi do a.r'= lrzl.lvlcos0" = a.b.
l l ll trr
-2 a -:-. -, :'.
Dac biQt u = Lt.Lt = A'; coscr= i.j ; ttong do i; j
li c6c vecto d<m vi vd g6c gita chring bing cr .
C6 th6 lap luin khi gap d4ng to6n 1i6n quan tdi d0 ddi, tich ciic d0 ddi, binh phuong dQ ddi vd d9 lcrn g6c thi ngudi ta su dpng hinh hoc d6ng dang. DiAu ndy c6 thd gidi thich ld tu hinh hgc d6ng dpng vd doi hinh ld trucrng hgp ri6ng cira n6, c6 th6 chimg minh rlinh lf Pythagore. Tt
dinh lf ndy c6 thii chimg minh dinh l1f c6sintrong tam gi5c. Tri dinh lf cdsin c6 th6 suy ra trong tam gi5c. Tri dinh lf cdsin c6 th6 suy ra dinh lf sin trong tam gi6c vd tu d6 xdy dpg
dugc c6c kiiSn thric vC nq tnric lugng.
DAO.IAM
(GV Khoa Tbdn, Dqi hqc Wnh)Thi dg l.l. Xit LABC c'i gic' BAC: u thay Thi dg l.l. Xit LABC c'i gic' BAC: u thay
aAi (O'< cr < 180o';, cac canh AB =21 AC =5 .
D tit cti€m thttQc ntba mdt phdng co bd ld du'd'ng
thang BC kh6ng chaa drlnh A sao cho ABCDdetr. Hiit'rac dinh a dA aO dai doqn AD ton detr. Hiit'rac dinh a dA aO dai doqn AD ton nhiit.
C6 th6 clinh hucrng theo hai c6ch gi6i bdi to6ntr6n nhu sau: tr6n nhu sau:
Cdch 1
D
Hinh 1
Bi6u thi AD qua bi6u thric php thuQc o:
lD = f (a). Bii toiln x6c dinh d0 ddi AD n€n
theo nhAn xdt tr6n ta sE su dpng dinh llf c6sin cho c6c tam gi6c thich hqp (h.1). Ap dgng dinh
llf c6sin cho LABC ta c6 B(j =29-2}ccsa (l).
Ei$ ABC = P. Khi d6 nhd dinh li c6sin 6p dpng
cho LABC ta co cosB=-.$ ' Ap
'129 -20coso-dlrng clinh l1f c6sin cho LABD ta c6: dlrng clinh l1f c6sin cho LABD ta c6:
ADz :33 - 20 coscr - 4J29 10 cos a
x (cos60ocosB-sin600 sinP) (2)
Thay sinp =
vdo (2) ta nh6n dugc
AD 1& nh6to AD2
ADz = 29 -20 cos(600 + cr). lcrn nh6t. Di6u d6 xity ra
^ ^ TORN HOC
khi vi chi khi 20cos(600 +cr) bd nhAt, khi cl6a =1200 vd gi6 tri lon nh6t cira AD =7 . a =1200 vd gi6 tri lon nh6t cira AD =7 .
Cdch 2. NhQn xet. Bdi to6n li6n quan d6n d0 ddi n€n ta sri dgng phdp doi hinh (h.1).
Thgc hiOn phdp quay Q c6 t6,m C, g6c quaybing 600 (cung chi€u kim tl6ng hO). Khi d6 bing 600 (cung chi€u kim tl6ng hO). Khi d6
qua phdp qtay Q:o' 6nh cira D lir B; trfr_ cia A
ld, A1. Tir d6 6nh cria doqn AD ld AtB vdr
AD=AP. Sir dgng BDT tam gi6c ta c6
AD = AtB < AB + AA, ; cting thirc xay ra
eAeBA' Khi d6 do 6hr=60' n6nnl,C = 1200 vd BAt l6n ntr6t ttri vd chi khi nl,C = 1200 vd BAt l6n ntr6t ttri vd chi khi
AD l6n nfr6t. rri d6 AD nhf,n gi6 tri lcrn nhAt
bing 7. DThi du Thi du
"1.2. Cho tam gidc ABC vdi AB + AC.
