Soal Olimpiade Matematika SMA Tingkat Nasional Dan Pembahasannya

 

7- Jhigmcgm    rgfg lnmcgm ¥  g j-

   

b- 7 l- : n- 0 :- Jgmlk rnigi~ jnzeg|g jk`kmc- R~g|~ `gzh lhg jnzeg|g entglg |n|gmccgmxg) Gmlh ? ²Tgihmc |hlge rgig` rg|~ lhgm|gzg eh|g |hlge tnzmg` jnzjk`kmc-¼ Lgzh hmakzfgrh hmh eh|g fnzgrg tgr|h jg`sg ¥g- Gmlh rnigi~ jnzjk`kmc j- Gmlh rnrnegih jnzjk`kmc b- Gmlh rnigi~ jnzeg|g jnmgz l- Gmlh rnrnegih jnzeg|g jnmgz n- Gmlh |hlge tnzmg` jnzeg|g gtg t~m 2- Jhigmcgm m |nzjnrgz rn`hmccg g- 0 j- :: b- :8 l- << n- 00

 fnfjgch  glgig` ¥-

<- Tnzmxg|ggm fgmgeg` xgmc jnmgz ; g- Oheg p 9 4 fgeg p: = p j- Oheg p: = 4 fgeg p = 4 b- Oheg p: = p fgeg p = 4 l- Oheg p: = p fgeg p 9 4 n- Oheg p 9 7 fgeg p: 9 p >- Fhrgiegm



lnmcgm¥g jbl-





rgfg lnmcgm

 ~m|~e rn|hgt jhigmcgm zngi p- Fgeg 

                                      

n- J~egm lhgm|gzg G) J) B lgm L

N&fghi ? sgig~[`gjhr[|nzgmcDzkben|fghi-bkf sgig~[`gjhr[|nzgmcDzkben|fghi-bkf(40>201443133 (40>201443133 + 40>22::44432









rgfg

1- Ihfg nekz egfjhmc fnfgegm z~ft~| rni~gr > egih ~e~zgm igtgmcgm jkig lgigf > `gzh- Jnzgtg `gzh xgmc lhtnzi~egm lhtnzi ~egm kin` 2 nekz egfjhmc ~m|~e fnmc`gjhregm fnmc`gjhreg m z~ft~| rni~gr 2 egih igtgmcgm jkig ; g- : j- 2 b- < l- > n- 1 3- Oheg ~m|~e rn|hgt p) x jhigmcgm zngi jnzige~ p , x 6 px æ p % x fgeg (p % x,(p æ x rgfg lnmcgm ¥g- p: æ x: % :p j- p: æ x: æ :p b- p: æ x: % :x l- p: æ x: æ :x n- p: æ x: 0- Jnzgtg jgmxge tgrgmc jhigmcgm j~ig| tkrh|ha (g)j xgmc fnfnm~`h g jbln-

7 : 2 < >

   



8- ^m|~e mhigh g xgmc fgmgeg` cgzhr i~z~r x 6 1p fnfk|kmc tgzgjkig x 6 p : % g |ntg| lh rg|~ |h|he; g- 3 j- 0 b- 8 l- 74 n- 77 74- Lhch| 7) 8) 8) 0 lgigf 7880 fnft~mxgh o~fig` |k|gi 7 % 8 % 8 % 0 6 :3- Jhigmcgm jnzhe~|mxg xgmc fnft~mxgh o~fig` lhch| :3 |nzoglh lh gm|gzg |g`~m ¥g- :>44 lgm :344 j- :347 lgm :844 b- :847 lgm 2744 l- 2747 lgm 8844 n- 8847 lgm 8888


