MMHMCTapCTB0 npoceeTe, HayKe M TeXH0n0WKOr pa1eoja APYWTBO MATEMATWfAPA CP6~JE

OnWTMHCK0 TaKMM'letbe 1.13 MaTeMaTMKe y'leHMKa 0CH0BHMX WK0na

02.03.2019

Ill pa3peA

1 nyKa je nowao Aa ce caCTaHe ca APYroM v1saHoM. nyKa je 1113 Kyne · 1113awao y 16h 45min, a v1saH y 16h 35min. ~aAa je v1_saH crnrao Ha

MeCTo caCTaHKa, car je noKa3111sao 17h 26mm. ny_Ka Je 61110 Ha TOM Mecry 111 4eKao v1saHa sen 12 M111Hyra. Ko11111Ko Je speMeHa 6111110 norpe6HO nyKll1, a KOJ1111KO v1BaHy Aa CTl!1rHY Ha MeCTO caCTaHKa?

2. Aew 1114>pyj ca6111patbe (111CTe ~111q>pe cy 3aMetbeHe 111cT111M, a pa3111114111re

pa311"14111Tl/1M CJ10B1.-1Ma) . AA + BB+ CC= ABC.

3. 6ojaHa je 3aM1.-1C111.-111a jeAaH 6poj, AOAana MY 5, A6°6~jeH1.-1 J6111.~

noAe11w1a ca 3, pe3ymar noMHO>K1.-111a ca 4, OA AO 111JeHo_r poJ OAY3ena 6 na A06111jeHy pa3Jlll1KY n0Ae1111111a ca 7 111 Ha raJ Hallll1H A0611111a pe3ymar 2. Koj111 6poj je 6ojaHa 3aM111rn11111a?

4. 3an 111 w111 cse p111McKe 6pojese Matbe op, 100 Koj111 ce 3an111cyjy noMony

ABe ~wppe.

s. AKo je a • b = 160, 1113pa4yHaj: a) a . (b . 4); 6) (a : 4) • b; s)(a: 2) · (b · 2) ;

CsaK111 3a,qaraK ce 6op,yje ca no 20 6op,osa. l,13pa,qa 3a,qaraKa rpaje 120 M111Hyra. Pewetbe csaKor 3a,qarKa KpaTKO 111 jacHo 06pa311o>K111rn.

r) (a• 9) · (b : 3).

111 PA3PEA np1.13Haen1.1 ceaK1.1 Ta1.1aH nocrynaK Koj1.1 ce pa3n1.1t<yje 0A Klby1.1a .

60,qoeatbe np1,maro,q1.1T1.1 K0HKpeTH0M Ha1.11.1Hy pewaeatba.

1. (Mn S 1-3) nyKa je crnrao y 17h 14min [S noeHa], na MY je rpe6a110 15min + 14min = 29min [S noeHa]. v1saHy je rpe6a110 25min + 26min = 51 min [10 noeHa].

2 . A = 1, B = 9, C = 8, rj . peweH>e pe6yca je 11 + 99 + 88 = 198 [20 noeHa] .

3. (Mn S3-2) npe noCJ1ep,H>er p,e!beH>a ca 7 pe3ymar je 61110 14 [ 4 noeHa], npe op,y3111MaH>a 6poja 6 je 61110 20 [4 noeHa], npe MHO>KeH>a ca 4 je 61110 5 [4 noeHa], npe .QelbeH>a ca 3 61110 je 15 [4 noeHa], a npe ,qo,qasatba 6poja 5 61110 je 10 [4 noeHa] . 6ojaHa je 3aM111CJ111111a 6poj 10.

4. v1Ma 12 raKs111x 6pojesa: II, IV, VI, IX, XI, XV, XX, XL, LI, LV, LX, XC.

6op,osarn Ka KO je Hasep,eHo y ra6enw

Ta4HO HaBe,Q'eHll1X 6pojesa 0-2 3

noeHa 0 1

S. a) a· (b · 4) = 160 • 4 = 640 [S noeHa]; 6) (a : 4) · b = 160 : 4 = 40 [S noeHa]; s) (a: 2) · (b • 2) = 160 [S noeHa];

4

3

r) (a· 9) · (b : 3) = 160 • 3 = 480 [S noeHa].

