7/21/2019 THTT So 278 Thang 08 Nam 2000

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.

{ HOC

VrEr

NAi/&

ffi

C&,

;ffiffi

:.'

,i

r.

{'

[_.

.

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'

$

,,

'=

=:'

f

[=

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1

.tl

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E

:

i-

'iw

c0NG

THUc

TiNH

rdxc

wffi

rewffiffiffi

wssffire ffi

Cdc ban

nh6

ch5c rAt

tUng

tfng khi

gap

nh0ng

ph6p

tinh

tdng

phtlc

tap, chdng

hAn

tinh A

=

1+2+3+4+5+6+7+B+9+tO. ruhfng

viOn

Oic6 thd

gifp

ich cho ban, gi6ng

nhucdc

nhA

nghi6n

cuu

toan

thoi

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i]

1-1

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.

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bi

nhu

hinh

1

vd

l5y c6c

bdng

giAy

ngdn

cdch

ch0ng

tfri

oe dang

thAy

1+2+3+4+5*o

=

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=*

=r,

22

TU

d6

c6 thd tdng

qudt

h6a

thirnh

cOng

thtlc tinh

tdng

1

1+2+3+...+n='D(n*1).

2',

.

Hlnh

Anh

b6n

girip

cdc ban

d5 nhd

c6ng

thr1c,

d6ng

thdi

cUng

goi

y

c6ch

chfng

minh cOng

thtlc

Hinlt

l

DANH CHO

BAN

DOC

1) H)nh

2

goi

cho

ban c6ng

thrlc

gi

? Ban

c6 cdch

x6p

bi niro khdc

O6

tim duo.c

c6ng

thtlc

do

kh6ng

?

2)

H6i nhutr6n

otii voi

hlnh

3.

Ndm

phdn

thudng

danh cho

cdc

ban

g&i

v6 sdm

va c6

y

tLrdng hay.

rrataa

I

t

I

I

I

I

I

ttta

ii

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at

ot

at

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t:

*

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1r

tl

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ctAt

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3

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dinh

li Ole

dein

ndm

"c&o

hidm"

MOt

s6 ban dd

chfng

minh duoc dinh li Ole. Cdc ban dbu lh|y :

Ca.D

=

2C vd

Cm.M

=

2C.

TiJ

d6

se

cd,1*

1

-Ql

M

-c+2

=

1o1.

N6u

ca.

c->3rhi

-

-*1.1*1=l

rarthuAn.

CdCm2C2CC2-'CoCm442

maCa

Cn23,suyra Ct=3hoicCm=3.Khi Co=gthi

+=1-1nen3

x+y+z

Gie sfr:

r>

y>

z.Xt:

(x*y\(*-y\

+

1y-z)02-

z2)

+ (z-x)(22-

*)

>

0

e

2{rt+f

+;7

2

ryz+xz2+y*+yz2+z*+zy2

(l)

C$ng

hai vd

cria

(1)

vd,i

f+y3+23

vil

bidn

ddi

c6 :

3(f+y3+f)

>

(x+y+z)(f+y2+22)

>-

>

(r+y+e)

iW*

+

f+y3+23

>

x+y+Z.

DPcm.

D&ng

thric

xty ra 0vd

ab+bc+ca

=

abc.

.{ar+Zb, .{br+Zc, .{C+Zo,

r

+

,

+-/-YJ.

a0 0c ca

Gi.di. DLtl

=

r,

1

I

' a

U= i-=z'

Ta

c6

:

x,

,

z

> 0

vI

x+y+q,=

|

Ta

phf,i

chring

minh :

-lll

Xt

u'=

(x,

r'lZy

);

v'=

( ,

tl?z);

t

'=

(1,

"12

x)

lll

tht u'+

v'+

t

'=

(x+y+z;',12y

+

"l2z

+

l'l?x)

hay lx

f+

?+

/=

(t;

O)

vi lrl+flf

+trl

>

tt+f+rl

non

ta

c6

dpcm.

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theo

bdt

df,ng

thric

Bunhiacdpxki

:

.lW6>

x+y+y:

'ltO4*+A

>

y+z+z:

{164r,

+xz)

>

z+x+x,

tr)

d6

cung c6 dpcm.

4.

DH

N0ng

nghiQp

1

Cho

a,

b, c

>

0vd

abc

=

l.

Tim minP

vdi

-bccaab

=ffi+

bh+b6+

cza+czb

Gi.d.i.

^tru6c

hdt

ta

c6

: N6u r,

y,

e

>

0 thi

:

*

*1

+I+

1+3+

1

y+z

z+x

x+y

| / 1 I 1.9

=

,(x+l+l+z+z+x)

(

r.,

*

yn*

,n

)4,

Hav:a*I*3->?.

'

y+z

z+x

x+y

'z

DEng

thrlc

xhy

ru

a lr.y=z

(day

lI

bdt

ding

thfc

Nasobit)

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aDc

Ta

c6

:

x, ,2>

0;

ry4

=

1 vi

'

P=

* *L*L

y+z

z+x

x+y

xk+y+z\

yk+v+d

zk+y+zl

=r+-+--(r+y+i/

+z

z+x

x+y

=e+y+z)(

r

*

I

*'

-l)>

Y.

'\

)+z

z+x

x+y

/

2.

