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100 Elementarnih zadataka
**** MLADEN SRAGA ****
POTPUNO RIJEŠENI ZADACI PRIRUČNIK ZA SAMOSTALNO UČENJE
100 ELEMENTARNIH ZADATAKA koje bi svatko trebao znati riješiti
I
POTENCIJE
a a aa a a
aa
a a a
a bc a b c
a a
ab
ab
ab
ba
ba
aa a
aa
n m n m
n m n m
n
mn m n m
n n n n
n m n m
n n
n
n n n
n
⋅ =
=
= =
=
=
FHGIKJ =
FHGIKJ = FHG
IKJ =
=
=
=
+
−
−
⋅
−
−
:
:
b gd i
0
1
1
1
1
1aa
nn=−
ALGEBARSKI IZRAZI
a b a b a b a ab b
a b b a
a b a b a b a ab b
a b b a
a b a b
a b a b a b
a b a a b ab b
a b a a b ab b
a b a b a ab b
a b
+ = + ⋅ + = + +
+ = +
− = − ⋅ − = − +
− = −
− − = +
− ⋅ + = −
+ = + + +
− = − + −
− = − ⋅ + +
+
b g b g b gb g b g
b g b g b gb g b gb g b gb g b gb gb g
b g d i
2 2 2
2 2
2 2 2
2 2
2 2
2 2
3 3 2 2 3
3 3 2 2 3
3 3 2 2
3
2
2
3 3
3 3
3 2 2
2 2 2 2 2 2 2
2
2 2
= + ⋅ − +
+ + = + + + + +
RSTUVW
R U+ + =
+ =
⋅ == + ⋅ +
+ =
a b a ab b
a b c a b c ab ac bc
x px qm n p
m n qx m x n
m n b
b g d ib g
a f a f
ST VW⋅ = ⋅+ + = = + + + =
m n a cax bx c ax mx nx c . . .
α
TEL. 01-4578-431
M.I.M.-SRAGA 1999./2004.
M I M SRAGA 1999 / 2004
2
100 Elementarnih zadataka
Autor: MLADEN SRAGA Grafička obrada MLADEN SRAGA
Procijena i ocijena znanjaako se spremate :
Za prijemne na sve Za prijemne na sve EKONOMSKE fakultete TEHNIČKE fakultete
0 50 0 0 50 0
51 75 1 51 81 176 85 2 82 86 286 91 3 87 90 392 95 4 91 94 4
96 100 5 9
− = − =
− = − =− = − =− = − =− = − =− =
hitno tražite instruktora
trebali bi ozbiljno poćeti vježbati
Vi znate nešto ali ima tu puno rupa
5 100 5
Moj komentar:ako ste dobili ocjenu 0 -ako ste dobili ocjenu 1 - ako ste dobili ocjenu 2 - ako st
− =
dobro možete krenut na pripreme
vi ste spremni za zadatke sa prijemnih
odlično- vama zadaci s prijemnog
e dobili ocjenu 3 - ako ste dobili ocjenu 4 - ako ste dobili ocjenu 5 - nebi trebali predstavljat problem
Prvo riješite zadatke pa tek onda gledajte u rješenja jer su tamo kompletno riješeni....
Potpunu garanciju na zadatke i rješenja daje: centar za dopisnu poduku M.I.M.-SRAGA -dakle sve što vam se čini nejasno krivo ili sumnjivo - zovite 01-4578-431 ili 01-4579-130 i tražite dodatne upute i objašnjenja ...
Zadatci su u potpunosti naš autorski rad , ako se neki zadatak već pojavio u nekoj drugoj zbirci ...nije namjerno ponovljen s naše strane.
