P2 ISMC 2018 (W Answers)

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC >297 (Rrjeiry >) 9 gfur 12 ejcuths

Jcstruatjfcs tf pirtjajpicts 9. Mf cft fphc tgh nf nfflkht flkht uctjk yfu irh tfkm tf mf sf sf.. >. Itthept IKK >8 quhstjfcs. 1. _rjth y yfur fur ics icswhrs whrs chitky jc tgh Icswhr Qghht prfv prfvjmhm. jmhm. =. Eirls irh iwirmhm `fr afrrhat icswhrs fcky. 8. Ikk `jourhs irh cft mriwc tf saikh. 0. Chjtghr eitgheitjaik tinkhs cfr aikaukitfrs eiy nh ushm. Yuhstjfcs jc Qhatjfc I airry > eirls hiag, quhstjfcs jc Qhatjfc N airry = eirls hiag icm quhstjfcs jc Qhatjfc A airry nhtwhhc 0 tf 92 eirls hiag.

Dfjctky froicjshm ny

9

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC Qhatjfc I4

Hiag f` tgh quhstjfcs 9 tf 92 airrjhs > eirls. 9.

18 njayakhs njayakhs irh pirlhm jc i strijogt kjch. kjch. ]ghrh irh 90 njayakhs njayakhs pirlhm jc `rfct f` Ictfcy‚s njayakh icm 9= njayakhs nhgjcm Nrucf‚s njayakh. Gfw eicy njayakhs irh tghrh nhtwhhc  Ictfcy‚s njayakh icm Nrucf‚s njayakh6 18 ‑ 90 ‑ 9= ‑ > 5 1 ]ghrh irh 1 njayakhs nhtwhhc Ictfcy‚s njayakh icm Nrucf‚s njayakh. 

>.  Ici js 7 yhirs fkm. fkm. Jc 0 yhirs‚ tjeh, Ici‚s Ici‚s eftghr w wjkk jkk nh 1 tjehs is fkm is Ici. Gfw eicy yhirs fkmhr tgic Ici js Ici‚s eftghr 6 7 + 0 5 9= 9= ß > 5 >7  Ici‚s eftghr js >7 yhirs fkmhr.

1.

]gh tinkh nhkfw sgfws tgh tftik cuenhr f` airs i sikhseic sfkm `rfe Duky tf Mhahenhr. @jcm tgh tftik cuenhr f` airs sfkm jc tgh efctg tgit gh shkks tgh efst airs icm tgh efctg tgit gh shkks tgh khist airs.  airs.  Duky  Iuoust Qhpthenhr

92 9> :

Fatfnhr Cfvhenhr Mhahenhr

99 7 0

9> + 0 5 97 ]gh tftik cuenhr f` airs jc tgfsh twf efctgs js 97.

=.

Citgicjhk irricohs f` mj``hrhct mj``hrhct afkfurstgh jc iyhkkfw rfw `rfe kh`t kh`t tf chxt rjogt. rjogt.tfGh pkiahs tgh rhm hcvhkfph chxt tf8ihcvhkfphs nkuh hcvhkfph, wgjkh shttjco hcvhkfph tgh fricoh fch. ]ghc, gh puts tgh orhhc hcvhkfph nhtwhhc tgh nkuh icm yhkkfw hcvhkfphs. J` tgh fricoh hcvhkfph js tgh `jrst hcvhkfph `rfe tgh rjogt, wgit js tgh afkfur f` tgh hcvhkfph tgit js shafcm `rfe tgh kh`t6 Rfssjnkh irricohehcts f` hcvhkfphs `rfe kh`t tf rjogt4 Hjtghr F, Z, O, N, ^ fr ^, N, O, Z, F Qjcah tgh F^ICOH hcvhkfph js tgh `jrst `rfe tgh rjogt, tgh NKXH hcvhkfph wjkk nh shafcm `rfe tgh kh`t.

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JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

8.

Qtumy tgh pjaturh ojvhc. 8

J` tgh pfsjtjfc f`

js ( 8 , 1 ) icm

=

tgh pfsjtjfc f`

1

tghc tgh pfsjtjfc f`

js (

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icm tgh pfsjtjfc f`

js (

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js ( 9 , 8 ) ,

9 9

0.

