Duong thingAC cltBQ taiD... Tucrng.[r]
(1)xuflr eiil rU rgo+
2015 s6 457
rap cxi Rn HAruc rxAruc - NAM r-nU 52
oAruH cHo rRUNG xoc pxd rHOruc vA rnuruc uoc co s6
Tru s6: 1B7B Gi6ng Vo, Ha NOi
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Email:toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre
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N H A lt U AT- gA N sGJ hti "D U"B*:V-I ET*I(I A IVi
ffig,qr,
w& e-dsv& KeB, tur*?ei-€#UYffireff,
qO?siich "Tii liQu chuyOn Toin Trung hgc co so"
2lilcatdt c& cudn cho cdc lfu 6.7 ,8,9 m6i lcp
gb* 2 tap: tdp mOt la &ai sdi& ldp 6, iap mQi ti Sd hqc), t6p hai th Ffinh fuoc N$i dung moi cudr: ddu b6rT"l sit cl"r*arg irir:h To6n & c6c ldp THCS M5i c**n ddu c6 tiai ph5n: phSn Cdc chuyArt dd
c* b&r' {vidt theo trlrig chuang cia SCK} vi phAn Ff6t sd chz*y€.re dd n&*g cco Ilh*n dAu m6i
chu'y€n dd Ia tom t6t cdc kidn thirc !i thuy€1 c*a chxy6n dd m& HS c*n ndm vkg ti6'p thec gidi thiCu mot sd vt dir n:inh haq v& sau di la cac bdi iap dd nghi dd HS luyQn tqp itherg chuy€n dd ndng ca* r:h&r:t gi*p HS hidu shu vi r$ng hcr:: *hr?:g r:6i dl:ng c6 1i6n quan d6n ci:xsng irinh llhr*"lg d dv
vi cac h&i t6p trinh b&y trung sSch dr:o, c c*c i6c gii ch*n ioc kh* k? iu&:g, d'6 c& nhi6u vi du, bli
tip li r':hr*g ir*i thi !"r*c slnh gi*i Toin Qr:dc gia, thi Olya:pic ?cdn Bang vi€c doc, gili nhi*-rg vi dq v* bdi tQp nly, HS sE drtgc m& rOng, d-ro siu
nhl*'rg kidn thrit d6 h*c qua d* d'rr c rdn iuy€r: khi
n&r:g l*di:y To*n hoc CuOi sach sd la phAn H*o*g
d#n g;{i; cho cac bii t?p duoc dm ra & cudi mdi
ant.:uen de
Nh6nr tiic gi6 ciia bO sdch id nhtlrg thAy co gi6o dE
hc*c dang Ei&ns d+y & c6c iop chi.ry€n To6n ctia c5c
t*l*:-rg ilai hoc vd cdc tr*ifi-rE Trung hgc phd th0ng c6 *y tfn, l* nhulrg chuy€n gia To6n &Tap chi Tadn hac & TuS; tr* r,'d Nhd xudt b&n CiSo dyc ViO Nam
ching han nhu, NGND Va H** Binh {ch* bian},
NGLt" lYgu5,6r": ?cnt Srvn, nh& giSo Dam Hidu
Chid* PCS.TS" V#n lVhir Cr*dng, PCS.TS ildm Vdn ffhi, TS Ph*rn Thi Bccf, Ngqc, PCS.TS Ld Q*dc ffc{n, TS Trdn H#* lV*rn "
Hi vcng rhng bE sdch sd li mdt tdi liqu thi6't thltc va bd ich girlp c*c em l":qc sinh iiidu s*u s5c nhtlrg kidn ihrlc Tcdn dE hoc, vi: g6p phAn v&o vi€c n&ng cao chdt tirqng d*o tao vi bdi dr-rong hpc sinh gi6i ToSn d cdp Trung hoc cd s&
tsqn dgc c6 thd d$t nrua s&*tr Qi c6c C&:rg ty Sich - Thi*l bi Tnrdng hgc & e&c dla phrrong
'$i!.'1i3#"r*ix*'.r-.:x
r}::lt iri-'iir iriil lFll:l il-'N |fr I ( - ft
T(JAN }(
ra" r" S- E
H?ru:'i Fr**
.-a,,.' ,, , ,
ro4} *
liil';rl **C
rSXKT==
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:
(3),l/6n toan o chwong trinh giao dac ph6
,//A th6ng hidn hanh cila nadc ta chi di cQp
d€n cidu hiAt cac s6 chia hijt cho 2; 3; vd
ll/godi ra, trong qua trinh tim tdi, hgc h6i, chung ta cdn dffqc biil d€n cdc diiu hiQu chia
h€t khac nhu ddu hiQu chia h1t cho 4; 6; 8; 1;
TrAn m\t tii di6n ddn todn hpc cfing c6 ddng
m\t vdi k€t qud xet diiu hi€u chia hiit cho'1
.,1
Vcti niem dam m€ toan hgc, t6i dd tim tdi va xin rrinh bdy m|t diiu hiQu chia hiit cho nhd cac