5 6 7 8

5 7 9 11

9 10 11 12

13 15 17 20

I

IV PAJP(A

OpMJMaen111 CNKM T .... " nocryna111 aoj• Cf pa111111q;, OA IUlty~. 6oAONtbf

np1,•mar-OAMTM KOM~TMOM Ma'4MMY ~-N~

1. ( Mn S 2 · 3 l 4e1w~ec e1 iiaoje iieue "'""" 84 t10r e. n.a i-tory n onMua ui.,,a 2 27 - 84 = 14 3 18 nc>eMal. Kana 6111 me e1on111ue "1MaM no tp14 HOH:. YKYMH 6poj Hory cronMua 614o 6111 l tJ. l = 116. Ka1<0 je 14 3 - 126 =- 17, 10 IW.a 17 nonMua c.a 4 Hore t 12 no@Ml}.

2. lM.n 52 -3 l J.lyll'WIKa Manoi npasoyr aOH1<tKa je naarry1 aeha OA tt>erose w14p111He. na je OOl'IM Mano, npa11oyraoH11111.a 6 ny,a 8(>h111 OA UMpm1e. JlaMe, npaHMUe Manor npasoy,am1111K3 cy 4<m III S< m !10 noeMa] Crµa~uqe aenK11or npaeoyrao1mKa cy Sc rn 111 20cm, a ~ , o~ OOVIM jP S6c.m IS ~• J C 1pa1114ua KBaAJ)aT a c.a TMM o6uMOM je l4{.m \S no~M•l I A.110 y-.e1◄ 1,11< norix-wHo Orux'A" crpam~ue w.anor npaeoyrao· >ilAK3 , a1w1 nalh{' , c 1a1<11 111M non.ai.iw>l>cl . itpaewmo ypa.nu APY™ !lt"O J.aJla11<a . Ao6uja oi:v oaapajy1w 1ioe11e )

3 Ila ry fbl4X rµoj l4U3 YPM14M 1iaM>J1l'WM 6poj 3-aAataKa Ma~. yicynaH 6poj ypaf)e1◄ux l4A31aM 01~0 6M n, (4 7 ◄ 39 ➔ S8): 22S - 144 = 81 16 n~Ha] KaKO 6111 y TOM

cnr~ajy opo1 ypal}t'l1111X lc\A,l l tlK<l :I.a (s.lKO[ Oil lt,.MX 6Mo jei:it◄aK, lO JH<ll.4111 A3 6111 CP.<ll<.O QA .,..,,. 11Mc'l0 81 3 : 27 ypaf)(>tti.x la,1.llaKa 18 no ... a] 6poj 3cll'ICT3

ypaf)<"HI-IX \ll.MlM,a )<' 6"0'. np&01 l7 + t. 7 = 7t. . APYTOf l7 + 39 = 66. rpehl"r n + S8 = B'> !no 2 nof'Ma .\3 ta3Kl4 u~,◄ onroaop).

'4 Ka 1<.o 1ct11a11 6po) 1111opa no~1,11wrn l.l14¢,poM p.ll/1111-.1«10M OJl Hyne, a"' 1aapwasarn C<' 14nOM u1114>poM. UH'/l lll A<l cpem•wtt.a 4144ipa M0l'<t 6urn Haje111we 3 !'4 no..-a) Moryl\"' < y rn<"n<'h" u,y,iaj('el'I: 11 a1<.o !<' c P<"AIIIW1t,a Ll"4Jpa 3, nonoj" /<"JldH Tc'lMB 6poj 10301 . ]j clKO j(' c pe)ll-1Wlt..l U144ipa 2, nOCTOjM )t>AdH Tat<.ae 6poj 10]01 ; 31 a1<.o )<" cpt>/l111Wtt>ll l.lll!4,p,1 1. nonoje rp111 rd1<.sa 6poja: 10101 , 111 11 . 20 102.: ,n a.1<.o je cpeJluw, .. a Ull!4lpa o. nocrojt> rpu 1a1<.aa 6poJa 10001 , 11011 , 20002 /lallvl<'. y11ym◄ o noc,0) 1-1 8 6pojNla ca Hc'lBeJl<'HOM oco6111HOM Ila csa"'"' U'-1 1◄ 0

OAP<"f>et◄ 111 6poj 2 n oe' l◄ i}. a la CMKH norpeumo ttaee.Qe♦◄ 111 6poJ - 1 noet◄ . c ll'IM Acl y1<yndH 6poj nOl"l◄3 ♦◄ (' 6yA<" H('f'c'l!l4B.lH).

