103 Geometry Selected Exercise.pdf

FrNImaRtviPaKkñúgbøg;

eroberogeday lwm plÁún TMBr½ 1

CMBUkTI1

kRmglMhat;eRCIserIs

!> eK[RtIekaN ABC mYymanRCug bAC,aBC nig cAB . knøHbnÞat;BuHkñúgénmMu A kat; BC Rtg; L . cUrRsayfa

b

c

LC

BL rYcKNna BL nig LC CaGnuKmn_én c,b,a .

@> eK[RtIekaN ABC mYy . knøHbnÞat;BuHkñúgénmMu A kat;rgVg;carwkeRkAén RtIekaNRtg;cMnuc K . cUrRsayfa

2

Acos2

cbAK

Edl cAB,bAC .

#> eK[RtIekaN ABC mYymanRCug bAC,aBC nig cAB . knøHbnÞat;BuHkñúgénmMu A kat; BC Rtg; L . cUrRsayfa ])

cb

a(1[bcAL 22

.

$> eK[RtIekaNsmgS½ ABC mYyehIy Q CacMnucmYyénRCug ]BC[ bnÞat; )AQ( kat;rgVg;carwkeRkAénRtIekaN ABC Rtg; P . cUrRsaybBa¢ak;fa

PQ

1

PC

1

PB

1 .



%> eK[kaer ABCD mYy ehIy P CacMnucmYyenAkñúgkaerEdlbMeBjlkçxNÐ o15PBCPAB . cUrbgðajfa PCD CaRtIekaNsmgS½ ? ^> eKtag 1r nig 2r CakaMénrgVg;p©it A Edlb:HnwgRCug BC én ABC erogKñaRtg; B nig C . cUrRsayfa 2

21 Rr.r ¬R kaMrgVg;carwkeRkAén ABC ¦ &> eK[ctuekaNe)a:g ABCD mYymanRCug cCD,bBC,aAB nig dDA carwkkñúgrgVg;mYy . cUrRsayfa

cdab

bcad

BD

AC

. *> ebIctuekaNe)a:g ABCD mYymanRCug cCD,bBC,aAB nig dDA carwkkñúgrgVg;mYy ehIycarwkeRkArgVg;mYyeTotenaHbgðajfa épÞRkLa S rbs;vaKW abcdS . (>ebIctuekaNe)a:g ABCD mYymanRCug cCD,bBC,aAB nig dDA carwkkñúgrgVg;mYymankaM R enaHbgðajfaépÞRkLa S rbs;va kMNt;eday

22

R16

)cdab)(bdac)(adbc(S

.

!0> tag CamMupÁMúedayGgát;RTUg AC nig BD rbs;ctuekaNe)a:g ABCD cUrRsayfaépÞRkLarbs;ctuekaNenHkMNt;eday sin.BD.AC

2

1S



!!> eK[RtIekaNsmBaØ ABC mYymanRCúg cAB,bAC,aBC eKtag p CaknøHbrimaRt r CakaMrgVg;cawikkñúg R CakaMrgVg;cawikeRkA nig S CaépÞRkLarbs;RtIekaN ehIy CBA r,r,r tagerogKñaCakaMrgVg; carwkeRkARtIekaNkñúgmMuerogKña C,B,A .

cUrRsaybBa¢ak;TMnak;TMngxageRkam ³ 1> r.R4rpcabcab 22 2> r.R8r2p2cba 22222 3>

p

r

2

Ctan

2

Btan.

2

Atan

4> 2ACCBBA prrrrrr

5> CBA r

1

r

1

r

1

r

1

6> Ar

1

r

1

S

a

7> CB r

1

r

1

S

a

8> R4

r

2

Csin

2

Bsin

2

Asin

9> p

rR4

2

Ctan

2

Btan

2

Atan

10> r

1

)ap)(cp(

1

)cp)(bp(

1

)bp)(ap(

1



11> R

pCsinBsinAsin

12> r

rrrp CBA2

13> R4

p

2

Ccos

2

Bcos

2

Acos

14> abc2

cba

c

Ccos

b

Bcos

a

Acos 222

15> Btan

Atan

acb

bca222

222

16> AcosBsinAsin2CsinBsinAsin 222 17>

r

p

2

Ccot

2

Bcot

2

Acot

18> CsinBsinAsinR2S 2 19> 3 222 CsinBsinAsincba

2

1S

20> 2R

S2C2sinB2sinA2sin

21> S.a)rr(rr CBA 22>

p

ap

r

r

A

23> rR4rrr CBA 24> )cp)(bp(r.r A



!@> eK[ c,b,a CaRCugénRtIekaNmYy . cUrRsaybBa¢ak;fa ³ k> )abcabc(4)cba()abcabc3 2 x> )