,w_l

'z

-2

7/21/2019 THTT So 278 Thang 08 Nam 2000

9/28

Vfy MinP

=

6=S=6=l

g

a=y=4=l

I

5.

Dai

hoc

M6

-

Dia

chflt

MBCc6:0 tgx

+

cotgx

(1)

Gini.

0)

0.

Dft:

er+cotgx=

t>

2.

(1)

0.

DltP=u)4.

Ta

chfng

minh

:

flu)

=

u3

-

7uz

+

l4u-

8

>

o voi

u>

4.

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b6ng bidn thiOn

c:0,a

flu)

ho[c

vidt

fla)

=

(u-l)(u-2)@a)

dd

suy

ra

dpcm.

7.

Dai

hoc

Ngoai

thuung

(TP

HCM)

Cho tarn

gidc

ABC.

Gid

stl

14

ld

m/t ilidm

thay

ddi ten

cenh BC.

Gpi

Rt, R2ldn

laqt

ld

bdn

kinh

vdng

trdn ngoqi

tidp cdc

tam

gidc

ABM

vd

ACM. Hdy

xdc dinh vi

trl crta

didm

M

sao

cho

ft,

+

R,

ld

nhI

nhdt.

Gini.

Ap

dung

dinh

li hlm

sd

sin cho

c6c

tam

gi6c

ABM

vl

ACM

ta c6 :

I

oat coo"

R,=

AB

:

R.=

'

ZsinZAMB'

L

Do

sin/AMB

=

sinZAMC

nln:

R,

+ R,

=

f,{ea*er,

#*r>

ltea*ee

Dtng

thrlc xby

ra

7/21/2019 THTT So 278 Thang 08 Nam 2000

10/28

cAc

cu0c

rHl

roAnt

GOM

NHIEU

CAU

HOI,

TRA

LOI

NHANH

NcuvEN

vEr

HAt

thio

luan

vdl

10 bAi

tqp

nh6

trong

oo

qlrlt

1OA

tni

ding

trong

THTT

s6

257,

thdng

1 1/1998).

CuQc

thi todn

0-vttng

Ftanders

thuQc.nydc

Bi

c63'ubng,lbngtgbrirtSt?,u\(\\iirtstg\\e.m\{rn

trong

90

phrlt,

vdng

2

g6m 30 cau

h6itrdc

nghiQm

ldm

trong

60

ph0t,

vdng

3

gOm

4

bli

tap

kh6.

CuQc

thi

Olympic

todn

d

ntdc

Bi

g61n

3

vdng

:

vdno thi

so

tuvdn

q6m

30

cAu

h6i

trdc

nghi6m,

von[

z

gom

eb

ca-u

nOi

tric

nghiOm

vd

bdi

t&p

nh6

ldm

trong

90

phtit,

vdng

3 thi

chgn

hQc

sinh

gi6i

qu6c gia

gom

4

bAi

tAP kh6.

CuOc

thitodn

Canguru (t6n lodi th0 c6

trii

s6.ng

d

Oxtralia)

d

Oxtralia

c6

tir

nhfing

nim

80

gOm

cdc

cAu

h6i trSc

nghiQm,

d6n nay

c6

tdi

nfia

triQu

hqc

sinh

tham

gia.

Cdc

nhA

todn

hqc

PhAp

vd

Oxtralia

dd bdn

bac

td chrlc

cuQc

thi

Canguru

khOng

bi6n

gidt

(Le

Kangourou

sans

lrontidres)

tir

ndm

1991,

d6n

ndrn

1993

da

lan

sang

qdc

nudc

chAu Au

vd

du-d. c

sq,

0ng

hQ

cria

Uy

ban

ch6u

Au

vd

UNESCO

trong

D6

dn 2000.

.SO

trrgng

thi

sinh

tham

gia

cuQc

thi

nAy

tdng

dAn :

1991

1993

1995

1996

6

Ph6p

1 20

000

420

000

500

000 600

000

d ngodi

40 000

200

000

500

000

nudc

Phdp

D5n

ndm

1996

da c6

hoc

sinh

c0a

26

nulc

tham

gia cuQc

thi Canguru

kh6ng

bi6n

gidt:

Phdp,

Drlc,

Anh,

T6Y

Ban

Nha,

ltalia,

Lucxembua,

HA

Lan,

Nga,

B6larut,

Ba

Lan,

S6c,

Sl6v6nia,

EstOnia,

Hunggari,

Max6d0nia,

M6ndavi,

Rumani

(chdu

Au),

Oxtralia

(chAu

Uc),

Bu6ckina

Phas6,

COt

Divoa,

Mar6c,

Madagasca,

MOritani,

SOnOgan,

Tuynidi,

(" ''i11.Phi), Libing

(chAu

A). Cdc

bqn mu6n

bi6t chi

ti6t

hdy gt]i

thu

d6n

dia

chi

:

Kangourou

Sans

Frontidres

50

Flue

des Ecoles,

75005

PARIS-FRANCE

ho{c

tr6n

lnternet

:

http

;

//Aaruw.mathkang'org'

Xin

gidt

thiQu

mQt

s6

cAu

h6i

trong

cu6c

thi

cZigiri

xi6is

oi6i

iia

ii^

rsgz.

iuoc

ini

td

Cac

cuQc

thi

todn

duo.

c

td

chfc

bdng

nhi6u

hlnh

thrlc

khdc

nhau.