>

1

=

8

5 ( 1, =), icm

5 ( =, 9) 9)  

]ghrh irh = pitgs tf oht tf . _gjag f` tgh `fkkfwjco pitgs wfukm tilh tgh sgfrthst tjeh6

 I

N

A

M

 I 5 19 ucjts, N 5 >1 ucjts, A 5 >: ucjts, M 5 >8 ucjts Ritg N tilhs tgh sgfrthst tjeh. ?.

Agrjstjch, Eirji, Fkjvhr icm ]jeftgy irh irh sgirjco i nfwk f` oriphs. Eirji Eirji hits 1 efrh oriphs tgic ]jeftgy, wgjkh Agrjstjch hits 1 `hwhr oriphs tgic Fkjvhr. J` Fkjvhr hits 8 efrh oriphs tgic ]jeftgy, wgf ith efrh oriphs, Eirji fr Fkjvhr6 Gfw eicy efrh6  VVVVVVVVVVV ith VVVVVVVVV V VVVVVVVV efrh oriphs tgic VVVVVVV VVVVVVVVVVV. VVVV. (Eirji, Fkjvhr) (Eirji, Fkjvhr) Fkjvhr ith > efrh oriphs tgic Eirji.

1

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

7.

9

Okfrji tffl  f` tgh cuenhr f` aghrrjhs jc i nio. Ikhx tffl 9= aghrrjhs icm Micjhk tffl tgh 1

rheijcjco 92 aghrrjhs. Gfw eicy aghrrjhs whrh jc tgh nio it `jrst6 9

>

1

1

9 ‑   5   > ucjts

↞

9= + 92 5 >= ÿ

↞

9 ucjt >= ß > 1 9>  1 5 9> 10 ]ghrh whrh 10 aghrrjhs jc tgh nio it `jrst. ↞

:.

^fnhrt sphcms tgh i`thrcffc pjaljco pjaljco ippkhs it it ic fragirm. It tgh hcm f` tgh miy, gh rhturcs gfeh wjtg i nio f` >2 rhm ippkhs icm shvhrik orhhc ippkhs. ]gh chxt efrcjco, ^fnhrt ghims 9 tf tgh kfaik eirlht tf shkks tgh ippkhs. I`thr 9> orhhc ippkhs icm  tgh cuenhr f` rhm ippkhs > whrh sfkm, tghrh wis ic hquik cuenhr f` rhm icm orhhc ippkhs kh`t. Gfw eicy efrh orhhc ippkhs tgic rhm ippkhs mjm ^fnhrt pjal `rfe tgh fragirm6 >2 ÿ > 5 92 9> ‑ 92 5 > ^fnhrt pjalhm > efrh orhhc ippkhs tgic rhm ippkhs.

92. ]gh mjiories nhkfw irh tilhc `rfe pirts f` tgh sieh aikhcmir efctg. _git js tgh vikuh f`  +   ‑ ‑  6 0

99

 

91

 

9: >7

 5 98,

 

1

:

 5 0 icm

 5

97

98 + 0 ‑ 97 5 1

=

9=

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC Qhatjfc N

Hiag f` tgh quhstjfcs 99 tf >2 airrjhs = eirls. 99. I akiss f` 1? pupjks irh shithm shithm jc = rfws iaafrmjco tf tghjr akiss kjkjst st cuenhr is sgfwc nhkfw. 9

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1

=

8

0

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Jc wgjag rfw irh tghrh tgh `hwhst cuenhr f` pupjks wgfsh akiss kjst kjst cuenhr js i euktjpkh f` 86 ^fws 9, > icm 1 hiag givh > euktjpkhs f` 8 hxahpt ^fw =.

9>. Jc W7 ‑ 7W 5 W, tgh sieh mjojt js ejssjco `rfe nftg nftg cuenhrs. _git js tgh ejssjco mjojt6 :7 ‑ 7: 5 : ]gh ejssjco mjojt js :.

