Tạp chí Toán học và Tuổi trẻ Số 457 (Tháng 72015) gồm khoảng 28 bài viết trong các chuyên mục: Dành cho Trung học cơ sở, diễn đàn dạy học toán, diễn đàn phương pháp giải toán, đề ra kì này, giải bài kì trước, sai lầm ở đâu, Toán học và đời sống, tìm hiểu sâu hơn Toán học sơ cấp. Mời các bạn cùng đón đọc.
Trang 1xuflr eiil rU rgo+
2015
s6 457
rap cxi Rn HAruc rxAruc - NAM r-nU 52oAruH cHo rRUNG xoc pxd rHOruc vA rnuruc uoc co s6
Tru s6: 1B7B Gi6ng Vo, Ha NOi
DT Bi6n tap: (04) 35121607; DT - Fax Ph5t hdnh, Tri su: (04) 35121606 Email:toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre
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i
Trang 2#r_, ,n t +G ''.fi* ,*e ,€f+ '* 7
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& 8 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$i dung trong moi cudr:
ddu b6rT"l sit cl"r*arg irir:h To6n & c6c ldp THCS
M5i c**n ddu c6 tiai ph5n: phSn Cdc chuyArt dd
c* b&r' {vidt theo trlrig chuang cia SCK} vi phAn
Ff6t sd chz*y€.re dd n&*g cco Ilh*n dAu trong m6i
chu'y€n dd Ia tom t6t cdc kidn thirc !i thuy€1 c*a
chxy6n dd m& HS c*n ndm vkg ti6'p thec gidi
thiCu mot sd vt dir n:inh haq v& sau di la cac bdi iap
dd nghi dd HS luyQn tqp itherg chuy€n dd ndng
ca* r:h&r:t gi*p HS hidu shu vi r$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, trong 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$t nrua s&*tr Qi c6c C&:rg ty Sich - Thi*l bi Tnrdng hgc & e&c dla phrrong
Trang 3,l/6n toan o chwong trinh giao dac ph6
,//A th6ng hidn hanh cila nadc ta chi di cQp
d€n cidu hiAt cac s6 chia hijt cho 2; 3; 5 vd 9.
ll/godi ra, trong qua trinh tim tdi, hgc h6i,
chung ta cdn dffqc biil d€n cdc diiu hiQu chia
h€t khac nhu ddu hiQu chia h1t cho 4; 6; 8; 1 1;
TrAn m\t tii di6n ddn todn hpc cfing c6 ddng
m\t vdi k€t qud khi xet diiu hi€u chia hiit cho'1
.,1
Vcti niem dam m€ toan hgc, t6i dd tim tdi va xin
rrinh bdy m|t diiu hiQu chia hiit cho 7 nhd cac
bQ s6 duqc sd:p theo tha try md t\i sd dqt tAn
rrong phin trinh bdy dwdi ddy.
I Nh$n x6t m& tlflu
Xet cdc s6 co d4ng 10r vdi k:1,2,3 vit
m6l sti niy khi chia cho 7 co cdc k6t qui sau:
Nhu vfly, voi citc s6 tri 10' cl6n 106 khi chia cho
7 thi s6 du l6n luot theo thu t.u ld: 3;2; 6; 4; 5;
1 Di0u thri vi ld c6c sd ti6p theo tu 107d6n 1012
khi chia cho 7 cflng l[p lAi c5c sO du theo b0 sd
(3;2;6;4; 5; l).
0iltltllfllln iltT (lt0 I
"'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 ndyli€n quan gi d6n "d6u hi6u chia h6t cho 7"?
II Nhfln x6t tdng qu6tKhi chia cac s6 cd dqng lOk vit l0k*6 lvbi k: l; 2;3; 4; ) cho 7 thi chting c6 cilng tij do.
Chang minh:
o6 thay 10r kh6ng chia hi5t cho 7 (l0r :2k.5\.
Gi6 sir chia lOt cho 7 dugc thuong sO ld 4 vd sO
Ta cAn chimg minh khi chia 10r*6 cho 7 cflngdugc s5 du ld r ThQt vfly, tu (1) ttit trqp voinhpn xdt md rlAu ta c6: 10r*6 : 10r.106
3.b -2.a chia h6t cho 7 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 7.
t, nrr,r-roru, T?3I#?E
I
Trang 4Edt 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 7 khi 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 1 img v6i chfr sd hdng rlcm vil hC
sO 3 img v6i cht sd hdng chgc e, hQ s6 2 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 ra "Sd fi6 nhiAn c6 6
chir si| A: ,brdrf chia hdt cho 7 khi vd chi khi
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 6 chti sd ,brff chia h1t cho 7
khi vd chi khi 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 0 thi khi d6 (3) van clirng.