p

abcp(

35

36cba 2222 Edl

2

cbap

K> 3ba

c

ac

b

cb

a

2

3

X> 8

abc)cp)(bp)(ap(

!#> RtIekaN ABC manRCug c,b,a ehIytag r CakaMrgVg;carwkkñúg nig p CaknøHbrimaRténRtIekaN . cUrRsayfa

2222 r

1

)cp(

1

)bp(

1

)ap(

1

.

!$> cUrRsaybBa¢ak;fa ³ k> abc

2

3)cp(c)bp(b)ap(aabc 222

x> )cp)(bp)(ap(48)ba(ab)ac(ca)cb(bc K> abcp)cp(c)bp(b)ap(a 333 Edl c,b,a CaRCugénRtIekaNmYy nig p CaknøHbrimaRt . !%> ebI c,b,a CaRCugénRtIekaNmYyenaHcUrRsayfa ³ 0)ac(ac)cb(cb)ba(ba 222 etIeBlNaeTIbeK)ansmPaB .



!^> eKtag c,b,a CaRCugénRtIekaNmYy ehIy 2

cbap

.

cUrbgðajfa ³ k> 2223 cba27)cp)(bp)(ap(p64 x>

222 c

1

b

1

a

1

abc

p2

K> p

9

cp

1

bp

1

ap

1

X> 2

9

ba

cp

ac

bp

cb

ap

4

15

g> p3cpbpapp !&> eK[RtIekaN ABC mYymanRCug c,b,a . tag R CakaMrgVg;carwkkñúg énRtIekaNenH . k> Rsayfa )CcosBcosAcos1(R8cba 2222 x> edayeRbIRTwsþIbTLibniccUrTajfa

8

1CcosBcosAcos .

K> ebI C,B,A CamMuRsYcenaH 2

33CsinBsinAsin2 .

!*> kñúgRtIekaN ABC mYycUrRsayfa ³ k>

R

S2CcoscBcosbAcosa

x> C2sinB2sinA2sinCsinBsinAsin



!(> eK[RtIekaN ABC mYymanRCug c,b,a . k> Rsayfa

bc2

a

2

Asin rYcTajfa

8

1

2

Csin

2

Bsin

2

Asin .

x> tag r nig R CakaMrgVg;carwkkñúg nig carwkeRkAénRtIekaN ABC . bgðajfa

R4

r

2

Csin

2

Bsin

2

Asin rYcTajfa r2R .

@0> eK[RtIekaN ABC mYy . cUrbgðajfa 2)CcosB(cos2Acos ? etIeBlNaeTIbsmPaBenHkøayCasmPaB ? @!> kñúgRKb;RtIekaN ABC cUrRsayfa ³

2

1)CcosB(cosAcos2

RKb;cMnYnBit 0 . @@> eK[ z,y,x CacMnYnBitEdl 0xyz . cUrRsayfa )

y

zx

x

yz

z

xy(

2

1CcoszBcosyAcosx

Edl C,B,A CamMukñúgénRtIekaNmYy . @#> kñúgRKb;RtIekaN ABC cUrRsaybBa¢ak;fa ³ k> CcosBcosAcos24)AC(cos)CB(cos)BA(cos 222 x> 3

2

Ctan

2

Btan

2

Atan

K> 12

Ctan

2

Btan

2

Atan 222



@$> kñúgRKb;RtIekaN ABC cUrRsayfa ³ k> 345

2

Atan

2

Ctan5

2

Ctan

2

Btan5

2

Btan

2

Atan

x> 9

1

2

Ctan

2

Btan

2

Atan 666

K> 3CcotBcotAcot @%> kñúgRKb;RtIekaN ABC cUrRsayfa ³ k> 33

2

Ccot

2

Bcot

2

Acot

x> 92

Ccot

2

Bcot

2

Acot 222

@^> kñúgRKb;RtIekaN ABC cUrRsayfa ³ 3

CsinBsinAsin

CcosBcosAcos1

etIeK)ansmPaBenAeBlNa ? @&> eK[RtIekaN ABC mYymanRCug c,b,a . RsayfaebI

2

cba

enaH

2

CBA

.