91. Dial icm Djkk irh i pijr f` nrftghr icm sjsthr. Hiag f` Djkk‚s sjsthrs givh > ftghr sjsthrs icm hiag f` Dial‚s nrftghr gis 9 ftghr nrftghr. Gfw eicy sjnkjcos irh tghrh iktfohtghr jc tghjr `iejky6 ]ghrh irh 1 ojrks icm > nfys. ]ftik f` 8 sjnkjcos jc tgh `iejky.

9=. @jvh `rjhcms, Agirkftth, Hejky, Rhray, Rhray, Eiry icm Icch of tf tgh sieh sagffk. Eiry js jc i mj``hrhct akiss `rfe Icch icm Hejky, icm Rhray js cft jc tgh sieh akiss is Agirkftth icm Hejky. J` tgrhh f` tghe irh jc Akiss I icm twf f` tghe irh jc Akiss N, wgf irh jc Akiss N6 Akiss I4 Agirkftth, Hejky icm Icch Akiss N4 Rhray icm Eiry 8

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

98. ]gh tinkh nhkfw sgfws gfw euag `fur phfpkh sphct fvhr `fur efctgs. Gfw euag efrh efchy mjm Quej sphcm jc tgh efctgs f` Dicuiry tf Iprjk tgic ^icj sphcm jc tgh `fur efctgs6 Dicuiry @hnruiry Eirag  Iprjk

Riuk $=?? $8?= $801 $=12

^icj $118 $900 $:0 $99>9

Quej $901 $92>? $809 $?12

]je $981 $>>> $:72 $921=

118 + 900 + :0 + 99>9 5 9?97 901 + 92>? + 809 + ?12 5 >=79 >=79 ‑ 9?97 5 ?01 Quej sphct $?01 efrh tgic ^icj `rfe Dicuiry tf Iprjk.

90. ]gh cuenhrs 187, 8>: icm ?79 irh rhprhshcthm is rhsphatjvhky. _git cuenhr mfhs

,  icm  

 rhprhshct6

 5 ?9:  ?9: 

9?. Er ]ic sht gjs ikire akfal tf rjco hvhry >2 ejcuths. Jt rico `fr tgh `jrst tjeh it 1 f‚akfal.   It wgit tjeh mjm tgh ikire akfal rjco `fr tgh 92 tg tjeh6 1422 14>2 14=2 =422 Jt wjkk rjco it 0 f‚akfal.

=4>2

=4=2

8422

84>2

84=2

0422

97. Ritrjaji shkks chwspiphrs chwspiphrs ikfco i nusy strhht jc tgh efrcjco icm tgh i`thrcffc. ]fmiy, sgh sfkm >= chwspiphrs nh`frh cffc, icm

9

 f` tgh rheijcjco chwspiphrs ny tgh hcm f` tgh miy. J`

1

tghrh irh = chwspiphrs rheijcjco it tgh hcm f` tgh miy, gfw eicy chwspiphrs mjm Ritrjaji givh it tgh stirt f` tgh miy6 9 ‑ 9 5 >  1

1

>

= ÿ  5 0 1

0 + >= 5 12 Ritrjaji gim 12 chwspiphrs it tgh stirt f` tgh miy.

9:. _gjag fch f` tgh `fkkfwjco airms wis cft ushm tf `fre tgh `jourh nhkfw6

0

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

 I

N

A

M

H

M js cft pirt f` tgh `jourh.

>2. Ehryjc gis sfeh $82, $92 icm $> cfths jc gjs wik wikkht. kht. ]ghrh irh iktfohtghr 92 cfths icm tghjr tftik iefuct js $>>7. Gfw eicy $92 cfths mfhs Ehrvjc givh6 Eixjeue cuenhr f` hiag cfth ojvhc tgit tgh tftik cuenhr f` cfths js 924 @fur $82 cfths, twf $92 cfths icm `fur $> cfths. Ehrvjc gis twf $92 cfths.

Qhatjfc A

Yuhstjfcs >9, >>, >1, >= icm >8 airry 0, ?, 7, : icm 92 eirls rhsphatjvhky. >9. Gfw eicy =-sjmhm sgiphs irh tghrh jc tgh ojvhc `jourh6  `jourh6  ]ghrh irh 7 `fur-sjmhm sgiphs jc tgh `jourh.