bQ s6 duqc sd:p theo tha try md t\i sd dqt tAn rrong phin trinh bdy dwdi ddy
I Nh$n x6t m& tlflu
Xet cdc s6 co d4ng 10r vdi k:1,2,3 vit m6l sti niy khi chia cho co cdc k6t qui sau:
10 :7.1 + 3
l0::7.14 + 2
103 :7 .142 +
10r:7 t428+
105:7 14285 +
n6:i.142857+I
101 :'7 1428571 +
108 : 7 142857t4 + t}e:1 t42857t42+ 6
loro:7 142857t428 +
101r :7 14285714285 +
Nhu vfly, voi citc s6 tri 10' cl6n 106 chia cho thi s6 du l6n luot theo thu t.u ld: 3;2; 6; 4; 5; Di0u thri vi ld c6c sd ti6p theo tu 107d6n 1012
khi chia cho cflng l[p lAi c5c sO du theo b0 sd
(3;2;6;4; 5; l).
0iltltllfllln iltT (lt0 I
{1:- , .* : NGUYEN VAN HIEU
"'a (dv fiUCS Wnh Htrng Phu Lic, Th*a Thiln Hud)
Di6u ndy d6n <l6n c6c dp do6n ph6i chdng c6 s1r
l[p lai thir vi ndo tl6 ,rC sO duZ C6c b0 s6 ndy li€n quan gi d6n "d6u hi6u chia h6t cho 7"?
II Nhfln x6t tdng qu6t
Khi chia cac s6 cd dqng lOk vit l0k*6 lvbi k: l;
2;3; 4; ) cho thi chting c6 cilng tij do
Chang minh:
o6 thay 10r kh6ng chia hi5t cho (l0r :2k.5\.
Gi6 sir chia lOt cho dugc thuong sO ld 4 vd sO
duldr: 10k:7q +r(1) (q,reN.vd0 <r<7).
Ta cAn chimg minh khi chia 10r*6 cho cflng dugc s5 du ld r ThQt vfly, tu (1) ttit trqp voi nhpn xdt md rlAu ta c6: 10r*6 : 10r.106
: (7q + r)(7.142857 + 1) : 7X+ r (X e N.)
III Quy ur6rc c6ch ggi t6n
1) C6c bQ s6 sAp thu t.u: (1;3;2; _l; _3; -2),
(l;3;2; -1; -3; -2; l), (l; 3;2; -1; -3; -2; I;3),
(1; 3; 2; -1; -3:' -2; l; 3; 2) dusc goi li c6c
"bQ s6 tuAn hodn" tucrng img v6i c6c sd
abcdef , abcdefg , abcdefgh, abcdefghi,
2) Kh6ng -At tioh t6ng qu6t, xdt sO tu nhi6n abcdef , d6 n€u t6ng f + 3.e + 2.d * l.c
-3.b -2.a chia h6t cho thi ta n6i "c6c cht s6 cira s6 abcdef thoa mdn bQ s6 tudn hodn tvong ring"
IV D6u hiQu chia tr6t ctro Z
l) Tradc hdt u fim diiu ki€n cdn vd dti dA s6 c6
6 chir s6 dqng abcdef chia h1t cho
(4)Edt A: abcdef : a(7.14285 + 5) + b(7.1428 + 4) + c(7.142 + 6) + d (7.t4 + 2) + e(7.1+ 3) + f :7(14285a + 1428b + l42c + l4d + e) + 5a + 4b + 6c + 2d + 3e + f : 7(14285a + 1428b + l42c + l4d + e) + 7(a + b + c) - 2a - 3b * c + 2d + 3e +l Nhu vQy A chiah6t cho vd chi
khit6ng R: 2a-3b-c+2d+3e+f chia
h5t cho 7 (2)
Sip x6p lai c6c hQ s6 -2; -3; -l;2;3; I c6 m[t
trong t6ng tr6n theo b0 (1; 3;2; -l; -3; -2) v1i thri qu: hQ sd img v6i chfr sd hdng rlcm vil hC
i^, I '
sO img v6i cht sd hdng chgc e, hQ s6 img
vcri cht s5 heng t5m d,hQ sO -1 img vdi chir sti
hdng nghin c, hQ si5 -3 ung v6i cht sO hdng chpc nghin b, h0 st5 I tmg v6i cht sii neng trdm nghin a Tn Q) suy "Sd fi6 nhiAn c6
chir si| A: ,brdrf chia hdt cho vd chi
f + 3e + 2d - c - 3b -2a chia hdt cho 7" (3)
V6i quy u6c cSch goi t6n nhu trOn, hic ndy (3) di6n dat nhu sau:
"Sii tq nhiAn cd chti sd ,brff chia h1t cho
khi vd chi cdc chfr s6 ctia n6 thda mdn b6 sd
;,
tudn hodn tu(mg il,ng 0 : 3: 2: -l; -3; -2)"
Chri ]i ring, trong chimg minh tr6n c6c cht sd
a, b, c, d, e, .f c6 th6 d6ng thoi bing 0 ho[c kh6ng rl6ng thoi bing thi d6 (3) van clirng
Thi d1r Kh6ng ding phdp chia, xdt xem s5 3780 c6 chia hiit cho hay kh6ng?
