J

N\ffi

Bo crAo DUC vA DAo

|l

TAP

---lG

rAo * Hgr roAN Hec vrEr NAM

ffi

4poz1 1994

cui ne NcAy rs uANc rsANc

0 ffier mFo trH6 D€ @E#r efrl Eo6N 5 f,GH

Ar,eb. o Dinh li lon Fermat at

aA

r\

va vi6c tim toi cf fn!. minh so cflp cua n0 t,

6 ffittH ffi$fl ffi{}a A'trt

$$[

ffiUHffi

'^P\t

?A #,&flI[ $[AI

'$$Y ,/ $$ mail ?ff.{H

f,l,tt $& I

uHai effis$ki Thd,y uit. trb trudng.PTTH Quynh

Luu 3,NghA An udi Tap chi Todn hgc uit. Tudi trd -

#Ht$#ffiffi, ffffitn*r*

mUsT

ia

s Ddm den

ffi# fru'- iffif! uumr'eimrorug,*,*'u

TOAI{ HQC VA TUbI TRE MATHEMATICS AND YOUTH

MUC LUC Trang

o

-

ctfrc bgn trung hgc co st For Lower Secondary School Leuel Friends Vu Hilu Binh : MQt mgo nh6 dd giAi biri to6n v6 diQn tich d l6P 8

Ddnh cho

Tdng biAn fiP

NGUYEN CANH TOAN

Phqm Brio ; Hlnh hqc hda nQi dung vd cdch giAi mOt s6 biri to6n dai sd o Gidi bdi l$ trrtic Solutions of problems in preuious issue - c6c bai tt T1/198 d6n T10/198, L1/198, L2ll98 o Db ra hil ruit Problems in this issue - Cilcbai tt Tll2O2 ddn T10/202,LU202,L21202 10

-

o 6ng kinh cdi ctfrch dqY vd hPc totin

KaLidascope : Refonn of Marhs Teaching and' I*arning Nguydn Drtc Td,n; V6 mQt tidt day to6n 6

-

lOi ctrgn, l6p

chuY6n

11

thi vdo dqi hoc (Jniuersity Entrance Exam Preparert For College and, sin-h vio tuydl thi Dd Nguyfin Vd'n Mfuu: (Kh6i A) L2 1993 nam He NOi hgp Tdng Oiiiq"

o

-

Ddnh cho cdc bun chudn bi

Bgn c6 biit Do you know ? Nguydn Cd.nh Tod.n: Dinh lY Fermat vI viec tlm tbi chrlng minh so cdp ctra A"d" Ngq. S"".' Hai con s6 k}, la

o -

-

- Nguy€n Dung;

Chuong

nti

15

BIa 3

trinh rrit gon tinh

s6liQt Fibonacci

Bia 4

o Gidi tri todn hpc

-

:

Phd tdng bidn tdP : NGO DAT TLI HOANG CHUNG

Hor ooNG BIEN TAP

:

Nguy6n CAnh Todn, Hodng Chung, Ng6 Dat Tr1, LO Kh6c

BAo, Nguy6n HUY

Doan,

Nguy6n Vi6t Hei, Dinh Quang HAo, Nguy6n XuAn HuY, Phan Huy KhAi, Vri Thanh Khi6t, Lo Hei Khoi, Ng.ty6+ Ven Mau, Hodngl6 Minh, NguY6n Kh6c

Minh, Trdn Van

Nhung,

Nguy6n DEng Phdt, Phan Thanh Quang, Ta H6ng QuAng, D4ng tr{tng Thing, Vr1

Drrong Thuy, Trdn T[enh

Trai, L6 Bri Khdnh Trinh, Ng6 ViQt Trung, Dang Quan Vi6n.

Fun with Mathematics Binh Phuong; GiAi dap bei Con s6 73 ki la Nguydn Dinh Titng: Ddm ddn

Trq. sd tba soqn :

81 Trdn Hung Dqo, HA NQi 231 Nguy6n Ven Cfi. TP Hd Chi

DT: 260786

Minh DT:

356111

'

tri stl : \16 XfU T$UY rrinn bdy: DOAN nbNc

Bian @p uit,

a x6t bii to6n diQn tich sau thu6c chrrong trinh ldp 8 : E Cho tam gi6c ABC. Qua di€m. D thuQc cqnh BC kA cdr dudng thd.ng song song udi cdc cqnh cfia tant gid.c tao thiuth hai tam gid.c nh6 c6 diQn tich 4cm2 uir.9cm2. Tinh d.iQn tich tam gid,c AtsC.

cD

(L

sd 0

V6i hoc sinh l6p 9, giai bbi todn tr6n kh6ng khd khen l6m : BD DC

TacdrBC+BC=1

D+t dt (ABC) = S. Cric tam giric EBD ddng dang v6i AABC n6n :

E U

,BDr2 4',DCtz /-:-\-='-\--\ac)-s'\.Bcl

!trr

IF

z

E (U1. EI A r!

Fq

rl--

q9

D

c[r

Gi

k € E

-L ts{ d h

B.

€

e) r$

a o €$

sid ,B

a

9

\/

Tt

c

(2) (3)

15

1. Do d
cta (1)

:

=,

BD.DC

_-

BC2 -t

1

(no DC \2 s6 BD.DC rt(2),irc.BCl =sr* u", =

trI trI

(4), (5)

(4)

(5)

ls6 : U *2.8= l. Do ddS = ZS (cm2)

sd di phai 'di drrdng vbng" nhrr trdn vi 6 ldp 8 chua ccj khdi ni6m vc {s. pd th6o 96 vu6ng m6c niy chi cdn m6t meo nh6 : dat s = d.2. Nhtr vdy ldi giei d6! vdi l6p 8 crlng rdt ng6n gon :

0 qT n

4 ,DC,2 I BD Z DC J ar' \ac) =-;nan BC=e' BC=A' BDDC23 Ta lai c6 : U o AC= r* i*i = 1 + d = 5. Vay 'r'aco.,BDr2 (BC) =

cE

S = d.2 = 25 (cmz). Tdng qurit ctia bdi todn tron : ndu di6n tich cic tam gi6c nh6 la a2 va b2 thi di6n tlch tam gi6c ABC bing (a + b)2. Bqn hay dirng phuonc ph6p tr6n dd giai bai to6n sau v6i ki6n thiic l6p g : Cho tam gii,c ABC vi mQt didm O nim trong tam gidc. eua O v6 c6c drrdng thing song song v6i c6c canh cta tam'gidc, chia nd ra ba tam gidc nh6 vd ba hinh binh hdnh. Tlnh di6n tlch tam gld,c ABC bidt cti6n tich ba tam girlc nh6 bing : a) 4cn* , gcmz , l1cmz ;

il cUo

E Dr

0 fr

FDC

V4yS =25(cm2) Hoc sinh ldp B chrra hoc cdn bdc hai ndn dd giei bai to6n tr6n thudng ph&i

S'S'-'

o

Ee

,*23*

4 9

G

&

(1) vn (3)

._-L._t9

il

e4

Tt

= {-s 'BC

BC

vb.

-s

ti*l'. (#)'nr '*#

G

b a

-BD2DC3 =

suY ra

bidn ddi nhu sau : Binh phrrong hai vd

il

s$

(1)

b) a2,, b2 ,

li

c2.

Ddp s6 : a) B1cm2 b) (o + b * c)2,

T=rong hinh h'oc cdc tinh ctrdt dfnh luong cria E mQt hinh thrJdng duoc thd hiQn bang cic h€ thrfc. Ch&ng h4n, nhtr cr{c edng thtlc dd tinh cgnh vA diQn tlch cria mQt tam gi6c : a2 = b2 + cz - 4bc cosA,

1_

S= tlp(p- a) {P;-bcsinA=

&)

(p*c),

v.v...

N6u str dgng phrrong phap &a d0 dd nghiGn crlu thi cric khrii niQm nhu didrn, dudng thing, dudng trbn, m6t ph&ng, mat cdu.., ttdu duqc thd hiQn bBng phuong trlnh. Trong ir6 tqa d6 Db ca.c ctia kh6ng gian h,ai chi6u mQt didm drrgc bidu di6n bing mQt cap sd sip th* tu

(r, y), dttdng th&ng li rn6t t$p hqp didm bidu di6n bdi mQt phrrong trinh d4ng ax * by * c = 0, dttbng

n*irum

suuv * sa,rmlr

111 ytt-r)"i"6V+ 1^ <

Trong kh6rig gian 3 chi6u, rn6t didm dr:qc bidu di6n bing rn6t b$ 3 sS sip th13 tU (x, y, z). iludng th&ng l6L {(r, y, e)} mh

x'-r, !*lo &bc

z-2,

r*4t phingl8r ift,.y, z)i rnh.A.r+By* Cz*D = {) troec rn?icllu ld {(r, y, e)} rnir (x - to)2 + (.y - lo)z 4 (z - z,rlz = R:'

Do dti, v6i rnQt sd bii torin dai sfi, t'try theo cr{c dt ki€n ndu ta "li6n tddng" dd nh&n ra ciic

r}{c didm hinh hqc cria nd, cd thd ta sE tim dr.tgc nhirng c6ch giAi ng6n gqn, dQc drio. Sau day le m6t sS vi dg. Vi Arp I z Chzt O 4 x, !, z < 1. Chung mirthbdt d&ngttulcx(1* y)a'y?* z)+z(1-x) < I (2) Gidri: Mdi i;ich d vd trrii crla (2) cd thd xem [i tich hai c+nh cria rnQt tam giric vb vl sg nhrr nhau n6n ta hey l6y mQt tam gi6c c6nh

b&ng J. TrBn c6c cqnh AB, BC vh CA ldn lr-tqlt ldy cdc didm M, N vb P sao choAM = r, EI{

=e,CP=y.Tacdbdt

1

il G- a)sinGff

cfiar.vtud€tcd: -'-:--

r/-+"oFr"u"V + sin:(.r 1) i-> zr + (Dd ,;hi tuydn sinh vio dai hqc. sd 45) Gi6i : Cdc c6n thrlc trdn gdi cho ta ngti ngay d6n c6ng thdc dO dai cria mdt dc4nth&ng' Troirg met ph&ng tga d0 Oxy ldy cic didm M (Zcosx cosy, sin(r - / )) vn

\iitnffir+-i"]-LF[

nmn ruor muNG vA cAcH GIAI Hoc "^r{i-r----d mor s6 mne r0AN sal sO

trbn lh rnQt tflp hop didm bi6u di6n bdi mQt phrrong trinh dane- {x - a}2 + (y - bF = R2...

d&ng thtlc diQn

-t{t|-*)sin60e+

g . lrsin6U' VAy.r(l -J) + z(1 - il +Y (J- - zt < l Vi AB 2 z Chftng minh ritng udi mai gid tri

PH.+.M

d6u ABC crj

+ s&vcP
tich

;

EAo

N (2sinr siny, sin(r - Y)) ; !gl,!

:

oM=d4coffi

*si#(r -y).