Cac di€m M,N di d6ng ftAn cdc tia BA,CA
sao cho BM : CIn. Chmg minh riing dudngtrung trlrc cia doan thiing MN lu6n di qua mQt trung trlrc cia doan thiing MN lu6n di qua mQt
-.; ,: -. ,
dtem co dtnh. Cdch gidi (h.2)
Phwong thwc2.Di6n dat lpi gin tni6t vir t<6t tu4n cria biri to6n vO d4ng sdng t6 duqc cich k6t n6i
vrit tri thrirc dfl c6 iI6 ph6t hi6n hur5mg giii. C6 th6 ldm s5ng tb c6c phucrng thirc ndy qua thi dg sau ddy:
Thi dB 2.1. Cho 2k+l sij tu nhi€n d6i m\t
khac nhau (k>l;teN- ) rhoa mdn t6ng cila
k+l sii tu nhian bifu @ ftong cac tii da ,ho l6nhon tdng cua k sd con lai c6ng vdi 2016. Chang hon tdng cua k sd con lai c6ng vdi 2016. Chang minh riing *6t s6 trr nhiAn tuong Zk+ldd cho
l6n hon k+2016.
Cdch gidi. R6 rdng v6i m6i k2l c6 vO sO UO
2k+l s6 th6a mdn di6u kiQn c15 cho. Tir d6 hqc
sinh (HS) gip hh6 khdn khi tim ldi gi6i cho mqi bQ 2k+1 th6a mdn clidu kiQn bdi to6n.
oo m6i bQ 2k+1 s6 tu nhi6n hiru h4n n6n c6
th6 sdp x6p chring theo thri tp tbng dAn:
atlaz<a3...<azr,4azt*t (1). Tt d6 c6 th6diSn dpt lai bdi to6n nhu sau; Cho 2k+I s6 W diSn dpt lai bdi to6n nhu sau; Cho 2k+I s6 W nhiAn thda mdn diiu ki€n (l) vd fing ctia k+l
tA niit ki hy trong (l) tdn hon dng cila k sii,
cdn lgi cQng vcti 2016. Chilmg minh rdng m6i s6 trong cdc tii tiiy ft (l) lon hon k+2016.
V6i c6ch di6n dat tr6n sE xu6t hiQn y tucrngchimg minh le chi ra ar>k+2016. N6i c6ch chimg minh le chi ra ar>k+2016. N6i c6ch
kh6c, phii chimg t6 sO be nh6t ton hcrn
k+2016. Khi d6 ta c6 k! thu4t chimg minhnhu sau: %*z + a** *...+ (h*u +T16 < q + q +... + q+l nhu sau: %*z + a** *...+ (h*u +T16 < q + q +... + q+l
e (ao*, - q) + (qu - %) +... + (aro*, - ao*) +2416 < q.Do m5i sd h4ng trong d6ru ngo{c o vi5 tr6i 16n Do m5i sd h4ng trong d6ru ngo{c o vi5 tr6i 16n hon ho{c bing 1 (hiQu hai s5 t.u nhi6n khac
nhau) do c6 fr s0 h4ng trong ngoflc n€n
ar>k+2016.4
Thi dg 2.2, Tim cac s6 nguy€n daong a,b,c,dthda mdn rl +8 +C +8 +\2<ab+3b+tu+M (l) thda mdn rl +8 +C +8 +\2<ab+3b+tu+M (l) Ctich gidi. Vi c6c s6 cAn dm nguy€n, ducrng
^ ,.- t ,,. i
n6n vd tr6i vd vC phii cria BDT (1) ld nhirng s0 nguy€n, duong, sai thSc nhau it nhdt mQt tton
vi. Khi d6 c6 th6 diOn dat lai bii toin: Tim cdc
,!^
so nguyAn, drong a, b. c, d thda mdn:
az +bz +c2 +d2 +13< ab+3b+6c+2d (2)
Khi d6 (2) o a'+b' +c2 +dz +13
-ab-3b-6c-2d<0 (3)
Vii trai cira BDT (3) gqi cho HS 1i0n tu&ng tluavA dang tdngcfucbinh phuong thing: vA dang tdngcfucbinh phuong thing:
A
Hinh 2
Di6u kiQn cira bii toSn 1i6n quan t6i d0 ddi c6c doqn thing BM vd CN (BM: CN) vd chring
thugc hai clulng thdng kh6c phucrng n€n c6 m6t ph6p quay,bi0n B thdnh Cvd MthiLnhN.
