1. DVODIMENZIONALNI NIZOVI (MATRICE) - Programming language C
Tehnička škola
Takovska, 22 Pirot
DVODIMENZIONALNI NIZOVI
(MATRICE)
- Programming language C
- English language
- Mathematics
Struktura prezentacije
1. Čas - Opšti pojmovi, primeri i uvežbavanja
2. Čas - Osnovne operacije i njihova primena
3. Čas - Utvrđivanje, samostalna provera i smernice za dalji rad

2. ČAS 1. DEFINICIJA DVODIMENZIONALNOG NIZA - M A T R I C E
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- Primeri matričnog (tabelarnog) uređenja podataka i primene ovakve strukture
- Matematička definicija strukture i funkcionalnosti
- Primena u vežbi na engleskom jeziku
- Matrice u programiranju
(Programski jezik C)

3. Primeri (Examples) Šahovska tabla
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4. Tablica množenja
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Next x 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90
100

5. Matrični displej
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6. Tastatura
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7. I ko zna koliko još ...
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8. Brainstorming !!! (Ideas, please…)
Navedite bez velikog razmišljanja svoj primer matrične (tabelarne) strukture podataka iz okoline !
na primer:
Kalendar, Ukrštene reči, Lavirint,
Periodični sistem hemijskih elementa,
Šare pirotskog ćilima, ...
Need HELP ? Click here !!!
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9. Poveži sa nečim što već znaš !
Ređanjem više jednodimenzionalnih nizova podataka istih dimenzija i tipa podataka jedan do drugog dobijamo i drugu dimenziju .
II dimenzija
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Next a1 a2 a3 . an a1 a2 a3 . an a1 a2 a3 . an . . . a1 a2 a3 . an
I dimenzija

10. Struktura, osnovni pojmovi
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Elementi matrice - presek svakog reda (i) i svake kolone (j)
Kolone (COLUMNS)
a11
a12
a13
a14
a15
...
a1n
a21
a22
a23
a24
a25
a2n
a31
a32
a33
a34
a35
a3n ..
am1
am2
am3
am4
am5
amn
Redovi (ROWS)
aij

11. Terminologija (Basic terms)
(Rows) Redovi ili vrste su horizontale u matrici odnosno nizovi elemenata čiji je prvi indeks jednak
(Columns) Kolone su vertikale u matrici odnosno nizovi elemenata čiji je drugi indeks jednak
(Size) Veličina odnosno dimenzije matrice određene su brojem redova (m) i brojem kolona (n) i zapisuje se kao m x n
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12. Skraćeni zapis matrice (Mathematics)
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A = {aij}, i=1,m ; j= 1,n

13. Vežba - PREPOZNAJ SEBE U MATRICI
Spojiti klupe i formirati matricu dimenzija 5x6 (ili sličnu)
Svaki učenik na pratećem obrascu u zadatim poljima upisuje svoju poziciju u učionici <click za dalje>
Svaki učenik na pratećem obrascu u zadatom polju upisuje svoju visinu u cm <click za dalje>
Neka ustanu svi učenici zadatog reda/kolone
redom (1, 4, 2, 3, 5) <click za dalje>
Neka ustanu redom svi učenici kod kojih je
i>j, i<j, i=j
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14. Vežba - OSMOSMERKA
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U narednoj vežbi uvežbavaju se osnovni pojmovi vezani za strukturu matrice. Vežba je na engleskom jeziku i podrazumeva znanja na temu "Nepravilni glagoli" odnosno njihovi oblici u infinitivu, prošlom vremenu i prošlom participu.
Rezultate vežbe učenici upisuju u odgovarajuća polja na priloženom obrascu za praćenje časa. Ovi rezultati koriste se i u narednim vežbama.