@*> eK[RtIekaN ABC mYymanRCug c,b,a . tag R CakaMrgVg;carwkkñúg nig S CaépÞRklarbs;RtIekaNenH . cUrRsaybBa¢ak;fa

R

S32CsincBsinbAsina



@(> RtIekaN ABC mYymanmMu CBA . tag R CakaMrgVg;carwkkñúg ehIy r Cacm¶ayrvagp©itrgVg;carwkkñúg nig p©it rgVg;carwkeRkAénRtIekaN . cUrRsaybBa¢ak;fa ³ CcosR2dRBcosR2dRAcosR2 #0> eK[RtIekaN ABC mYymanRCug c,b,a . tag R CakaMrgVg;carwkkñúgehIy r CakaMrgVg;carwkeRkAénRtIekaN . cUrRsaybBa¢ak;fa

2

R9CsincBsinbAsinar9 .

#!> eK[RtIekaN ABC mYymanRCug c,b,a . tag S CaépÞRklaénRtIekaNenH . cUrRsaybBa¢ak;fa ³ k> S34cba 222 x> 2444 S16cba K> 2222222 S16baaccb X>

cba

abc9S34

g> 32 )3

S4()abc(

#@> eK[RtIekaN ABC mYymanRCug c,b,a . tag S CaépÞRklaénRtIekaNenH . cUrRsaybBa¢ak;fa ³ 6222222222222 )S4()cba()bac()acb(27



##> eK[RtIekaN ABC mYymanRCug c,b,a .

tag S CaépÞRklaénRtIekaNenH ehIyyk 2

cbau

222

nig abcabcv .

Rsayfa 12

)vu(S

12

)v5u3)(vu( 22

. #$> tag S CaépÞRkLa nig c,b,a CaRCugénRtIekaNmYy .

RKb; 0n bgðajfa 3nnn

3

cba

4

3S

.

#%> eK[RtIekaN ABC mYymanRCug c,b,a . eKtag r nig R erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . yk O Cap©itrgVg;carwkeRkA I Cap©itrgVg;carwkkñúg nig H CaGrtUsg; énRtIekaN ABC . Rsayfa )cba(R9OH 22222 nig CcosBcosAcosR4r2HI 222 . #^> eKtag r nig R erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN mYyEdlman p CaknøHbrimaRt . cUrRsayfa ³ k> 22 )rR4(p3)R4r(r9 x> 222 )R2r(2Rp2)R4r(r6 K> 22 )rR4(R)rR2(p2 X> 222 r3Rr4R4p)r5R16(r



#&> RtIekaN ABCmYymanRCug c,b,a eKtag p CaknøHbrimaRt r nig R erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN. cUrRsayfa ³ k> 22 r27p x> rR27p2 2 K> 22222 R9cbar36

X> 22222

2 R2r4

cba)rrR2(3

g> 22 R9abcabcr36 c> 22 )Rr(

4

abcabcrrR5

q> Rr9)cp(c)bp(b)ap(a C> 22 Rr)16312(rR8abc Q>

r2

3

c

1

b

1

a

1

R

3

j> )rR(2

33

c

1

b

1

a

1

d> 22 r4

1

ab

1

ca

1

bc

1

R

1

z> cba

abcr4 2

D> 3)3R(abc



#*> eK[RtIekaNsmBaØ ABC mYymanRCúg cAB,bAC,aBC eKtag p CaknøHbrimaRt r CakaMrgVg;cawikkñúg R CakaMrgVg;cawikeRkA nig S CaépÞRkLarbs;RtIekaN ehIy CBA r,r,r tagerogKñaCakaMrgVg; carwkeRkARtIekaNkñúgmMuerogKña C,B,A . cUrRsaybBa¢ak;fa ³

k> 4rr

c

rr

b

rr

a

BA

2

AC

2

CB

2

x> 3

rR4ap A

K> ap

rR4

S2

)cb(a3 A

22

X> 2C

2B

2A

2 rrrp g>

8

33

c

r

b

r

a

r CBA

c> 4

3

abcabc

rrrrrr BAACCB

q> R2

9rrrr9 CBA

C> 22C

2B

2A R

4

27rrr

Q> 22C

2B

2A

2 R7rrrr j> 2)rR(3S



#(> eK[RtIekaNsmBaØ ABC mYymanRCúg cAB,bAC,aBC eKtag p CaknøHbrimaRt r CakaMrgVg;cawikkñúg R CakaMrgVg;cawikeRkA nig S CaépÞRkLarbs;RtIekaN ehIy cba m,m,m tagerogKñaCaemdüan nig cba h,h,h Cakm<s;KUsecjBIkMBUlerogKña C,B,A . cUrRsaybBa¢ak;fa ³ k> R