Centar za poduku M.I.M.-Sraga zadržava sva prava na komercijalno iskorištavanje ovih zadataka. Svako kopiranje umnažanje za vlastite potrebe i potrebe poduke i nastave je dozvoljeno i dobrodošlo. Cijena ovih zadataka je 0 kn tj. besplatni su ...i tako treba i ostati... Sve kritike i dobre i loše rado ćemo saslušati ... javite nam se na tel. 01-4578-431 ili e-mailom : [email protected]
M I M SRAGA 1999 / 2004
3
100 Elementarnih zadataka
STO ELEMENTARNIH ZADATAKA KOJE BI SVI TREBALI ZNATI RIJEŠITI
Koristeći ove formule trebali bi riješiti zadane zadatke:
( )
( )
( )( )
0
1
1
1
1
1
POTENCIJE
:
:
1
1
1
1 1
1 1
n m n m
n m n m
nn m n m
m
n n n n
mn n m
zmn n m z
n n
n
n n n
n
nn
nn
nn
n
mn mn
mn
m n mn
a a aa a a
a a a aaabc a b c
a a
a a
a ab b
a b bb a a
aa a
aa
aa
a a
aaa
a a
aaa
+
−
−
⋅
⋅ ⋅
−
−
−
−
−
⋅ =
=
= =
=
=
=
=
= =
=
=
=
=
=
= =
=
= =
( )
( )( )( )( ) ( )
( )( ) ( )
( )
( )
2 3
2 4 6
2 523 2
4 2
7 2 3
7 8 9
8
3
32
543
26 52 2
35 52 4
32
42 3
322 3 2
1
2 3
1)2)
3)4) :5) : :6) :
7 )
8)
9)
10)
11) :
12) 2
213)3
214) : 23
15)
16)
x xx x x
x x xx xx x xx x xxx
x
x
x x
x x
ab
x y
a b ab
xy
x yy x
−
−
⋅ =
⋅ ⋅ =
⋅ ⋅ =
=
=
⋅ =
=
=
=
⋅ =
=
=
=
=
=
⋅ =
3 222 2
3 3 4
1 2 3
10 0 0 1
1 10 1 3
17 ) :
2 1 118)3 3 2
119) 27 23
1 120) 2 22 5
− − −
−
−− −
⋅ =
+ − =
⋅ ⋅ ⋅ ⋅ =
⋅ ⋅ ⋅ ⋅ =
a y yy a a
x y
y
M I M SRAGA 1999 / 2004
4
100 Elementarnih zadataka
12
12
23
34
12
14
12
14
12
32
3 3
4 4
5 2 5 3
6 5 6 2
21)
22)
23)
24)
25) 36
26) 16
2527 )49
1628)81
6429)49
930)16
31) 10 10
32) 3 9
33) 2 8
34)
35) :
x
x
x
x
x x
x x
−
−
−
−
=
=
=
=
=
=
=
=
=
=
⋅ =
⋅ =
⋅ =
⋅ =
=
( )
( )( )( )
0
1
1
1
1
1
POTENCIJE
KORIJENI
:
:
1
1
1
1 1
1 1
n m n m
n m n m
nn m n m
m
n n n n
mn n m
zmn n m z
n n
n
n n n
n
nn
nn
nn
n
mn mn
mn
m n mn
n n n
n n
a a aa a a
a a a aaabc a b c
a a
a a
a ab b
a b bb a a
aa a
aa
aa
a a
aaa
a a
aaa
a b a b
aa b
+
−
−
⋅
⋅ ⋅
−
−
−
−
−
⋅ =
=
= =
=
=
=
=
= =
=
=
=
=
=
= =
=
= =
⋅ = ⋅
⋅ = nb
M I M SRAGA 1999 / 2004
5
100 Elementarnih zadataka
Koristeći formule koje smo upravo obradili i ove nove koje vam dajem ovdje riješite sljedeće zadatke: 36
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )( ) ( )
( ) ( )( )( )
( ) ( )( ) ( )
( )
2 2 2
2 2
2 2 2
2 2
2 2
2 2
3 3 2 2 3
3 3 2 2 3
3 3 2 2
3 3 2 2
2 2 2 2
ALGEBARSKI IZRAZI
2
2
3 3
3 3
2 2
a b a b a b a ab b
a b b a
a b a b a b a ab b
a b b a
a b a b
a b a b a b
a b a a b ab b
a b a a b ab b
a b a b a ab b
a b a b a ab b
a b c a b c ab a
+ = + ⋅ + = + +
+ = +
− = − ⋅ − = − +
− = −
− − = +
− ⋅ + = −
+ = + + +
− = − + −
− = − ⋅ + +
+ = + ⋅ − +
+ + = + + + + 2c bc+
M I M SRAGA 1999 / 2004
37 38 39 50 51 52
( )
( )
( )
( )
( )
( ) ( )
( ) (
( )
( ) ( )
( )
2
2
2
23
23 4
2
2
2 3 2 3
2 3 4 5 6 2 3 4 5 6
3
2
3
) 3
) 2 3
) 3 4
1 2)2 3
2 340)3 4
41) 5 6
42) 2 3
43) 2 5 2 5
2 4 2 444)3 5 3 5
45) 3 4 3 4
46) 3
47 ) 2 2
48) 2 3
49
−
+ =
+ =
− =
+ =
− =
− =
− − =
− ⋅ + =
− ⋅ + =
− ⋅ −
+ =
+ ⋅ + =
− =
x
x y
x y
x y
x y
x y
a y
x x
x y x y
x y z a b x y z a b
x
x x
y z
( )
( ) ( )
3
32 3
32 3 4
3 3
1 1)2 3
2 3)3 2
) 3 2
)
+ =
− =
− =
+ − − =
x
x y
x y z
x y x y
) =
6
100 Elementarnih zadataka
P
( ) ( ) ( ) ( ) ( ) ( )( )
( ) ( )
( ) ( )
( ) ( )
3 6
3 3 23 23 6 3 3 3 2 2 2 2 2
333 3 2
2
2 2 2 4
3 3 2 2Prepoznaj:
rimjer:27 8 ?