15. Word Search Puzzle - Irregular verbs1 2 3
ATE
GOTTEN
TAUGHT
BROKEN
GROWN
THOUGHT
DONE
LET
TORE 4
DRANK
RIDDEN
WORE 5
FED
SLEPT
WRITTEN 1 2 3 4 5 6 7 8 9 10 L D C N X K M A T E I R H O F U S G B P J W Z V
Find the following irregular verbs in the MATRIX to the right.
Give a list of (i,j) positions of the elements in the matrix.
state whether the verb is in the past or past participle form (II and III column in the list of irregular verbs)
supply the missing two.
Example
Been b (4,1) e (3,2) e (2,3) n (1,4)
been –past participle
be - was/were - been
<click>
<clear>
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16. Word Search Puzzle - SOLUTION
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Next 1 2 3
ATE
GOTTEN
TAUGHT
BROKEN
GROWN
THOUGHT
DONE
LET
TORE 4
DRANK
RIDDEN
WORE 5
FED
SLEPT
WRITTEN 1 2 3 4 5 6 7 8 9 10 L D C N X K M A T E I R H O F U S G B P J W Z V
ATE
(8,6) (7,6) (6,6)
BROKEN
(7,2) (6,2) (5,2) (4,2) (3,2) (2,2)
DONE
(6,9) (6,8) (6,7) (6,6)
DRANK
(4,6) (4,5) (4,4) (4,3) (4,2)
FED
(3,4) (2,3) (1,2)
GOTTEN
(9,8) (8,7) (7,6) (6,5) (5,4) (4,3)
GROWN
(10,7) (9,7) (8,7) (7,7) (6,7)
LET
(9,4) (8,5) (7,6)
RIDDEN
(2,7) (2,6) (2,5) (2,4) (2,3) (2,2)
SLEPT
(10,4) (9,4) (8,4) (7,4) (6,4)
TAUGHT
(1,10) (2,9) (3,8) (4,7) (5,6) (6,5)
THOUGHT (7,10) (6,10) (5,10) (4,10) (3,10) (2,10) (1,10)
TORE
(5,1) (5,2) (5,3) (5,4)
WORE
(8,1) (8,2) (8,3) (8,4)
WRITTEN (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)(6,7)

17. Matrice u programskom jeziku C
Deklaracija matrice:
tip Ident[m][n]; tip - tip podataka (Data type)
elemenata matrice
Ident - ime matrice
m - broj redova
n - broj kolona
Primer:
int M[5][6]; ili uz inicijalizaciju (dodelu početnih
vrednosti elemenata matrice):
float R[2][3] = { {2.3,4,0.45}, {3,3.4,5.6} };
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18. Matrice u programskom jeziku C
Specifičnost programskog jezika C:
indeksi redova i kolona su iz opsega
i = m-1 (ukupno m)
j = n-1 (ukupno n)
Primer:
Ukoliko je deklarisana matrica na sledeći način:
int M[5][6];
to znači da se indeksi vrste i kolone kreću u sledećim granicama:
i = , j =
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19. Pristup elementima u matrici
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Pristup elementima ("upotrebi ili dodeli vrednost") matrice je operacija u kojoj koristimo određeni element/e matrice sa ciljem korišćenja njegove vrednosti u drugim naredbama/funkcijama ili dodele vrednosti datom elementu.
Ident[i][j]
Primer:
int M[5][6], a, b=4;
M[0][3] = b;
M[2][5] = 25;
a = M[0][3] + 2;
if( a> M[2][5]) printf("Vrednost promenljive A je veća !!!);

20. Vežba - Napiši svoj program
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Svoje rezultate iz osmosmerke prikazati programski po ugledu na zadati primer. Svoj program zapisati u zadato polje na pratećen obrascu.
#include<stdio.h>
void main() {
int A[10][10] = {{...} ...};
printf("%d %d ...", A[4][5], A[4][6], ...);
printf("Infinitiv glagola je: ....");
printf("Zadati glagol ... je iz 2/3 kolone."); }