2

9mmmr9 cba

x> 2

R9hhhr9 cba

K> 2cba )mmm(

27

3S

X> S33mmm 2c

2b

2a

g> rR4mmm cba $0> eK[RtIekaNsmBaØ ABC mYymanRCúg cAB,bAC,aBC tag CBA ,, erogKñaCaknøHbnÞat;BuHkñúgénmMu C,B,A nig S Ca épÞRkLaénRtIekaN . cUrRsayfa ³ k> S332

C2

B2

A x> )cba(9)(16 4444

C4

B4

A K> S.p.. CBA Edl

2

cbap

.



$!> eKyk P CacMnucenAkñúgRtIekaN ABC ehIy F,E,D CaRbsBVrvag CP,BP,AP CamYyRCug AB,CA,BC erogKña . cUrRsayfa

2

9

CP

CF

BP

BE

AP

AD .

$@> rgVg;carwkkñúgénRtIekaN ABC b:HnwgRCug AB,CA,BC erogKñaRtg; cMNuc F,E,D .

Rsayfa 12DE

AB

FD

CA

EF

BC 222

$#> eK[RtIekaN ABC mYymankm<s; CF,BE,AD .

Rsayfa 4

3

AB

DE

CA

FD

BC

EF 222

$$> eKmanRtIekaN ABC mYymanRCug c,b,a nigépÞRkla S . tag T CaépÞRkLaénRtIekaNsmgS½carwkkñúgRtIekaN ABCxagelI .

cUrRsayfa S34cba

S32T

222

2

.

$%> eKmanRtIekaNBIr ABC nig XYZ carwkkñúgrgVg;EtmYy . eKdwgfa AB//CZ,CA//BY,BC//AX . tag c,b,a CaRCugén ABC nig z,y,x CaRCugén XYZ . Rsayfa 1

cba

zyx

nig 3

c

z

b

y

a

x .



$^> eK[RtIekaN ABC mYyEkgRtg; A nigmankm<s; hAH . tag r CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa

2

1

h

r12 ?

$&> eK[RtIekaN ABC mYyEkgRtg; A . eKtag r CakaMrgVg;carwkkñúgénRtIekaN . cUrRsaybBa¢ak;vismPaB ³

2

BCAC.AB

2

2r . etIeBlNaeTIbeK)ansmPaB ?

$*> eKtag P CacMnucmYyenAkñúgRtIekaN ABC ehIyyk F,E,D CacMnuc RbsBVrvagbnÞat; CP,BP,AP CamYyRCug AB,CA,BC erogKña . cUrRsayfa 6

PF

CP

PE

BP

PD

AP nig 8

PF

CP.

PE

BP.

PD

AP

$(> eKtag 111 C,B,A CabIcMnucsßitenAelIRCug AB,CA,BC én ABC cUrbgðajfa

2

3

CABCAB

CCBBAA

2

1 111

. %0> kñúgRtIekaN ABC mYycUrRsayfa ³

R

r1

AcosCcos

AcosCcos

CcosBcos

CcosBcos

BcosAcos

BcosAcos 222222

%!> eKmanRtIekaN ABC mYyEkgRtg; A . tag M P,N, CacMnucb:H rvagrgVg;carwkkñúgCamYyRCug AB,CA,BC . cUrRsayfa ³ BC

9

34MPMN .



%@> RtIekaN ABCmYycarwkkñúgrgVg;p©it O kaM R . tag 321 R,R,R CakaMrgVg;carwkeRkAénRtIekaN OCA,OBC,OAB erogKña . Rsayfa

R

323

R

1

R

1

R

1

321

?