0N:

rj4sinTisin? Dgng hinh

Y

hlnh

binh

ONPM

--

(xem 2sin{r!)

hinh 2) Ta

---z'

,aii,,i

-r/ z

cci

OM+ON =

-/,/l

'

an(e-y)

OM+PM>-AP

Cfing d6 thdy P cd toa dd li {2 (cosr cosy +

2cosa

+sirusiny); %in

(rl)l;

cosY 2si,zfrstn)/

Hinh 2

do d
* sinz(r -Y) + > rl4*"oF(r -.yi

+' qsLF-irr

* -v) > 2

Vi dq 3 ; Ctw 3 sd x, y, z > 0 th&:. t'^zcn iw th;Zc :

xyz(x*Y4z) = I

{3)

Tint gid tri nh.6 nhdt cila (x + y) {x + z). Giii : HQ thrlc (3) la tich cira 4 s6 gQi cho ta "Iigu tU0ng--ddn-qQUg-t]rlc H6-r6ng 5 = ff1p - a)@ - b)(p--c.;. Ndu datx : P - d, y _: p - b, z =p - c tlti x + Y +7 = P v&

Sz=xJzk+y+z)=i

Tr6n hinh 3 cdc s6 x, !, z ia d0 dai c'1c ti6p tuy6n k6 trl dinh ctra tam giric ddn drldng trbn ndi ti6p. Cqnh A-B

*,

*Y, AC = x *

z.

a_

-2ABAC

' S"U dri hinh.:.. pfurong'Z v6 c6c d&ng thrlc trOn '. lrr .r' ', ..: tr"O ,, . I*z - .,2 "i + x2 + u2 = u2 *y2 = I vd til l*": lttL =.yz

n€nAB.AC=

thay u

" LABC 1

=-rABACsinA 1

xu'+1,u:0n6-ur I y.h.o4c

= (x*y)x

x(x*d>

>2s*ur=z I "li

F Hinh

&

3

[(r +y) (x * z)7 = 2 Vi du 4. Tint gia tri nhd nhdt crtu F = r[zF -2, *, +rF+@ + jl, +l + VAy min

+w

Gi6i : Tuong tg nhu vi du 2, ta vidt lai

f:

x=utaed xy*uu=0. ?rl cric vi

'

du

tr6n ta nh&n thAy

e*c bai tofn

d6

thitc chdt la nhfrng bhi todn

hinh hec nhung duoc ph6t bidu b&ng "ng6n ngu dai s6", ndn khi giei phai cd nhin

Ilinh 5 ra cdi "hdn hinh hoc" crla nei dd tt dd vin dung cdc kidn thrlc hinh hgc rni tim ra nhtng loi giAi ng6n ggn "tudng minh".

Trong met phing toa d6 Oxy ldy cdc di6'rn

M(x, y =

x),

AQ, 1), 8(--f , *f,1,

Sau dAy 1.

c(*+ ,**r,

rn6t sd dd todn dd qic ban luy6n

Tim giri tri nh6 nhdt cria hdm

sd

y=Gz-r+1+frurffi;+1

tacd: F = MA+MB+MC Cong d6 thdy tam gidcABC ddu vi F nh6 nhdt khi M tring vdi g6c O

2. Cho

tri

gi6

a.

Dd.PsdY=f; * h * c = 2 vd. ax *by * cz * 6 tim

nh6 nhdt cria

P = \fi6{ 4;ry + {{Sr a5zy + {{6V177

Dd.ps6:P:1A

vAminF'=3khi

r = 0 (xem hinh 4).

li

t+p:

3. Cho 3 s6 r, y, z thba m6n Ili.ruh A

l*2

lf

Yi a,l

5, Chrtng ninh rdrug ndu x2+ y2 = u3 *.uz =.7 (5r) udyy *yr-t = O (5a) thi x2 +u2 =

y'+u'=7uitx.y*uu=0.

Gi*.i : Tryng rnet ph&ng toa dQ Ory phrrang trinh .r2 * y2 = I li phrrong trinh dudng trdn cci tAm O(0, O) bdn kinh bing l. Nhung ddng nhdt nrat ph&ng tqa d6 Oxy vdimat phi.ng toa d6 Ouu thl dudng trbn dd erlng li clrrdng trln cri phuong trinh ld uz + u2 = J. Tr6n dudng trdn drilfly hai d$m M(x, y), Ntu, u) vd x6t hai vectd OM(x,g), QN(u,u). Tir (52) clia ta xu * yu .+€

= OM. ON = 0 n€n OM t

ON. Thdnh thrl, ndu M thuQc cung vu6ng I ihi N hoac nim trong cung vu6ng 11, hoEc nirn trong cung vu6ng /V, Nhrrng dir trong cung vu6ng nho ta cring cr5 lrl = lul va n6u r, u ctng ddu thi

w,ytrdiddu lul = lyl ho6cr,yctngdduthi y, u trrii ddu.

+*y *7,2 -

hQ

o2

+w *22:bZ

lz2 +'zx + x2 =

c2

tirrt, ,y *.yz *

zx

Dd,p sd,1rS, 4, GiA str

f

(r, y) =

S*r,

l* *yl {T +x-z tn +iz

C:hdng minh r&ug v6i mqi a, b, c La cd

(a,c) < P (o, b) +f(b, c). 5. Chrlng minh r&ng vdi moi a, b, c, x, y, z

f

ta cci

:

at *

by

*

cz *,17,7+ Or+

2

a\ti4 rra-4 ,

-g(o+6+c).xk*y-rz) Hudrug ddn :

Ding tich vd

hrrdng.

1t (---.Suvra: ht *t hk 111

hl'

-!-r--L

znzr'

bhzr

1111 Bei T1/X98. Chtng minh

rd,ng c6 uO sd sd

chia hdt cho Ldt*' mi, trong bidu d.i6n thQp phan crta cdc sd dd hhdng c6 cd.c chu s6 A, L, 2, 3 Ldi gifli z CiaVu Tdt Thd.ng (9CT, Nghia T&n, Tr) Li6m, He NOi vd.L€Vfi, Long,9T, Lam Son, Thanh Hcia.

.

Goi o € N md trong bidu di6n th$p phAn 7,2,3. RO rdng

etia nri khdng c6 edc chtr sd 0, cr!

_36 s6 sd o nhrr vAy" X6t dey

s6

a.aa..-.4,...4. #

- 1991

11

111
hl

NtrQn x6t : Chi c6 2 bqn girli sai th$t d6ng ti6c. C6c ban c6 ldi giAi 16r la , Db Hb Nga (9D, D6ng Da, He Nqi), Dodn Trung Tdng (9, Nghia TAn, HitNQi), Vu Thw Hudng (9, PTNK Hei Hug), Mai Thanh Blnh (7M, Mari Quyri, FIA NQi), Ngaydn Quay.g Hd (7A, Am Thuqng, Thanh Hda, Vinh Phn). Trinh Hodi Nam'(9't Nguy6n Binh Khi€m, Vinh I.ong), Trbn Ngty€n Ngpc (9K, Le LQi, Ha D6ng. HA Tay). i.;a D*c Thdnh, Phqm Huy Tilng (9A-8A. Bd VEn DAn. t{b NQi)...

*ls6a

tt'

=1*

" "' J- {2o - t)hl

Dem chia 7f'nn'+ I sd kh6c nhau ndy cho Lgtt*t. Theo nguy€n t6c Dirichl6 sE crf {t nhdt 2 sd khi chia l9s'"'cho cilng sd du. GiA srl dd "ho

Khong hie-r sao rdt

it bpn grri ld giAi bai nay r-E

rsdxc

?

NH-Ar

Ti/198 :^ Cho A ABC c6 ^ BAi A= 5W,B=7@,C=6F n|i tidp trong

duitrug trdn td.nr A bd.n kinh R.'Cd.c tia BO, CO cbt AC v&. AB ld.n luot tai D ud. E. Cqi H ld.

tr$c

td.we.

h,

ABC. Chrtng minh L, DEH ld

tant gidc dbu.

[,ei giAi : D6 ding chfng minh drroc c6c g1ec QDC vd-lQC caa-tai C^ ("i CDO = COD = 8P , CE\= CpE:^?*).Ve

trN'*u

hav d.1. a-0 .1

.T): Jd'''"

ffi";;*s)'

,*',

Nhung (10,

19)

=

suy ra d a: \,# m-n

hm

dudng cao C.F'. Suy ra Ct=Cz:Cz=2€. TrOn Ctr'ldy H vi tr€n CB ldy K th6a mdn

1#"n'

CK=CH=CO=CD:R,

J suyra ( lOl ,79stw 1 *

1

Jd'*'. vay tdn t?i v6 sd sd

chia h6t cho 1{'*'md trongbidu di6n thQp ph6n cria chring kh6ng c
Dodn Tnrng Tirng,gCl, Nghia Tdn,

Ftr6a vA

Ti

Liem,

He NOi de gi6i bdi toan thay .ci 195's3 bing s6 tU nhien bdt ki M me M, 10) = .1. C5c ban Pham L€ Hing,9A, Tnlng Nhi, He NQi ; Nguydn Thi Thanh 9Ar, H6ng Blng, hAi Phdng cflng c6 ldi giei tdt.

I

+. BaiT2/198. D@hn = l+ J+u . .+ zn_1. Chtng minh rd.ng : udi Vn e Z+ ta c6 :

11L1 --: *-: *-- +... +---------- ^ <2 (2n - 1) . h; hi 3hr2 ihi

Ini

ga,ili

Vtrong He

_-: 11 (2h - 1)h7

z

(c{nN guydn Anh T\i 9H PTCS Trung k e tr vA. h 2 2 ta a :

NO.i). V6i

tzh

-

r)(hk_r +

1

2p_) .h*

EKD=lSd-8e-6P=4U. Suv ra

KED = /n

704.

Me

I{EC=KCE:4Oo. ^

Suyra 6ie = 3oo Mat khric do If

vi D ddi xfng

Td NGUYEN

11

Ta ed L CDK ddu n6n DK = E (1) Do CF xtlng ctlta EB vi OK n6n A BEO = = L EBK (cgc) Suv ra ? 4 ^ EK = BO = "R (2) vd EKB = EOB =8f . (2) (1) ve suy ra LEKD edn tai K vd c6 Tt l& trr,rc ddi

nhau qua CO n6n

vi = ]! HEC- DEC=Sff.

E9-_

S"yruUPAd6u.