JM 11pn a 11 npsa KonoHa: 1 S noe1◄a ; o S. [ Ta-.1◄ 0 nony1t.<•1◄a nJ nara1<. Ta6et1e S noet◄ a . )

7 5 3 2 1 9 4 6 8

3 21 15 9 6 3 27 12 18 24 4 28 20 12 8 4 36 16 24 32 5 35 25 15 10 5 45 20 30 40 2 14 10 6 4 2 18 8 12 16

7 49 35 21 14 7 63 28 42 56 1 7 5 3 2 1 9 4 6 8 6 42 30 18 12 6 54 24 36 48

8 56 40 24 16 8 72 32 48 64

9 63 45 27 18 9 81 36 54 72

MMHMCTapCTBO npocaeTe, HayKe .., TeXH0n0WK0r puaoja

APYWTBO MATEMATM~APA CP6MJE

OnWTMHCKO TaKMMlfetbe 1113 MaTeMaTMKe ylfeHMKa 0CH0BHMX WK0na

02.03,2019 - IV palpeA

1. Y jeAHOj cam, VIMa 42 CT011Vll.le. HeKe VIMajy 3 Hore, a HeKe 4. KaA Ha CBaKy CT0111Al.lY ceAHe no jeAHO AeTe, y canVI VIMa 227 Hory (yKynHO, 111 Ae4jVIX VI CT011Vll.la) . K011VIKO VIMa CT011Vllla ca 4 Hore?

2. npaaoyraoHVIK Ha CJlVll.1111 cacraslbeH je 0A ner jeAHaKVIX npasoyraoHVIKa. O6111M caaKor Manor npaaoyraoHVIKa je 24cm. Ko11111Ka je crpaH1111.1a KBaApara K0jlA VIMa o6VIM jeAHaK 06V1MY aenVIK0r npaaoyraoHVIKa?

3. TpVI APyra cy 3a AeBeT AaHa ypaAVllla 225 3aAaTaKa . ,Qa Je npBIA ypaAVI0 47 JaAaTaKa Mat-t>e, APYrVI 39 3aAaTaKa Mat-t>e, a rpel'lVI 58 3aAaTaKa Mat-t>e·, OHAa 6111 ypaAVl111A IACTVI 6poj 3aAaTaKa. K011VIKO je CBc'.IK0 OA lhlAX ypaAVIO 3aAaTaKa 3a rnx Aeaer AaHa?

4. KonVIKO 111Ma nero1.1V1<f>peH111x 6pojeaa Koj111 ce 111cro 4111Tajy rneAaHo cnesa Ha AeCH0 "1 3AeCHa Ha neso, 4111jVI 36Vlp 1.1111<j>apa HIAje sel'IVI OA 5?

S. npe4praj Aary ra6eny Ha nanV1p Koj111 f'lew npeAaTVI, a 3aTVIM pacnopeAVI 6pojese 1, 2, 3, 4, 5, 6, 7, 8, 9 y npa3Ha noJba npse epcre, a raKof)e VI y npa3Ha nofba npee K0110He (He VICTVIM peAOM!), T3KO A3 ce MO>Ke T34HO nonyHVITVI rneAefia ra6111111.1a MHO>t<ett>a. 3arnM nonyH111 ry ra6n1111.1y.

49

C0aK111 3aAaTaK ce 6oAyje ca no 20 6oAOBa. ~3paAa 3aAaTaKa rpaje 120 M111Hyra.

9

25

4

16

I

Pewett>e caaKor 3aAaTKa KparKo 111 jaCHo 06pa3no>i<111rn.

16

16

1

81

- I

VPA3PEA np"13Haean1 ceaKM Ta'laH noc,ynaK Koj1,1 ce pun1.1Kyje 0A Klby'la .

60AOBalbe np1,1nar0A"1T"1 K0HKpeTH0M Ha'IMHY pewaealba.

1. (Mil 51 -3)3fa1p A061t1jeH111x pe3ymara rne rp1t1 AeBoj4 1114e jeAHaK je 4eTB0pocrpyK0M 36111py Tpll1 ll1B1114e KBaApa 1113 jeAHOr TeMeHa na je 36111p re rp111 111s1114e jeAHaK (29cm + 31 cm + 32cm) : 4 = 23c

1

m (1 o noeHa] . CneA111 Aa cy A,'}Kll1He rnx 111s1114a 29cm - 23cm = 6cm 31 cm - 23cm = 8cm 111 32cm - 23cm = 9cm [5 noeHa]. nospw1t1H~ KBaApa je 348cm2 (5 noeHa].