ješenje:
27 8 3 2 3 2 3 2 3 3 2 2
3 2
3 2
3 2 9 6 4
x y
x y x y x y x y x x y y
a x b y
a x b y
x y x xy y
a b a b a a b b
− =
− = − = − = − ⋅ + ⋅ + =
↓ ↓
↓ ↓
= =
= =
= − ⋅ + +
− = − ⋅ + ⋅ +
r Koristi gornji primjer i ove formule riješi sljedeće zadatke:
( ) ( )( ) ( )
3 3 2 2
3 3 2 2
ALGEBARSKI IZRAZI
a b a b a ab b
a b a b a ab b
− = − ⋅ + +
+ = + ⋅ − +
Rastav
3
3
6
6 6
3
9 6 3
i na faktore:
) 8
) 8 27
) 125
)
1) 127
) 64 125
x
y
x
x y
y
x y z
− =
− =
+ =
− =
+ =
+ =
53 54 55 56 57 58
M I M SRAGA 1999 / 2004
7
100 Elementarnih zadataka
( )2 2 2 2 2 2 2a b c a b c ab ac bc+ + = + + + + +Koristeći se formulom riješi sljedeće zadatke: ( )
( )
( )
2
2
2
) 2 3
) 2 3 4
) 3 3
x y
x y z
x y
+ + =
+ + =
+ − =
59 60 61 Kor 62 63 64 65 66 67 68
( ) ( )
( ) ( )
( ) ( )
2
2 2
4 6
4 4
2 4 8
2 2
22
2 2isteći se formulom:
rastavi na faktore:
) x 16
) 4 36
1 25)4 49
)
) 4
) 2
) 36 16
y x
x y
x y
x y z
x y x y
x y x y
a b a b a b
− =
− =
− =
− =
− =
+ − − =
+ − − =
− = − ⋅ +
M I M SRAGA 1999 / 2004
8
100 Elementarnih zadataka
Kor 69 70) 71 72 73 74
( ) ( )
2 2
2 2
6 3 4 8
4 6 2 3
2 4 8
2 2 4
2 22 2 2 2isteći se formulama:
slijedeće zadatke zapiši u obliku kvadrata binoma:
) 9 6
4 20 25
) 9 12 4
) 2
) 2
1) 3 94
2 2
x y y
x xy y
x x y y
x y x y
x xy y
a ab b
a ab b a b a ab b a b
+ + =
− + =
− + =
+ + =
− + − =
− + =
+ + = + − + = −
( ) ( )2
2 2
2
2
2
...