(3) I

lai c6 Ta --^ ^ = 3P. ^HBC = HEC (ddi xring). Suy ra HBC Md C = 600 n6n BH L AC, VQy Il li truc t6m crla & ABC (4) Tt (3) vn (4) suy ra dpcm. F{h$n x& : Cdc ban c6 ldi'giii tdt , Ngay'6n Dfic Trhn Dlnh

lri,

Drtng

Phong ChAu, Vinh Phri ; Bili Ngoc Anh Nam

Dinh, Phpm D*ng Hodn, Thanh Li6rg Nam HA ; Hodng Thanh Hdi, PT nlng khi6t, Trinh Tht thrg, H6i Duong, Hii Htng ; D(ng Thyc Trinh, NguyEn Bd lting NguyEn Anh Cudng Nguy|n bic Nguy€n Anh Til pTCS Tnrng -Phuong, Vudng, Phgn Nguytn Thu Trang Trbr Haong eulnh Ti Li€m, HA NQi ; te Vidt Hdi, Lam Son Thanh H6a, l-t Thi Nggc Hoa, DHSP Vinh ; Phan Thl Thanh'Thny, pT nlng khi6r Ha Tinh; Trbn Thanh Quang, Qudc hqc Hu€; Ptrgi DiQu Hdng QuCic hgc Quy Nhon ; Hodng Vi€t Ngtt, Hodng Xudn Anh Ddo, Bi€n Hda, Ddng Nai.

vO KIM THOY

(9A, Bd VIn Dirn, Hir NOi), l,e ngec Gidp (7A. D6ng S
Gi6 sfi m

* i chiachop duj v6i i, j e X

thim+i=p.h+iv6ifteiV ' "a Do dci : (m p-2

* i)b - jP =

(pk +j)P

-jp

='2 c, @kY-' jt + p . (pk) jp-t + jp -jp .

oj-

Ldi giai

(cria da sd cdc b4n). tg\ ) tri € (- ul2, nl2) : i : 7, ..., 13. Kh6ngmdttdngqurit, gi6stro, < a2< ...
x, * n. Do4n [r,, xi + nl duoc chid'thAnh

NduO
A

:

*0<

rn/)t(m+lP + . . . + {nt+p-lf = B

+ (p *Ly (modpz;. VI p t1t nhi6n 16 n6n : a = [1P. + (p - rY!+pr + 0o - 2Y] +

=

lP +

.[GY.(#y) Nhungv6ijE jp + (p -./)p = =

?ri{rco

{1

,2r...,

]

1

-

tgxi_r

* tgxi. tgxi_t < tgn/tz

a,-a,, fr/tz < , *'', '-' < tg 7 ai ai_, -(

thi

(modp21

Dodd: B : A (modp2; +A = 0 (modpz) Nh$n x6t : c6 1lB bpn gti ldi gi6i v€ vi ddu gi6i ddng. MQt Mi hqn chfing minh bing phtlong ph6p quy ,Ap. rue, bgn c6 stl dung dinh ii Phec-ma nh6 holc kh6ng dd y "a. rdi (?k)p-'u6i 0 < r < p- 2chia hdt chop2mhxoayquanh C! di! suy ra C! chia hdt chop thi cic b6n dAnh si dun&giA rhi6r

p nguy€n td.

C6c ban c6 nh{n x6t vi ldi giii tdt td : phqm L€ Mintt (9T, Lam Sdn, Thanh H6a), Chu Nguy€n Atnft (12A, DIISp Vinh) Irln Quang Thdnh (10T, Amsterdam. Hi Ndi), ?nftli Hodng (10Tr) Nguyen Binh Khi€m , Vinh Long), I,ii {inh D*c Sctn (10CT, Lrrdng V6n T!ry, Ninh Binlr), phsm liodng viCt (llA, Qudc hqc Quy nhdn, Binh Dinh),Ng6 Dttc Thdnh

(

h-h

< tg (x, + n - xn) < tgnlll <+0 < tg(x, -xrt) < tgnll-Z ar - als

tirt

pn-{,+z).i,.(-1y +yz.ir-t = o

n)-r,r-r

(1)

O

c=+o<

p-1

tgxi

N6u chi cd 0 < (x, +

?P . .

...

) thr

o < tg (xi- xi-r) < tg wl12

e+o

:

18

db4n bdi cric di6m i7, r.1, ,.., xr1. VAy t6n tai doan cii dQ dei < nf 3

t=o

Suy ra

o*

D4t o, =

jt +pzk.ir-t o 6

(modp2)

!

o' 1+"j%'1fr-_trT ,*E

D-2

.

,,dudnd'

lE rH6Nc rurrdr BAi T5.148. Cho 13 s6 thtc ar, ..., d,,khdl nhau d\i mQt. Chrtng minh rdng tbn td'i a,utt. ap,1 4i, k < 13 sao cho

t=o

:p2 . Z C, g-
d d€ ra thanh chfi

Hoan ngh€nh ph6t hiQn cria cac ban.

BAi T4l198 : Cha p ld, m1t s6 neuydn t6 ti. Chfing minh rd.ng tdig cd.c tay tntia \qc p cil@ p sd nguyen liAn fidp chia hdt cho p2. - Lati gi_6i : Ta chl cdn giA thi6t p ti s6 ttr nhi6n 16. X6t p sd nguy6n-li6n ti6'p-nr., m * i, ..! m + p - I khi chia chop duocp sd du kh6c nhau n6n tgp c6c s6 du ln

x={0;1;2;...;p-tl.

\d'

(:2)

1+"1"t3tgnl12 r--...:

M4t khdc ,r#.=

EG-|, . {

(1) ; (2) ve (3) cho dpcm.

ffi,r,

Nh$n x6t. 1) Kdt lu{n cria bai ro6n mAnh hdn (theo c6ch . ch0ng minh tr€n).

^ aj-au t8-Jt iT a',.k<

u<

2) MQt sd ban d chuy€n to6n Trdn Phri (Hai phdng) cdn cho c6ch gi6i kh6c : chia dudng rrdnddnvi thinh 13 cip (26 g6c), sau d6 c{ng sil dr,rng nguydn li_Diricle.

NGLIYEN VAN MAU

B}ri T6/198 z Gid.i phuong trinh : 12(,{k +IY+ 1 +,[F +@UW> = t[1ee3 Ldi gi6i : Tr6n m6t phing v6i hQ tga dQ Decac vu6ng gdc ldy cric didm A (-1, *1) ; B(O, 31

U)

vdx6t didm X (x, 0) nim tr6n trpc hoinh,

Hidn nhi€n cti

;

XA + XB >- AB V X e tryc hoi,nh (1) Yi tung dQ cta A, B triti ddu n6n A, B nim kh6c phia ddi v6i truc holnh ; vd do d
(i) dudj:l+ng:_ \(;+-TrT-f . r' * (#)'

ry

" Vr €.R

2n +"*lrt Vr€-B

Di6u ndy chfng t6 phuong trinh da cho cti 31

= -A.

Nh$n x6t: l. Tda soan d5

nhQn duclc 163

cfc b4n hgc sinh d kh,p mgi mi6n ddt

ntrctc

lrli giii

cria

gili ve. Chn y6r,

cdc bpn de giei bei to6n theo mot trong ba cSch sau

:

a) St dUng phdp binh phrlc,ng hai vd cira phudng trinh

fidn otii ptrttong trinh da cho ve mot phudDg trinh b{c 2 ddi v6i.r. (CAch giii niy tuy khong ph0c t4p, nhung g{p phii Od

cdc tinh to6n c6ng k6nh !)

c) Srl dune bdr dins tnilc Mincdpxki crira

ya.b,c.d e

minh, d5

R

s:r] c.iung mQt

trtng cdc khing dlnh sai sau : i) f(x) : g(x) * f(x)z = g(x)z. (Chti i, : Ndi chung, ph|p blnh phuong hoi vd c*a plwong trtnh ld ph|p bidn ddi hQ qud)

ii) voi

a, 6, c, d

€

lChti

!:

+ {b +,t)r

+

-

ad hc. + g1f+F = voi q b, c, d e R thi +a? + cc dbng ttrbi ad = bc vi ac + bd > 0). Cf(b

{G +.f+

{;l-;;r

iii)17 I + b'-, =fr +fieavibcirngphu':,ng(c+ngtuydn). (Chti!: t-rj + tEj =fr +fl <+a vlb oing huting (nghia lir cung phrlcllrg va cnng chi&r)).

3. C5c ban co lrli giii tdt t Npy6n Thl Thanh {cdp li Hdng Blng, HAi Phdng) ; D6 nb Nga (PTCS D6ng Da, IJA NQi) ; Nga,udn Quang Nghia, Fhqm L€ Hdng (P1'CS 'I'nlng Nhi, Ha NQi) ; Irln MirthAnh, Nguv4n AnhTil (PTCS Trung Vudng He N$i) : N6d Dtic Thdnh, Nguvin Phrt Bln& (PTCS 86 Ven Din, Hi Noi : ?Dn l,'dn l-ong (9PT'NK Thanh l-i€m, Nam HA) ; Phqrn'fudn Anfi (9T Phan BQi Chau, Ngh( An) ; Npytn trlbng Ttun, Khdng Dic Thien {P-[TH l{irng Vrrong, Vinh Phri) ; NguyEn Mbth Hdi (PTNK rlAi Hdng) ; Le Cbt,-q Scra (PTTH ViQt Ditc, HA NOi) ; Trinh Dltng Gicng (PTDL Marie Curie, Ha NOi) ; Ngtyen Trong HAu, Ngb Thdi tluo1g Dinh Thdnh T-rung (P'lC't DHI'FI Hi NQi) ; La Tudn Anh (P"ITH Le I4i, I{n 'rfry) ; La Tb Y€n Phil (PT'I'tl Sdn Tay, He Tey) ; Dinh Huy Alnft (PTlIl Hoang r*'trn "i'hu. Hda Binh) ; l'hern Thy l{dng (P1'TI-| Chuyen, Thdi Binh) ; Irt'n& Thd Huynh (PTTH t-e *{dng I'hring, Narn Dinh) : l}inh Thtt Tknv, Brti Hodng Cadng. l,''u lhic S'r-lrr (l'i"Ill l-rldng V[n TiJy, Ninh Rinh) ; L€ !'td Nfrit'tlz. i{adng'{tti 7i.vdr {PT'fll

6

=

=

,.

2n +'{) fl "o"21*

cos2x4col @

rsirw)"

.1,3,3 cosl.r +

,corx

+

,sin:x

+

,

3

gk) = ,Y x. Ndu fir) lir him tudn hoin chu ki {J tni "a tIQy

r [i=! i

tt t /rd lir hdm li6n tgc vb th6a mdn di6u kiQn d+t ra. chen

f(x)

Nh{n x6t. C6c ban gili ldi giii d'dn dfu cho lcli giSi dring. Ndr han chd trong l6p hem tudn hoAn chu ki z thi co ihe dung dudc l6p rdt ci c5c him s6 thtra men phudng trinh dA cho. Ndu chi h4n ch6 trong l6p hnm tudn hoAn chu ki z/3 thi nghiQm tren lA duy nhAt.

R rhi

,t7;P ,r[,riT = {@;3

. fi. + catlx +

*

*;) l cot @ *+) . .,, , + (rcow.l \fgsil:r)"+ gk) = cos:x 2 1 {5 . ., +(-rcosrxdt g(x)

v{y ndu

:

,tFiT +{7T7,,{G+,}*6;8 nhifu bqn, trong ldi giii

coix

tr6i bing tV$)13,

b) Sri dung phudng ph6p hinh hqc.