?

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

>>. _jkkjie, Qieuhk, Dfgc icm Ohfroh irh wikljco fc i cirrfw pitg fch nhgjcm tgh ftghr, wjtg _jkkjie wikljco mjrhatky nhgjcm Dfgc, icm Ohfroh nhtwhhc _jkkjie icm Qieuhk. Dfgc cftjahs i njrm rhstjco fc tgh pitg = e jc `rfct f` gje icm stfps tgh orfup. J` ikk `fur `rjhcms irh sticmjco ic hquik mjsticah iwiy `rfe hiag ftghr, icm Dfgc icm Ohfroh irh 1 e ipirt, gfw `ir iwiy js Qieuhk sticmjco `rfe tgh njrm6 Rfsjtjfc `rfe `rfct tf nial4 D, _, O, Q 1 ÿ > 5 9.8 e 9.8 + 9.8 + 9.8 + = 5 Qieuhk js 7.8 e iwiy `rfe tgh njrm.

>1. Nhc icm Hkijch irh ojvhc i tftik f` 18 rhm icm nkuh phcs tf shkk. shkk. I`thr 8 rhm phcs icm gik` tgh cuenhr f` nkuh phcs irh sfkm, tgh cuenhr f` nkuh icm rhm phcs kh`t js tgh sieh. J` Nhc sfkm gik` tgh cuenhr f` phcs tgit Hkijch sfkm, gfw eicy phcs mjm Nhc shkk6 18 ‑ 8 5 12 12 ÿ 1 5 92 92 + 8 5 98 98 ÿ 1 5 8 Nhc sfkm 8 phcs.

>=. Diehs, Pjrojcji icm Hkkjft irh fut pjaljco pjaljco `kfwhrs jc i `jhkm. I`thr sfeh tjeh, tgh tgrhh f` tghe rhturc gfeh tfohtghr wjtg i nislht afctijcjco ikk tgh `kfwhrs tghy givh afkkhathm. Ikk hxahpt 8 f` tgh `kfwhrs whrh afkkhathm ny Diehs< ikk hxahpt 7 f` tghe whrh afkkhathm ny Pjrojcji icm ikk hxahpt ? f` tghe whrh afkkhathm ny Hkkjft. _gf afkkhathm tgh efst `kfwhrs tgit miy6 P icm H afkkhathm 8 `kfwhrs D icm H afkkhathm 7 `kfwhrs D icm P afkkhathm ? `kfwhrs P + H + D + H + D + P 5 8 + 7 + ? 5 >2 P, D icm H 5 >2 ÿ > 5 92  ‑ 5 8 D P 55 92 92 ‑ 8  7 5 > D 5 92 ‑ ? 5 1 Diehs afkkhathm tgh efst `kfwhrs tgit miy.

7

 

JC]H^CI]JFCIK QJCOIRF^H EI]GQ AFERH]J]JFC

>8. Dhrry ofhs tf tgh suphreirlht tf nuy sfeh orfahrjhs. orfahrjhs. ]gh mjiorie nhkfw sgfws tgh jthes jcsjmh nios I, N icm A `rfe tgh khist hxphcsjvh fc tgh kh`t tf tgh efst hxphcsjvh fc tgh rjogt.

Nio I

Nio N

Nio A

Khist hxphcsjvh

Efst hxphcsjvh

 Irricoh tgh `fkkfwjco pijrs f` jthes `rfe tgh khist hxphcsjvh fc tgh kh`t tf tgh efst hxphcsjvh fc tgh rjogt.

Rijr W

Rijr Z

Rijr T

 Icswhr4   Icswhr4  Khist hxphcsjvh

Efst hxphcsjvh

@rfe I icm A4

js efrh hxphcsjvh tgic

@rfe N icm A4

js efrh hxphcsjvh tgic

@rfe N icm A4

js efrh hxphcsjvh tgic

 Icswhr4 T W Z Hcm f` Riphr

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