Thi d1r 1 Kh6ng ding phdp chia, xdt xem s5
3780 c6 chia hiit cho 7 hay kh6ng?
Ta c6: 1.0 + 3.8 + 2.7 - 1.3 : 24 + 14 -3 : 35
chia ht5t cho 7, suy ra 3780 chia hi5t cho Z.
Thi d1t 2 36 120476 c6 tinh chit: 1.6 + 3.7 +
2) Bdy gid ta xdt diiu kiQn cin vd dil d€ cdc sd
cd nhidu hcrn 6 chtt s6 chia h€t cho 7 Tri nhf,nx6t m& t16u ta th6y c6 sU l6p lai vO sd du khichia c6c sii c6 dpng 10r cho 7, do do khdng ldmm5t tintr t6ng qu6t ta xdt s5 c6 7 chir si5 c6 dpngabcdefg 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 c6abcdefg chia hi5t cho 7 khi vd chi khi 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 7 khi
vd chi khi cdc chtr sd cfia n6 thda mdn "bQ si5
tuiin hodn" cd n sd tuong ftngz
Trang 5Hrldng ddn giii oE'rul ruvft*t srNH vAo rdp 10 cHuyErrr roAru
Bni l a) PT c6 hai nghi6m ph6n biQt
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 nhau n€n IC ctng ld ti6ptuy6n @i C cir' (Q, suy ra 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 ra E
ld trung diiSm cria D 1n O"g chta C), E c6dinh 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 ra AE ti€pxirc v6i <ludng trdn ngo4i ti€'p MCD tqiA
Bei 4 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 siichua tdi gi6n
Trang 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
Ciu II Q diAfi.1) Tim ciic sii nguyCn x, y kh6ng
nh6 hcrn 2 sao 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 6 vu6ng 2015 x 2015.
Ki hiQu 6 (r,j) li 6 0 hdng tht i, cQt th? j.Tavi6t
,i.
c6c sd nguy6n duong hr 1 d6n 2015 vdo c6c 6 cua
bing theo quy tic sau:
i) 56 1 duqc vi6t veo 6 (1,1),
ii)Ni5usokduscvi6tvdoO (r,j), v6i i >1, this6k+ 1 dugc vi6t vdo 6 (,-1,j+1),
iii)N6us6kdugcvi6tvdoo (1,j) thisii ft+l dugc vi6t vdo 6 (.i + 1,1) (Xem hinh 1)
Hinh 1
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 < 4 Chimg minh ring
a2 +b2 +c2 +a+b+c>2(ab+bc+ca)
VONG 2 (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 = 5
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 2 Tlm gi6
tri nh6 nh6t cria bi6u thuc
,/y;;- Jz+ x4 ,txirz
CAu III Q diA@ Cho tam gi6c nhgnABCkh6ng
c6n v6i AB < AC Gpi M ld trung ditim cira tlo4n
N sao cho AN =2ll4H.
1) Chung minh ring BN = AC.
2) Gqi 0 ld di6m ttOi xrmg v6i A qua N DuongthingAC 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 cingnim tr6n m6t duong thing Chimg minh ring c6 it
nhfit 2Ol5 <ludng thing ph6n biQt mi m6i ducrngthing di qua it nh6t hai di6m cira S.
NGUYfN Vfr LUCING-PHAM VaX nr)NC
(GV THPT chuyan KHTN - EHOG Hd N|i)
Trang 7V6i 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
Tu d6 s6 phric cAn tim ld z = -l 55 -1t.
Ciu 3 PT da cho <> sin2x-J3cos2x=Zcosx
Ciu 6 .K6 SF/ LAC (H eAC)
Do (SAC) -L (A-BC)n6n SH L(ABC)
2
0 Tinh tr=
ldv=ddx lv=e"
Trang 8' -!
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 =
OOi ctri6u vdi DK , , J5 ta chon x :5.