2. Rdt

:

(4)

31

duy nhdt nghi6m x

BAi T7l198. Hdi c6 tbn tqi hay hhdng mQt f : R * R sao cho : o2n,fi afQ) f{x +;\f@ - rnx)f - tf(x *;)tt -

hitnv sd liAn tuc

-tf(x

Khi dd, theo (2) vi (3), ddu "=" d (4) x6y I.a €=aI

NGUYEN K}{AC MINH

+'rl

(3)

Vi6t lai

{

Le LOi, PTTI I t-am S
NGUYEN VAN MAU

BAi T8/198 ; Cho sd nguy€n duorug n > 1. Giri sft e6 n sd nguy€n rt, rz, ...., {n Thay tdt

cd cdr sd niLy tuon! l1*g bdi cdr sd xn.l+xn fin*xl

xt +x,2 x) +x3

2 '

2 ""'

2

'

2

v6i

lai litm nhu thd, u.u... Hdy tint dibu ki€n cd.n ud dil d6i u6i x1, x2t ..',xn d.d trong tod.n bQ Qud. tri.nh thrlc hiQn uiQc thay sd ffen tu lu\n chi nhgti, duqc cd.c sd nguyan. Lbi giai : Y6i m4i k e N* ki hi€u rfk), rft<), . . .r,!*) li c6c sd nhAn drlgc sau Idn cdc sd m,6i nhQn duoc

thay s6 thd A. Dat

min

:

., rl,k)], ft e N'

lrltl,.rrrr<1, ,, "= [ M,.n - iltax I,r!^),,,r(k),..,rtk)|, "] & € NI rrx,.

..

)

Vdi m6i A € N* goi so !i sd tdt eA eric sd cci gi6 tri bang itlu *,ia?ay |1tl . 'qtl . , ,,\k)' Tir tinh chdt cir* r.rr;ng binh. c0ng1 d6 dnng stty ra :

11

-..Hoq. nita, d5. thdy, ndu Mk * M**, vA r!k), r!tt) , . . . , Ak) khong ddng tfidi rinfr'rihau thl so ) sk*1. Suy ra, ndu r[k) ,xy),...,x!*) kh6ng ddng thdi b&ng nhau thi tdn tai s € N., I < s ( ap, s&o cha Mu > Mk+". Do drj : ndu vdi rn6i & €N. d6u corf) , ,!k)l'.". . , r[k) kho*g dQlg lhdi bing nhau thi sE tdn tai d6y lMk] *i = J Ii day giArn ng6t v& bi chan du6i

b6i;1. (1) 1) Gia st v6i m5i k e N* c6c s6 , xf) , . . . , x(k) d6u nguy6n. Khi dci, .hidn nhi6n, Mo 4guy6n V k e ry-. Ma trong doan xlk)

[mr, M ] thl

(l)

c
hrru han c6c s6 nguy6n n6n tr]

suy ra phdi tdn tai , € N* sao cho xlt') = xt) = . . . = r,!l). Gqi f., lA sd b6 nhdt trong,c6c sd_t nhu vfly. Hidn nhi€n tn )/ tr. " GiA st to, 1. Khi dd : xlt"- t1 + xf"- 1) ,9,* ,) +r$t" - 1) 2

,{1"* rl

c.ics6nguyOnVfr€N*. tr'Ifly : Trong tohn b6 qur{ trinh th{c hi6n vi6c thay s6 ta ludn ehi nhAn drloc c6c s6 nguy6n khi vi chi khi cric sd nguy€n xr, fi2,..., rn hof,c th6a (5) ndu n 16, hodc th6a (6) ndu n chin.

Nh{n x6t : Hdu hdt c6c b4n grri ldi grei r6i tda soan chi

difu ki$n cdnvd dt ldxl: y2 - ... = xo I MOt sd bqn kh6c b6 s6t di€u kiQn.rl = .r2 (mod2) khi n ch5n. Duy nhdl ban Pkdng Son Ldm (12^[, Phan BOi Chiu, NghB An) m lrti giii ra

hoin chinh.

, NGUYF.N KHAC MINI{ BAi T9/198. Cho tant. giac ABC nhgn. Ching miruh rdng :

3fR+r)
Ldi gi6i. Ta chring minh bdt d&ng thrlc m4nh hon

c I{6u

ra 16

2R(t +

thi x$'"- 1)=xl"- t)=...=x{!"- r) mAu thudn vdi cSch chgn /". o N6u z ch6n thi : xf"'- t) = x9,-1) -.. . = *#nr 1) (2) vd :

xf"- r) = xt.-l) = . . . = *,9"-l) (B) GiA str ,f" - ,) + x$"- 1) (*). Khi dd, v6i quy rldc ,1") = xi, Vi = 1, n, tt (2) c6

*f"-r) +xf,-z) *t"-r) +x!,-21

2

-

-"'-

2

x$qz) +xf"-z)

: :ryt"-21 +x9,*2) *... +r[t" --2) = 2x$t"-1) @) Mat kh6c, tt (3) cei *9.- z) +r$t. -"z1 rf"- z) *rf" -2) =

:

(2+t!21(B+r) l n.\[2 '2' r

4' L :

r[r" -2). Suy ra

-----Z-:

z2

,t"-z) * x\t,-21

xt,-2).

Suy ra

=

=

uqi G) (do (*)).

= Zx,t,- 1), mau thuf,n Miu thudn niy cho ta

----Z-* . . . +rrGr, -l ,f,-z)

,f, - r) : xf":r) vi do d6 * ,f'-,) = xt"-l) ".. =rf;o-i), mAu thudn vdi c6ch chgn f". Tcim lqi, ph&i cti to = l vd trl dri suy ra 0r, = x2= ...= rr, (5) p6-u;r ld.

:

ffn va

2) Nguqc l4i, d6 thdy, v6i cdc s6 nguyon 11, x2t ....t rn th6a m6n (5) ndu n 16, hoec th6a (6) ndu n chin, ta sE cri rft<) ,ry) , . ." ,4t) ddu ln

*r{t' - i1

Suy ra

Orl = x3=,.. : f*1, x2 = x4= ... = = xz (mod2) (6) n6u n ch&n.

(1)

V)

sirrA +sinB +sinC riz €TosA (dtrng dinh L u; + cosBt cosc ' r r! hirm sd sin vi dang tht?c quen thuQc cosA

*

cosB

*

cosC

=I

+

trong AABC , tac6 A

r J?)

li

*

gie sit A lA goc l6n nhdt

> !o ; nann[ta

AA.BC nhon,

ncnf
< f,
eos,A+cosB+cosC= A cosA + 2sir5

ThAt vAy, bidn ddi 6 vd trlli sinB

\o''

* sinC

.ZsinB+Ccos B-C vl cosB * cosC = = ZlB+C B_C Zcos sau 'dd thay r--cosl-, . B+T cos B+C lfin ludt bing sln l-_ ---, AA. cos,

, sin

ddi, duoc

rut gr;n vi bidn 3A B_C (3) c+q6e- (cos-7-- - 1) > O @)

7

rdi nhAn

chr3o;

w 3"{ 3x Ti (2), ta lai cd ,< z'z 3A B-/(*a,=

*=Zt < 0, cbn

-

1) ththrlen nhien

non

<

0,

do dci (4) dring, tray (3) drlng. Xdt hdm sd f(x)

x

sinr * 2co5 cosr +

x

x6c dinh tr6n do4n

2sin,

ln

n1

p; zl "e

CIBCD, OCDA, ODAB, OABC vdV'1,V'2, V'3, V'o ldn lirgt Id thd tich cric hinh chc5p cut cci rndt aay ld BCD, cDA, DAB, ABC. Do cac tt diQn AtsCD , A, B 1, C r D I d6ng d4ng n6n :

\,+vt_

(*,)' _(y:n!:\' _= A]i ,l v =[-] =i, = (y::'\ = (, .#)'= (,.?)'

3

-1+sin7 c6f'(x) =

( < 0) hayf(r) nghlch

lcosr +_2sin|)2 bidn tr6n

sinA

W,;)

,lt Jt. €t Vre\r;r)hay

A

* 2co5

cosA +

) \AH hayv*v'r=v(1 +f,f = vz, fi, * v + svt * t ;+4. oo d6 v't = Svr

,t**

vdi moi A €

2sin,

K6t hop di6u nAy v6i (3), ta ra dpcm.

V,;)

cci (1) dring, suy

Nh{n x6t: 1. C6 68 ban tham gia giAi. trong s6 d6 c6 63 ban giii dring. C6c bgn sau dAy c6 k3i giii t6t : LA NguyAn

Chdt (ll'l - lamSoq Thanh I{6a), C/ru NpyAn Bnh (l2ADHSP Vinh), Nguy€n Thn Caong (11A, Ly TrJ Trpng - Cdn Tnay Nguv|n Minh Hdi (12 toin TrAn Phri - Hii Hung). 2. B}n Hodng Thi Tuy€t (l2T Lam Sdnq Thanh H6a) chilng minh "2 (ZR + r) a + b + c" rdi suy ra dpcm. B4n Ngd Dtc buy (10T - PTNK - Hdi Hung) gqi OI, Ol, OK.ldn hlqt lir khoang c5ch tit o a6nfc, CA, AB r6i dlng bdt ding thtc Ts€-bu-sdp dd c6 ol.a + oIJ.b +

oKc <

!


* ot + oK)

(a + b + c) (r). Ma vd tr6i crla 1'; Ulng 25 (vi O nlm trong LABC do A niy nhqn) n€n Mng r-(a + b + c), suy ra 3r < OI + OJ + O( vi dtrng di€u nay d6 chilng minh bAi toin.

Bai 10/198. Cho ttl di€n ABCD, udi didm tng udi m\.t BCD, nguiri tq ldy

O d b1n ffong.

O, ddi xilnk udi O ddi udi mQt phd.rug (BCD) rbi ne qua O, mQt ph&.ng song song udi nrQt phd.ng @CD).1v[At ihd.ng uita hd tqo udi ntQt BCD u?t, cdt mQt phd.ng chta cac nt'Qt cbn lai cila fi di€n thd.ni mqt hinh chdp c4t. D6i udi cd.c rnq.t cdrl lqi, nguiti ta citng lilm tuong tu uit. thu duqc th1m ba hinh ch6p cut. Xd.c dinn ui ri cila d.idm O sao cho tdig thd tich cira b6n h"inh ch6p ctqt tao thdnh lit nh6 nhd.t. Ldi giii. Goi .81, Ct, Dr ldn lrtot Ii giao didm cria c6c tia AE, .dC, AD vdi m4t ph&ng qua O vi song song v6i (BCD); f ld giao didm cria OOt v6i (BCD).H4 AH.t L(BCD) c6t (BCD), (BlCtDl) tuongdngtqicd;c

d.idm

I{, Hr. K

hi€uY, V1,V2,V3, V. lan ludt le thd t#n cac t irrt ttr diQn ABCD,

I

J--

v'v2'

MQt c6ch tudng ttr,

nrdrQng :V'.