Ta c6: 1.0 + 3.8 + 2.7 - 1.3 : 24 + 14 -3 : 35
chia ht5t cho 7, suy 3780 chia hi5t cho Z Thi d1t 36 120476 c6 tinh chit: 1.6 + 3.7 + 2.4 - 1.0 - 3.2 * 2.1 :27 kh6ng chia h6t cho
n6n s6 12}476kh6ng chiah6t cho7
Th{ dtt Tim gi6 ta cira chft sd x d€ s0
70x026 chia h6t cho
X6t t6ng: 1.6 + 3.2 + 2.0 - lx - 3.0 - 2.7
: -2 * x, s6 70x026 chia h6t cho 7 e-2 - x
chia hi5t cho 7 e r: 5
2) Bdy gid ta xdt diiu kiQn cin vd dil d€ cdc sd cd nhidu hcrn chtt s6 chia h€t cho Tri nhf,n x6t m& t16u ta th6y c6 sU l6p lai vO sd du chia c6c sii c6 dpng 10r cho 7, do khdng ldm m5t tintr t6ng qu6t ta xdt s5 c6 7 chir si5 c6 dpng
abcdefg Do abcdefg: a.106 + bcdefg , a6 i
c6ch phdn tich 106 nhu 0 mqc NhQn xdt md diu vd k6t qui chimg minh tr€n cho sd Ucae@ ta c6
abcdefg chia hi5t cho vd chi sd
1.g + 3f + 2e - ld - 3c - 2b + l.a chia h6t cho
7 Ta thdy xu6t hiQn bA sd ru6n hodn: (l; 3; 2; -l; -3; -2; l).
Chimg minh tuong tU nhu tr€n ta c6 bO sd tuin hodn t6ng qu6t sau: (l;3;2; -l; _3; 2; 1;3 ;2;
-l; -3; -2; )
3) D6u hiQu chia h6t cho Z
SO tg nhiAn
"r%%r""r-q chia h& cho khi
vd chi cdc chtr sd cfia n6 thda mdn "bQ si5 tuiin hodn" cd n sd tuong ftngz
(l; 3; 2;-1; -3; -2; 1; ; 2; -1; 1; *2; 1; ) Thi dlt 4 SO 10197824 co chia hi5t cho hay khdng? Ta c6:
1.4 + 3.2 + 2.8 - t.7 - 3.9 - 2.1 + 1.0 + 3.1 : 4 + 6+ l6- 7 -27 -2 + + : -7.
vfly s6 t0t97824 chia h6t cho
V Ap dgng
V6i d6u hiQu nhfln bi6t mQt st5 chia hiit cho
n6i tr6n, ta c6 th6 ki6m chimg dugc c5c k6t qui
sau:
1) Cric si5 c6 cht s6, 12 cht si5, 18 cht s5,
gi6ng thi chia h6t cho Nhu 111111; 222222222222; 999999999999999999 ;
2,s31#E:,*
(5)Hrldng ddn giii oE'rul ruvft*t srNH vAo rdp 10 cHuyErrr roAru
Bni l a) PT c6 hai nghi6m ph6n biQt
,:
-t -,lzJz -2 vd t + tlzJz -2.
b) Theo dinh li Vidte, ta c6
xt + x2 = 2, xrx, = (m -1)2 ,
n6n (4 -\)' =(q + *r)' -4*rh = - 4(m -l)' < 4, suy ra lx - xrl<z
Blri Ta co D =Yl]
x'+l
eDx2 -4x+D-3:0 (1)
+NtiuD:0thi *=+.