21. ČAS 2 - OBILAZAK MATRICE
Podsetimo se obilaska ("prolaza kroz") jednodimenzionalnih nizova za početak.
Obilazak matrice je osnovna operacija u matrici.
Realizuje se nadogradnjom algoritma obilaska nizova.
Naučimo i uvežbajmo načine obilaska "red po red" i "kolonu po kolonu".
Analizom primera upotrebe operacije obilaska matrice uočiti koje složenije operacije u matrici se zasnivaju na obilasku.
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22. Obilazak jednodimenzionalnog niza (da počnemo od poznatog)
Ova operacija u nizu izvodi se tako što se redom pristupa svim elementima niza počevši od prvog prema poslednjem
Pristup elementu niza znači korišćenje vrednosti datog elementa niza ili dodelu vrednosti datom elementu u nizu
Pristup elementu je operacija koja se ponavlja onoliko puta koliko ima elemenata u nizu
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23. Algoritamska / programska realizacija
Ponavljanje pristupa elementu niza realizujemo brojačkim ciklusom (petljom)
Primer:
for(i=0; i<n; i++)
Niz[i] = i;
ili
printf("%d", Niz[i]);
j = 1, M
Niz[i] = i
j = 1, M
Niz[i]
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24. Vežba - Obilazak niza podataka
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Opis algoritma uz demonstraciju u I redu učenika:
Prvog učenika u nizu proglašavamo najvišim in on stoji a ostali sede
indeks niza (brojač) počinje od vrednosti 2 i završava se brojem učenika u I redu
Upoređuju se visine do sada najvišeg učenika (koji stoji) i onaj do kog se stiglo indeksom (brojačem u petlji) koji takođe ustaje
Ukoliko je dostignuti učenik viši od do sada najvišeg on ostaje da stoji
indeks niza (brojač) se uvećava za 1 i ponavlja se korak 3 i 4
ukoliko je dostignut poslednji, učenik koji stoji je najviši i nalazi se na zadatoj poziciji

25. Vežba - "Play roles & demonstrate"
Using the following questions, answers and statements demonstrate the previously described algorithm.
My position in the array is ___
I am ___ cm tall.
What is your position in the array?
How tall are you?
I am taller and from now on : “I am the tallest in the array so far”
Who is the next student in the array to compare with?
Did we reach the end of the array?
I am the tallest in the array. I am ___ cm tall and my position is ___ in the array!
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26. Brainstorming !!! (Ideas, please!)
Kako proširiti prethodni algoritam da bi odredili najvišeg učenika u celom odeljenju (klupe čine matricu)?
Koliko puta treba ponoviti prethodni algoritam za celu matricu?
Šta i kako iskoristiti za to ponavljanje u programskoj realizaciji?
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27. Obilazak matrice a11 a21 a31 am1 a12 a22 a32 am2 a13 a23 a33 am3 . a1n
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U svakom prolazu kroz red indeks elemenata tog reda se kreće od 1 do n. Prelaskom u naredni red indeks elemenata se vraća na vrednost 1 i ponovo kreće do n.
U svakom prolazu kroz kolonu indeks elemenata te kolone se kreće od 1 do m. Prelaskom u narednu kolonu indeks elemenata se vraća na vrednost 1 i ponovo kreće do m.
"Row by row"
<click>
"Column by column"
<click>
a11
a21
a31 .
am1
a12
a22
a32 .
am2
a13
a23
a33 .
am3 .
a1n
a2n
a3n .
amn
a11
a21
a31 .
am1
a12
a22
a32 .
am2
a13
a23
a33 .
am3 .
a1n
a2n
a3n .
amn

28. Primer upotrebe obilaska matrice (Tablica množenja)
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Na primeru formiranja vrednosti elemenata matrice M(10,10) po ugledu na tablicu množenja demostrirati dva načina obilaska matrice: "red po red" i "kolonu po kolonu"
"red po red"
main () {
int i, j, M[10][10];
for(i=0; i<10; i++)
for(j=0; j<10; j++)
M[i][j] = i*j;
"kolona po kolona"
main () {
int i, j, M[10][10];
for(j=0; j<10; j++)
for(i=0; i<10; i++)
M[i][j] = i*j;