C6

hinh

thtlc

thi

dua

ra

cdc

bdi

ioan

tro,rrg

dOi

kh6,

ldl

gi6i

ld

mOt

chu6i

suy

luQn

phtc

tap,

dbr hbr

tirnh

0Q

phhn

Yrch

\6ng

hgp

kh5

cao.

C6c

bdi

todn

loqi

ndy

dugc

ddng

trdn

bdo

chi

dd

c6 thdi

gian

dAi

suy

nghl

gi&i

d6p

hoic

dirng

trong

cdc

LuQc

lhi

chgn.hqc

srnh

grdi'

Hinh

thrrc-khe;la

bai

thi

gOm

nhiou

cdu

h6i,

bdi

tQp

khOng

qud

kh6

vdl

y6u

cAu

tri

ldi

nhanh'

Loai

nav

doi hOi

nqc

sinnbiot

dp

dqng

dung

va

nhanh

cai;

fien

th(c

todn

pnd

tfrong

vdo

cdc

trudng

hgp

khdc

nhau,

cdc

tinh

hu6ng

thtlc

t6, tap

dugt

suy

luAn lOoic.

Mdt

khdc

bai thi nhiiiu

cdu

h6i

n6n

c6

thd

ba;

g6m'cdc

phAn

khdc

nhau

cta

chLrcmg

irinh,

kid"m

ira

sLr'nidu

oi6t

rQng

v6

ki5n

thrlc

todn.

Trong

cdc

cuQc

thi

ndy

thf

sinh

khOng

duoc

s0

dqng

cdc

loai

sdch

vd,

mdy

tfnh,

thudtc

t(nh,

gi6y

v6-dO

thi

mA

chi

duo.c

ddng

thudc

k6-thdng,

6o,irpa.

Bai

thi

cho

trong

dqng

cAu

h6i

tric

nqhiOm

(thi

sinh

dtrgc

chgn

mQt

trong

cdc

ddp dn

c6o

s5nj

hofc

cdc

bAi

tQp nh6

(thi

sinh

phAi giAi

bdi

tap

nhurng

chi

ghi

ddp

sO mA

khOng

trinh

bAy

ldi

qidi),

di6u

ndy

girlp

cho

viQc

chAm

thi

nhanh

ch6ng

(c6

thd ch6m

bdng

mdy

tinh),

dOng

thdi

vugt

qua

duo. c

hAng

rdo

ngon

ngfr

gitla

cdc nudc

khdc nhau. Vdi nh0ng dqc di6m tr6n hinh

thtlc

thi

todn

gOm

nniou

car]

h6i,

tra

ldi

nhanh

c6

thd

ddnh

cho

d1ng

dAo

hqg

sinh

&

nhiu

vdng,

nhl6u

ntdc,

dflp

r4g

nhu

cAu

giao

lur

todn-.hgc

gi0a

cdc

virng mi6n

khdc

nhau.

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ta

t6 chfc

thi

cUng

mQt

dO

bai vAo

cirng

mQt

thdi

di6m,

sau

d6

gAt

bAi

gidi

ngay

qua

p&t

drpn

(trong

tudng

lai

c6

thd

qua

thu

di6n

tit)

v6

cho

Ban

t6 cht?c

cuQc

thi,

Hinh

th0c

thi

nAy

d5 duo.c

td

chrlc

d My,

Oxtralia,

Bi,

Phep,

cdc

nudc

D6ng

Nam

A

vi

mQt

s0

nudc

chAu Au.

d nhi6u

nudc,

cuQc

thi

ndy

duo.

c coi

ld vdng

so

tuydn

cho

cuQc

thi

Olympic

to6n

qu6c gia,

Sau

dAy

lA mQt

s0

cuQc

thi

:

Trung tAm

gi6o dr;c

khoa

hq-c

vA

todn

hqc

DOng

Nam

A

(RESCAM)

da

td

chrlc

thi

todn

nhi6u

c6u

h6i,

trd ldl

nhanh

d

Malaixia

(7/1996)

vd

cho

9 nudc

EOng

Nam

A

1sltsoa1.

CuOc

thi

nAy

c6

3 vdng

:

v0ng

1

ddnh

cho

c6

nhdn

gdm

20

ciu

h6i

ngin

trong

90

phtlt'

vdng

2

ddnh

6ho

cd

nhdn

gOm

5

bdi

tap

nhd

trong

180

phtit,

vdng

3

cho

ph6p

cA 4

tht

sinh

trong

dQi

7/21/2019 THTT So 278 Thang 08 Nam 2000

11/28

chfc

ngay 2113/1997

cho 2 trinh

dQ

:

cdc

tdp dQ trt,

dQ tam

(tuong

duong

ldp

I, ldp

9 THCS)

vd cdc

tdp

lyc6es

(tuong

duong

cAp

3 THPT)

Thdi

gian:

75

phtit.

OC

tni

gOm

30

cdu

h6i

:

Cdc

cAu

h6itit

l d6n

10

duo.

c

3 didm

m5i

cdu

;

cdc cdu

h6i tir

11 d6n

20

Ouqc

+ didm m6i

cAu

i

cdc

cAu

h6i rU

21

d6;

ao

duo. c 5

Oidm

m6i cdu. Thf

sinh

duo. c chon

1

cdu

tri

loi

d0ng nhdt

trong 5

ddp

dn A, B,

C,

D,

E.

cAc

Lop rucs

CAu

6. Cho

ba didm

A,

B,

C khOng thEng

hdng.