2. (Mil 53-3) ,[l,ara jeAHaKoCT ce M0>Ke Han1,1carn y 06n1t1Ky _p_ = 3 · 673

3-5 n (6poj 673 je npocr) [5 noeHa]. KaKo 6poj p rpe6a ,Qa 6y,Qe npocr, nocroje rp111 MoryliHocrn: 1 °) p = 3, n = 3 • 5 · 673 = 10095; 2°) p = 5, n = 9 • 673 = 6057; 3°) p = 673, n = 45. [CsaK111 TaYH0 HaBe,QeH111 nap no 5 noeHa, MaKap 61,10 ,Qo61,1jeH 1,1 ,,npo6att>eM"; csaK111 norpewHo HaBe,QeH111 nap -2 noeHa, c rnM ,Qa yKynaH 6poj noeHa He 6y,Qe HerarnsaH.]

3. 3 nyra: caM 6poj 2019 1,1 jow ABa nyra Kao Kpaj je,QHor 111 noYeraK Hape,QHor 6poja: 19201921 (nocne,Ql-be ,QBe 4111q>pe 6poja 1920 111 npse ABe 41,1q>pe 6poja 1921) 111 92019202 (nocne,Ql-be rp1,1 4111q>pe 6poja 9201 111 npsa 4111q>pa 6poja 9202) [20 noeHa; 3a owosop 2 (ca np1t1MepoM): 10 noeHa].

4. KaKo je 2019 = 224 • 9 + 3, HajMal-b1t1 raKaB 6poj je 399 ... 99 (224 ,Qese­TKe) [10 noeHa], a ,Qpyrn no sem1Y111H111 je 4899 ... 99 (223 ,QeBerKe) [10 noeHa] .

5. L.(1t1q>pe Mory 6111m 2, 3, 5 1,1111,1 7. l-b111xos 36111p rpe6a ,Qa je ,Qelb1t1B a 9, 3a wra nocroje ABe MoryJiHocrn: 2 + 2 + 5 111111,1 3 + 3 + 3 (10 noeHa]. nocrnje Yernp1,1 raKBa 6poja: 225, 252, 522 1..1333 (10 noeHa]. [3a rp111 raYHO HaBe,QeHa 6poja: 8 noeHa; 3a ABa: 4 noeHa; 3a 1 raYaH: 2 noeHa.J

MMHMCTapCTBO npoceeTe, HayKe M TeXH0n0WK0r pa3eoja ,QPYWTBO MATEMAT"1YAPA CP6"1JE

OnwTMHCKO TaKMM'letbe "13 MaTeMaTMKe y11eHMKa 0CH0BHMX WK0na

02.03.2019.

Vpa3peA

1. AHa, 6e4a 1,1 Be4a cy Mep11111e AY>f<1t1HY 1,1s1,14a APBeHor KBa,Qpa. Cse rp1,1 cy 1t13Mep11111e AY>Kli1HY rp111 111s1..14e 1113 1,1cror reMeHa 111 csaKa no jeAHY OA 1,1s1,14a 1113 HeKor .Qpyror reMeHa. CsaKa je ca6pana Yernp1,1 1113MepeHe AY>f<111He 111 A06111ne cy 29cm, 31 cm 111 32cm. v13paYyHaj nospw1t1Hy rnr KBa,Qpa.

2. O.Qpe.Q1t1 cse napose (p, n), r,Qe je p npocr 6poj 111 n np111po,QaH 6poj, . p 2019

TaKBe,QaJe -=--. 15 n

3. 3an111caH1t1 cy pe,QoM np111po,QH111 6pojes111 OA 1 AO 10000, 6e3 pa3MaKa 1113Meljy 6pojesa. Kon1t1Ko ce nyra y TOM H1113Y nojas.Jbyje H1113 OA Yernp1,1 4111q>pe 2019?

4. O.Qpe,Q1t1 ABa HajMal-ba np111po.QHa 6poja Y111j 111 je 36111p 4111q>apa je,QHaK 2019.

5. Hase,Q1t1 rne rpo41,1cj>peHe np111po,QHe 6pojese ,Qe.Jb 111Be ca 9 Koj111Ma cy cse 4111q>pe npocrn 6pojes1,1.

CsaK111 3a.QaTaK ce 60,Qyje ca no 20 60,Qosa. lt13pa,Qa 3a,QaraKa rpaje 120 M111Hyra. Pewett>e ma Kor 3a,QaTKa Kpar Ko 111 j acHo o6pa3110}t<~m,.

MMHMCTapcreo npoceeTe, HayKe M TeXHonowKor pa3eoja APYWTBO MATEMAT"1lfAPA CP6"1JE

OnWTMHCKO TaKMM'4etbe M3 MaTeMan1Ke y11eHMKa 0CH0BHMX WK0na

02.03.2019.