:

?Vi+3

suy ra

i*i i=t

0l

V

r

I

=

ta cri c6ng thrlc

v?4

si4

+ .$ = 1, 2, 3, 4). - l

-.1 +

i:l

3rn L w

?"f i=l

*

11 , f,Z r=r

>3V+V 4*r, lA= 3VV61 = 3V+ 4 +G=GY

Id vi

sd di 4*4+4+vlr| Bunhiac6pxki-C6si, ta

oANc vtBN

didm

4f,

_

theo

cci

(4 + Vr+4 +fi) x {tz + 12 + tz + t2) > > (Vt+V2+V3*ri'=tP;

cbn

{ +4+4*4rfi

,u vi ap duns

Bunhiacdpxki-COsi sau d
< 4 . (vl +4,

+4 * 4)U,

+vz + v3 + vi <

. a.a{4+4+4nfl>y

Suy ra tdng dang xdt nh6 nhdt khi vi chi Y1 - v2 = V, = V, haY O li trong tAm trl

khi

diQn ABCD. Nh{n x6t : C6

70 bgn

giii

bAi ndyvir tdt

ci d€u giii dung

Loi giAi t6t gdm c6 : Hd Thanh Hdi ( 12T t-am Sctn, Thanh H6aj ;-Bilr iudn (lttA^ chuy€n Th6i Binh), Kbu Htu Dfittg 1r2i - chuyen Fiirng Vuong, Phri Thq, Vinh Phri), Drnn rrung Hdns (lOM Marie curie' Hd *3.1*o

,r,a* cd'ch mQi didm mQt citng Ta Bhi L1/198. ndm di thdi dbng ta ngudi h d,d.t mQi klz.cd.ng

hai hq.t theo phuong nd.m ngang uE hai chibu nguqc nhau udi uQn tdc tuong ilng ld, urvh, ur. Chtng mink rdng hhi khod.ng cd.ch hai hqt lit. X - (ur+u) {ipfi thi udcto uQn t6c cia hai hq.t d thiti di€m d.y se uudrug g6c udi nhau udi g lit. gia tdc trgng trilimg. 86 qua stc cd.n cila khong ktti. Hudng d6n gi6i. Sau khi phAn tich chuy6'n ddng cria trlng hat (drroc n6m di theo phrrong nim ngang), di d6n nhQn xdt : d mgi thdi didm hai h4t d6u cirng nim tr6n mQt dtldng thing nim ngang vA khoAng crich gita chring d m6t thdi didm , bdt ki sau khi n6m sE bing X = ui * urt = (urt u2)t, til dci suy ra thdi Oidm r khi kioAng Lacn-x = (r, * u) ,[i,u1g

x

{;-

Y to ) t = -* ur*u,= '

vi

I

(

1). Sau d
p mn hai vecto vdn t6c cria hai hat hgp vdi phtrong nim ngang ; drra vdo (1) I suy ra tga = do dd a * f = 90o. Tt dri rrit gcic

a

gdc

tgp, ra di6u cdn phAi chrlng minh.

NhSn x6t. C6 58 em de gui bai giii, trong sti d6 co 53 em c6 ldi

llB

giii

dring,

dic bi€t li

c5c em : Truong Ddng Nghla,

Qudc hsc Quy NhdrL Binh Dinh ; Ngtydn D*c ViQt r2CA, PTTH Bni Thi Xuan, tp Hd Chi Minh; Va Thi Blch I/a 11 PTTH chuy€nThSi B\nh;HodngDtc PhnongBol0b DHTH He Ndi; Phan HodngViet ILA Qudc hoc Quy Nhdn, Binh Dinh.

Bai L2l198.

MAI OANT"I Cho m4ch diQn nhrr hinh v6,

Rt 'R.; trong UO *r: *,

a) Chtng minh rd.ng cuitng dQ dbng diQn chqy qua ngubn (Er, r) khdng phu thuOc uito sudt dien dQng uit, dian ffd trong cila ngubn (Er, rr). b) Vidt bidu thttc tinh cOng sudt ileu thu tdng cQng ffAn b6n diQn trd R1, R2, R, ud Ro. Ldi giai : a) Dbng diQn trong c6c doan m4ch drrgc ki hiQu vi tinh theo chi6u vE tr6n hinh. Ta c6 I, = I, * I' ;

UAB

= Ueu * = RrIt

UMB = Ry't + + Rzgr +

R/z =

I')

(1)

;

vd'Ir=Is-I'; UAB

= UeN *

=

UNB

= RjI3 * RrI, =

RzIz + R4Q3

- I')

(2)

Nhan (l) vii R.. * Ro vA nhAn (2) vdi Rt + R2, sau dd c6ng lai ta drJoc :

(Rl+R)+RJ+Ra)Uas: =(Rr +Rr(R3 +R4X/r +/3) + @frt- RLR4)I' Theo giA thi6t Rfi,s - RtRt = 0, do dri (Er +Rr(.B: +Ra) +13=I) "AB R.+ R2+ R3+ R4 /(vi.I, ff

Mat kh6c UAB = E'

-

- rrL Do d
Er

f

(Er + R)(R3 + R4)

-_

Er +R2 +83

+& +rl

Nhrr vay ddng di6n qua ngudn kh6ng php thuQc vdo Ervd. rr. b)

Tt

gie thi6t

dd ta lai cri

(Er,r)

Rl R3 .Rl R2 Uo = R? *, "rr, "u.4 = *0,

:

E2 I=

(Rr

+Rr(Er+

Ra)

Rl +/?3 + R2+ R4

*12

Nhu v6y cdng sudt tdng cQng t6a ra tr6n bdn di6n trd RrRrR,rRrld:

E/

+E2I'+rrtr2, -r2I'2 Thay c6c bidu thfc d6 tim

P=

c:0:a

I

vd.

I'

ta

drrOc

El

D_

rgr-1"2(& LR, +

t"g

+.*zj2 Rz+Rt+R4 ' 't)

(Rr + R2)(R3 + R4)

Rr+R2+R3+R4 El

(ar+Er(]?2+n4)

r_ T (E,+R,XR-+R,* Rfi+Rr+RA *tJ L a,+ny6r**

Nhqn x6t. C6 11 em da grii bei gi6i, trong d6 c6 2 em m lcli giAi ding: Truong Hdm YAng, PTTH Li TU Trqng, Cdn 'lhc1, Vu Th|Blch Hd, llq PTTH chuy€n, Th6j Binh.

MAI TUNG

Bhi T7l202 : Cho tam thrlc bQc hai : f,(x) = atJ +' 616 * c vdi c5c h0 sei drrong vi a*b*c=7. Chrlng minh

&thi;

f(x) '-

Cric ldp PTCS

BAi Tt/202 : Tim cric s6 nguy6n td dang 2tee4d q t7 @ eAr; bidu di6n drroc drr6i dang

hiQu cdc ldp phrrong ctra hai sd tU nhi6n.

ab*ac*bc>oua!+a+ !ro ob ac !" NGUYE,N DE

T4t2O2 : Cho ba s6 nguy€n a,b, e, a ) O,ac *b2 = P = pl " p2.. . pm, trong d6 pt , . . . , pm lir c6c s6 nguyGn t6 kh6c nhau. Goi M(n) li sd cric e4p s6 r,gaybn (r, y)

Bai

th6a min

rRAN vAN HANH

:

1)f(n+r)>f(n)Vz€N* 2)

fff(n)) = rr.+ 1994 V n c N* NGIJYBN MINH DUC

B}ri Tgi202: Quayhinh vu6ngABCD quanh g' , @ < g < 9U dtqc hinh vu6ng A'B'C'D'. Ta ggi giao didm cl0,a AB vdA'B'ldM, BC vdB'C'laA', CD vdC'D'ldP,

DAvdD'A'l

"q.

a) Tinh diQn tich

a=ABvitg.

ti

gi6e

MNPQ theo

b) V6i g6c A nirc thi chu vi phdn chung hai hinh vu6ng bd nhdt. DAO TRUONG GIANG

BAi T10/202 : Cho tri di6n d6uABCD. Tim

a*+2bxy*ciF=n

D€ dg uydn thi to6n qudc td n5m 1993

B,di T51202: GiAi hQ phuong trinh

cria trl diQn bang k2 cho tnrdc. NGUYBN MINH HA

Cdc dd VAt

li

Bai Lt/202 : Anh sringm4t trdi chidu song thing dtlng vd tao vdi phrrong thing drlng gcic nhon a. Dtlng tnrdc tudng, dirng m6t gudng phing, mQt em b6 chi6u rlnh s6ng vu6ng gde v6i m+t tudng. Khi dd m6t phing ctia guong t4o v6i m4t ddt mQt gdc bing bao nhi6u ?

song v6i mdt btlc tudng

Chrtng minh rang M(n) ld hfru han M(n): M(fi .n) vdi mgi k > 0

vi

:

*l= ur g ca$?Yx2

NGUYEN DONG

BAi LZ|ZOZ: C
r/3

*z: g co$ tt x3 xl= {3 g cosrtxl

dd do crrUng dQ ddng diQn trong mQt do4n m4ch.

t
Ndu srl

It=2,95

{r

,,t= g cagfixl TO XUAN TIAI

Bhi TGI?02 : GiAi phrrong trinh I

arc cos sinx = -: x I NGI.JYEN VAN MAU

10

.k

tf('\x)l-

quf tich nhtng didm M sao cho tdng binh phrrong cric khoAng c6ch tir dd d6n c6c mat

C6c l6p PTTH

*

duong

tAm O cira nrj mQt gdc

BAi T3/202 : Cho tam gSac ABC nQi tidp drrdng trbn (O). Dgdng phdn gi6c trong cria g6e A c6t duang triin (O) t4i didm thrl hai D. MQt dudng trdn thay ddi lu6n lu6n di qua A, D c* c6c dtrdng th&ng AB, AC t4i cric didm tudng itng M, N. Tim t4p hqp trung didm I cta MN. onNc vr6N

arc sin cosx

;k._

sd nguyOn

Bei T8/202 : Ki hi€u N* la tAp c6c sd nguy6n drrong. Tim tdt cA cdc him /;.1[*+N* th6a m6n ddng thdi c6c di6u kiQn sau

ouc rAru B.Ci T2l2O2: Chtlng minh rhng di6u kiQn cdn vd dn dd c6c s6 a, b, c ctng ddu ld : NGLTYEN

ring vdi moi

TO GIANG

Chn 5, : - M5i bai gi6i vidt ri6ng tr€n mQt mAnh gidy. (ihi sd.eda bAi 6 gtic tr6n b6n tr6i ; hg t6n vi ldp, trtrdng, huyQn, tinh d grfc tr6n b6n ph6i,

Fon f,ower Secondary Schools Tllz0lL.Find all prime numbers of the forrr 219ef + 17 (n € AI) which can be writteni,as a difference of cubes of two natural numbers

'

.:.. ...,

..