Th6m

didm thf

tutren

mat

ph&ng

chia

A, B-,

C sa-o

cho

4

didm

d6 lQp

thAnh

rnQt

ninn

Uinn hdnh.

C6

bao nhi6u

didm v6

thOm th6a

m6n di6u tr6n ?

A:1 B:2

C:3

D:4

E:6

Ciu 10.

Budi

sdng. LOra

dang trang

didm,

lloeng

nhin

thdy tron

guoJrg

cni66

o6ndno

qud

lSctre.o

phia sau lung.

M{tdOng

hO khOng

ghi

bing

chrf

s0 ArQp.

C0 b6 k6u

l6n

"DOng

ho.ch6t

r6i

Bdy

gid

4

gid

k6m

5

phrit

A ?'. Lora

da nhAm.

H6i

dOng

h0

l0c

d6

chi mdy

gid

?

A:8h05

B:4h55

C:7h55

C6u 12.

M6i

c4nh c0a

hinh

ch0

nhit

duoc chia

thAnh

3

doqn

bdng nhau

vA cdc

didm

chia

duo. c

nOi vdi nhau

di

qua

tdm

nhu

hinh ben.

H6i di6n

tfch

phdn

Od trdng

bdng

bio

nhi6u

phan

c6

cdc'.,dAu.

chAm ?

nqoAi hdng

rAo

md huou

c6 thd an

duo. c

(tfnh

theo

m')?

A:96

8:99,14

C:1O2,28

D:105,42

E:108,56

CAC LOP

.

C6u

3j Sdu

nga

duOtng

dAn d6n

m6t

ddo

trdn

(bUng

binh),

trong

dd

c6

2

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mQt

chi6u

(chi

duqc

di

vdo).

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dio

trdn

phii

di nguoc chi6u

kim

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H6i

c6

bao

nhi6u

cdch vdo

-

ra khdc nhau mA

kh6ng di

qu6

1

vdng

quanh

dAo trdn

?

A:12 B:48

C:24 D:30

E:28

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Xem cdu

13 THCS

Ciu 1 1.

Xem cAu 1

2 THCS

GAu

20.

(c0ng

ld

cdu22

THCS)

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st5

1O...O,.trong

cdch

viOt cO

tggZ

ch0

sti

0 vi5t sau

chf

s5

1,

c-ho

15

thi

duo.

c

du

ld

bao

nhi6u ?

A:1

8:6 C:9

D:10 E:12

CAu 21.

Xem cAu 30 THCS

THPT

,=NlP\*,',

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B:1/2

C:1/3

D:1/4

E:2/3.

CAu 26. Nau

r(f(x))

nhi6u

?

A:

(x)

=

-2x+3

C:

(x)=+fi-s

E: f(x)

=

-4x+

1

=

4x-3

thi

(,

bdng

bao

B:

f(x)

=

2^'lx

-

3

D:f(x)

=2x-3

D:ar/5

E:

phu

C6u 13.

Trong dudng

trdn

bdn kfnh

3cm

vE mOt

hinh

ch0 nhAt

ABCD.

Ggi

l,

J,

K,

H

ld

trung

didm

cdc

canh

cia

hinh chrl

nhAt

dd.

H6i chu

vi

c0a

hinh

thoi

IJKH

bdng

bao

nhi6u

(cm)

?

A:6

B:9

C:12

-

.t

Vua

qua,

Bg

Gi6o

duc

vi

Dlo

tao

dl td chrrc

trong thd

L0

kr niOm

90

nlm ngly

sinh cfra

Gg

T4

Quang

Bfru

(23.7,1910).

Bd

tru0ng Nguy0n Minh

Hidn

Aa Adc Uii

Oien

vtrn

ddnh

gi6

cdng lao

tg lon

cfla

CS

Ta

Quarg

Buu

trong

c6c

hnh

vuc

kiroa hoc,

quOc

phdng,

ngo4r

giao

vi

dic

biQt li

rong

su

nghi0p.gi6o

dUc. Nhdn

dlp

nhy,

THTT

rrqn trgng

girii

thi6u

bhi

vi0t

crla

GS.

TSKH

NgO ViCr

Trung

v6

nhlng

k niirn

vcn

CS. Ta

Quang

Buu

(trang

2y.

*

_Chiiu

ngiy

1.8.2000,

Dar

hqc

Qu6c

gia

He NOi

da rd

chric

Lf

tuy6n duoug

khen

thuong

c6c

hoc

sinh dat

gihi

cao

trong

c6c

lii

thi

Qlyftpic

quiSc

t6,-Ciao su

-

Vign

ii

ilguy6n

VIn

DAo,

gi6m

d6c DHQG

Ha NQi

da

nhiOt li0t

hoan nghOnh

vi

chric m-ing q6c

.em

h}t

sinh

gilnh

thdng lE uong

iac

H

thi Olympic

qu6c

t0

u&

v0,

D{c

bitt, diy

li

16n

diu

tin

ci,3

em

d4t

huy

chwng Ving

m1n Toin

cfia do?tn ViQt

Nan du

lA

h7c

sinh

khdi

chuyn Toin

-

Tin

hgc

crta

twng

DHK\TN

thuQc

DHQG

HA

NAi,

Cir6o su

dI

d6nh

gi6

cao ss

c0

glng

cila

c6c em

hoc

sinh cr)ng sU

d6ng

g6p

cfra c6c

nhi

gilo,

sq

chdm s6c

cfra

cdc

vi

phu

huynh.