Vlpa3pe,q

1. Kom,-,Ko 1t1Ma rpoyrnosa 061t1Ma 49 K0A Koj1t1x cy AY>K1t1He crpaHm,1a npocrn 6pojes1t1?

2. AKo je a+ b = -0,7, nopeljaj no sen1t141t1H1t1 speAH0CT1111t13pa3a I a+ b - 0,51, 1-2 - a - bl 1t1 I a+ 0,3 + b - 1 j.

3. HeKa ce rnMerpana KpaKa AC jeAHaKoKpaKor rpoyrna ABC 1t1 rnMerpana yrna BCA ceKy y ra4Kl!1 Koja np1t1naAa KpaKy AB. OApeA1t1 yrnose rpoyrna ABC.

4. Ha On1t1Mn111jaA1t1 y4ecrsyje 2019 raKM1t14apa . .[l0Ka>1<1t1 Aa nocrojt.1 3eMJba 1,13 Koje je A0wno HajMat-be 64 raKM1t14apa, 1t1n1t1 Ha Om,1Mn1t1ja.Q111 y4ecrnyjy raKM1t14ap1t1 1,13 HajMaH>e 33 3eMJbe.

5. Ha KOfllt1K0 ce Ha41t1Ha 6poj 450 M0>Ke npe.QCTaBlt1Tlt1 Kao npo1t13B0A weer pa3n1,141,1rnx L.(en1t1x 6pojesa 41t1ja anconyrHa spe.QHOCT H1t1je Bena OA 6? (np01t13BO.Qe He CMarpaMO pa3nlt141t1-Tlt1M aKo ce pa3n1t1Kyjy caMo pe.Qocne.QoM 41t1H1t1nal..(a.)

CsaK1t1 3a.QaraK ce 60.Qyje ca no 20 60,Qosa. l-13pa.Qa 3a.QaraKa rpaje 120 M1t1Hyra. PeweH>e csaKor 3a,QarKa KparKo 1,1 jaCHo 06pa3no>1<1t1rn.

VIPA3PE.L\ npM3Haean, ceaKM -ralfaH nocrynaK KOjM ce punMK}'je OA KJby1ta.

60,qoeatbe npMnaro,qMTM KOHKpeTHOM HalfMHY pewaeatba.

1. (Mn s 1-2) HajAy>i<a apaH1A4a rpoyrna Mopa 61Arn sena 0A rpenlAHe 061AMa IA MaH:>a 0A H:>erose nonoBIAHe, y 0B0M cny4ajy Mopa 61Arn IA3Meljy 16 1A 24. npocrn 6pojeBIA y TOM IAHTepaany cy 17, 19 1A 23 . v1cnlArnBaH:>eM rnx cny4ajeaa A061AjaMo 5 peweH:>a: 23 + 23 + 3, 23 + 19 + 7, 23 + 13 + 13, 19 + 19 + 11 IA 19 + 17 + 13. [CaaKo peweH:>e no 4 noeHa; caaKa Hera4H0 HaBeAeHa rpojKa (Hnp. 41 + 5 + 3 IA en.) 6oAOBarn ca - 3 noeHa.]

2. (M/1 53-3) BpeAH0CTIA Aarnx 1A3pa3a cy, peA0M, 1-0,7 - 0,51 = 1,2 [6 noeHa], l-2 - (-0,7)1 = 1,3 [6 noeHa] IA l-0,7 - 0,71 = 1,4 [6 noeHa], na je I a+ b - 0,51 < 1- 2 - a - bl < I a+ 0,3 + b- 1 I [2 noeHa].

3. O3Ha4111Mo yrnose rpoyrna ca a, f3 111 y Ha yo61,14ajeH111 Ha4111H, ca D cpeA111wre KpaKa AC 111 ca E ra4KY KpaKa AB y Kojoj ce ceKy rnMerpane noMeHyre y 3aAaTKY (cn111Ka). v13 D..ADE = .6.CDE (CYC) (111n1.1 1113 csojcrna

CIIIMerpane AY>KIII) cne,QIII ,Qa je a= r [15 noeHa], 2

0,QH0CH0 /3 = y = 2a, na je Sa= 180°, a= 36° 111 /3 = y = 72° [S noeHa].

A

4. AKO HllljeAaH 0A HaBeAeHIIIX ycnosa He 6111 fa.to 1.1cnyH>eH, Hajsen111 Moryfi111 6poj raKMll14apa 61Ao 61A 32 • 63 = 2016 < 2019, cynporno no,QarKy IA3 3a,QaTKa. 3aro Mopa 61Arn 1Acnyl-t>eH 6ap je.QaH 0A HaBe,QeH111x ycnosa [20 noeHa].