Tt d6 IA-5 o (a-Z)z +(-a-3)2 =25
Trang 9/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 do 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 nhau li
cloan.thing ngin nh6t trong 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)
I{inh I
Vi (ABB'A')\|(DCC'D') n€n BMIID'N Tucrng
binhhdnh Ke MH IBD' thi
sr*o,* =2.Srro, =2.!.ao'.rua = BD'.MH
2
lt eu6c UAN(GV Khoa Todn - EH Wnh, NghQ.An)
Ydy Sr*o,* dat gi6 tri nh6 nh6t khi vd chi khi
MH dqt gi6trinh6 nh6t, nghia 1d khi vd chi khi
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 dQnglai 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 2 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 choA'M = AN Tim vi tr{ ctia M,N dA d0 ddi doqnthdng MN dqt gid tr! nhd nhiit
Loi gi,rti (h.2)
B
Hinh 2CuOi tOi gihi, ta sE th6y ring khi ilQ ddi doanthFng 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
Trang 10Tt 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
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 trong 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 3 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 sao cho MN = b
(voi b ld d0 ddi cho truoc) Tim gid tri ldn
nhdt cua thd tich ttr didn ABMN .
Ldi gidi
vHinh 3
Theo ilinh lli Pythagore c6
BM2 = AB2 + AM2 = MNz -BNz suy ra
AM2 + BN2 = MN2 - AB2 = b2 - a2 Tri d6
OThf dq 4 Cho ta di€n ABCD vd mQt di€m M
chuydn d6ng trong /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 thacfAM fBM reM DM - ,
' = I *o' * l rg * lw * tl ua aqr statr! nhd nhiit
Ldi sirti b.a)
Do d6 minP: 12, dqt clugc khi vd chi khi
B -tuagw
Trang 11MA, MB, MC, MD, I
-=-=-=-==;
, hay M ld trOng
AA,BB,CC,DD,4,-._J
tdm cira tu diQn ABCD
b) Eat Vupco = a2,Vutco = b2 ,Vr*o = c2,
trgng t6m cria tu diQn ABCD (Bpn <19c hdy
chimg minh chi ti6t nhan x6t ndy)
Chrt i) M6t s6 bdi to6n cgc tri hinh hqc g6p trong
c6c k!,thi hoc sinh gi6i hay V6 dich to6n, nhi6u khi
phai t.u xdy drmg MOt li thuyAt mdt (so v6i hiriu bit5t
th6ng thudng) m6i gi6i tlugc chirng.
OThf dq 5 Cho t* di€n ABCD Tim cdc di€m
X,Y,Z,T thuQc cdc mfi phdng (BCD), (CDA),
(DAB),(ABC) sao cho tdng sau ddy dqt gia tri
nhd nhdt W2 +Y,2 +fl2 +YZ2 +Wz +2T2
Ldi gidi D€ gi6i bii to6n ndy, cAn dua vdo c6c
kh6i niem vd k6t qui sau
Dinh nghTa Cho tu diQn ABCD, M ld trung
di6m canh CD Mra/-phdng (MAB) tluqc ggi ld
m\t trung diQn xuiltphdt tu cqnh AB cria tu diQn.
Mgt tu di6n c6 s6u m{t trung di6n D6 th6y s6u
m{t trung diQn cua tu diQn tl6ng quy tpi trgng
tAm cira tu diQn.
Dinh nghia Cho tu di€n ABCD Gqi (a),(0)
tucrng tmg ld m{t phdn gi6c vd m6t trung di6n
xu6t ph6t tu c4nh AB ci.r- tu diQn Col (7) h
6nh cira (f) q""phdp d6i xtmg mflt (a) rcri
att (y) dugc goi ld mqt diii trung xu6t ph6t tucqnh AB cta tu diQn.
MOt trr di6n c6 s6u mdt dOi trung Ngudi ta d5chimg minh clugc ring s6u mpt d6i trung cia tudiQn tl6ng quy tpi m6t clii5m Di6m ndy tlugc ggi
ld di€m diii trgng tdm crtaff diQn.
Dinhl!, Chotu di€n ABCD vd M ldm6tdi€mnim trong tu diqn Gqi H,I,J,K tucrng tmg li
hinh chii5u ci:r M *u6ng c5c m{t phing (BCD),(CDA), (DAB),(ABC) Khi d6 c6c didu kiQn sau tucrng duong:
i) M lddiem dOi trgng tdm cua tu didn ABCD;
j.:\ Vrrro
-Vroro -Vr*o -V**, .