NhAn dip niy

DHQG

Hi

NQ

di

tuy0n durrng khen

thudng

cdc emhoc

sinh

dat

giii

vd

nhiOu

nhl

gi6o

c6 thhnh tich

lroug

vi6c

boi duoug

hoc

si,h

gidi'

THTT

thuoc

h)nh

chO nhAt

CAu

26.

Nguoi

ta

gdp

doi

m6t

td

giAy

hinh

chff

nhat sao

cho

hai chi6u

rQng

trUng.nhau.

Trong

m6i

?n

ggp

doi

ti6p

theo

cAn

phdi

gap

vu61g

g6c

vdi

I_an

gap

trudc.

Sau

5 lAn

gAp

ngudi

ta

cit

mOt

ch0t

o4

dinh hinh

ch0

nhQt

d.5

gip.

H6i

c6

bao

nhi6u

t6

thtng

d

b6n

trong td

giAy

duoc

m0 ra

nhu ltic

ban

dAu

?

A:4 B:9

C:18

D:20

E:21

CAu 30.

MOt

con huou cao

cd

dtrong

mOt mAnh

dAt

hinh tam

gidc

vdi d0

ddi ba

c.anh lA

20m,

16m

vA 1

2m. MAnh dat

c6 hAng

rAo thdp

nhung

huou c6

cd

dai nen

c6 thd dn

c6 xahh ngon.b6n

ngoai

manh

Odt

cacn

hAng rdo

2m.

Trong-cdc

sO siu

d6y, s6

ndo

bi6u thi

gan

dUng

nhdt

di6n

tich

phAn

c6 moc

7/21/2019 THTT So 278 Thang 08 Nam 2000

12/28

THI

TOAN

Ki

tniTodn

qutSc

to

1tuo1

lAn

th0

41 duo. c

td chfc

tqi

Taejon

(HAn

Qu

7/21/2019 THTT So 278 Thang 08 Nam 2000

13/28

Bni 3.

Cho

tnltrc

s6

nguyOn duong

n>

2.Ll6c

dtu

c6

n

cnr,

bq

ch6t &

tren

mQt

dr:Ong

thtng

nim

ngang sao cho

khOng

phti

tdt

c6 chring

ddP

0

tai cing

mQl.

didm.

Vdri

m6i

sd

thuc duong

1,, mOt bwfic chuydn

c6c con

bq

ch6t

duqc

dinh

nghia

nhu

sau

:

Chqn

hai

con

bg

ch6t

nlm 0

hai

didmA

vl B

trong

d6

A

nXm

bOn

tr6i B.

Sau

d6

dd con

bq ch6t

O a-nney ffi

di6m C

nim

trOn

ihdng

thtng

nly

vir

0 bOn

phtri

B vfi

BCIAB

=X.

X6c

dinh

tdt cL

cdc

gi6

t4 dtrcrng

l.

sao

cho

vdi

mgi dim

M n[m

ren

du0ng thtng

tren

v]

vdi

mgi

vi

tri

ban diu

ctra

n

con

bq

ch6t,

ta d0u

c6 thd thuc

hign

dugc

mOt sd hrru

han c6c budc

chuydn

dd Crn

to}n b0

c6c

con

bq

ch6t

tdi

cdc vi

tri

nlmbOn

phAi

dimM.

NGAYTHUHAI

Thdi

gian

ldm

bdi

:

4

gid

j0

phrlt

Bei

4.

MQt

nha

[o thu0t

c6

mOt

tr[m

tdm

thO

d6nh sd

rir

1 tdi

100.

Ong

ta

s{p

xdp

tdt

ct

chring

vtro ba

chidc

hQp, mQt

hQp

son

mdu

d6,

mQt hQp

son

mlu

tr{ng

vtr

mQt hQp son

mtru xanh sao cho

mdi hOp ddu c6

chria

it

nhdt mQt tdm th6.

MQt

kh6n

gil

dugc

d0

nghi

chgn

hai hQp tty

f

trong ba

hQp

n6i

trOn

rdi tir

mdi

hQp

dugc

chqn

rut

ra

mQt

tdm

th6,

cQng

hai

sd

tren

hai

tdm th6

dx

rut

ra

vt

thOng

b6o

k6t

quA.

Chi ctn bidt

duqc

tdng

nly,

nh}

6o thuflt c6

thd

n6i

chinh

x6c

chi6c

hQp mh kh6n

gifl

d6 dl

khOng

chqn.

H6i

c6

tdt

ce

bao

nhiOu c6ch

s{p xdp

100

tdm

thO

nly vtro

ba

chidc

hQp trOn

dd

6o

thuflt

nly

luOn

thtrnh cOng

?

(Hai

c6ch sfp

xdp

thO

duqc

coi

ld

kh6c

nhau

ndu trong

hai

cdch

xdp nly

c6

it

nhdt mQt

t{m

thO

duqc

d{t

0

hai hQp kh4c nhau).

Bni

5.