S. v13 450 = 2 • 32 • 52 A06111jajy ce 4 pewett>a:

(- 5) · (- 3) · 1 · 2 · 3 · 5, (- 5) · (-3) · (-2) · (-1) · 3 · 5, (- 5) · (- 1) · 1 · 3 · 5 · 6, (-6) · (-5) · (-3) · (- 1) · 1 · 5.

[CsaKo ra4H0 pewett>e no S noeHa, csaKo HeTa4Ho -3 noeHa, c rnM Aa yKynaH 6poj noeHa He 6yAe Herar111saH.]

MHHHCTapCTB0 npocBeTe, HayKe 11 TeXH0n0WK0r pa3BOja

APYWTBO MATEMAT"1'-IAPA CP6"1JE

OnWTl1HCKO TaKMl1'1etbe 113 MaTeMaTl1Ke y'leH11Ka 0CH0BHl1X WKona

02.03.2019.

VII pupeA

1. OApeALi1 npL11p0AaH 6poj n Ta KO Aa Ba>KL'l:

3.59 +2 ·59 312 ·9 11 :2?1° a) 56 5n; 6) 3n 3.

2 . OApeAL11 HajMat-bL'l Lil Hajsen1.1 npLilpoAaH 6poj x 3a KOjLil je 6poj

a=.J66-✓x+1 raKol)e np1t1poAaH.

3. Ko111t1Ko rpe6a Aa 6yAe pa3110MaKa y Li13pa3y x 2

• X 3 X 3 • x• x• • x 5

x x2 X

3

Aa 61t1 1-berosa speAHOCT 61.111a x2010?

4. Y yHyrpawl-bOCTl.1 npasoyraoH1.1Ka ABCD cMewreH je npasoyraoHl'1K A'B'C'D' 41.1je cy apaH1t1L\e napanentte apaH1t1L\aMa npaso­yraoH1t1Ka ABCD (B1'1A1'1 CJ11t1Ky). AKo je AB= 10cm, BC= 7cm, A'B' = 4cm 1t1 B'C' = 3cm, OAPeA1'1 WTa je Bene - 361.1p noBpW1.1Ha rpane3a ABB'A' 1,1 CDD'C' 1,1111,1 361,1p nospw1-1Ha rpane3a ADD'A' 1.1 BCC'B'.

5. Aar je rpoyrao v1 Ha csaKoj OA H>erosv1x crpaHt.1L\a 1,13a6paHe cy Yernp1,1 Ta4Ke, pa3111,141,1re OA TeMeHa rpoyrna . Ko11v1Ko 1t1Ma YeTBopoyrnosa 41,1ja cy TeMeHa He Ke OA 12 1.13a6paHv1x Ta Ya Ka?

CsaK1.1 3a,[laTaK ce 60,llyje ca no 20 6oAOBa. v13pa,[1a 3a,[laTaKa rpaje 120 Mt.1Hyra. Pewet-be CBaKor 3a,[\aTKa KparKo 1,1 jarno 06pa3110>1<1.1rn.

VII PA3PE.Q np1113HaBaTIII CBaKIII Ta~aH noaynaK KOjlll ce pun111K}'je OA Klby-.a.

60AOBalbe np111narOAIIITIII KOHKpeTHOM Ha-.111Hy pewaBalba.

1. (Mil 53-2) a) n = 4 [10 noeHa];

6 312. 911: 2?10 312. 322 : 330

) 3

n 3

n 34-n=3 3a4-n=l , Tj.n=3[10noeHa].

2. (M/153-2) 3a x = 1 je 0 = ✓66-✓2 , a 3a x= 2 je a = ✓66-✓3, WTO HIIICY

np11 po.QHl1 6pojeB11. 3a x = 3 je a = ./64 = 8, na je x = 3 HajMal-ba OA

rpa>1<eH11x Bpe.QHOCTl1 [10 noeHa]. HajBerie x ce ,Qo611ja Ka,Qa je ~ = 65,

rj . x = 4224, y KOM CJJyYajy je a= 1 [10 noeHa] .