Lt' sj s; s3 s;
ii1) M ld trqng tAm cira tu diQn HIJK .
O day ,SA ,,SB ,,Sc ,,SD ld ki hiQu dien tich cbc mf;t
(rco),(coe),(oen),(eac) cira tu diQn ABC D.(Bpn tlqc h6y chtmg minh k6t qui ndy)
Bdy gid ta trd lqi Thi dw 5 Gqi G ld trgng tdm
Chf i ring, vi G ld trgng t6m ctra tu diQn
WZI ndn Wz +Yz +Yz +YZz +W2 +ZI2
=4(GX2 +GYz +GZz *GTr) Tt d6
W +Y +W +YZ +YP +7r2 ru > 'si +q 36VfuD+s:
+sB 'Dlng thric xtty ra e G nim trong tu di€nABCDsao cho 6,ffi,U,,Gf tucrng img vudng g6cv6i c6c mflt phing (nco), (cot), (odn),(eac)
Trang 12sao cho GX,GY,GZ,GI tuong
tdm G cila t* di€n, cdt SA,SB,SC tuong ung
tqi D,E,F- Tim gid tri l6n nhdt cfia bi6u ththc
111
D= ' ' sD.sE ' sE.sF ' sF.sD'
OThi dq 7 Cho ttb di€n SABC vdi SA = a,
SB=b,SC=c M1t m1t phdng (a) thay d6i
lu6n di qua trqng tdm G cfia ta di€n, cdtSA,SB,SC tuong ung tqi D,E,F.Tim gid tr!
nh6 nhrit cila bi€u thtrc p =#.#.#
Ldi gi,rti &.6) LQp lufln tuong t.u thi du 6 nhfn
tluschetr*c
$+S*#=o hay 9a!a9=4
trong tl6 x = SD,y = SE,z=,SF, 0 1 x I a,
0<y< b,01z4c (1) Theo BDT Bunyakovsky,rc=( \ x y z) r.! + 0.1*r.1)' <( a2 +u*., t[1*1*1) '\r y' r)
img w6ng g6c v6i c6c mat phing (nCO), (Con),
e GX,GY,GZ,GT tuong img vudng g6c vdi
cbc mdt phrng (nco),(coe),(oea),(eac)
vir G ld di6m tl5i trqng tdm cta tu diQn
ABCD >X,Y,Z,T tucvng img ld hinh chitiu
vudng g6c cria di6m d6i trgng t6m cria tu diQn
ABCD l6n c5c mpt phing (nCo), (Coe),
(oea), (enc) (*) vfy t6ng
Wz +Y} +XI2 +YZ2 +YT? +ZT2 nh6 nh6,t
khi vd chi khi X,Y,Z,T th6a mdn (*) vd gi|trl
nh6 nh6t d6 bing 36VkcD
sj +sf +s] +Sfr
3 Phu'ong phip vecto
MQt s6 bdi todn cgc tri hinh hqc <tugc gi6i ggn
hcrn ni5u ta bi6t sir dung c6ng cg vecto thich
hqp Ngodi nhfrng ki6n thirc quen thuQc tl6 hqc
o bOc THPT, bpn dsc cin bi6t th6m kh6i niQm
vit circ tinh ch6t cira trgng t6m mQt h0 di6m,
c6ng.thric Lagrange - Jacobi, titmfi,c1r cria mQt
hQ tli0m, Dinh li "Con nhim" cho kh6i tu diQn
Dlnh ngh1a Gi6 sit A1,A2, ,A* ld mQt h0 m
-,: ! i.
cli6m sap x€p tuy j trong khOng gian khdng
phdn biQt thir t.u Di6m G dugc ggi ld trqng
tdm c'bahQ di6m tr6n n6u iO+ =O .
OC ttr5y trong tdm cria mQt he di6m lu6n t6n tpi
vi duy nhat Hon nta n6u G ld trgng tdm cta
h9 diCm A1,4, ,A^ thi v6i mgi dii5m M
1 trong khong gian, c6 MG =:- mf, )',tWA,.t
m-Dinh lit (Cdng thtc Lagrange - Jacobi ) Gi6 su
G ld trgng t0m cria hQ di6m Ar,,4r, ,A* vd
M ldmQtdi€m tuy f trong khdng gian Tht5 thi
tml
MG2 =i-\ ull r= t AA2
m? ' m2.-?, t r
r= I t:t <j:m
OThi dg 6 Cho tn diAn SABC vdi SA-SB=,ff:l.