Tt d6 di6m cd tlinh rlugc dg tlo6n li t6m cira
phdp quay n6i tr6n. Ph6p quay bil5n B thenh C
vd MthitnhNn6n tdm cta ph6p quay tl6 li giao
cia cdc trung tr.uc cria dopn BC vit do1t NM. Gi6 sir S ld giao cria hai dudng trung truc n6i tr6n, khi d6 NBM = ASCN (c.c.c). Tt d6
SBM=SCN. Khi d6 SBA=SCA, suy ra S
thuQccluongtrdn (c) ngopiti6p LABC. VflyS
ld giao ctn (e) vd trung tryc cua rlopn BC n€n S
cO Oinfr. O
TORN HOC ,.,.
(, *(,-;4 *Gr-r)' +(.-:;' +(a-r)' < o
ea=t;b=2;c=3;d=1. E
Phwo'ng thri'c 3. Di6n tl4t crii di cho, tlidu cAn
tim, cin chriL'ng minh theo cic ngdn ngfr'khicnhau vi lqla chgn ngdn ngir thuin tiQn cho nhau vi lqla chgn ngdn ngir thuin tiQn cho viQc k6t n6i v,i'i tri thri'c tli c6.
Thi dU 3.1. Tim gid tri nho nhdt ct)u ltiOu thti'c
Jr-i-
r|_B
llinh 3
Qua trii nghi€m cta HS gini bdi to5n tr6n choth6y h9 gpp kh6 khdn khi sir d\mg <lao him cli5 th6y h9 gpp kh6 khdn khi sir d\mg <lao him cli5
gi6i quyiit bdi to6n. C6 th6 hudng d6n HS nh{nxet chc bitiu thic trong cln ducrng. Voi hai gi6 xet chc bitiu thic trong cln ducrng. Voi hai gi6 tri cira x <li5i nhau thi voi x A"m gi6 tri ctra bi6u
thirc 16n hotr khi x duong.
Vfy gi6 tri nh6 nh6t cira bii5u thirc chi dat v6i x
ducrng. Tt d6 c6 th6 hu6ng d6n HS li6n tuong
Jl+f; D dO ddi cqnh BD cua tam giSc
ABD c6 AB --l; AD = x. C6 th6 ki€m tra diAu
d6 nhd su dpng dinh l)? c6sin cho A'ABD c6
ffi =600 G.3). Tucrng tu + + rt-ffi ra aq ddi cpnh CD cin LACD c6 AC =l; AD =x vi
dli =300. Tri d6 nhd BDT tam giSc ta c6
BD + DC > BC, ding thric xiy ra khi vi chi khi
DeBC. Khi d6 BD+DC=BC=J2. fri 0O
gi6 tri nh6 nhat cira bitiu thric n6i tr0n bing O.Thi dU 3.2. Cho tant giat' diu ABL', trin canh Thi dU 3.2. Cho tant giat' diu ABL', trin canh
BC ttiv ccic diim A,.A. sao choBA, = 4,X. = AtC. TrAn cclnh CA tiv cric diiim BA, = 4,X. = AtC. TrAn cclnh CA tiv cric diiim Br.B.sao c:ho CBr=8,8.:BzA. TrAn AB tdy
, =.;
c'dc' diint C, ,C, ,sao chrt AC, = C,C, = 9.3. Chirn54 minh rdng cat' giao dii:m t'tta t'uc cQSt
dad'n g t hang: (BB. ; C C, ), (AA,; BB, ), (AA,:CC,), (CC.;BB,), (BB,;AA), (AA.:CC,), ld cdc dilnh ct)a m6t lq.tc giac c6 cac drd'ng c:heo ding qtrt,.