X6c dinh

xem

c6 tdn

t4i

hay kh0ng

sd

nguyen

duong

n

thba man ddng thrli hai

didu

kiQn sau :

n

cO

dring

2000

udc

nguy0n

td

ph0n

biQt

vl X

+

I

ctuahdt

cho n.

Bei

6. Gil srlAIll,

BHz,

CH3ltr ba du0ng cao

ctra

mQt tam

gi6c

nhgn

ABC.

Dtrlng

tron

noi

tidp

tam gi6c

ABC

trp

xric

vdi

c6c

canh

BC,

CA,

AB

tuong

r-mg

14i

71, 72,

73. Gqi

h

ll

dr$ng

thlng

ddi

xtmg

v|i

H2H3

qua

7273,

/2

lI

tludng

thflng

cldi

xung vdi

H3Hl

qua

I3I1, 13

lI

sin(cosx).

Solution,

Because

of

periodicity,

we need

consider

only

values

of

.r

belonging to some

intdrval

of

Iength 22. Thus,

assume

that

-nl2

3

x

0.

Hence

cos(sinr)

>

sin(cosr)

for

nlT < x

n.

xn_rlxn+x)

xn(xix2)

T41278. T\e

similar ftiangles ABC

and

APtCr

satisfy the

conditions

:

A1

lies

on the

ray

CB,

B,

lies

on the ny

AC,

C,

lies

on

the

ny

BA.

Prove that

the orthoccnlcr

of tliangle

APP.

coincides with

the

circumcenlcr

ol

triangle

ABC.

TSl278.

The vertices

of a convex

hexagon

are

labelled clockwise

by

6

consecutive

even

numbers.

We

make

a

change of

labelling

by

choosing

a side

of

the

hexagon

and

adding an

integer

to

each

number

written

at the

extremities of this

side.

Is

this

possible

that

after

a finite

nurnber

of

such changes,

the

numbers

written

at the vertices

ol

the

hexagon

becomc

all

equal

?

FOR

UPPER

SECONDARY

SCHOOLS

T61278.

The

sequence

of

numbers

(u)

(n =

0,

1,2,...)

is

defined

by :

uo

=.1,

ut

=

l,

iln+z

=

I999un*1-

un

for

every n

=

0,

7,2,

...

Find all natural

numbers

n such

that

a,

is

primc.

'l7l27t].

Prove

thal

h/t

sirul'+sin2l+sin3r-2(en+1)+

+.jf,l

+5

L

nam

tim

7/21/2019 THTT So 278 Thang 08 Nam 2000

19/28

ctAr

eAr

ri rnuoc

*

Xdt

htrm

seiflx)

=

2x.arctgx

-

ln(l

+

x2)

c6f'(x)=2.arctgx.

Vdi.r

>

0

thi/'(-r)

>

0;

vdi

x 1+ln(1+-r2)

=)rr*rn(l

+

xz)27=ffiffi

.irrr.

Tn'( 1)

vh

(2)

suyradidu

cdnphtri

chring

minh.

Nhin

x6L 1)

MOt

sdb4n

chrlng minh

bdt ding thrlc

(1)

qu6 phfc

tap.

2)

Tdt

cd c6c

ban ddu

gili

ddng

vtr rdt

d6ng

khen.

Clm

on c6c ban.

lErr6NcNsAr

Bei

T71278.

Tim

nghi\m

duottg

cia

phuury

tritth

,,1

I

rrn(r*1)'*i-

itn( t*1)'?

=t-x

x)

\-

f)

Lcri

giii. (cia

Trdn

Tudn

Anh,

tdp

12

To6n,

tnrdng t

Qu

DOn, Nha

Trang, I(hdnh

Hdu

Trdn

Dinh Nguyn,

PTNK

DHQG

TP H6

Chi Minh

vt

mor

sd ban

kh6c).

Vd'i.t

>

0,

dua

phtrung

trinh da

cho

vd

d4ng

tuong

duorlg

sau

:

(x+r)rr(,

.1

)

-

(x3+x)hr(

t

+

i)

=,

-,

e(.r+l)ln(r+1)-r=

=

r[

(.r2+1)

rn(

r

+

i

l_

,

]

o)

Dar/(.x)

=

x[(r+1)rn(r

+*)-,

],

tr,i

rrl

c6

d4ng

11x1

=

f(*)

Q)

'tacof

'(x)=

(2x+1)ln(

,11-z

=(z,r+1)[rn(r+*l

=]

r:i

2

D{t

s(x)

=

*( t

.i

)-

l,

voi.r>

0.

,*;

DE

thdv

s'(x)

=

I

I

uw

vtqJ

6

\^,,,

-

,(r*D

*

_r___+

4

4x(x+l)

(l+x+l)

< 0 v->0

VQy

S(x)

l)

hdm

gi6m

khi,r>0, mtr

rims(.r)

=

rmI

m( r

+

i

)

-

*

]

=

o

r-)l{

x-+rob'

\

^'

**;'

+

g(r)

>

g(0)

Vx

>

0

Do

2x+I

>

0

Vx

>

0,

nOn

tt

(3)

suy ra

f

'(x)

> 0

V->0,

tric

llfl.r)

lh

him

t6ng

khi

>0.