MQt mfit phdng (a) thay d6i tuAn di qua trong
ro T?[l#B! '.*
Trang 131 Cho hinh lap phucrng ABCD.A'B'C'D' Hai
di€m M,N chuytin il6ng trOn AB vit DD' sao
cho AM =DN.
a) Chrmg minh c6c tlucrng thingMN ludn song
song v6i mQt m{t phing cO Ointr.
b) X6c dinh vi tri ctra M,N dC cloan thing
MN co dO ddi nh6 nhAt.
2 Cho tu diQn ABCD vd m6t tli6m M
chuy6n dQng trong +? diQn C6c.tlucrng thdng
AM,BM,CM,DM cdt circ mdt d6i di€n cria tu
diQn Dn lugt tai A',B',C',D' Tim gi5 tri nh6
nhAt cria biiSu thfc
( ue\' ( ue.t' ( ruc\' ( uo\'
lm ) .[r".,1 [.".J [rr.J
3 Cho tu diQn ABCD vd mdt di6m M chuy6n
d6ng trong mdt (BCD) X6c dinh vi tri M dC
t6ng c6c binh phucrng kho6ng cilchti M d6n
ba mdt cdn lai dat gi6tanh6 nh6t.
4 Cho hai ducrng thing ,Lr,By ch6o nhau c6
AB = a ld ducrng r.udng g6c chung Hai di€m
M,N chuy6n ilQng tr€n Ax,By tucrng img sao
c) X6c dinh vi tri M,N AC Aign tich todn phdn
cria tu di€n ABMN dqt gi|tri nh6 nh6t.
5 Trong khdng gian cho ducrng th[ng
a '- =- Va mat pnang
t2-2
(a) :2x -2y + z *3 = O c1t nhau t4i A Tr€n d
6y mQt di0m B cO einn sao cho AB = a Xet
6* Cho tu diQn d€u ABCD Tim c5c cli6m
X,Y,Z,T thuQc c5c m[t phdng (nCO),(COe),
(OeA),(enC) sao cho t6ng dO ddi cdc cpnh
cira tu diQn WZI dqt gi|tri nh6 nh6t.
7* Cho tu diQn ABCD sao cho c6c cpnhAB,BC,CA tl€u nh6 hon c6c canh DA,DB,
DC Tlm gi6 talcyn nh6t vd nhd nhdt cua PD,
trong tl6 di6m P th6a mdn di6u kien
PDz =PAz +PB2 +PC2
8* Cho tu dien ABCD vudng <y D Goi
a,f,/ tucrng img ld g6c gifta <ludng cao
DH voi c6c cpnh DA,DB,DC Tim gi6 tri nh6
nh6t cta bi6u thric p=x|! *Y!z +z+x
trong <16 x = cosd,y = cosp,z = cosy .
9* Xdt cilc ttt diQn ArArA.,4, cirng ngoai titipmQt m{t cAu cho tru6c M6i ti6p diQn oiua mgtcdu, song song v6i mOt mlt cria tu diQn ndy, cit
ra kh6i tu diqn d6 mOt tu diqn nh6 Goi
v,(i=1,2,34) ta tn6 tfch cira kh6i tu diQn nh6
c6 clinh A, vd V ld the tich cta tu dien
Ar44Ao.Tim gi6 tq nh6 nh6t cirayt+v1 +y1+vt
-tr vA x6c dinh nhirng tu diQnAr,4rArA, nhu thi5.
10* Cho tu diQn ABCD L6y mQt di6m Qnim trong tu dien Ggi Q' ld di6m cl6i ximg v6iqua m{t phing (nco) eua e' dr,rng m6t
phing song song v6i m{t phdng (nCO), m4t
phing niry clt c6c cludng thing AB,AC,ADtucrng img tqi At,Bl,Ct t4o thdnh hinh ch6p
cytt BCDBTCTD, Bing cilchlly diOm dOi ximg
v6i qua mpt mat cdn lpi cria tu diQn r6i dlmg
cic mf;t ph[ng song song tuong t.u nhu tr6n, tadugc ba hinh ch6p cpt nta
Hdy xilc dinh vi tri cli6m dC t6ng c6c th6 tichcria b6n ch6p cut tao thdnh dqt gi|tri nh6 nh6t.