::

.. i

,

:

arcsincosx

1'

* arccostinr: de NClIYF,N VAN M.\IJ

mPAL"Iet f(x) = ax2 +bx * c withq

NGIJYEN DE

circumscribed circle (O). The inbisector of A meets (O) again at a point D. A vafiable circle passing through A, D intersects AB aw)" AC at M and N, respectively. Find the locus cf the midpoints .I of J}fN.

b.

TBl202. Denote by N* the set of all positive integers. Find all functions /.'N**N* such that

1)f(n+1)>f(n)Vra€N*

T4l2O2. Let a, b, c be given integers with a ) 0, ac -b2 = P = pt...p,r,, where P1., . . . , Pn are distinct prime numbers. Let M(rt) denote the number of pairs of integers (x, y) for which o.x2 + 2bxy * cy2 = 77. Prove that M(n) is finite and fuI(n) = M(F Q for every integer k > o IMO31. GI:O 3.

{5 COS Jt X)

2) f(f(n)) -- tt + 1994 V

n € N*

NGI]YEN MiNII DI]C

Tgl202. Let A'B'C'D' be the image of the square ABCD by the rotation of angie p (0 < g < 9e\ about its center O. AB, BC, CD and DAintersectA?', B'C', C'D' and D'A' at M, N, P and Q. respectively. a) Calculate the area of MNPQ in c. : A,B, g b) Determine the value of ,p such ttrat the perimeter of the common part of two squares ls nilnlmum.

Xt,Y= -'i-

DAO'IRL,ONC Glz\NG

TlAn02. Let be given a

{3 I"=:COSfirz9r cos 7t x,1

'VT cos lt g

la,

TRAN VAN }IAN}I

For Upper Secondary Schools

= {,g

c>0

f(x) > fft2h)l'r for every positive integer

DANG VIE,N

xl:

I

ando+b*c:l.Provethat

TB|2O2. Let be given a triangle ABC and its

xs

:

'

TZ|2O2. Show that 3 numbers a, b, c are of the same siga if and only if ob * bc * ca > 0 and

T5l2A2. Solve the system

.

$::0F,':THIST t$Sil$H

T8l2O2. Solve the equation

NGUYEN DUC TAN

111 --= *;- +- > 0 ao ac cq

: '. :

pm

xl

regular

tetrahedron ABCD. Find the locus of points M such that the sum ofsquares of distances from M to each face of the tetrahedron ABCD is equal to a given value k2 . NG(]YEN MINII I'IA

TO XUAN FIAI

il #N{}KfiN T cAt cAcH DAY vA HQC TOAN

\*nr{

VH MOT TIET DAY TOAN 6 l6p chon, l6p chuy6n

cho /l-T d5 dtJrlc dUa vao lnot s6 ti6t trong chrtdng trinh toarn ri 6 c5c lop chr-ry6n, lrip chon. ThUc ra, toirn b6 clrc ciiiu hi&r chia hdt n6i tren cri thd thitc hiOn trqn ven trong mot ti6r day : Ddu hi€u chia h6t cho 2k

Ddu hifu chia h6t cho 2k vi sk 1t e q li su tdng qudt cta c6c d6u hi&: chia hdt cho 2, cho -5. cho./, cho 25, cho 8,

.5k

(k € AD

Sfj N= rrr"-1...r-+lrr--t viat dtroc tlU,fi dang N=r"r"-l-rt+1.10k+o*-.o, ma 10k : 2k.5k chia htlt cho 2k vir 5k do vay N ; 2k ( hay ,5k ; ndr vir chi ndr ,rr*u2k tha.v -lk ; va chi nhfing s6 dn mrti chia hdt "r: cho ?k thay -sk ;. X6t cac tntdng hqp cv rhd k = I ta co ddu hieu chia h6t cho J, cho.5

k : 2 ta c6 ddu hifu chia hdt cho J, cho 2-5 k = 3 ta c6 ddu hiOu chia hdr cho 8. cho .11-s la ci-ic cldu

-

Vie.c dpy hec loen c6 khi bit ddu tri cdi cu rhe dd di ddn ciii tdng quit nhtlng cring c6 khi giAi quydt cii t6ng qudt ta se dUOc cai cU thri mA c6 thd rtit ng[n duc,c thdi gian. Xin n€u vi dp :

vi

hi€r.r

chia hdt thrttrng sit dung.

I-ly vgng n'ing thqc hi6n

em

-hr2c

tidt d4y

n2ry s6

dc'm den cho c6c

sinh nhidu didu bd ich.

NGUl'I]N DIJC'I.,iN

ll

tdnh cho ctfic bgn churin

br.

tht vtio ilat taoc

\Al2\

HOC DAI VAO SINH DE THI TUYEN '-----N ,r v . ^Tr6nc HqP HA Nfi ruAnrt tges (KHOI A) -

Phdn b6t buQc : Cd,u L X6t hdm sd : y = (, - t)z (x - a)2 (L) 1) Kh6o srit s1r bidn thi6n vd vE dd th! cria

.A

fng vdia = 0. 2) Xdc dlnh o d6' dd thi cta hdm sd (1) cd

him

s6 (1)

didm crre dai. 3) Chrlng minh ring vdi moi o, dd th! hdm s6 (1) lu6n lu6n cci truc ddi xrlng song song vdi tryc tung, Cdu II. 1) Xdc dlnh nz dd cdc bdt phrrong trinh sau c
[r2-zr*m-t
1) Chrlng minh ring AB drrdng vudng gdc chung

+{74

Stg 2x - 4tg 3x : tg27xtgzx 2, Chrfuig minh ring trong mgi tam giric ABC ta d6u cri :

B

-

ABC cosS *cos *cosV , C'A B sin r+sinV+sinV

Bai giai

Cd.uI:1)Khio=0thly = qx2 - x)z y' =2(x2 - xSQx - 1) ; !'=o + ['r=o'J1 =o

lri

= 1" = 2'lz=TG lr"=1.v^=0 [

y"

,

'v5

:12(x2-r+|);r'i

= o+

BAng bidn thi6n

Phdn tg chgn : Thi sinh chon mQt

2) Cho hinh l4p phtrongABCDA'B'C'D'e6 er{c c4nh bing 1, Tr6n c.ic canh BB', CD, A'D' ldn lugt ldy edc didm M, N, P sao cho :

B'M=CN=D'P=a(0
--' - +AD + (o-i)AA' MN = -aAB *b) f{nh c6c tich v6 hddng Intn . fr' ua MP . AC' Tt dd cd thd nrii gi v6 vi tr{ cira AC'd6i vdi mat phing LMNPI ? Cd.u IV (b) : Cho tf diQn ABCD cd AB = x, CD = b, cric c?nh cdn lai bing nhau vd bdng a. Ggi .8, F ldn lrrqt lh trung didm cria AB vd CD.

1,2

x4,s:i (t=*) ll,s=

(vA ehi m6t) trong hai cdu fV (a) vi IV (b). Cd.u IV (a)' : l\ Tinh diQn tich hinh ph&ng gi6i han bdi c6c dudng :

y = | lSx I ;! = O;x= 1110;x=70.

t CD vit EF ld AB vn CD. Tinh

E[theo x, o., b. fDlhir" r dd hai mat ph&ng tACDl vdlBCDl *,uHg gcic vdi nhau. Chrlng minh ring khi dti td diQn ABCD cd thd tlch l6n nhdt.

J

2) Giei frd Uat phrrong trinh fr{ax a < x >+ l{tTE Cd.u III. i'. Ciai phtrong trinh :

cliua

Ddthi: r= 0,! = 0

v- 0,x = 0,L

1

s6

2) m ?xe sdi crla ra dtrong n6n dd thi hdm sd cri didrn cgc dai khi vd chi khi phrrong trinh !' = A cri 3 nghi6m phdn bi6t.

!' J'

= 2 {x - 1)'(x

-

a) (2x - a - 1) = 0 cd 3 nghiQrn phAn bi6t <+

cos2x

*

0, cos\x

*

0

{.*)

:

* 0) vi r'6u n$tqc Xai thi ti

phtrong trirrh ( 1)

ta suy ratg 2x: tg 3x "+ f, * tg2?x: O (v6 li). Ta cri : - tp2x - ts?x = tg3r *-7t*r tg3x (I) e=+J =

I - *1 l"

3)Dat x=X+o!-1 2

!=YthiY=

ki|n:

StgZx - 4tg\x = tg23x. tg2x (tgZx - tgsx) = tgSx tl + tgsx " tSZx) (L) NhQn x6t ring moi ngh,iQm cria phuong trinh lu6n ludn th6a m6n hO thrlc 1 * tg?x tgZx * O (=>

cosx

*a*1

la*l la+1

Ta cci

1) Di6u

*3

Ia+1

L*o lo

III.

Cdu

[r'- (+)1'

Ddy lA hdm s6 chSn n6n X = 0 li. truc ddi xring

iT t$rt-C,r; tlf* .. *-stgx '{* e2tgx " = 1 - 3tg2x ftgx : 0

(stg2x- J) =

0

.= | I-g a*l Itsx= *l u - 2 li, truc ddi xrlng. Cd.u II : 1) Di6u kiQn dd cric bdt phrrong V=kn(ke4 c= | trlnh cri nghiOm ld I-t [r=+arctS\l S+kr(keZ) )'[L',:2*nt>o ca hai hC nghiQm ndy ddu th6qm6n di6u ki6n (x)

-hay r

:

1u',=

7-2m>oom<2

( 0<+ 1+{, -nL x2 +4x *2m -3 < o€r.J ( x ( xq i 13,4 = -z+'[7=E'n

2) Vr

Khi dd x2 -2x +nl. -l rl ( r =< 12 i xt.z= -

Cdc bdt phtrong

*l*, -lr,L

(

rJ

trinh

nghigm chung < -2 + < 1 +{2

_

.- fl -{Z -rn

-i-3
rn=Zi

(t) Qi

-*

Di6u ki6n (2) lu6n lu6n th6a mdnVm < 2. Vav gg !4t ph.rglgj4Sh cd nghiem chung khi

s

<,{z=i

+t[7 -zm

ere< e-3m+z{@=n4(-Ei -111,

4

O

*11*ro

L]mz+44m-s6
c+nz( -22+{s4o

(th6d man di6u ki6n m < 2) 2) Di6u kiQn :

1*, = Binh phrrong cric vd cta ttng bdt phuong . trlnh, ta duoc h6 tuong drrong Iz

l-
u

l-(r(5 oz7 < >ot= 1n^ r x<4' -Er+z < te

l;ffi,1r"

lx2

,inf, ,

,

nu^

ABC

cosV*cos

r*cosV B .A stnV*sinV*sinVC

"o,!*"or'2* "o,$ .

z

B /.4 <+(srn,+stn,-cos,)

< 2e

fin!* C,

rrnf,* rt"f,1

+

C A. , /. B .r(smr+srnr-cosr)+ +

(sin

,t

f, * ",r;- -

Di6u niy ddu drrong

"o" #) dfng vi m6i bidu thric trong ngoec

C A+B cosT=Sln 2

=

B

B

.A C = slnTcosz +slnzcos, < AB < sinT+sin7. TttOng trr

17

* rinB, *

c
r, -l -2 -{j=Zi -------L < [3 t2 -i +{7 -zm <

sin!