Do

vfly

Q)

a

*=

12

y

>

0 thi

ln

|

,2'-1"

'yx+y

2)

Trong

sd 21

ban

gili

sai,

thi sai

ldm

chri

ydu

lb

:

-

NgQ

nhfln

mQt

htrm

sd

ll

d6ng bidn.

-

Tinh

todn

dgo

htrm

cria

c6c

him

siu

viQt

sai.

Dic

biet

cd mQr vli

ban bidn

ddi

;

,l

1

tnIt+i)=tnt.tni(tt)

\

^x-

/

x'

3) C6c ban

sau d0y

c6

ldi

gitri

tdt.

Hi

Ndi:

NgryEn

Hodng Thgch,

10T,

Ha

NOi

-

Arnsterdaq

NguyZn Tutn Duong,

l0T,

Trdn

Tdt Dqt.

l2B

to6n,

DHKHTN-DHQG

Hr

Noi;

Tp Hd

Chi

Minh

: Hodng

Thanh Ldm,12 I'of,n,

Luong Thd Nltdn,

1l

To6n, DHQG Tp Hd

Chi

Minh;

Thrii

Nguyn:

Vd

Quang

Dftc

vd

Trdn Dfirc

Mgnh,

11

To6n

NK

Th6i

Nguy0n

Vinh

Phric: Nguy4n Ti6t

Thinh, llAl,

Nguy4n

Trmg

ljp,l2A,

chuyOn Vinh Phric

;

Bdc Ninlr: E6 Arth Dfir, Nguy1n Ngpc

Cudng,

H6a 11,

n

7/21/2019 THTT So 278 Thang 08 Nam 2000

20/28

crAr

BAI

r'irnuoc

Todn

11,

Trudng

TI{PT

NK

B6c

Ninh;

tlli

Dtrong:

Nguy1n

Thanh

Hdi,

11 To6n,

THPT

Nguy6n

Trai,

Htri

Duong;

Thanh

H6al

Phan Vdn

Tidn,

l2T,

TIIPT Lam

Son,

Thanh

H6a;

Nghg

Ant

DinhThanhThudng'

11A,

THPT

Hermann Gmeiner,

NguyZn

Thanh Son,

11

Tin,

Khdi

PTCT

-

Tin DHSP

Vinh; L0m

D6ngt

Phan

Thi

ThanhVdn,

ll

To6n,

THPT

chuy0n

L[mDdng;

Ninh

Binh:

Er)ift

Htu

Tiep,

11

'l'ohn,

THPT

Luong

Vtrn

T\ry;

Vfing

Tiru:

Trdn

Quang

Vinh, ll

T2,

THPT

t

e

Quli

Dtin.

PHANHUY

KHAI

Biti T81274.

Cho

tam

Sidc

ABC. Chtmg

minh

rdng

ABC

1

+cof

I

+cosf

1

+cos;

-

^'.-G-t.

,-

>

3\5

trong

dd

cdc gdc

A,

B,

C

do bdng radian.

Loi

gi6i.

Cdch

1.

@rta

TO

Minh

Hodng,

11T, THPT

chuy0n

Hf,i

Duung

vtr

nhidu ban

kh6c).

X6t hlm

sd

f(t)

=

tgt+

sinr

-

2t,

t

.lO,t),,{O)

=

O

Ta

c6

f

'(t)

=

*,.cos,

-

2,f

'(0)=

0,

,

[O,l)

os-,

f

"(t)

=*.-

sin

>

o,

I

e

[0,

|

)

cos-,

Suy

ra

f

'(t)

>

"f

'(0)

=

0,

/

e

ddng

bidn trong

I

o,;]

Suy

raflr)

>flO)

=

0,

M[t

khic,

trong

MBC ta

luOn c6

ABC

T

=

cotl+

cotg,

+

cotS

ABC

=

cot\cotgrcotg,

,r;@=3{7,

nOn

Z> 3{t. Thay

vio

(1)

ta

dugc

dpcm.

Cdch

2.

(crla

da

sd cric

bgn).

vdi x

e

(0,

;

)thi

sinxcot|

DOi

vdri

clc

g6c

B,

C

ta c{lng c6 bdt

ding

thric

tuong

tU

(*).

Do

vfy

ABC

I

+co5

I

+co5

I

+cos,

+-

--

ABC

ABC

>

cotf+

cotE+

cots;.

(l)

ta

c6

(*)

18

7/21/2019 THTT So 278 Thang 08 Nam 2000

21/28

ctAr

eAr

ri rnuroc

Vdn

Tidn,

1147,

TTIPT

Kinh

MOn, T6 Minh Hodng,

Phqm

Thdnh Trung,

Ya

Xudn

Nam,

Chu

Ngpc Hung,

NgA Xudn Bdch,

NguyAn Tidn

ViQt

Hwng,

l0T,

Brti

Drry Thlnh, 11T, TIIPT

chuyOn Nguy6n Trai;

Be Rla

-

Vring

Tiu: Nguy4n

Vdn

Thdnh, Phan

Hodng

Va,

12T,

TIIPT

chuyOn

LC

Qu

DOn;

Hud:

Nguy4n Du

Thdi,

1lT, DHKI{

Hud;

Hn

N6i:

Ddo

Tudn

Son,

NguyAn

Hodng

Thgclt, lOT,

D(ng

Nggc

Minh,

llT,

THPT Amstedarn"