+ N6u D + 0 thi (1) phai c6 nghigmx, vi th6
A=16-4D(D-3)>0
e 4D2 +l2D + 16 > 0 e (2D -3)2 <25
<>-5<2D-3 <5<> -l<D<4.
)
Khi D: -1, (l) c6 nghidm kdp x - - - -2. D Khi D : 4, (1) c6 nghidm kdp x = 1= D2+
1
Vay maxD : khi * =;, minD : -l khi x: -2. Bni
a) Do MCN =90" n6n MN liL tluong kinh cira
(Q, kdo theo O' ld trung didmci.r MN
K6 ti€p tuyi5n tpi C clilra (O), cit ,4A k6o ddi tpi L
Vi (O), (C) ti6p xtrc n€n IC ctng ld ti6p tuy6n @i C cir' (Q, suy dfifu =1efu;
ICM = CBA * CNM =CBA = MN l/ AB
b) Ta c6 O'D LAB n€n O'D LMN Do d6
MD = ND hay CD liL tia phdn giSc ctn ACB
K6o ddi CD cit (O) taidiOm thf hai E Suy E ld trung diiSm cria D 1n O"g chta C), E c6 dinh Vfly CD lu6n ili qua di6mE cO einfr c) Ta c6 F,,tlD=AA (ch6n hai cung AE = BE) Xdt ducrng trdn ngoai ti€p MCD, frD chhn
AD, tia AE nLm ngodi MCD, suy AE ti€p xirc v6i <ludng trdn ngo4i ti€'p MCD tqiA
Bei Gqi d =UCLN(,a' +4, n+5).
Do n2 + 4=(r+ S)(n -5)+29
ndn d = UCLN (n + 5,29) Ta c6 p ld phdn sii chua tdi gi6n
e d >l e n+5 i 29 (vi 29 ld sd nguy€n t6)
en:29k-5(keZ).
Theo gi6 thi6t < n<2015
el< zgk-5<2ot5o 6 <k=2020
29 29
Md fr e Z n€n k : 1, 2, 3, , 69 YQy n = 29k - 5 voi k : 1, 2, 3, ,.69
CAo TRAN rtlnAr (Ninh ThuQn) Gi.it thiQu
(6)', xr rryr6ru srrur rrA.o r-0e ro
ilpr cnuytn g0A HQc r{ uNtru DHqG nA nfi ulu nsc zou-zors
VONG t (120 phrit) C0u I (3 diAm) l) Gid su a, b ld hai s6 thpc ph6n
biQt th6a mdn a2 +3a = b2 +3b =2. a) Chrmg minh ring a + b: -3 b) Chung minh rdng a3 +b3 =45.
[?, -r- ?., : 5 -,
2) Giei hQ phuong trinhJ -^ ' "' ""r
l4x2 +y2 =5ry2
Ciu II Q diAfi.1) Tim ciic sii nguyCn x, y kh6ng nh6 hcrn cho xy -l chiah6t cho (-r-1)(y-1) 2) Voi x, y ld nhirng s6 thUc th6a mdn ding thric
x2y2 +2y+1=0, tim gi6 td lon nh6t vd nhO nh6t cua bi6u thuc P = ,4
3v+l
Cffu III (3 di€m), Cho tam gi6c nhgn ABC thdng cdn c6 t6m dudng trdn nQi ti6p h di6m1 Dudng
thingU cit aC t4i D Ggi E,F ldnluqt ld c6c di6m ttOi xung cria D qua IC ,lB.