Ggi M,N,P,Q,,R,S theo thf t.u li giao dii5m
cia cic c[p dudng thing n€u tr6n. Khi gi6i bdi ndy nhi6u HS ngQ nhfln luc giilc MNPQRS lit
llrc gi6c tl€u vd chimg tb cbc tlucrng ch6o cl6ng
quy, viQc ngQ nhfn nhu vfly li sai vi cbc c{p canh ddi cua lgc gi5c ndy thuQc hai cludng
thi.ng cit nhau.
Sau ddy li c6ch gi6i clugc diSn dat theo ng6n
ngt gin gfli voi HS.
sffic
Hinh 4
Do tam gi6c ddu c6 ba cluong cao ld ba truc ddi ximg cira tam gi6c d6. Xdt duong cao A,4, qua ph6p ddi ximg trpc A,4o errh cua A lit chinh n6; 6nh cria B ld C; inh cira C liL B. Khi d6 do phdp
OOi xrmg b6o tdin g6c vd dO dei dopn thing n6n inh cria C2 ld" 4; 6nh cira 4 ld, Cz; 6nh cua Bz ld Ct; inh cira Cl ld 82. TiI d6 suy ra c6c di6m M vd Q qta phdp d6i ximg tr.uc AAoc6 anh le chinh n6 @.4). Lfp luQn tucrng t.u P vd S
li hai di€m thuQc tr.uc d5i xrmg BBr; N vd R thuQc tnlc O5i xung CC,. Tn cl6 do c6c cludng
cao ddng quy n6n suy ra c5c tlulng ch6o cira luc gi6c MNPQRS ddng quy tai tryc tdm H ctrr- tam gi6c ABC.
TrOn tl6y chimg tdi trinh bdy ba phuong thfc
giirp dinh hu6ng tim tdi ldi gi6i c6c bii toin,bpn dgc c6 th6 nghiCn cuu dd xu6t th6m c5c bpn dgc c6 th6 nghiCn cuu dd xu6t th6m c5c phucrng thric kh6c giup cho viqc dinh hudng tim
tdi ldi giii c6c bdi to6n o truong ph6 ttrOngth6m phong phir hon. th6m phong phir hon.
30 t931#E:
:-:
ffisffiffi! s$&Pd
0t bAi to6n trong ki thi hgc sinh gioi
qu6c gia trung hgc ph6 th6ng mdn To5n ndm2016 (Bdi 7) da d0 cap cl6n mQt v6n AO Kra
kinh diOn trong sd hgc, d6 ld sd hodn h6o (hay cdn goi li si5 hodn chinh), mQt chir AO ft lang qu6n lai dugc chgn trong d6 thi ndm nay. DC gi6i bdi to6n niry, ngodi c6c dinh li da biet vC so
hodn h6o, thi sinh cdn phii bi€t vQn dpng tinh ch6t nhdn tinh cira hdm o(n) (hdm t6ng c6c udc s5 cin n) li c6 thiS gi6i quy6t dugc. NhAn
bdi to6n n6u trOn, chirng t6i xin gi6i thi6u v6i
b4n clgc ldm quen mQt s6 tinh ch6t ctra s6 hodn h6o cirng cilcvi dg minh hoa.
Ta ild bi6t tdn tai v6 han s6 t.u nhi6n n mi tdng
c6c udc s5lducrngl cta n (kh6ngtinhn) ldnho
hcrn n. Ching h4n c6c s6 nhu vay ld c6c sdnguy6n t6 vd lty thria cria chring. Cflng tOn tpi nguy6n t6 vd lty thria cria chring. Cflng tOn tpi
vd han c6c s5 Lu nhiOn n md t6ng tuong ring
l6n hcrn n. Vi du c5c sd c6 dpng n=Lk'3,v1i
k=2,3,... Tuy nhi0n ta chua bi6t c6 tdn tai v6 han c6c s6 t.u nhi€n z md t6ng c6c u6c s6 cira
n (kh6ng tinh n) bing n hay kh6ng. C5c si5 c6tinh ch6t ndy tlugc goi li c6c si5 hodn h6o. Cp tinh ch6t ndy tlugc goi li c6c si5 hodn h6o. Cp
th(5, 6 =I+2+3 , 28=I+2+4+7 +I4,
56 =l+2+4+7 +14+28 . T6ng c6c u6c sd cria
n dtgc ki hiQu ld o(n). Do <16 m6t sd t.u nhi6n ld hodn h6o khi vd chi Lhi o(n) =/n .