29. Algoritmi obilaska matrice Demonstracija u odeljenju-matrici
Koristeći dva učenika kao indekse i i j izvršiti prozivku učenika prethodno opisanim obilascima matrice! Učenici "indeksi" stoje sa strane (i) i ispred klupa (j) i prate promene koje u algoritmu čine brojačke promenljive i i j diktirane spoljnom i unutrašnjom for petljom. "Prozvani" učenik izgovara svoju visinu u cm. Uočiti i komentarisati dinamiku brojača unutrašnje i spoljne for petlje.
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30. Vežba: Primer obilaska matrice (Broj pojavljivanja podatka u matrici)
Algorithm counting the given data P in the matrix using matrix iteration“row by row”
Set the starting value of the counter No=0 (No=0 at the beginning means that maybe there is no given data in the matrix)
Set the row counter of the matrix as i=1 and increase it in every iteration for 1 increment (i). If the bordering value is exceeded, go to the step 7. (i= n+1- the number of rows in the matrix + 1)
Set an element counter within the row j=1 and increase it in every iteration for 1 increment (j). If the bordering value is exceeded , go to the step 6. (j=m+1 – the number of columns in the matrix plus 1)
Compare the achieved element of the matrix with the given data M (i,j)=P? If that is the case, increase the number of occurrence - Increment (Br)
Go back to step 3.
Go back to step 2.
Show the value of Br which represents the number of occurrence of the data N in the matrix.
Answer the question:
What could we count in the classroom using this method? Choose the most appropriate examples.
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31. ČAS 3. - Proveri šta si naučio!
Obnovi stečena/uvežbana znanja i usvojene pojmove sa prethodnih časova radeći na tekstu na engleskom jeziku.
Prepoznaj celine u kojima se ne snalaziš dobro! Vrati se u ličnom pratećem materijalu i podseti se rešenja!
Pokušaj da prepoznaš najbitnije i kritički se osvrneš na moguće primene.
Zapitaj se "Šta sledi?" i pokušaj da nađeš prave odgovore!
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32. Pročitaj pažljivo tekst!
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Two-dimensional data arrays- MATRICES
Data are usually organized in the form of a table which suits the given data best. There is a rich variety of examples in our surroundings: school register, weekly attendance records, charts of weekly city temperatures, calendar, chess, “Reveal the pairs” game, results of laboratory measurements, etc.
If we take several one–dimensional data arrays of the same length (number of elements) and write them one below the other, thus forming a table, we get the so called two-dimensional array or “an array of arrays” which is usually referred to as a matrix. The number of one-dimensional arrays (rows) represents the first dimension of the matrix and the number of elements in each of the arrays (column) represents the second dimension of the matrix. These numbers mxn are called matrix dimensions.
The position (place) of the data within the matrix is determined by the number of the row and the number of the column of the particular data (intersection of the given column and the given row). These numbers are usually referred to as row subscript and column subscript according to the element subscript in one-dimensional array.
All the matrix elements represent the data of the same type (uniformity of data type) and they carry the same name which is at the same time the name of the whole matrix. (uniqueness of the identifiers)

33. "Row by row" "Column by column"
Matrix of the dimensions mxn can be mathematically expressed in the following way:
or the short form A=(aij), i=1,m j=1,n.
The basic operation that can be performed in the matrix is the so called matrix iteration or passing round matrix. Essentially, this operation represents the access to all matrix elements “row by row” or “column by column”.
In programming matrix iteration is used for solving the following tasks which are usually found when solving complex problems:
Assigning values to matrix elements
(using the keyboard or the assigning operator “=”)
Matrix illustration
Searching for the given data in the matrix
Determining of the smallest/biggest matrix element
Determining of sum/average value of the matrix elements
Row/column sorting in the matrix
"Row by row"
"Column by column"
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34. Test - Answer the Questions!
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Odgovori na sledeća pitanja na osnovu prethodno pročitanog teksta i sadržaja na slajdovima.
(Umesto testa na narednim slajdovima moguće je iskoristiti interaktivnu verziju testa u prilogu!)

35. Test - Part I
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How do we call the horizontal and vertical wholes in a matrix?
What determines the position of every element in a matrix?
Matrix elements are data of the __________ data type.
Element subscripts in the matrix are ________ which determine in which ________ and which _______ the given element is positioned.
Matrix dimensions are determined by (circle all the correct answers):
The values of the neighboring elements in the matrix
The number of columns in the matrix
The position of elements in the bottom right corner of the matrix
The number of rows in the matrix
Element inscription order
What does the uniformity of data type in a matrix mean?
What determines a specific name (identifier) of every matrix element?
If the matrix dimensions are mxn, what is the spectrum in which subscript of column j and subscript of row i occurs?

36. Test - Part II What is the short form of matrix in mathematics?
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What is the short form of matrix in mathematics?
Matrix iteration can also be called _________ matrix.
In what ways can we perform matrix iteration?
Which of the following examples of practical usage of matrix is not an adequate one and why?
Chess
Multiplication table
Winning lottery numbers
Calendar
State some examples of “matrix iteration” which you can use for more complex operations.
How many elements are there in Amxn matrix?
In which part of the matrix are the elements for whose subscripts i=j?
Elements of the same matrix row have the same subscript j. Yes / No?