77- Tglg r~g|~ rnch|hcg rnch| hcg GJB) r~l~| r~l~| B |hcg egih jnrgz r~l~| G lgm lgm r~l~| J l~g egih jnrgz jnrgz r~l~| GG- Jnzgtgeg` Jnzgtgeg ` tnzjgmlhmcgm (zgrhk gm|gzg tgmogmc GJ lnmcgm JB ; 7:- Jgmlk lgm Jgmlh hmchm fnmcnbg| tgcgz) Jgmlk lgtg| fnmxnin fnmxninrghegm rghegm tnmcnbg|g tnmcnbg|gm m tgcgz kin` lhzhmxg rnmlhzh lgigf sge|~ 2 ogf) rnlgmcegm Jgmlh lgtg| fnmxninrghegmmxg lgigf < ogf- Tglg t~e~i 7:?44 rhgmc fnzneg f~igh fnmcnbg| tgcgz jnzrgfg&rgfgjnzrgfg&rgfg- Gegm Gegm |n|gth tglg tglg r~g|~ en|heg fnzneg jnz|nmcegzjnz|nmcegz- Fnzneg Fnzneg jnz|nmcegz rnigfg 74 fnmh| lgm lgigf fgrg h|~ |hlge rg|~t~m xgmc fnige~egm tnmcnbg|gm- Rn|nig` tnz|nmcegzgm |nzrnj~| Jgmlh tnzch lgm Jgmlk fnxninrghegm tnmcnbg|gm tgcgz rnmlhzhgm- Oheg O heg Jgmlk fnmxninrghegm tnmcnbg|gm tnmcnbg|gm tglg t~e~i 7) 7) tglg t~e~i jnzgtgeg` tnz|nmcegzgm lhf~igh ; 72- Jnzgtgeg` o~fig` lhch| · lhch| jhigmcgm : :44:  >:442; 7<- Jnzgtg jgmxge jhigmcgm jhigmcgm tkrh|ha tkrh|ha xgmc e~zgmc lgzh 74-444 74-444 xgmc jnzjnm|~e p 0 % x0 ~m|~e r~g|~ jhigmcgm j~ig| p = 4 lgm x = 4 ; 7>- \nm|~egm jhigmcgm m |nzenbhi rn`hmccg rn|hgt r~j`hft~mgm r~j`hft~mgm lgzh u7) :) 2)  ) :4v xgmc jnzgmcck|gegm m ~mr~z tgr|h fnmcgml~mc l~g gmcck|g xgmc rnihrh`mxg 0' 71- Cgzhr GJ lgm BL rnogogz lgm jnzogzge < rg|~gm- Fhrgiegm GL fnfk|kmc JB lh |h|he T lhgm|gzg enl~g cgzhr- Oheg GJ 6 < lgm BL 6 7:) jnzgtg og~` T lgzh cgzhr BL ; 73- Fhrgiegm g lgm j jhigmcgm zngi xgmc jnzjnlg rn`hmccg         \nm|~egm mhigh '

    



70- Jhigmcgm j~ig| tkrh|ha t  : lhrnj~| jhigmcgm tzhfg oheg hg `gmxg fnft~mxgh age|kz 7 lgm t- \nm|~egm mhigh tnmo~fig`gm rnf~g jhigmcgm tzhfg lhgm|gzg 7 lgm 744 xgmc rnegihc~r jnzrhag| ? rg|~ injh`mxg lgzh r~g|~ jhigmcgm enihtg|gm > lgm rg|~ e~zgmcmxg lgzh r~g|~ jhigmcgm enihtg|gm 1 ' 78- Fhrgiegm 

Lgm

              

               

\nm|~egm jhigmcgm j~ig| xgmc mhighmxg tgihmc lneg| en g · j' :4- R~g|~ tnzrnch tgmogmc jnz~e~zgm 0 egih   fnft~mxgh |h|he t~rg| xgmc rgfg lnmcgm r~g|~ ihmcegzgm jnzogzh&ogzh :- Jnzgtgeg` i~gr lgnzg` hzhrgm gm|gzg tnzrnch tgmogmc lgm ihmcegzgm |nzrnj~| ;


TNFJG@GRGM

7- (Ogsgjgm ? B

       



:- (Ogsgjgm ? J Hmcegzgm lgzh ? Tgihmc |hlge rgig` rg|~ lh gm|gzg eh|g |hlge tnzmg` jnzjk`kmc glgig` ?  Enl~g&l~gmxg tnzmg` jnzjk`kmc 2- (Ogsgjgm ? B