3. Pa3Jl0ML\11 y 113pa3y cy pe,Q0M je,QHaKl1 x4, X5

, x6, ... AKO je noCJJe,Ql-bl,1 OA 1-bl1X x", 0H,Qa je Bpe,QHOCT 1,13pa3a x4+s+ .+n, WTO je jeAHaK0 x2°10 a KO je 4 + 5 + ... + n = 2010 [8 noeHa] . no3HarnM noaynKOM ce A0611ja ,Qa je 4 + 5 + ... + n =

~n+O . (1 + 2 + ... + n) - 6 = ---- 6, na ce noCJ1e,Ql-ba Je,QHaK0CT CB0Al.1 Ha

2

n(n + l) -6 = 201 O [6 noeHa], 0,QHOCH0 n(n + 1) = 4032, WTO je 1,1cnyl-be-H0 2

3a n = 63 [5 noeHa] . 6poj pa3110MaKa y 1,13pa3y je 63 - 3 = 60 (1 noeH].

4 . O3Ha411Mo Bl1rnHy rpane3a ABB'A' 113 TeMeHa A' ca x, a s11rnHy rpane3a BCC'B' 113 reMeHa B' ca y. Ta,Qa je s1.1rnHa rpane3a CDD'C 1n TeMeHa D' je,QHaKa 4 - X, a Bl1Cl1Ha Tpane3a ADD'A' 113 TeMeHa A' je,QHaKa 6 - y (8

. 10+4 10+4 noeHa]. 3aro Je PABB'A' +Pcoo·c =--x+--(4-x)=28, Psccs· +PAoo·A· =

2 2 7+3 7+3 . --y +--(6- y) = 30 [12 noeHa], na je ,Qpyrn 361,1p sen11 OA npsor.

2 2 5. Te Yernopoyrnose M0>KeMo no,Qen11rn y ,QBe rpyne:

1 °) 0H11 Koj11 11Majy no ,QBa TeMeHa Ha ,QBe OA rp11 apaH111.1e rpoyrna; 2°) 0Hl1 KOjl1 11Majy ,QBa TeMeHa Ha je,QH0j CTpaH"1L\11 11 no je,[\HO TeMe Ha CBaKoj 0A ,Qpyre ,QBe apaH111.1e [4 noeHa]. Y npsoj rpyn11, crpaH111.1e Ha Koj11Ma cy ,QBa reMeHa Mo>KeMo 113a6parn Ha 3 Ha411Ha, a 3arnM Ha CBaKoj 0A 1-bl1X 2 reMeHa Ha 6 Ha411Ha, na rpoyrnosa y osoj rpyn11 11Ma 3 · 6 · 6 = 108 [8 noeHa]. Y ,Qpyroj rpyn111, crpaH111.1y Ha Kojoj cy ,QBa reMeHa M0>KeMo 113a6parn Ha 3 Ha411Ha 11 Ha tt>oj ra ,QBa reMeHa Ha 6 Ha411Ha; AaJbe, Ha csaKoj 0A npeocrane ,QBe crpaH"1L\e M0>KeMo je,QHO reMe 113a6parn Ha 4 Ha4111Ha . 6poj Yernopoyrnosa y .Qpyroj rpyn11 je 3 • 6 • 4 • 4 = 288 [8 noeHa] . YKynaH 6poj 4eTsopoyrnosa je 108 + 288 = 396.

MHHHCTapCTB0 npoceeTe, HayKe M TeXH0n0WK0r pa3eoja

.QPYWTBO MATEMATWfAPA CP61,1JE

OnWTMHCKO TaKMMlfetbe M3 MaTeMaTMKe

ylfeHMKa 0CH0BHMX WKona

02.03.2019.

VIII pupe,q

2. ,[l0Ka>1<1-1 Aa je 6poj s = 1 + 7 + 72 + 73 + ... + 72018

Ae/bl-18 ca 19.

3. nospw1-1Ha OMOTa4a npas1-111He weaocrpaHe np1-13Me je 648cm2, a A1-1jaroHa11a 6o4He apaHe je 15cm. 1113pa4yHaj nospw1-1Hy np1t13Me.

4. ,ll0Ka>1<1-1 Aa je npo1-13BOA 4ernp1,1 y3acronHa HenapHa np1t1pOAHa 6poja ysenaH 3a 16 jeAHaK KBaApary np1-1poAHOr 6poja.

S. Kpy>t<H1-14e k1 1,1 ki jeAHaK111x no11ynpe4H1t1Ka, ca 4eHrp111Ma A 1,1 8, pe,QoM, ceKy ce y ra4KaMa C 111 D. nonynpase AC 111 AD ceKy Kpy>t<H1t14y ki jow y ra4KaMa f 111 F. AKo je zKAD = 75°, OAPeA111 4.EBF.

CsaK111 3a,QaraK ce 6oMje ca no 20 60AOBa. v13pa.£1a 3a,llaTaKa rpaje 120 M111Hyra. PeweH>e CBaKor 3a.QaTKa KparKo 111 jaCHo o6pa3no>1<1t1rn.