:

ABC cos, <

sinT +sin7

;

BCA cos; ---2 z < sin;2 +sin-

13

: AD N€N AF I CD BC=BDn6nBF tCD YQy CD t (ABD + CD -L AIi vA CD t EF DoBC =AC+CE rABmiAts tCD+

Cd,u

10

Cd.u

=

IV(a)

:S

=j

1110 ltgxlax= - !

-J

I

tgxldx =

0.1

tgxd.x+

!

tgxd,

I 0,1 Y\ tgx = lge . lnx, ta chi cdn tlnh nguy6n hdm cria him s6 f(x) - lnx.la c6 : {x ln x)' = lnx * 1 vdy lnx - (r ln x - x)' Suy ra : 110 s = tse = txtn& t*t", *., 0,1

l-

IV (b) : 7) Do AC

ABl-(CDE)*ABLEF.

Y$y EF ln drrdng vu6ng gcic chung c&a AB vd CD.

-Ol)

),,

=9,9-8,t.lge. fr =4, rt) =4,

2) a) Dat

rt'=4

)

D,

,) + rinh

,,

,6{' - Atr - AE me

62

AF! -- ot -Z

i

.y2

rb

er2

62 x2 + EF)) = az) -Z -l

=7

+-+++

Ta cti : = .lfN = MB +BC + CN '+ -4 -t B'B h{B = (1 = (a * 1)et .

")---t -+ --) BC=AD=ez +++ CN=aCD=-
-'++-")*r*er*(q-l)et +MN= -t++*AD +(a *1)AA' hay MN = -a,AB

b) Theo a) thi MN = -a e, * er* (a - l)et +..+-t+ mdAC' = €t*er*e, -+-t--'+-''+ -)1 * MN . AC' = (-ocr+ ei! @-!21@r* er* 4\=

-g+1+J-1=Q+1$C'1MN

MP=MB,+B'A',+A',P ++ mdMB': aBB'= dE3 ++€ B'A'= BA= -et +++

(l -a)A'D! = (L -a)e, + + + +MP*ae3 er+(l-a)ez A'P =

+

(0
* ry .AC' =n(--er + (1 * a)er*aer)x * (r!+ ez!) = -I + (1 * a) * a = a+ + AC, LMP Tt 14

dd suy ra AC'

t

(MNP).

-/-\

2) Theo 1), ilo CD L (ABn n6n A.FB ln g
khi EF

=+*'UrtW=V=l =;*

Khi dd Veaco = =

*r,

V6ABF

. sasr *f,o, .

tt

*

sABF

VDeBF

=

=

j

= i sourer + DF) = snuo. cD mit CD = b kh6ng ddi n6n Veaco l6n nhdt khi ln FA. FB. sin AFB l6n nhdt. Sorolln l,hdt

i

-.-t

++

r, = r{ae=V--::F

Do

1

rh = FP = ,7'{W;F

kh6ng ddi nen

Vnacoldn nhdt

cesin

^AIIB=lofrE=9N. NGUYEN VAN MAU

Tqp chi Tll&TT s6 9, 1993 th6ng brio ring "dinh li ldn Ferm,at dd drroc chdng minh". Ti6p do, trong sd t, 1994, lai th6ng b6o ti6p v6 vi6c de ph6t hi6n ra m6t ch6 hcing lrln trong chfng minh dri n6n dinh li niy v€rn tidp tuc ld m6t thrich thrie dfii vdi todn hoc. O sd g, 1993, t6c giA th6ng brio vidt ; "... Sau chtlng moi. rrid- hi th6t bai trong vi6c tim iai chrtng *inn uo qu€n
Khi n = 2,ta

c
dinh li quen bi6t sau d6y:

sirrh Ii l" Khi n = 2 thi diau

kiQn cb.n ud di dd ko, !u, z) trd thanh nt6t ldy gia tri u2 + u2 cbn xo uit yn chia iii" f"i gid tri u2 - u2 t:d. Zuu, u uir u lit. hai sd nguyAn duong ltiy tity !,nti6n ld ching nguy\n fi citng nhau uir chdn li khdc nhau. F{hfn xdt : zo bao gid crlng 16 va ldn hdn xo , J, c6n trong hai sd r, , }, thi mot ch6n, mQt !6. Dd cho ti€n, ta s€ goi sd chf,n li ro, Thd thi : nghiQm crta

(1") Ld zo

(zj = Zu,d, lo = u2 * L)2, zo = u2 + u2 Bay gid ta x6t ddn cdc tnrdng hqp n > z vit ehrlng minh r&ng (1) kh6ng cd x,n

nghi6tn.

Trrrdng hop don giAh nhdt lA tnrbng hgp n = 4. Khi dd, det x2 = X, y2 = y, z2 = Z tht (1) trd thinh : y2 qya = g2 (3) Ndu (1) cd mQt nghiQm (x, !r,, 1) tni (B) sc cci nghi6m (Xr,yo,Zo) v6i Xo = xj, Y, = tZ , Zo = 2f,. Theo dinh li I thi 6t Ia :

X, = Zpq,

Yo = p2

-

Q2,

Zo = p2 + q2

trong dd p, q la hai sd nguy€n duong ndo dri nguyOn td cirng nhau *d ch5n. nhau. Thay Ii, bingyj, ta cci : y2o+q2

tr6

khdc

=p2;

lo, p , g nguy6n td ctng nhau vi moi rrdc sd chung cria chring phai ln u6c sd chung ctrap, e mip, g nguy6n t6 ctng nhau. VAy (!o,p,q) ln mOt nghiOm cira (1) (v6i n = 2). Vi yo 16 (do f, ld) n6n q phAi chin. VAy, theo dirrh li tr, ta c6 : q !q6, lo

= a2

-b2, p = a2 +b2, trong d6 a, bli

=

hai sd,guy6n dttong nio dci nguvdn td

cirng nhau vd ch6n, ld khdc nhau. V$y : lt

-

*3 = Zpq

:

4ab@2

+

b21.

rli6u drf buQca6(a2

+ b21phfri le m6t chinh phuong. Nhr.rng ob vb. a2 * b2 nguyen td- cing nhau vi rring n6u d ld m6t rrdc sd chung ciia chring thi d s6 *nI" rra,, a2 + b2 * Zab, ttlc Ii chia h6t a t b vt do dcj chia h6t (a + b) + (a - b) ttlc li chia

hdt 2a , 2b ; d kh6ng thd le 2 vi ncf chia hdt s6 16 o.2 + b2 (do o, 6 chin, 16 khic nhau), vdy d ld udc sd chung cta a, b n6n d = J. vi ab vd a2 + 62 nguyBn td cung

F

,g

r d m

a iJ

md ffin

ffiffitq

mil diB q4q

f) 0r

cE.

0\

+

3nm

d

Em CD

rE E

d d

sr # € ES,

€ mt

"m5 trB FI& fftd

a

I, Ed ffih

ffi. EME €? ffi rrd

deS

bq q

I

ffi

ffi ffi

ed

d gffiB ffi Hd M

us#

w

,{m tuq

ffi! ffi$r.

m s

s? ffi"r ffie

ry ffi

ffi* 15

nhau n6n mu6n cho ab(a2 +b1ld m6t chinh phttong thi ob vd a2 +b? d6u phAi ld chinh ptruong. Nhtrng o, b cring nguy6n td cirng nhau ,r6n o, & d6u phAi li chinh phttong ; vfy 6t ph6i cd nhtrng sd nguy6n drrong r, .s, f sao eho :

a=fr,b=s2,a2+b2=t2 Dodd:r1 +sl=t2

n = 5, chrlng minh dd dga vdo dinh Iy sau day : Ndu x5 * 15 = z5 thi mQt trong c6c s6 r, y, z phli chia h6t cho 5. Dinh lf niy dtrgc md rQng nhu sau : n6,t n ld m6t sd nguyOn td vd 2n * I cflng nguy6n td thi tit * +f = zn s€ suy ra ho6c r, ho6cy, hoQic z chia hdt cho n' Sophie Germ.ain cbn md rQng hon nta dinh ly YOi

D6n dAy, ta chri Y rang vi o, b nguY6n t6 cirng nhau vA ch6n, 16 kh6c nhau n6n r, s ctng vQy. M4t khde, trong lQp luQn d tr6n, ta chtta sr? dgng giA thidt dngf Id luy thrla b{c 4 md

tr6n vd dirng dfnh ly cria minh dd chrlng minh ring ndu n lA m6t sd nguyOn td nh6 hdn 100 thl phrrong trinh (1) kh6ng cd nghi6m dang (xo, lo, zo) trong d6 xo, ! o, zrd6u kh6ng chia

chi m6i stt dsng giA thi6t zltattty thia b4c 2'

hdt cho n.

toin bQ lAp luSn d tr6n ddi vdi ,1,y1 va @?)2 cri thd lap tei irodn toin d6i vdi 14 , s4 vd l. Nntrng 14 +s4 < xt+yt vi vd thrl

Vi

vQy

nhdt bing a2 +b2 tfe

p

< p2

*

q2

: 2r.

li

bing p md

z', =

4 +fi = x! +y[.

R6t cu6e, tt ch6 giA thidt ring (1) cd nghiQm (4o,!o, zo) khi n = 4, ta suy ra dttgc srJ tdn tai crla r, s, / cflng th6a mdn (1) ( mi6n lh nhin vd thrl hai chl nhrr mQt chinh phuong chrl kh6ng phAi la trrly thrla bdc 4), chl khac ta

,

14 +

"4

.

*2 + y[. Lai tidp t$c l4p lu6n

dci,

/

+s4 < 11 +s4 v.v... vh crl thd mdi. Di6u dci li v6 lf trong phpm vi c6c s6 nguy6n duong khi ta xudt ph6t til cric s6 r,, lo htu han. V6Y ta cci :

ta sE di d6n

Dinh li Il : Phuong'trinh (L) kh6ng cd nghiQmkhin=4 EIO qu6 : phrrong trinh (1) kh6ng c
n = 74. Dirichlet chtlng minh n6m 1832 n = 7. Lrinti chrlng minh nam 1839.