Phan

Nhdt

Thdng, 11A1,

PmL

TOn

Dric Thdng, L

Chi

KiAn,11T,

Hodng

Tidn

Mqilh,

Nguy1n

Kim

Thanh,

l0B, Hodng Ttng,

L2A, Ngi

Qudc

Anh,

11A, Nguy4n

Quang

Hdi, 10AT,

DHKHTN-DHS,

Ngry1n

Trung KiAn, 10A2, Hdn

Thd Anh, 11A1, DHS;

Hi Tinh:

Nguy4n

Thta

Thdng,

1lT, TIIPT

chuyn; NghO

An:

NguyAn

Dinh

Trung,

NguyAn Thanh

Son,

lLT,

Vfr NguyAn

Bdc,

lt Tdt

Thdng,

11A,

DHSP Vinh;

Ninh

Thugn: LA Tidn

Trung, L2A,

Ddo

Dqi

Duong,11A2,

THPT

Chu Vdn

An;

Lam

Ddngt Phan

Dinh Hdi

Soz, 11A10; Khtinh

IJin:

12

Th Khdnh

Hiln,llT,

THPT

I.

Quf

Don;

Gia

Lai: D(ng

Tudn

HiQp,llC3,

THPT Hing

Vuong;

'

Quing

Ngdi:

Trdn

Thdi

An

Nghia;

Quing

Tri:

.

Hodrtg Minh Phyng,

llT, Bqch

Nggc

CAng

D*c, l2T,

THPT

chuyn; Thanh H6a:

Trinh Khdc Tudn,

M,

THPT

LC Hodn, Hodng

Minh

Tidn,

1lB,

THPT Bim

Son,

Bri

Ngpc Hdn,

l0T,

Phan

Vdn

Tidn,

LZT, Lanr

San,

N

guydn

Vdn Trun

g,

I 1A1, TIIPT Hflu

LQc

1

;

Phri

Tho:

Hodng

Tudn

Anh,

llH, TIIPT

COng

nghiQp,

Nguydn

Dinh

H)a,1041,

chuy0n

Hing

Vuong;

Vinh

Phict

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4-6'.7

MOT SO

NGHICH LI

CAA

XAC

SUAT

I{GrIICffi'

r,i

r}E[rEfifr

NGUYENDUY

n6N

(EHKHTN-DHUG

HdN\i)

Nghich

lf:

Khi

tung

4

ldn

mQt

con s:(,c

sic

can

ddi

thi x6c

sudt

dd it ntrdt

mOt

ldn

m6t I

xuft

hien

ldn

hon

Il2. Trong

hic

d6 khi tung

24ldn hai

con

sric sdc

cOn

cldi

thi x6c

sudt

dd

it

nhft

mQt

lin

hai

con ddu xudt

hiOn

m6t

1

b

hon

ll2. Di6u

nly

thflt

d6ng nggc

nhiOn,

bdi vi

khA

nlng

nhfln

dugc

mQt

sd

1

gdp

6 l6n

khl

ndng

xuft

hign

hai

sd 1,

cdn

sd

24

thi

gdp

6

ldn

sd 4.

Giii thich:

Ndu

tung

mQt

con

sric

sdc c0n

ddi

k

lin

thi

sd

kh6

ning

c6

tlrd

c6

bing

6ft.

Trong

sd

d6

c6

5ft

sd

kh6

n[ng

khOng

xudt

hiqn

mflt 1,

vi

do d6 xdc

sudt

0d xuft

hiqn

ft

nhdt

mQt

ldn

mat

I

blng

:

pk

=

r-f

I)*.

\6/

D6

kidm

tra

lai rlng

P1

> tl2

nu

k

>

4.

Trrrng

tg x6c

sudt

xuft

hiQn

it

nhdt

mOt

lin

hai

m[t

1

khi tung

,t ldn

hai

con

s:f,c sdc

b]ng

:

e*=t-f-)-

\36l

8*

ll2

v1t k

> 25.

yi

viy

"gi6

tri

gidi

hpn" cria

sd

lin

tung

dd

x6y

ra

didu

trOn

ddi

vdi

m$t

con

sric sdc lI4

'rl

ddi vdi

hai

con sric

sdc

ltr

25.

Chti

thich.

*

Nghich

li

tr0n

lI

thSc

mdc

cta

Dd M0r0

(De

Mdrd)

khi

rrao

ddi v0i Pascan

(pascal),

nhi hqc

gi6

vi

dai

cria

thd ki

17.

N[m

1654

Pascal

vi Phecma

(Fermat)

cting

tim

ra

clclrr

gili

thfch

trOn.

Tuy vQy,

eich

gifu

thich

nly

khOng

lim

De M6rd

th6a

mtn

vi

6ng cho

ring

"kh6ng

p_hn

hqp

vdi nguy0n

li

d

le

cfia

gi6

tri

ti0u

chudn".

Nlm 1718

Moavro

(Moivre)

da

chi

ra

rlng

"Nguy0n

li

ti

lg

cta

cdc

gi6

tri

ti0u

chudn" kh6 gin

vdi

chin li,

b0i

vi,

ndu

p

ld,

xic

sudt

cria sU kiQn nlo

d6,

thi

gi6

rri

riu

chudn

k

c6

thd

tim

tr)

phr:ong

trinh

:

(1-

p)*

=

ll2

(phuong

trinh ntry c6 nghiQm

ndu

0