l) Chrmg minh ringEF song song v6iBC
2) GqiM,N,"r hn luqt li trung tliiSm cria c6c doqn thingDE,DF,EF.Ducrng trdn ngo4i ti6p tam gi6c
AEM cit dudng trdn ngopi ti6p tam gi6c AFN tqi P
LJndc A Chrmg minh rlng b6n di6mM,P,N,"r cirng thuQc mQt dudng tron
3) Chung minh ring ba di6m A, J ,P thing hdng Cdu IV Q die@.1) Cho b6ng vu6ng 2015 x 2015 Ki hiQu 6 (r,j) li 6 hdng tht i, cQt th? j.Tavi6t
,i
c6c sd nguy6n duong hr d6n 2015 vdo c6c cua bing theo quy tic sau:
i) 56 duqc vi6t (1,1),
ii)Ni5usokduscvi6tvdoO (r,j), v6i i >1, this6 k+ dugc vi6t vdo (,-1,j+1),
iii)N6us6kdugcvi6tvdoo (1,j) thisii ft+l dugc vi6t vdo (.i + 1,1) (Xem hinh 1)
Hinh
Khi d6, sii zOts dugc vi6t viro 6 (m,n) Hdy x6c
dithmvdn
D Gie s'iJ a, b, c ld cic sO ttrqrc ducrng th6a mdn ab + bc + ca + abc < Chimg minh ring
a2 +b2 +c2 +a+b+c>2(ab+bc+ca) VONG (150 phitt)
Cnu I (3 diim).1)Yoi a, b, c ldcSc s6 thUc th6a thilngBC.Gqi,F/ldhinhchi6uvu6ngg6ccua B tr6n mdn (3d+3b+3c)3 tloqn thing AM Trdn tia d6i cua tia AM 16y di6m
=24+(3a+b-c)3 +(3b + c - a)3 + (3c + a - b)3
Chrmg minh ring (a + 2b)(b + 2c)(c + 2a) = l. D Giei hQ phuong trinh
{2x +2y +.ry =
f271x+y)+y3 +7 =26x3 +27x2 +9x
CAu II Q diA@.1) Tim s6 t.u nhi6n n d6 n+5 vit
n + 30 tt6u ld s6 chinh phuong (s6 chinh phucmg ld sli bing binh phuong cria mQt s<i nguy6n)
2) Tim x, y nguydn th6a mdn tting thric
t+rtx+y+3 =./7*Jy.
3) Gi6 si x, y, z ld nhtng sO tfrgc l6n hon Tlm gi6
tri nh6 nh6t cria bi6u thuc
D: x - ) r z
,/y;;- Jz+ x4 ,txirz CAu III Q diA@ Cho tam gi6c nhgnABCkh6ng c6n v6i AB < AC Gpi M ld trung ditim cira tlo4n
N cho AN =2ll4H
1) Chung minh ring BN = AC
2) Gqi 0 ld di6m ttOi xrmg v6i A qua N Duong thingAC cltBQ taiD Chimg minh ring b6n ili6ni
B, D, N, C cirng thuQc mQt tludng trdn, ggi tludng trdn ndy le (O)
3) Duong trdn ngopi tii5p tam gi6c AQD clt (O) tqi C kh6c D Chrmg minh ring NG song song vli BC Ciu IV Q die@
Kf hiQu S ld tip hqp g6m 2015 di6m ph6n biQt tr6n mat phing Gi6 sri tit cd cilc tli6m cta S kh6ng cing nim tr6n m6t duong thing Chimg minh ring c6 it nhfit 2Ol5 <ludng thing ph6n biQt mi m6i ducrng
thing di qua it nh6t hai di6m cira S
NGUYfN Vfr LUCING-PHAM VaX nr)NC
(GV THPT chuyan KHTN - EHOG Hd N|i)
Gidi thiAu
I J t0
2
4
7
TORN HOC
(7)itu
Cffu a) Bpn c19c t.u gi6i
b) Ta c6 J' =3mx2 -6mx,),'=0e['= 2 Nm*ol.
lx=2'
Do y'AOi Oau khi qua x =}vdx =2n€n AO tni hdm s5 c6 2 di€mcgc tri
V6i x =0 th\ y=tm-l):, vfi x=2 thi y=-y-3.
Do vai trd ctra A vd B nhu nhau, kh6ng m5t t6ng qu6t gi6 sirA(0; 3m-3),8(2; -m-3). Tac6 2AB2 -(OAz +OB2)=98
e 9(m - l)z + + (m + 3)z - 2(4 + I6mz ) = -99
ellm2 +6m-56=0 <> m=2hodc *=-4.
11'
Cirl2 a) T :4030.
b) DAt z = a+bi (x,y elR) = 7 = a-bi.Ta c6
I i, - :l = V - z - rl e l-u - z + all =la - 2 + (b - lll
= a - _2b _l ) y = (_2b _t) + bi
nghiQm, suy PT (1) v6 nghi6m
)
Vcri x a -;, ,t (l ) tucrng duong v6i
|4 2 t8
2s-919- -:-+;;
Dat + =, (0.,
= +l,Pr (3) ffd thdnh
' x2 \ 4/
o(t-'O -+r)= zr+-!!l- lo
36(t -2) 2(t -2)(r + 4)
l+J9-4t L+r
ett-D( ' (,1+J9-4r l>-*4.)=o (4)
l+t )
v6i o .t<2 co Lr4 ,a
4 l+J9-4r
t+4 3 18 " 18 t+4
'' '-lr " <4<-: nen: >0.
1+r l+r 4 1+J9-+t [+r
Dod6 (4)e t=2.Suyra x=-;.0 Tird6 , X?\e"ri
ONIG DANI GIAI .DE SO 9
(3)
Ding thric Ta co lzl=
,Fil.+=+
xiy ra kJli b = -?.