Kvadratni trinom
Koristeći se gornjim formulama rastavi na faktore slijedeće kvadratne trinome:
75) x 2 15
76) 3 2
77 )
ax bx c ax mx nx c
m n px px q x m x n
m n q
m n b
m n a c
x
x x
x x
+ + = = + + + =
+ =+ + = = + ⋅ +
⋅ =
+ =
⋅ = ⋅
+ − =
+ − =
− 12− =
M I M SRAGA 1999 / 2004
9
100 Elementarnih zadataka
S 78 79 80 81 82 83 84 85 86
( )
( )
2 2
2 2
4 4
2 2
2
2 2
2
4 4
3 3
2 2
3
Pogledaj u formule za ove brojnike u 81) i 82)
krati razlomke koristeći sve što smo izvježbali do sada:
4 6)6 9
)2
)2
)
)
)3 3
)
x yx y
x yx xy y
a ba ab b
x yy x
y xx y
x yx y
x
−=
−
−=
+ +
−=
− +
−=
−
+=
−
−=
−
( )
3
3
2
2
2 2
2 2
3 18 27)2 18
2 3)3 6 3
yx y
x xx
x xy yx xy y
+=
+
+ +=
−
− +=
+ +
M I M SRAGA 1999 / 2004
10
100 Elementarnih zadataka
87 87 88 89 90 91
=
( )
( )
2
2
2
2 2 4 4
2 2 2
3 2)2 3 4
1 2)2 2
2 3 3 2)1 1 1
1 1)3 3
1)
)
x x x
x x
xx x x
y y
yx y x y
y yx y y x
+ −− + =
− =+ −
−+ −
− + −
− =− −
− =+ −
− =− −
Iz 92 93 94
( )
2 2 2 2
3 3 2
4 4 6 3 3 6
3 3 4 2 2
4
2
računaj:
4 4 4)
2)
1 2 1)1 1 9 1
x y x xy yx y y x
x y x x y yx y x x y
xx x x
− + +⋅ =
− +
− + +⋅ =
+ +
− + ⋅ = + − −
M I M SRAGA 1999 / 2004
11
100 Elementarnih zadataka
( )( )
2 2
2 2
2 2 2
2 22
3 2
2 2
2 3
2
3 18 27 9) :6 24 4 4
2) :
1 1 3 2 2)1 1 1 6 9
111)
111
111 2)111 2
00)4 4
x y xy y xx y y x x
x y y yx yxz yz zx y z
x x x xx x x x x
y
y
xxxx
x y x yx y x yx y xy
xy x y y
+ + −=
− + +
+ − +=
− −− −
− + + − + ⋅ − + − −
+−
=−
+
+−
− =+
+−
− +−
+ −=
−− + −
95
96
97
=
98
99
1
M I M SRAGA 1999 / 2004
12
100 Elementarnih zadataka
Kompletna rješenja s postupkom
( )
( )( )( )( ) ( )
2 3 2 3 5
2 4 6 2 4 6 12
2 2 2 6 5 32 2 5 315 22 3 23 3 2 62
4 2 4 2 2
7 2 3 7 2 3 2
7 8 9 7 8 9 6
88 3 8 3 5
3
32 2 3 6
543 3 4 5 60
26 52 2 2 6
1)2)
3)4) :5) : :6) :
7 ) :
8)
9)
10)
+
+ +
⋅ + ⋅ + ⋅+ + ⋅
−
− −
+ −
−
⋅
⋅ ⋅
⋅ ⋅
⋅ = =
⋅ ⋅ = =
⋅ ⋅ = = =
= =
= =
⋅ = =
= = =
= =
= =
⋅ =
x x x xx x x x x
x x x x x xx x x xx x x x xx x x x xx x x x xx
x x x
x x x
x x x
( )( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )( )
2 2 5 24 10 24 10 34
35 52 4 2 5 3 4 5 30 20 30 20 10
3 32 3 3 2 3 2 3 3 6
4 4 4 42 3 2 3 8 124
3 32 3 3 22 3 2 2 3 2 2 2 6 9 23
11) : : :
12) 2 2 8 8
2 2 1613)3 3 81
2 2 814) : 2 : 2 : 43 3 27
⋅ +
⋅ ⋅ ⋅ −
⋅
⋅ = ⋅ = =
= = = =
= ⋅ ⋅ = ⋅ ⋅ =
= ⋅ ⋅ =
= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅
x x x x x
x x x x x x x x
ab a b a b a b
x y x y x y
a b ab a b a b a b a( )
( )( ) ( )
( )( )
4
6 9 2 4 4 5 4 5
2 4 2 4
1
2 3 2 3 2 3 2 3 5
3 2 3 2 3 5
3 23 2 2 222 2 2 6
3 2 23 3 4 93 3 4
8 1 4 2 1 227 4 27 4 27
15)
16)
17 ) : :
−
− +
+
⋅ =
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ = ⋅ =
⋅ ⋅ ⋅ ⋅
=
⋅ = ⋅ = ⋅ = =