%#> ebI ABCD CaRbelLÚRkamenaHbgðajfa BD.AC|BCAB| 22 . %$> ebI c,b,a CaRbEvgRCugénRtIekaNmYy ehIy cba m,m,m Caemdüan nig R CakaMrgVg;carwkkñúgenaHbgðajfa ³

k> R12m

ac

m

cb

m

ba

b

22

a

22

c

22

x> 0)cab(m)bca(m)abc(m 2c

2b

2a

%%> (APMO, 1996) tag c,b,a CaRbEvgRCugénRtIekaNmYy . cUrbgðajfa ³ cbabacacbcba %^> (IMO, 1964) tag c,b,a CaRbEvgRCugénRtIekaNmYy . cUrbgðajfa ³ abc3)cba(c)bac(b)acb(a 222 %&> (IMO, 1983) tag c,b,a CaRbEvgRCugénRtIekaNmYy . cUrbgðajfa ³ 0)ac(ac)cb(cb)ba(ba 222 . %*> eK[RtIekaN ABC mYyEkgRtg; A . cMeBaHRKb;cMnYnKt; 2n cUrRsayfa nnn BCACAB .



%(> tag cba h,h,h CaRbEvgkm<s;énRtIekaN ABC mYyEdlcarwkeRkA rgVg;p©it I nigkaM r . cUrRsayfa ³ k>

cba h

1

h

1

h

1

r

1 x> r9hhh cba

^0> (Korea, 1995)eK[RtIekaN ABC mYyehIytag N,M,L CacMnucsßitelIRCug AB,CA,BC erogKña . tag Q,P nig R CacMNucRbsBVrvagbnÞat; BM,AL nig CN CamYyrgVg;carwkeRkAén ABC erogKña . cUrRsayfa 9

NR

CN

MQ

BM

LP

AL .

^!> (Shortlist IMO, 1997)RbEvgRCugénqekaN ABCDEF epÞógpÞat; DECD,BCAB nig FAEF . cUrRsayfa

2

3

FC

FA

DA

DE

BE

BC .

^@> eKyk CF,BE,AD Cakm<s;énRtIekaN ABC ehIy PS,PR,PQ Cacm¶ayBIcMnuc P eTARCug AB,CA,BC erogKña . cUrbgðajfa 9

PS

CF

PR

BE

PQ

AD .

^#> snμtfargVg;carwkkñúgénRtIekaN ABC b:HeTAnwgRCug AD,CA,BC

erogKñaRtg; F,E,D . bgðajfa 3

pDEFDEF

2222

Edl p CaknøHbrimaRtén ABC .



^$> RtIekaN ABC mYymanRCug c,b,a nigtag SCaépÞRklaén ABC . cUrRsaybBa¢ak;fa ³ k> S34)ac()cb()ba(cba 222222 x> S34cabcab K> S34

cabcab

abc)cba(3

^%> tag c,b,a CaRCug nig R CakaMrgVg;carwkeRkAénRtIekaNmYy .

cUrRsayfa R33cba

c

bac

b

acb

a 222

.

^^> ctuekaNe)a:g ABCD mYymanRCug d,c,b,a . tag S CaépÞRkLa rbs;ctuekaNenH . cUrRsayfa ³ k>

2

cdabS

x> 2

bdacS

K>

2

db

2

caS

^&> ebI P CacMnucenAkñúgRtIekaN ABC Edl PCn,PBm,PA enaHcUrbgðajfa ncmba)nm)(nmnm( 222 Edl c,b,a CaRCugén ABC .



^*> eKtag O Cap©itrgVg;carwkkñúg nig G CaTIRbCMuTm¶n;én ABC mYy . eKtag r nig R erogKñaCakaMrgVg;carwkkñúg nig carwkeRkAénRtIekaN . cUrRsayfa )r2R(ROG . (Balkan, 1996) ^(> (Taiwan, 1997) eK[RtIekaN ABC mYycarwkkñúgrgVg;p©it O nigkaM R . RsayfaebI AO kat;rgVg;carwkeRkAén OBC Rtg; D ehIy BO kat;rgVg; carwkeRkAén OCA Rtg; E ehIy CO kat;rgVg;carwkeRkAén OAB Rtg; F enaH 3R8PF.OE.OD . &0> (APMO, 1997)eK[ ABC CaRtIekaNmYy . bnÞat;BuHkñúgénmMu A CYbGgát; BCRtg; XnigCYbrgVg;carwkeRkAén ABC Rtg; Y . tag

AY

AXLa .

eKkMNt; bL nig cL tamrebobdUcKñaEdr . cUrRsayfa 3

Csin

L

Bsin

L

Asin

L2c

2b

2a ?