)
Tu d6 s6 phric cAn tim ld z = -l -1t.
55
Ciu 3 PT da cho <> sin2x-J3cos2x=Zcosx
.( n) (n )
<>srnl 2y l=srnl;-J l.
[ 3' tL )
Ttr d6 suy nghiQm cira PT rl5 cho li
5n k2n
c =4+k2n (k ezt' r=18+
3 'r
CAu PT thf hai cria hQ tucrng duong voi
(2x-l)3 =(r- y)3 =!=l-x. Tlruy y =l- x vdo PT thir nh6t cira hQ clugc
25* *gJg*, -+ =' *
"8!
, (1)
x x'2 +7
)
Giai (1) DK: lxl>:.
)
Vcri x >:, PT ( l) tucrng <lucrng vdi
3
,r*'$'11 =l*-!- (z)
x x' xt+I
I r r\
IVT(2) > 251 *>i
I
Tac6] \ J) nenPT(2)vo
lvrtzi ,|*ff.2s
D{t r=,ffi ud6 ,,=-.j:J*=)ttt-ltt. vay 1 =- i(oL.rr-rr)
Ciu 6 .K6 SF/ LAC (H eAC) Do (SAC) -L (A-BC)n6n SH L(ABC)
2
0 Tinh tr=
[{x-zYax.
1
2
2
lu=x-2 fdu=dx Dat i =<
ldv=ddx lv=e"
TOAN HQC
(8)B
Ta c6 SH = SA.sin.ilZ =zoJl.)= ali.
t
rH =!es.ec.sH
YAy Vs.ABc = JS* ,
1
=l3a.4a.ali =2a, JT
.Kd HD L BC (D e BC), HK L SD (I( e SD)
Khid6 HK=d.(H;(SBC) Yi AH =SA.cos.ilD
=zoJ|.$=3a, ndnAC =4HC
= d(A;(SBC)) = 4d(H;(SBC)) = 4Hx.
mctt
ffi= UE=ro=+. Tt d6 d(A;(SBC)) = 4HK
4SH.HD
=-=
JSH,TMT
aali.3a -5
CAu Mpt ciu(S)c6 t0m1(1; -3 -Z)vi b6n kinhR = Vi d(I, (a)) = < R suy clpcm
Ggi FI vd r lin luqt ld tdm vd b6n kinh cira
tlunng trdn giao tuy5n, ta thhy H ld hinh chi6u vu6ng g6c cria I tr€n (a) Ta c6
IH =d(1,(a))=4, r=JF:|frT =2J5.
Tqa d0 di6m H ld nghiQm cua HPT [x = I + 2t; y = -3-2t; z = -2-t
1-l2*-2y-z+2=0
t ( _1._Z)
=r=-; =Hl-*:J \ : 3' 3)'
Cflu 56 hAng khdng chria x ld
clr{-t)424 :7920.
cdu lo .Dat ' la =.Ear' y J xz !" * lor'*!y' Ta c6
u >L( z* *?\*L( z" *?\ = Z( r *, *1 *1')
Js[ x) Js\" y) Js[ x Y) (Theo BDT Bunyakovsky)
4(- r ) 4( - r -)
> *l
"/.t";l +-* | = +l 4,,!x1 + 3Jry I
J5[" ,l-y) Js['" ,l*y " )
' x2 +l y'+1
Ar- x Y
'"-7-:-l) 3-r ^ l\
l*'* o)*4 l'"* o)*4
-xv4x4v '
-!
" =-LL
- -., ' ,3 4x+3' 4y+3
** + Y*4
Hcm nta: 4x * 4Y =z- l(-f-. l )
4x+3' 4y+3-' "[4x+3 ' ay+3)
a44
,2-3
+r *+y *O<2-3.m = 5 Do tl6
4 ^ ^E
-rt-;(2) Tu(l) vd(2) suy P>16-
,-I -4
Khi "r =y=; thi p=2J5-S VaV
minP = 2Ji - L\J -!'
(GV rH P r chuy an N gay 6n ffi#:;'Hl',i#
6aJ1
Ciu 7 Dulng trdn(C) c6 6m I(2; kinhR =Ji, Ae(A)= A(a:-a-l).
Eit IA=x (x>O),fu=2.athi sina =
AB JV= , , /;.
cosa = IA = * (vor -x > V5 )
1
Ta c6 S* - 8 <+
1AB.AC.sin2a =8
e AB.ACsind.cosa = 8 Suy
(x'-5)./5(x'-5) _,
-, =8=5(x:-5)3 =64xa
o 5x6 -l39xa +375x2 -625 =0
a (xz -25)(5xa -14x2 +25) =0 = x = *5 OOi ctri6u vdi DK , , J5 ta chon x :5.