⋅ = ⋅ =
b
a b a a b b a ba b a b
x yy x
x y y y y y y yy x x x x x x x
a ya y y y ay a a yy a a
2 4 6 2 8
6 8 9 6 4
2 8 8 8
9 2 2 9 2 11
1 2 3 2 3
10 0 0 1
10 1
:
2 1 1 3 3 2 3 3 3 2 518) 9 8 13 3 2 2 1 1 2 2 2 2
1 319) 27 2 1 1 1 2 2 3 63 1
1 120) 22 5
+
− − −
−
−
⋅ = ⋅ ⋅ =
⋅= = =
⋅ ⋅
+ + − = + − = + − = + = =
⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ =
⋅ ⋅ ⋅
y y a y aa a y a y
y a a ay y y y y
x y
y1
33
1 1 5 1 1 5 1 52 12 2 1 2 2 2 8 32
−− ⋅ ⋅
⋅ = ⋅ ⋅ ⋅ ⋅ = = ⋅ ⋅
M I M SRAGA 1999 / 2004
13
100 Elementarnih zadataka
( )
( )
12
12
12
2 1 12 2 3 23 3 3
34
3 1 3434 4
122
14 444
12
1 4444 44
4
1 12 2
)
1 1)
)
1 1 1)
) 36 36 6 6
) 16 16 2 2
25 25 25 5)49 49 749
16 16 2 2 2)81 81 3 3 3
64 49 49)49 64
−
⋅
−
−
=
= =
= = =
= = =
= = =
= = =
= = =
= = = =
= =
x x
xxx
x x x x
xxxx
333 3 32 2 32 3222 2 2
2 3
2
3 33 3 33
4 44 4 44
5 2 5 3 2 3 5 55
6 5
49 764 864
9 16 4 4 4 4 4)16 9 3 3 3 3 3 27
) 10 10 10 10 10 10
) 3 9 3 9 27 3 3
) 2 8 2 8 16 2 2
)
) :
− ⋅
= =
= = = = = = =
⋅ = ⋅ = =
⋅ = ⋅ = = =
⋅ = ⋅ = = =
⋅ = ⋅ = =x x x x x x
x 2 36 5 2 6 5 2 6 3 3 26 : ⋅−= = = = = =x x x x x x x x
64
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
M I M SRAGA 1999 / 2004
( )
( ) ( ) ( )
( ) ( ) ( )
2 2 2
2 2 2 2 2 2 2
2 2 2 2 2 2 2
2 2 2 23 3 3 2 3
2
2
2 2
2 2
36) 3 2 3 3
37 ) 2 3 2 2 2 3 3 2 12 3
38) 3 4 3 2 3 4 4 3 24 4
1 2 1 1 2 2 1 239) 22 3 2 2 3 3 32
6 9
4 12 9
9 24 16
x x x
x y x x y y x xy y
x y x x y y x xy y
x y x x y y x xy
x x
x xy y
x xy y
+ = + ⋅ ⋅ + =
+ = + ⋅ ⋅ + = + + =
− = − ⋅ ⋅ + = − + =
+ = + ⋅ ⋅ + = + +
+ +
+ +
− +
( )
( ) ( )
( )( ) ( ) ( )
( ) ( )( ) ( )
2 23 32
2 2 2 2 22 23 4 3 3 4 4 3 3 4 32 2
2
2 2 2
22
2 6
6 3 4 8
2 2
23
2 2 32 3 2 2 3 3 2 340) 23 4 3 3 4 4 3 43 4
1 141) 5 65 6 5 2 5 6 6
42) 2 3 1 2 3 1
1 2 44 3 9
4 99 16
125 60 36
y
x y x x y y x x y x
x yx y x x y y
a y a y
x xy y
x x y y
x xy y−
=
⋅ ⋅ − = − ⋅ ⋅ + = − ⋅ ⋅ + = ⋅
=
− = = =− − ⋅ ⋅ +
− − = − ⋅ + = −
+ +
− +
− +
( ) ( ) ( )( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
2 2 2 2
2 2 2 2
2 2 2 22 22 3 2 3 2 3 2 32 2
22 3 4 5 6 2 3 4 5 6 2 3 4
2 2
2
4 6
2 3 1 2 2 2 3 3
43) 2 5 2 5 2 5 2 25
2 4 2 4 2 4 2 444)3 5 3 5 3 5 3 5
45) 3 4 3 4 3 4
4 12 9
4 25
4 169 25
a y a a y y
x x x x
x y x y x y x y
x y z a b x y z a b x y z
a ay y
x
x y
⋅ + = ⋅ + ⋅ ⋅ + =
− ⋅ + = − = − =
− ⋅ + = − = − =
− ⋅ − = −
+ +
−
−
( )( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
25 6
2 2 2 2 22 2 3 4 2 5 6
3 3 2 2 3 3 2
2 2 1 2 1 3 3 2 2 3
3 3 2 2
4 6 8 10 12
3 2
3 2
3 4
46) 3 3 3 3 3 3 9 3 9 27
47 ) 2 2 2 2 2 2 3 2 3 2 2
48) 2 3 2 3 2 3 3 2 3 3
9 16
9 27 27
6 12
a b
x y z a b
x x x x x x x
x x x x x x x x x