Nhrrng nhd to6n hqc h)ng danh da dd c6ng srlc ra mh trong khoAng mOt th6 kj', cring chi

d4t ddn nhtng kdt quA khi6m t6n nhrr v63'' Nrii "Khi6m tdn" vi ti nhtng kdt quA dri cho ddn ch5 chtlng minh trgn vgn (v6i mqi n) cbn ld xa vdi. Hai chfr "thdt bai" n6u ra d sd 9, 1993 ctra TH&TT n6n hidu nhu thd. Trong thQp kf

40 cira thd ki XIX )t ki6n srl drlng sd phrlc dd phdn tich duqc/' * jt rathiLnh tich cta z thrla s6 tuydn tinh duoc d6 xudt vi thAo luAn. Tr6n co sE dr5, Kumm,er xdy dtrng n6n lf thuydt cira minh. Day ln c6i m6c md tt d6 cdc nhi to6n hgc chuy6n nghiQp hdu nhrr tin ch6c ring kh6ng thd ed chrlng minh so c6p eho dinh ly l6n Fermat Ndi "hdu nhd" cci nghia Id cung chda khang dlnh hin. Bdi 16 Fe.rmat da ghi

ring 6ng ta tim drrgc chrlng minh nhrlng ti6c ring 16 srich kh6ng dt ch6 dd ghi. VA chang, trong llch stt todn hqc, de tr)ng cri nhrlng bdi torin nhu bdi to6n Varing md ldi giAi so cdp lai ddn muQn hon c6c ldi giai cao cdp : nanr lTTA,Vating xrr6ng l6n mi khdng chdng minh dfnh ly sau ddy ; mgi sd nguyAn N '> 1 c6 thd bidu d.i6n thimh mQt t6ng h sd hq.ng, m.6i s6 han g ld. ntQt luy thita bQc n cfta m.Qt s6 nguyAn duong, s6 h nity pht4. thuQc udo n .' fy' =

q+...+oft. Tt

1909 ddn 1934, nhi6u

nhi to6n

hqc

danh (nhrt Hilbert, VinOgradOu) da tham gia giAi quydt bdi torin vdi nhtlng lti giai cao cdp' Nam 1942, Linnhic mdi tim ra ldi giAi so cdp. Dd kdt thric, xin ndi th6m rhng : chAn li thudng don giAn nhrrng con dubng tim d6n chAn l{ thudng quanh co, Phtlc t4P. NCUY6,N CANH TOAN

16

ltng

LTS : K{ su QUAN NGQC SON nd.m nay 61 tudi, kh6ng may bi bQnh, h6ng cd hai mdt, dd. nghi huu tit nam 1982. Tuy d.d. "nhibu nd.nt sdng trong b6ng t6i hod.n tod.n", nhung 6ng "ud.n gid.nh nhiDu thiti gian suy nghi, tim. tbi, phd.t hiQn nhibu quy luQt li thn cfia cd.c con sd' (theo thu 6ng giti Tba soqn ngiry 291911993).

2. Sd 987654321 Day ld "sd l6n ngugc" cira s6 123456789 de dudc gi6i thi6u d sd brio Ul9g2. Crlng gidng nhrr s6 niy, sd 987654321 cd dac tinh trrong

Cd.m dQng tru6c td,m guong lao dQng,lbng uit. c6 thd n6i lit. yAu diti, yAu tudi tri cila kl su,.tba soqn xin gidi thi€u biti bdo thi hai crta ki su udi ban doc.

Khi nhAn s6 987654321 vdi c6c sd nhdn nh6 hon 100 ddng thdi th6a m6n 2 di6u ki6n sau :

ydu khoa hgc,

Trong mQt sd b5o trudc, s6 U1992 tOi de cd dlp gi6i thi6u con sd ki la 123456789. Ki ndy xin dugc gidi thi6u hai con sd ki la khric cring cd thu6c t{nh trrong tu. t. Sd 12345679. Ch6c nhi6u bpn doc da bidt t6i con sti ndy vi m6t dac tinh cring kh6 ki Ia cria nri.

Khi nhAn con sd tr6n v6i cdc s6 nhdn bing 9 vd b6i s6 ctra g nh6 hdn 82, thi kdt quA s6 ld mQt sd cri chin cht sd gidng nhau. Thi du : 12345679 x 9 : 111 111 111 12345675 x 63 = 777 777 777 Tuy nhi6n con s6 tr6n cdn cd m6t dac tinh thltc ki la khric nila rnd it ai ngd t6i. Dac tinh kl la ndy duoc phrit bi6'u nhrr sau : Khi nhAn s612345679 v6i cric sd nhdn nh6 hon 82 ddng thdi kh6ng phAi ld sd 3 vi bdi sd ctia 3, thi kdt quA s6 li mOt sd cd 8 ho[c 9 chu s6 md kh6ng mOt chrf s6 nio tritng nhau". O aay cring xin mr& ngodc ncii th6m, sd di c
Vids:

12345679x 12345679

7

=86419753;

x 25 =

12345679 x'?4

=

308641975 913580246

;

w... Ta thdy trong tdng sd 81 sd nguy6n ddu ti6n d5 c6 tdi 54 sd nhAn cho kdt quA dnc bi6t nhrr vAy trlc li dat ti 16 2l3.Ti 16 rdt cao nay chrlng t6 dAy kh6ng phAi ld m6t srr ng6u nh:6n md th{c srJ ld m6t quy luQt ddy bi dn.

tu nhung phfc tap hon. D4c tinh ndy drroc di6n dat nhtt sau

:

a. 56 dd khOng phf,i ld 3 vd b6i s6 cria 3. b. Tdng cac chri sd

cta

sd dci nh6 hon 9.

Thi kct

quA s6ld m6t sd cri mudi ho?" 11udi m6t chr/ s6.

tit

Ndu ld mtrdi ch(I s6 thi dci sO ld cric cht sd 0 ddn 9, kh6ng m6t chrt sd nAo tring nhau"

Ndu ln mudi mOt chir sd thi tdt y6u phAi cri t6i thidu hai chrr s6 trirng nhau. Ta goi day In

kdt qud thqc. Ta thrrc hi6n thuAt to6n don giAn nhtl sau : dem s6 ddu cria kdt quA thuc cOng vdi s6 cudi ctia chinh nd ta drloc mrJdi chil sd nri ta goi ld kdt qud. dd. dibu chinh. Va khi dd

mddi chrl sd niy lai trd thinh mudi chil sd kh6c nhau til 0 ddn 9. Vi du : 987654327 X 8 = 7901234568 987654321 x 13 = 12839506173 (kdt quA thtlc) 2839506173 * 1 = 2839506174 (kdt quA di6u chinh) 987654321 x 61 : 60246913581 (kdt quA thrtc) 0246913581 di6u chinh)

* 6:

0246913587 (kdt quA

Ta thdy trong tdng s6 99 sd nguydn nh6 hon 100 dd cci 33 sd nhdn cho k6t quA dac bi6t, ttlc chidm ti l9 1/3. Ddy ln mdt ti 16 kh6ng nh6. Vd nhri vAy m6t ldn nta, ldn thrl ba, ta thdy 16 rdng dAy khOng phAi la mQt s{ ng6u nhi6n.

Ddn ddy, chhc cdc ban cring nhu t6i ddu rdt thich thu ndu drtoc doc crlng tr6n tap chf niy ldi giAi thich v6 nhtng quy luAt ddy sE

bi dn dd.

cHudNg rRiNH RUr GgN rirun s6 uEr FtBoNAccl Trong b5o THTT s6 6 - 93, muc Tin hoc, tdc 976 d6 n6u chuong trinh sau dAy dd tinh cdc sd hang cria sd liQt Fibonacci. 1

Gidi ddp bdi

M+

Con sd 73

+ -

kI le

GiA stl a vd b ld hai phdn cria mQt sd N ndo dd md ld bOi s6 cria s6 n phAi llm. N = ffi . Gid s{, N = 100a * b (1). Ta binh phrrong (1) l6n dtac 10.000a2 + 2o0ab + b2 Q). (2) cring la b6i s6 cria n. Nhdn (1) v6i 200a

dtrgc : 20.000a2 + 200ab (3) ; (3) cfing ld boi s6 cl8,a n. Ldy (3) trrf cho (2) ta c6 : 10.000a2 - b2 @); (4) ciing ln bqi sd cia n. Bi6t ring ta phAi tim o vi b sao cho a2 +b2 6) bQi s6 cria z. Md (4) + (5) = 10-001a2 fi la ring li bQi s6 cliua n. VAy n li m6t udc s6 cira 10,007. Ba rtdc s6 cta 10.001Id : n : I (ldi giAi tdm thudng) ; n = 73 (ldi giAi da bi6t) vA n = 137 Odi giei cdn tim). Ta thtl nghiQm : Ldy N = 325 x 137 = 44.525. Trncli o - 445, b = 25 c6 a2 + b2 = 198.025 + 625 = 198650 = 1.450 x 137. eiNH pHrJoNc

offnn DEN

MRC MRC

MM+

(

I

L4p lai hai ddng cudi M6i dbng cta chuong trinh ldn lrrgt cho cic sd hang clia s6 liQt Fibonacci : 1, 1, 2,3, 5, 8,.... Kdt thric m6i ddng ta cd mQt sd h4ng tr€n khung s6. Sd h4ng ldn nhdt thu dudc tron m{iy tinh b6 tfi th6 so li ur, : 63245986. B4n Ng6 Kim Anh, l6p 12 42, PTTH Vi6t Tri da goi y chrrong trinh ndy cti thd rrit gqn vd dtta ra mQt chrrong trinh khde ng6n hon.

lu-

1

l*MRC I

lai 2 dbng cudi

"ap

Nhrrng tidc ring chuong trinh niy khOng phn hop v6i m6t s6 mriy th6 so. Vi du m6y SANYO OX 110. Tuy v6y y ki6n cria b4n Ngd Kim Anh de girip tdc gSh cAi tidn chrrong trinh tr6n thdnh chrrong trlnh sau dAy : M+

DOm thu

trlng

s6ng, gid reo

Ph6 phrrdng nhOn nhip, dbn treo sring ngdi Vui chdn dao d6m ddn choi Hon ba trdm nggn, h6i ngtliri cd haY ? K6t n6m trdn sd ddn niry Bdy dbn kdt mQt, thi hai ngqn thia Kdt chin cbn bdn nggn du Ngdn ngo em ddm tlt mil dEn sao HOi ngrtdi tri sring, tii cao Dtng chdn t{nh girip cri bao nhi6u dbn ? NcuYEN oiNll ruNc

MRC M+

L4p lai hai ddng cu6i

{

,tF

Chuong trinh'dE drrgc nit gqn ro rQt, kdt thric m5i dbng ta vin drrqc mQt sd h4ng crla sd li6t. MOt chi ti6t nh6 ld : sd hang Idn nhdt thu dugc bay gid kh6ng phAi ld u, mi lA u:s : 39088169 Tdc gSn hoan nghOnh vA ch6n thdnh crim on ban NgO Kim Anh dd e6p !. NGUYEN DONG

ISSN: 0866 -

8035.

M6 sd : 8BT04M4

Chi sd 12884 In tai Xtr&ng Ch6bAn in Nhd xudt bAn GiSo duc. In xong vi grli lrru chidu thdng 4 11994

Gi6:

1200d

MQt nghln hai trdm tlbng

r { ]i

i