Tt d6 IA-5 o (a-Z)z +(-a-3)2 =25
o az +a-6 = <) a=Zhodca=-3.
Yoi a=2 thi A(2: -3); vbi a=1 thl A(1;2).
2)vd bilrr
IB J5
-:-IA x'
9a2
5a"* _
TONN HOC
(9)rlrftro {r},,*"hl
ffi
n1r s6 pyorontq P?(AP
cnh lifu ToTN CUC TBI TBoNG
HINHH$CI{IIONO0IAN
/Whi gidi cdc bdi to6n cr,rc tri trong hinh hgc
'Ja/khdnggian, ta thucrng g6p bdi to6n tim gi6 trt lon nh6t hay nh6 nhAt cria s6 mQt g6c, clQ
ddi mQt dopn th6ng, 9ien tich cira mQt hinh phdng, th6 tich c5c kh6i tla diQn hay kh6i trdn xoay Tpp chi TH&TT dA gi6i thiQu mQt sti
phuong ph6p gini l6p to6n ndy dga vin c6ng cq t7a dQ; Trong bdi b5o niy, chirng t6i trinh bdy mQt sd phuong ph6p cht,y6u sri dlmg c6ng cq hinh hqc thudn tily, c6 th6 kdt hqp vm c6ng cqt
dai s6hay c6ng cq vecto
1 Phuung phdp sri'dung thuin tiry cril: rlinh
li hinh hqrt:
C6c tlinh lf quen thuQc thuong dugc str dpng lh: ducrng vu6ng g6c ngln hcrn moi dudng xi6n, m6i cpnh tam gi6c nho hcrn t6ng hai c4nh cdn
lai vd lon hon hi6u cria chfng, dunng vu6ng g6c chung cta hai ducrng thdng ch6o li cloan.thing ngin nh6t c5c tto4n thing c6 hai ddu mrit ndm tr6n hai <ludng thdng dy, OThi dU l; Cho hinh lQp phmng ArcD.A:B'CD
vd mQt di€m M chuy€n dQng ffAn cqnh AA' Mqtphdng (nttO') cdt CC' teti N Timvitri di€m M ai aien tich thi& diQn BMD'N dqt gia tri nhd nhdt
Ldi sidi (h.t)
A' D'
I{inh I
Vi (ABB'A')\|(DCC'D') n€n BMIID'N Tucrng
tv, MD'llBN Vay th gi6c BMD'N ld hinh
binhhdnh Ke MH IBD' thi
sr*o,* =2.Srro, =2.!.ao'.rua = BD'.MH
lt eu6c UAN
(GV Khoa Todn - EH Wnh, NghQ.An)
Ydy Sr*o,* dat gi6 tri nh6 nh6t vd chi
MH dqt gi6trinh6 nh6t, nghia 1d vd chi MH ld tluong lu6ng g6c chung cia AA' vit
BD' (hay M ld trung ditim cira AA'vd H lir
trung diiim cua BD')
Chfi ! O ddy, di6m M phii nim tr€n doqn thiing
M' vit N phii nim tr6n doqn thdng CC'.Liti giii
tr6n th6a mdn ci hai didu kiQn 6y Tuy nhi6n kh6ng ph6i bao gio ta ctng gdp may mdn dy Trong mQt s6
bdi to6n, cbc chdn duong vu6ng g6c chung ctra hai ito4n thing lpi nim tr6n c6c tlopn tbing 6y kdo ddi
Trong mQt s6 bdi to6n l&6c, cdc di6m chuy6n dQng
lai phdi th6a m6n th6m.mQt sO aiiu kiQn b6 sung, n6n do4n thing ng6n nh6t chua hEn ld ttudng vu6ng
g6c chung Khi d6 ta phdi sri dpng k! thuft kh5c Thi dp sau ddy minh hga cho nhpn x6t d6
OThi dq Cho hinh lQp phuong ABCD.A'B'C'D'
vd hai di€m M,N ldn luqt chuydn dQng ffAn
cdc dudng chdo A'B vd AC sao cho A'M = AN Tim vi tr{ ctia M,N dA d0 ddi doqn thdng MN dqt gid tr! nhd nhiit
Loi gi,rti (h.2)
B
Hinh
CuOi tOi gihi, ta sE th6y ring khi ilQ ddi doan thFng MN dqt gi6 tri nh6 nha\ MN kh6ng phdi ld tlucrng ludng g6c chung c:0;a A'B vir