VIit PAJPfA npM3H38aTM CHKM TalfaH nocryna• KojN c.e ~ o,- ui.,..a. 6o~

npMnarOAKTM KOHK~ KallflOfY pftla~.

9 ( 3)1

1. 36or x 1 - 3x+ 4= x - 2 , 4 - 2✓3 =(✓3 - TY ~ ✓3 - 1> 0 ~lolK! cecso11111

Ha /x-~/= ✓3 [12 noeHa ]. H>eHa pewe1t>o cy ~ ... ✓3 "' ~ -✓3 'lnoeNat '. <np~ 2 2 2

5 ~ 1 - --1,1 a Ko cy pewett>a Jan111caHa y 06n1,11<y - + v 4 - 2✓3 . - - .J 4 - 2✓3 l [Alw raKMi.Nilp

2 2

HJOCTaBH JHaK anconyTHe spe.QHOCTIII KO,Q Ix -ii MM .no(Mje caA,fO je.a:HO DE.>'.llett><:'

nocne,Qtt>e je,QHa4HHe, 60.Qyje ce ca O noeHa.]

2 . (Mn S3-3) I peweH>e. 1 + 7 + 71 + 73 + ... + 7'°" + 110•1

... 1= = (1 + 7 + ?7) + 73

• (1 + 7 + 71) + ... + 710

'6

. (1 + 7 + 71) = 57 . (1 + 71 ... . - 7.ll'i. wro je ,QeJbHBO ca 19 jep je 5 7 = 19 · 3 [20 noeHa. AKo ce KOHCTa;yje Ai! ,e 1 - 7 - 7 = 57 ,QeJbHBO ca 19, 6eJ HJBol)eH>a ,QaJbHx JaKJby4aKa: 7 noeKa).

II peweH>e. Ocra1.1111 neneHa 6poja 7 np111 ,Qelbett>y ca 19 cy. ~.OOM. 7 11 l 7 11 1 . 7 noeHa] . 3aro J61,1p 6111no Koja Tp111 yJaaonHa crenetta aa,e cx-r aTaK 7 - 11 - 1 = 19, r1. ,QeJbHB je ca 19 [7 noeHa]. KaKo je 3 / 2019, ro rpym4Catt>i'M cao•paJ<a .QaTOr 36,tpa (Koj1,1 HMa 2019 ca6111pa1<a) y rpyne no TPH yJacrnntta, AOOK)clMO .!l<I Je 19 ' 5 ,6 ..,._..;

3 . (Mn S3-2) AKO ca a OJHa4111Mo OCHOBHY 111s1111.1y. a ca H BM0 1tty npM.3Me i0 y(Jl08d

Ja,QarKa ,Qofa.,jaMo ,Qa je 6aH = 648 1,1 a1 + H2 = 225 [ 4 no.tQ~. 1,13 2aH = 2 l 6 111 ~ .. '1_. = 225 cne,Qi.1 .Qa je (a+ H) 2 = a1 + H1 + 2aH = 225 + 216 = 441 ,-,.3 ,ea.,. H = ?1 i4 IIOftla ,

v13 a+ H = 21 111 aH = 108 .Qo6111ja ce .Qa nOCToje ,Qse M<>ryf'IHOCTK:

1 °)a = 12, H = 9 [3 noeHa]. 1,1 2°) a = 9. H = 12 [3 noflta :. Y nps:OM Cfty'l3jy nospw ... ...a 1e

(432✓3 + 648)cm2

[3 noeHa]. a y .QpyroM ( 243-✓3 ~ 648-Jcm ' [3 nofttil:.

4. (2n - 3)(2n - 1 )(2n + 1 )(2n + 3) + 16 [8 noetw! = (.in-' - ll(~ - 9) ... l o = l6n4

- 40n 2 + 25 [6 noeH.11] = (4n.: - Sr 33 n ?: 2 [6 ~ :

S. Ka Ko je ACBD poM6 . .QaKJ1e napanellOfpat.t, TO je MCB = 4CAO = 75° (cn11~) [6 noetta)

l'::. BCE je j e.QHa11.01<pa11.1111poyrao, na je 4BEA = // \'

i<. BEC = i<.ECB = 75°. CmNHO je 4. BFA = 75" (7 ~ ,-4--\ - -"'8 noeHa]. n peMa roMe je 4EBF = 3'.60" - 3 75" = 135° [7 noeHa]. tnp.n1H<'rn " ai,.o ce OJ.Ux'At1 1

Ko11KasaH i<.EBF = 225°.1 0

F

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