1. Mgt s6 tinh ch6t cria s6 hohn heo
Dinh li 1.1 (Euclide-Euler). 56 nguy0n ducrng
chin n ld s6 hodn h6o n6u vi chi n6u
n=2'-t(r- -r), trong d6 m ld s5 nguycn
ducrrg sao cho 2^ -l lds,5 nguy6n t6.
il)Nquu/Nq
NGUYEN CUU HUY - NGI YEN DiNH MI}IH
(GV THPT chuyAn LA QuY D6n, Dd Ndng)
Dinh lf 1.2 (Euler). Ni5u n ld s6 hodn hdo 16 thi
n phiti c6 dpng n- pa'*1m2, YOi p ld s6
nguy6n t6 dqng 4k+l vd m ld s6 nguY6nducrng kh6ng chia hi.lt cho p. ducrng kh6ng chia hi.lt cho p.
Dlnh li 1.3 (Catatdi-Fermat). N6u 2' - 1 ld sdnguy€n t6 th\ m ld sd nguy6n t6. nguy€n t6 th\ m ld sd nguy6n t6.
k
Dinh lf 1.4. NOu n=pi'pi'"'pfo :flp,l' le
i=l
ph6p phin tich nh6n tu cira n thinh c5c sdnguy€n t5, thi nguy€n t5, thi
o(r) =
-f-t(,.
p, + p? + "' + pi' ) =n+i
i=1
Dinh li 1.5. NCu n ld mQt s5 hodn heo le, thi
n = p" q?9, d9, ... qz\,, v6i o, F, .Z* ; p, et, ..., e,
lir cilc s6 nguy€n t6 16 phAn biet virp =o"=1 (moda). p =o"=1 (moda).
IIQ qun 1.6. Mqi sO hoen h6o chin cho bdicdng thirc 2r: (T - 1) , vdi p vit 2p - i ld c5c cdng thirc 2r: (T - 1) , vdi p vit 2p - i ld c5c
sd nguy6n t6.
HQ qun 1.7. NOu n li s6 hodn h6o thi
.t
t+ = 2, trong d6 d ld c6c u6c ducrng c:0;a n
-db d
khdng t<C t va chinh n.
Tt dinh li 1.1 suy ra viQc tim t6t ch cdc sti hodn h6o chBn tucrng tluong v6i viQc tim t6t c6 c6c sd
Mersenne (le cdc sO nguydn t6 c6 dang
M,=2'-l,neN,n >L). C6c dinh li vd hQ qui
tr€n c6 the chimg minh kh6 ngin gqn dua virodinh nghia s6 hodn h6o, dinh li ccr bin ctra sti dinh nghia s6 hodn h6o, dinh li ccr bin ctra sti hgc vd c6c tinh ch6t cira hdm o(n ) .
re n.u,r-ror., T?EilrHE! 3 1
Tinh d6n nay, ngudi ta <16 tim ra dugc 48 sd
hodn h6o, tdt ci chring c16u ld s6 chin vi vdn chua bi6t c6 tdn tai s5 hodn h6o 16 hay kh0ng. Ndm 1993, R. P. Brent, G. L. Cohen vd H. J. J. te Riele trong bdi b6o cria minh c16 chimg minh tlugc r6ng n6u t6n tai c6c sd hodn h6o ld n,thi n6 phii
ld n> 10300. Ducvi ddy h mQt sd vi dp khai thSc
tu c6c tinh chdtcias5 hoan hio n€u tr6n. 2. C6c vi dg
Thi dg 2.1. Chimg minh rdng n€u n ld,mQt s6
hodn hdo chdn thi s6 8n+1 ld s6 chinh
phaong.
Ldi gi,rti. Vi n ld sd hodn h6o ch6n, n6n ta c6