&!> (Balkan, 1999)eK[ ABC CaRtIekaNmYymanmMukñúgCamMuRsYc ehIytag N,M,L CaeCIgéncMeNalEkgBITIRbCMuTm¶n; G énRtIekaN ABC eTAelI RCug AB,CA,BC erogKña . cUrbfðajfa

4

1

S

S

27

4

ABC

LMN ?



&@> tag D nig E CacMnucenAelIRCug AB nig CA énRtIekaN ABC edaydwgfa DE Rsbnwg BC ehIy DE b:HnwgrgVg;carwkkñúgén ABC . cUrbgðajfa

8

CABCABDE

. (Italy, 1999)

&#> (Mediterranean, 2000) tag S,R,Q,P CacMnuckNþal erogKña AB,DA,CD,BC énctuekaNe)a:g ABCD mYy . cUrRsayfa ³ )DACDBCAB(5)DSCRBQAP(4 22222222 &$> eKtag R nig r CakaMrgVg;carwkeRkA nig kaMrgVg;carwkkñúgén ABC . snμtfa A CamMuFMCageKénRtIekaN ABC . yk M CacMNuckNþal én BCehIy X CaRbsBVrvagbnÞat;b:HeTAnwgrgVg;carwkeRkARtIekaNABC Rtg; B nig C . cUrRsayfa

AX

AM

R

r (Korea, 2004)

&%> eK[RtIekaN ABC mYyehIyyk O CacMNucmYyenAkñúgRtIekaNenH . bnÞat; OC,OB,OA CYbRCugénRtIekaNRtg; 111 C,B,A erogKña . tag 321 R,R,R CakaMrgVg;carwkeRkAénRtIekaN OAB,OCA,OBC erogKñaehIy R CakaMénrgVg;carwkeRkAén ABC . cUrRsayfa RR.

CC

OCR.

BB

OBR.

AA

OA3

1

12

1

11

1

1 (Romania, 2004)



&^>mMukñúgénRtIekaN ABC mYyCamMuRsYc ehIy ABC2ACB . D CacMnucmYysßitelIRCug BC ehIyepÞógpÞat; ABCBAD2 . cUrRsayfa

AC

1

AB

1

BD

1 ?

( Mathematical Olympiad in Poland 1999 ) &&> cMNuc D nig E sßitenAelIRCug BC nig AC énRtIekaN ABC erogKña bnÞat; AD nig BC kat;KñaRtg; P . K nig L CacMNucenAelIRCug BC nig AC erogKñaEdl CLPK CaRbelLÚRkam . cUrRsayfa

DK

BD

EL

AE (Mathematical Olympiad in Poland 2000 )

&*> bIcMNucxusKña C,B,A sßitenAelIrgVg; )O( . bnÞat;b:HrgVg; )O( Rtg;A nig B kat;KñaRtg; P. bnÞat;b:HrgVg; )O( Rtg;C kat;bnÞat; AB Rtg; Q . cUrRsaybBa¢ak;fa 222 QCPBPQ ? ¬Mathematical Olympiad in Poland 2002 ) &(>bnÞat;b:HeTAnwgrgVg;carwkkñúgénRtIekaNsmgS½ ABC kat;RCug AB nig AC Rtg; D nig E . cUrRsaybBa¢ak;fa 1

EC

AE

DB

AD ?

( Mathematical Olympiad in Poland 1995 )



*0> bnÞat;BuHkñúgénmMu C,B,A énRtIekaN ABC mYykat;RCugQmRtg; F,E,D erogKña ehIykat;rgVg;carwkeRkAénRtIekaN ABC Rtg; M,L,K erogKña . bgðajfa 9

FM

CF

EL

BE

DK

AD ?

( Mathematical Olympiad in Poland 1996 )

*!> kñúgRtIekaN ABC mYyman ACAB . D CacMNuckNþalén BC ehIy E sßitenAelI AC . P nig Q CaeCIgéncMeNalEkgBIcMNuc B nig E erogKñaeTelIbnÞat; AD . bgðajfa ACAEBE luHRtaEt PQAD ?