y z y y z y z
x y z a b
x x x
x x x+
=
= − =
+ = + ⋅ ⋅ + ⋅ ⋅ + = + + ⋅ ⋅ + =
+ ⋅ + = + ⋅ + = + = + = + ⋅ ⋅ + ⋅ ⋅ + =
− = − ⋅ ⋅ + ⋅ ⋅ −
−
+ + +
+ + +
( )3 3 3 2 2 2 2 3 3
3 3 2 2 3 3 32 3
3 3
32 3 2
3 2 2 3
2 3
2 9 2 6 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 149) 3 3 3 32 3 2 2 3 2 3 3 4 3 2 92 3
2 3 250)3 2 3
8 36 54 27
1 1 1 18 4 6 27
z y y z y z z
x x x x x x x
x y x
y y z yz z
x x x
= − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − =
=
+ = + ⋅ ⋅ + ⋅ ⋅ + = + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + =
=
− =
− + −
+ + +
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( )
3 2 2 32 3 2 3 3
3 2 2 33 2 2 32 2 3 2 3 33 2 2 3
3 3 2 2 32 3 4 2 3 2 3 4 2 3 4 4
3 3 23 2 3 2
6 4 3 2 6 9
2 3 2 3 33 33 2 3 2 2
2 2 3 3 33 223 3 2 2
51) 3 2 3 3 3 2 3 3 2 2
3 3 2 9
8 9 27227 2 8
x y x y y
x x y x y y
x y z x y x y z x y z z
x y x y
x x y x y y
− ⋅ ⋅ + ⋅ ⋅ − =
= − ⋅ ⋅ + ⋅ ⋅ − =
=
− = − ⋅ ⋅ + ⋅ ⋅ − =
= − ⋅ ⋅ ⋅ ⋅
− + −
( ) ( ) ( )
( ) ( ) ( )
2 2 33 4 2 3 4 3 4
3 3 3 2 2 3 3 2 2 3
3 2 2 3 3 2 2 3
6 9 4 6 2 3 8 12
2 3
9 4 2
52) 3 3 3 3
3 3 3 3
27 54 36 8
6 2
z x y z z
x y x y x x y xy y x x y xy y
x x y xy y x x y xy y
x y x y x y z z
x y y
⋅ + ⋅ ⋅ − =
=
+ − − = + + + − − + − =
= + + + − + − + =
− + −
+
14
100 Elementarnih zadataka
M I M SRAGA 1999 / 2004
8
15
100 Elementarnih zadataka
M I M SRAGA 1999 / 2004
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
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100 Elementarnih zadataka
M I M SRAGA 1999 / 2004 M I M SRAGA 1999 / 2004
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100 Elementarnih zadataka
M I M SRAGA 1999 / 2004
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100 Elementarnih zadataka
Ovo su uglavnom zadaci iz PRVOG razreda gimnazije oni rijetko dolaze u ovom obliku, već su sastavni dio težih zadataka ali ako ih ne znate riješiti nećete znati niti riješiti zadatke kojih su oni dio… Naš prijedlog provježbajte ih koristit će Vam !!!!!! U zadnjih 10-godina na Eko –prijemnom bio je barem po jedan zadatak ovog tipa…. Na TEH-fakultetima ovi zadatci dolaze uvijek u kombinaciji sa još nečim….. Moguće da je jedan zadatak krivo riješen ako ga pronađete javite nam: [email protected] Ako ne vidite dobro tekst od 15-20 strane zum namjestite na 150%
M I M SRAGA 1999 / 2004
![Iracionalnost vrijednosti Riemannove zeta funkcijepejkovic/files/zetairac.pdf · pokazati korištenjem elementarnih argumenata sličnih Čebiševljevim (v. reference u [Fi 04, §1])](https://img.dokumen.tips/doc/110x75/5ecbb414496524297a607bfe/iracionalnost-vrijednosti-riemannove-zeta-funkcije-pejkovicfiles-pokazati-koritenjem.jpg)