37. Test - Part III
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I’m 194 cm tall and I am __________ in the array. (Circle the correct answer) a) tallest b) the tallest c) taller
The position (place) of the data within the matrix is ___________ by intersection of the given column and the given row. (Circle the correct answer)
a) determination b) determines c) determined
If we take several one–dimensional data arrays of the same length
and write them one below the other, we get the so called ________________.
Is the following sentence grammatically correct? Why / Why not?
Data is usually organized in the form of a table which suits the given data best.
The number of elements in each of the arrays represents the second dimension of the matrix and it is called column / colon. Circle the correct answer.
What’s the plural form of the noun matrix?

38. Part I - CORRECT ANSWERS
Horizontal = ROWS Vertical = COLUMNS
ROW SUBSCRIPT AND COLUMN SUBSCRIPT
Matrix elements are data of the SAME data type.
Element subscripts in the matrix are NUMBERS which determine in which ROW and which COLUMN the given element is positioned.
Matrix dimensions are determined by (circle all the correct answers):
The values of the neighboring elements in the matrix
The number of columns in the matrix
The position of elements in the bottom right corner of the matrix
The number of rows in the matrix
Element inscription order
All the matrix elements represent the DATA OF THE SAME TYPE.
The NAME of the WHOLE MATRIX.
j = m i = n
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39. Part II - CORRECT ANSWERS
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A=(aij), i=1,m j=1,n
Matrix iteration can also be called PASSING ROUND MATRIX .
“row by row” or “column by column”
Which of the following examples of practical usage of matrix is not an adequate one and why?
Chess
Multiplication table
Winning lottery numbers - one dimensional array
Calendar
Assigning values to matrix elements (using the keyboard or the assigning operator “=”)
Matrix illustration
Searching for the given data in the matrix
Determining of the smallest/biggest matrix element
Determining of sum/average value of the matrix elements
Row/column sorting in the matrix
m x n
Main diagonal
Elements of the same matrix row have the same subscript j. Yes / No ?

40. Part III - CORRECT ANSWERS
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I’m 194 cm tall and I am the tallest in the array. (Circle the correct answer) a) tallest b) the tallest c) taller
The position (place) of the data within the matrix is determined by intersection of the given column and the given row. (Circle the correct answer)
a) determination b) determines c) determined
If we take several one–dimensional data arrays of the same length
and write them one below the other, we get the so called
two–dimensional data arrays - matrix.
Is the following sentence grammatically correct? Why / Why not?
Data is usually organized in the form of a table which suits the given data best.
The sentence is not correct because the noun data is in the plural form and it requires another form of the verb to be and that is are.
Data are usually organized in the form of a table which suits the given data best.
The number of elements in each of the arrays represents the second dimension of the matrix and it is called column / colon. Circle the correct answer.
MATRICES

41. VEŽBA - Uklopi u celinu (Programiraj)
Koristeći delove programa "razbacane" po slajdu izborom i pravilnim ređanjem uklopi u program zadate namene:
a) Prikaz elemenata matrice na ekranu
b) Sumu elemenata matrice
obilaskom matrice "kolonu po kolonu"
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42. for(i=0; i<m; i++) (4) suma += A[i][j]; (2)
} (8)
printf("\n"); (11)
printf("Suma matrice =%d\n", suma); (1)
int suma = 0; (13)
main() { (9)
{ (10)
} (6)
printf("%5d", A[i][j]); (7)
#include <stdio.h> (3)
for(j=0; j<n; j++) (12)
int i,j; (5)
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43. Prikaz elemenata matrice na ekranu
#include <stdio.h> (3)
main() { (9)
int i,j; (5)
for(i=0; i<m; i++) (4)
{ (10)
for(j=0; j<n; j++) (12)
printf("%5d", A[i][j]); (7)
printf("\n"); (11)
} (6)
} (8)
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44. Sumu elemenata matrice obilaskom matrice "kolonu po kolonu"
#include <stdio.h> (3)
main() { (9)
int suma = 0; (13)
int i,j; (5)
for(j=0; j<n; j++) (12)
for(i=0; i<m; i++) (4)
suma += A[i][j]; (2)
printf("Suma matrice =%d\n", suma); (1)
} (8)
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45. Popuni evaluacioni formular! Hvala na iskrenim odgovorima!
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46. The end
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