(Mathematical Olympiad in Poland 1997 )

*@>eKtag O Cap©itrgVg;carwkeRkA nig H CaGrtUsg;énRtIekaN ABC mYy EdlmanmMukñúgCamMuRsYc . cUrRsayfaépÞRkLamYyénRtIekaN COH,BOH,AOH esμInwgplbUk énépÞRkLaBIrepSgeTot ? (Asian–Pacific Mathematical Olympiad 2004 )



*#> kñúgRtIekaN ABC cMNuc M nig N sßitelIRCug ABnig ACerogKña edaydwgfa CNBCMB . tag r nig R erogKñaCakaMrgVg;carwkkñúg nig carwkeRkAénRtIekaN . cUrsresrkenSampleFob

BC

MN CaGnuKmn_én R nig r .

(Asian–Pacific Mathematical Olympiad 2005) *$> eK[ ABC CaRtIekaNmYymanmMukñúgCamMuRsYc ehIy o60BAC nig ACAB . yk I Cap©itrgVg;carwkkñúg nig H CaGrtUsg;én ABC cUrRsayfa ABC3AHI2 ? (Asian–Pacific Mathematical Olympiad 2007) *%> eKtag G CaTIRbCMuTm¶n;énRtIekaN ABC ehIy M CacMNuckNþalén RCug BC . bnÞat;KUsecjBI G Rsbnwg BC kat; AB Rtg; X ehIy Ktat; AC Rtg; Y . snμtfa XC nig GB kat;KñaRtg; Q ehIy YB nig GC kat;KñaRtg; P . cUrRsayfaRtIekaN MPQ dUcKñanwgRtIekaN ABC ? (Asian–Pacific Mathematical Olympiad 1991 )



*^> elIRCug AB nig BC énkaer ABCD eKedAcMNuc E nig F erogKña Edl BFBE . tag BN Cakm<s;énRtIekaN BCE . cUrRsayfa o90DNF ?

(Austrian–Polish Mathematical Competition 1979) *&> rgVg;carwkkñúgénRtIekaN ABC b:HRCug BC nig AC Rtg; Dnig E erogKña . bgðajfaebI BEAD enaH ABC CaRtIekaNsm)at ? (Austrian Mathematical Olympiad 2006) **> kñúgRtIekaN ABC mYyeKyk E CacMNuckNþalénRCug AC nig F CacMNuckNþalénRCug BC . tag G CacMeNalEkgBI C eTA AB . bgðajfa EFG CaRtIekaNsm)atluHRtaEt ABC sm)at . (Austrian Mathematical Olympiad 2008) *(> eKyk E,D nig F erogKñaCacMNuckNþalénRCug CA,BC nig AB énRtIekaN ABC . yk cba H,H,H CaeCIgéncMeNalEkgBI C,B,A eTARCugQmerogKña .yk R,Q,P CacMNuckNþalén cbHH acHH nig baHH erogKña . bgðajfa QE,PD nig RF RbsBVKña Rtg;cMnucmYy ? (Austrian Mathematical Olympiad 2009)



(0>kñúgctuekaNe)a:g ABCD mYyeKyk E CacMnucRbsBVrvagGgát;RTUg ehIy 21 S,S nig S CaépÞRkLaén CDE,ABE nig ABCDerogKña . cUrRsayfa SSS 21 ? (Austrian Mathematical Olympiad 1990) (!> bBa¢ekaNe)a:g ABCDE mYycarwkkñúgrgVg; . cm¶ayBI A eTAbnÞat; DE,CD,BC esμIerogKña c,b,a . cUrKNnacm¶ayBIkMBUl A eTAbnÞat; )BE( . (Austrian Mathematical Olympiad 1990)

(@> eK[RtIekaNsm)at ABC mYyEdl o90A . eKyk M CacMNuckNþalén AB . bnÞat;mYyKUsecjBI A ehIyEkg nwg CM kat;RCug BC Rtg; P . bgðajfa BMPAMC ?

(Baltic Way 2000) (#>eK[RtIekaN ABC mYymanmMu o120A . cMNuc K nig L sßitenAelIRCug AB nig AC erogKña . eKsg;RtIekaNsmgS½ BKP nig CLQ enAeRkARCugénRtIekaNABC . cUrRsayfa )ACAB(

2

3PQ ?