                             Egznmg 0 |hlge fnfjgch     fgeg ? 



mfger 6 :8

<- (Ogsgjgm ? G Lgrgz |nkzh ? Oheg p 9 4 fgeg p : = p Oheg 4 9 p 9 7 fgeg p: 9 p Oheg p = 7 fgeg p: = p g- Jnmgz j- Rgig` egznmg oheg p: = 4 lhf~mcehmegm lhf~mcehmeg m p 9 4 g|g~ p = 4 : b- Rgig`- Egznmg p = p fgeg p (pæ7 = 4 rn`hmccg p 9 4 g|g~ p = 7 l- Rgig` egznmg oheg p: = p lhf~mcehmegm lhf~mcehmeg m p 9 4 g|g~ p = 7 n- Rgig` egznmg ~m|~e p 9 4 fgeg p: = p :  Tnzmxg|ggm xgmc jnmgz glgig` ? oheg p 9 4 fgeg p = p >- (Ogsgjgm ? G (g2 æ j2 ( 6 g æ j(g : % gj % j: 

                                                



















1- (Ogsgjgm ? L Enbntg|gm fgegm ~m|~e 7 nekz egfjhmc) qe 6 7 igt- jkig+ Qe 6 7+> igt jkig+`gzh+egfjhmc - Jgmxgemxg z~ft~| z~ft~| xgmc lhfgegm) lhfg egm) m z lhz~f~regm lnmcgm ? mz 6 qe  m`gzh  megfjhmc 26 

 



m`gzh  2

m`gzh 6 > `gzh


3- (Ogsgjgm ? L (p % x , (p æ x 6 (p % x (p æ x æ (p % x % (pæ x (p % x , (p æ x 6 p : æ x: æ :x  0- (Ogsgjgm ? ;

   rn`hmccg g = 1 ¥¥- (7           Egznmg   fgeg                    ¥¥¥¥- (: Egznmg j = 4 fgeg







Egznmg g = 1 fgeg fgeg (j æ 1 = 4  (2 Egznmg g jhigmcgm jhigmcgm j~ig| fgeg (j æ1 glgig` age|kz age|kz lgzh 21 lgm egznmg (j æ 1 = 4 fgeg mhigh (j æ 1 xgmc fnfnm~`h glgig` 75 :5 25 <5 15 85 7:5 70 g|g~ 21^m|~e jæ167 jæ16: jæ162 jæ16< jæ161 j63 j60 j68 j 6 74 j 6 7: g 6 <: g 6 :< g 6 70 g 6 7> g 6 7: j·1 68 j æ 1 6 7: j æ 1 6 70 j æ 1 6 21 j 6 7> j 6 70 j 6 :< j 6 <: g 6 74 g 68 g 60 g 63 Tgrgmcgm jhigmcgm j~ig| (g) j xgmc fnfnm~`h glgig` ? u (3)<: 5 (0):< 5 (8)70 5 (74)7> 5 (7:)7: 5 (7>)74 5 (70)8 5 (:<)0 5 (<:)3 v  Fgeg jgmxgemxg tgrgmcgm (g) j xgmc fnfnm~`h glgig` 88- (Ogsgjgm ? B Egznmg 1p 6 p: % g fgeg p: æ 1p % g 6 4 Lhrez 6 1:æ <(7(g 6 21 æ Egznmg lhhmchmegm J rnenbhi&enbhimxg) fgeg (B % L `gz~r rnjnrgz&jnrgzmxg lgm egznmg J  85B  8 lgm L  8 fgeg fgeg (B % L fger 6 70 rn`hmccg J fhm 6 :> æ 70 6 3Fgeg |g`~m jnzhe~|mxg xgmc lhch|mxg lhch| mxg jnzo~fig` :3 glgig` :388  Fgeg |g`~m jnzhe~|mxg xgmc lhch|mxg jnzo~fig` :3 |nzoglh lh gm|gzg |g`~m :347 lgm :844


Soal Olimpiade Matematika SMA Tingkat Nasional Dan Pembahasannya

Recommend Documents