AC (tr ttAy, cli6u kiQn rdng buQc ld A'M = AN )
Kf hiQu canh hinh lf,p phucrng dd cho ld a, A, M = AN = x thi O<x<aJ2,MB=NC =ffi -x. Tr€n AB lAydi€m H saocho MH ll AA' rti
AH A,M AN
- = :
- = - n6n theo tlinh lli Thales tl6o
HB MB NC
HN IIBC
o
A
(10)Tt d6 tam gilc MHN rudng lai H Vi cac
tam gi6c AHN vd BHM vu6ng c0n t4i FI n6n
HN =Ay- =+, HM =ry=o'12 * vay
Jz Jz 'lz J2
MIP =MIP+NLP =
,)
tr
+lalz-x)-=f -al2x+d. MN dqt gi6 tri nh6 nh6t e x - ft e M,N
\IL
tucrng img ld trung di6m cia A'B,AC Khi d6
MN=BM=BN={ n6n BMN liL tam gi6c d6u,
l.l \L
do cl6 g6c gifta MN vd A'B b54g 60" Tucrng
tr, g6c giita MN vd AC cflng bdng 60"
2 Phuong phip thi6t Bp hQ thric xic tllnh Trong nhi6u bni to6n, d6 tim clrc tri ctia d0 ddi mQt doan thing, diQn tich cira mQt hinh phing hav th6 tich ciri mot kh6i tla diQn, tru6c hi5t ta pnai tnet lqp mQt sO t C thric li€n quan d6n c6c
ti6n chria c5c bi6u thric bi€u di6n cdc dqi lugng n6i tr6n Sau cl6 dtng c6c.BDT quen thuQc (Cauchy, Bunyakovsky ) dC tim cuc tri
gin cdc bii5u thric trOn
OThi dq Cho hai dudng thiing Lx,By cheo
nhau vd vu6ng gdc vdi nhau cd AB : a ld dadng vu6ng gdc chung Hai di6m M, N lin luqt chuydn d6ng tAn Ax, By cho MN = b
(voi b ld d0 ddi cho truoc) Tim gid tri ldn nhdt cua thd tich ttr didn ABMN
Ldi gidi
v
Hinh
DAt AM = u, BN = v Vi BN LAB, BN L AM theo gi6 thi6t, nen BN L(ABM),
Do tl6 BN LBM, n€nV* =Vn.o*=l.S*rm,'
= !.!.tu.en.BN = !.auv th.3\
32
Theo ilinh lli Pythagore c6
BM2 = AB2 + AM2 = MNz -BNz suy AM2 + BN2 = MN2 - AB2 = b2 - a2 Tri d6 ttz +v2 - b2 -az (kh6ng e5l1 ncn uv ltrnnhdt
t;t:F
(nghia ld VA^MN lcrn nhdt) eu-r=l';
a(u -a')
theo BDT Cauchy Vpy max Vor*r: )
lZ ,
E -^2
dattlugc eAM=BN= l;
OThf dq Cho ta di€n ABCD vd mQt di€m M chuydn d6ng /r diQn Cac dadng thdng
AM, BM,CM, DM cdt cac mdt (BCD), (ADC),
(ABD), (ABC) tqi A"B"C"D' taong rhng a) Xdc dinh vi tr{ di€m M dA bidu thftc
p=M nBM nCM,*DM, dat sia tri nho nhtit
' MA, ' W, MC, MD, -': o'-'
b) Xdc dinh vi tr{ di€m M dA bi6u thac
fAM fBM reM DM - ,
' = I *o' * l rg * lw * tl ua aqr sta tr! nhd nhiit
Ldi sirti b.a)
A
C
Hinh
ot-tj, MA _vr.rro MB' _vu.aco
" t ' " Au|t - vA
BCD' BB' vu.orr' MC' _vM.ABp .MD' _v*.ou, u;
cc' vc.ABp' DD' Vr.*,
V*.uro *Vr.oro *V*.*o *V*.*, =V*r* suy
MA'_MB'_MC'_MD' _l
AA,, BB, , CC, DD,
Theo BDT Cauchv , !' * BB' , * 9L ?4
MA , MB, , MC, ' MD, ( MA, MB, MC, MD,\
-r-r-i-r- |
-[aa' ' BB' cc' ' DD')
( AA' BB' cc' DD')
-lw.ffit+fu+ffi)'-'u
NtI BM CM DM
l'udo
MA,+ MB,* uc,* ul
_ AA' + BB' +CC' + DD' _4> 12
MA, MB, MC' MD'
Do d6 minP: 12, dqt clugc vd chi
TO6N HOC