ts nur,r-roru, T?EilrH?[ I I
Trang 14ffifiG lffi BIT D0llG lllUG", i.,
ll
grgiynirMt,lnilPrlfililrrl,l,tinrn il
OQei toan chimg minh b6t dlng thric (BDT)
UO co su d6nh gi|thay O6i so mfl cira c6c lf,y
thria thudng lim cho hgc sinh lung tung Bdi
vir5t giOi thiQu cho bpn <19c phucrng ph6p su
dung BDT Cauchy AC AiC" chinh sO mfl ctra c6c
lfry thira trong chimg minh BET ho{c tim gi|tri
lcrn nhat (GTLN) vit gi|tri nh6 nfr6t (CfXtN)
r PHUoNG psAp
Oe Aanf, gi6 tu lffy thria xo sang lfry thua xp
(v6i x ld sd thlrc ducrng), ta thudng ldm nhu sau:
X5c dinh <li6m roi cria BDT;
Dua vdo tliiSm rcri, dua vdo phAn tu php fuhAn
tu phu thuong ld hing s5 ho[c tdng qu6t h bi6u
thtc c6 dqng xt ), dua vdo BDT Cauchy chfng
ta thi6t hp d6nh gi6 sau:
mx" + nxt > (m + n).^*{l x^" .x'1
mq+nf
,A
(voi m, n la cdc s6 nguy6n duong) Tu (*) dC
c6 tl6nh gi5 tu x" sang xB ta cin chon c6c s5
nguy6nduong mvdnsaocho *o*nY
=9
m+n
rr MQr so rHi Dv
Thi dU l Gia sL x 1,, z, t s ld cac s6 thq'c
du'o'ng lha.v dot lhoa man:
Phdn tich Dd su dpng dugc gii thi5t, ta phii
thi6t l4p d6nh gi5 fii x7 sang x6, nghia ld gi6m si5
mff, ta chgn ph0n tu phu ld hing s5, dlra viro
cli6m roi x = ! = z =t =s =1, n6n ta chon phAn
ffi d6 h 1, khi d6 vdi m, n ld c6c sii nguy€n
ducrng thi theo BDT Cauchy ta c6:
Tucrng t.u, ta c6: 6y7 +lr-7 yu , 627 +l> 7 z6
6{ +l>7t6,6s1+1>7s6.
CQng c6c BDT cing chidu o tr6n, ta dugc:
6(x7 + y' + r' +t7 + s7)+5
>7(xu + yu + zu +tu + su1.
MAt kh6c, theo gi6 thi6t thi
x'+y'+z'+t7 +s7 <5
rz T?[l#E! ,,*
Trang 15Tir hai BDT trdn ta suy ra P <5 DA th6y khi
vi P=5 Vay maxP=5
Thi du 2 Cho ba s6 thu'c dttong thay dl)i a, b, c'
thoa mdn ab+bc+ca:3abc Tim GTLN cl)ct
sang 4, dlra vdo di6m roi a=b=c=|, ta cdn
dua vdo phAn tu phg ld 1 *f,i tl6 v6i m, n lit cftc
Thi dU 3 Cho cdc s6 thac daong a, b, c.
Chu'ng minh ring;
Trang 16+tr+oi( ' 'r[b+c a+c | * I * I a+b )-322-:=1 ) 2 2
Tri hai BDT 0 tr6n suy ra:
( a \j ( rt \j (c ); 3
I I
'l I
rl-l\.-lo+r) 'lo+") 'lo+b) -2J,
BDT da cho dugc chimg minh
Ding thric xiry rakhi vd chi khi
Ta cAn dbnh gi6 hi J; rrg Vi, nghia ld ta
Ldi gidi Theo'BDT Cauchy, ta c6:
Ji * Ji+ 1 > 3Vi > 2J;+ 1 > 3Vi.