(Baltic Way 2000)



($> kñúgRtIekaN ABC mYyeK[ BCACAB2 . cUrRsayfap©itrgVg; carwkkñúg ABC p©itrgVg;carwkeRkA ABC ehIycMNuckNþalén AC nig BC sßitenAelIrgVg;EtmYy . (Problems for the Team Competition Baltic Way 1999) (%>eK[ ABC nig DAC CaRtIekaNsm)atBIrEdlmanmuMkMBUl o20BAC nig o100ADC . cUrRsayfa CDBCAB ? (Flanders Mathematical Olympiad 1996) (^>eKtag R CakaMrgVg;carwkeRkA nig r CakaMrgVg;carwkkñúgén ABC mYy. S CacMnucmYyenAkñúg ABC ehIybnÞat; CS,BS,AS kat;RCugQm Rtg; Z,Y,X erogKña . cUrRsayfa

r

rR

CZ

BZ.AZ

BY

AY.CY

AX

CX.BX222

( Bosnia and Hercegovina Mathematical Olympiad 2000 )

(&> eK[RtIekaNEkgmYy ABC Edl o90ACB ehIyyk CH Edl ABH CakMBs;cMeBaHRCug AB ehIytag P nig QCacMNucb:Hrvag rgVg;carwkkñúgénRtIekaN ABCCamYyRCug AC nig BC erogKña . ebI HPAQ cUrkMNt;pleFob

BH

AH ? (Bulgarian Mathematical Olympiad 2006)



(*> eK[ctuekaNcarwkkñúgrgVg; ABCD mYy. bnÞat; AD nig BC CYbKña Rtg;cMNuc E ehIyGgát;RTUg AC nig BD CYbKñaRtg; F . ebI M nig N CacMNuckNþalén AB nig CD enaHcUrbgðajfa ³

AB

CD

CD

AB

2

1

EF

MN .

(Bulgarian Mathematical Olympiad 1997) ((> bBa©ekaNe)a:g ABCDE mYycarwkkñúgrgVg;kaM R . eKsMKal; XYZr Ca rgVas;kaMénrVg;carwkkñúgén XYZ . cUrRsaybBa¢ak;fa ³ k>

R

r1BCAcosABCcosCABcos ABC

x> ebI AEDABC rr nig AECABD rr enaH AEDABC (Bulgarian Mathematical Olympiad 1998)

!00> eK[bnÞat; )( kat;RCug AC nig BC énRtIekaN ABC erogKñaRtg; cMNuc E nig F . cUrRsayfabnÞat; )( kat;tamp©itrgVg;carwkkñúgénRtIekaN ABC luHRtaEt AB

CF

BF.AC

CE

AE.BC .

(Bulgarian Mathematical Olympiad 1973)



!0!> eKtag M CaTIRbCMuTm¶n;énRtIekaN ABCmYy . cUrRsayfavismPaB

3

2CBMsinCAMsin .

(Bulgarian Mathematical Olympiad 1997) !0@> ebI c,b,a CaRCug nig R CakaMrgVg;carwkeRkAénRtIekaN ABC

enaHbgðajfa 222

22

cb2a22

baR

. etIsmPaBmanenAeBlNa?

( Baltic Way 1998 ) !0#> eK[RtIekaN ABC mYyEkgRtg; A . D CacMNucmYyelIRCug BC Edl BAD2BDA . cUrRsaybBa¢ak;smPaB

CD

1

BD

1

2

1

AD

1 ( Baltic Way 1998 )

eroberogeday lwm pl:ún Tel: (017)768 246



CMBUkTI2

EpñkdMeNaHRsay lMhat;TI1 eK[RtIekaN ABC mYymanRCug bAC,aBC nig cAB . knøHbnÞat;BuHkñúgénmMu A kat; BC Rtg; L . cUrRsayfa

b

c

LC

BL rYcKNna BL nig LC CaGnuKmn_én c,b,a .

dMeNaHRsay Rsayfa

b

c

LC

BL

tamRTwsþIbTsIunUs )1(

2

Asin

BL

Bsin

AL

)2(

2

Asin

LC

Csin

AL

EcksmIkar )2(&)1( GgÁ nig GgÁeK)an LC

BL

Bsin

Csin Et

b

c

Bsin

Csin

dUcenH b

c

LC

BL .

A

B C L

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