Ding thric xiry rakhi vd chi khi
-=-OA=b=C a+c 2
, _l
Thi dII 4 Cho x, y, z ld cdc sd thac daong
Chu'ng minh ring;
Trang 17Thi du 5 Cho n ,;6 thtt'c rrr, -r., -r thcr.v d6i.
l: t: |,thoa miin 2) +2 2 + +22 ) n v6'i n ld s6
nguy1n du'o'ng cho trtrhc, va lu6n thda mdn diitr
kiin n>2 Tittt Cf.ry,ry t'trct hi<;tr tltirt':
x6t tuan Tu bdi virit chirng ta rit ra mot kinh
nghiCm cho viQc hoc to6n tl6 ld thucmg xuy6n
tim c6ch eC Ui6n AOi v6n dA, bi6n d6i c6c tinh
hu6ng oia cdc bdi to6n Tir d6 sE g6p phAn cho
chring ta giiti quy6t duoc nhi6u bdi to6n kh6 tu
cilcbiti toSn co b6n EC thdnh thao v6i phucrng
ph6p tr6n c6c em th? bht tay vdo gi6i c6c bdi
2 Gie s'iu a, b, c ld cdc si5 thgc duong th6a mdn
at + b5 + ct =729 TIrGTNN cta bi6u thric:
ai + b1 > 2 Chimg minh ring aE + bi > 2.
4 Cho c6c s6 thlrc ducrng a, b, c th6a mdn
6 Cho c6c s6 thuc duong x, !, Z thay tl6i th6a
man r'ooo + y'ooo + z'ooo >3 Tim GTNN cirabitiu thr?c: A= y2ots * y2ots + z2ots
\-lzo+t") 'lzc+ta) 'lza+3b) -s6'
8 Vdi a, b, c, d, eld c6c si5 thqrc ducrng thayaOi tim GTNN cria biiSu thfc sau:
Trang 18BitiT2t457 (L{rp 7) Tim c6c s6 nguy6n a, b, c
dc c6 la * ul+lo - cl+lc - al= 20t4" + 2015'
NGUYEN EUc rAN gr ua Chi Minh)
nguy6n vn fil) : 2 Chtmg minh ring ft7)
khdng the h sd chinh phu<rng.
NGUYENDINH SON (GV THPT Trdn Cao Vdn, Tam K), Qudng Nam)
Bili T41457 Cho tam gi6c nhgn ABC vbi cdc
tlucrng cao AH,BK Ggi M ld trwg di€m ctn AB
Dudng thing CM clt UX tai D Kd AL vudng
g6c v6i BD tai L Chimg minh ring dudng trdn
tli qua ba di6m C, K, L ti6p xric v6i rlucrng thing
BC,
I-P VIPT AN gt *o Thi€n - Hu€)
Bii T5/457 Giei he phucrng trinh:
for'+ y2 +J5-6y =6
LIJONG CAO VINH
(G[/ THPT COng Hiin, Wnh Bdo, Hdi Phdng)
CAC rOp THPT Bili T61457 Chofir) ld tta thfc bOc ba v6i hQ
s6 cao nh6t ta 2 vd th6a mdn:
f (2014) =20t5, f (2015) =20t6.
T itlh f (20t6) - f (2013) .
rArra quoc roAN
(GV THPT chuy€n Nguydn Thi Minh Khai, S6c Trdng)
BitiT7l457 Kf hiQu Se, V theo thr? t.u ld diQn
tich todn phin vd thti tich cira tu diQn ABCD
(r \3
Chrmg minh ring I \o ls,, - / I >Jir2.
LUC BiNH
(GV Truong Tnmg Vuong, D6ng Hd, Qudng Tri)
Bii T8/457 Cho da gi6c 16i r canh (n>4) 44.4.Chimgminhrang
n + sin A, + sin A, + + sin A,
Ding thilc xhy rakhi nio?
NGITYEN VIET HLING (GV THPT chuyAn KHTN, DHQG Hd NAi)
BitiT9l457 Tim t6t ci cbc b0 ba (x; y; p) g}m
hai s6 nguy6n ducrng x,y vdsd nguydn t6 p th6amdn p'-lP =1
NGU^1EN TUAN NGSC (GV THPT chuyAn Tiin Giang, TP Mj Tho, Tiin Giang)
Bni T10/457 Cho ft ld sd thuc l6n hon 1 Xdt
n dAu canbdchai
Chimg minh{x,} ld day hQi tq vd tim limx, .
n-)@
PHAM HUNG (Hd NAi)
Bni T11/457.Ybi m6i sO n nguy6n ducrng, <t{t
dd=z&.
dl,
1) Chrmg minh ring y\;n) ld him nhdn tinh,nghia li: rftab)=tfta)ly($ nil (a,b)=1
2) Cho / ld sd nguy6n ducrng 16 Chimg minh
nguy6n duong n.
NGTTYEN DIJ THAI
(Lop Todn K25, DHKH Hue)
rCT?!T#M