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Neka su $m$, $n$ i $k$ prirodni brojevi i neka su $p_1, p_2, \ldots, p_n$ brojevi $1, 2, \ldots, n$ u nekom poretku. Ako za svaki $i \in \{1, 2, \ldots, n\}$ vrijedi

$$k \mid (m + p_i - i),$$

dokaži da je barem jedan od brojeva $m$ i $n$ višekratnik broja $k$.

Ante je zapisao niz $a_1, a_2, \ldots, a_{2020}$ u kojem se svaki od brojeva $1, 2, \ldots, 2020$ pojavljuje točno jednom. Barbara želi saznati taj niz, a Ante joj daje informacije kroz niz razmjena.

U jednoj razmjeni Barbara na papir zapiše brojeve od $1$ do $2020$ te neke od njih spoji crtom i to tako da je svaki broj spojen s najviše jednim drugim i preda taj papir Anti. Ante zatim na drugi papir zapiše brojeve od $1$ do $2020$ te za svaki par brojeva $i$ i $j$ koje je Barbara spojila crtom, on spoji crtom brojeve $a_i$ i $a_j$, te na kraju preda taj papir Barbari.

Koliko je najmanje razmjena potrebno da Barbara može sa sigurnošću odrediti Antin niz?

Za permutaciju $(a_1, a_2, \ldots, a_n)$ skupa $\{1, 2, \ldots, n\}$ kažemo da je uravnotežena ako vrijedi $$a_1 \leq 2a_2 \leq \ldots \leq na_n.$$

Neka $S(n)$ označava broj uravnoteženih permutacija skupa $\{1, 2, \ldots, n\}$.

Odredi $S(20)$ i $S(21)$.

Odredi sve prirodne brojeve $n$ za koje pstoji permutacija $(d_1, d_2, \ldots, d_k)$ skupa svih pozitivnih djelitelja od $n$ takva da je, za svaki $i \in \{1,2,\ldots,k\}$, broj $$d_1 + d_2 + \ldots + d_i$$ kvadrat prirodnog broja.

Five students, $A, B, C, D, E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Let $p_n(k)$ be the number of permutations of the set $\{1, \ldots, n\}$, $n \geq 1$, which have exactly $k$ fixed points. Prove that

$$\sum_{k=0}^{n} k \cdot p_n(k) = n!.$$

(Remark: A permutation $f$ of a set $S$ is a one-to-one mapping of $S$ onto itself. An element $i$ in $S$ is called a fixed point of the permutation $f$ if $f(i) = i$.)

A permutation $(x_1, x_2, \ldots, x_m)$ of the set $\{1, 2, \ldots, 2n\}$, where $n$ is a positive integer, is said to have property $P$ if $|x_i - x_{i+1}| = n$ for at least one $i$ in $\{1, 2, \ldots, 2n-1\}$. Show that, for each $n$, there are more permutations with property $P$ than without.

Let $x_1, x_2, \ldots, x_n$ be real numbers satisfying the conditions

$$|x_1 + x_2 + \cdots + x_n| = 1$$

and

$$|x_i| \leq \frac{n+1}{2} \qquad i = 1, 2, \ldots, n.$$

Show that there exists a permutation $y_1, y_2, \ldots, y_n$ of $x_1, x_2, \ldots, x_n$ such that

$$|y_1 + 2y_2 + \cdots + ny_n| \leq \frac{n+1}{2}.$$

Let $n$ be an odd integer greater than 1, and let $k_1, k_2, \ldots, k_n$ be given integers. For each of the $n!$ permutations $a = (a_1, a_2, \ldots, a_n)$ of $1, 2, \ldots, n$, let

$$S(a) = \sum_{i=1}^{n} k_i a_i.$$

Prove that there are two permutations $b$ and $c, b \neq c$, such that $n!$ is a divisor of $S(b) - S(c)$.

Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let $M$ be a set of $n - 1$ positive integers not containing $s = a_1 + a_2 + \cdots + a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1, a_2, \ldots, a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a < k < b$.

There are $n\geq 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from $1$ to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the current number plates are swapped exactly once during the day.

How many different sequences of number plates are possible at the end of the day?

There are $n$ students standing in line in positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i - j|$ steps. Determine the maximal sum of steps of all students that they can achieve.

Neka je $n$ pozitivan cijeli broj veći od $1$. Koliko ima permutacija $(a_1, a_2, \ldots, a_n)$ brojeva $1, 2, \ldots, n$ takvih da postoji točno jedan indeks $i \in \{1, 2, \ldots, n-1\}$ za koji je $a_i > a_{i+1}$?

Prirodni brojevi od $1$ do $2003$ poredani su u niz. Na nizu vršimo ovu operaciju: ako je prvi broj u nizu jednak $k$, okrenemo poredak prvih $k$ brojeva. Dokazati da se nakon konačno uzastopnih primjena ove operacije broj $1$ pojavi na prvom mjestu, nezavisno od početnog rasporeda.

U utrci sudjeluje 200 biciklista. Na početku utrke biciklisti su poredani jedan iza drugoga. Kažemo da neki biciklist pretječe ako mijenja mjesto s biciklistom neposredno ispred sebe. Tijekom utrke poredak se mijenja samo kad neki biciklist pretječe.

Neka je $A$ broj svih mogućih poredaka na kraju utrke u kojoj je svaki biciklist pretjecao točno jednom, te neka je $B$ broj svih mogućih poredaka na kraju utrke u kojoj je svaki biciklist pretjecao najviše jednom. Dokaži da vrijedi $$2A = B.$$

Neka je $n \geqslant 2$ prirodan broj te neka je $(p_1, \ldots, p_n)$ neka permutacija skupa $\{1, 2, \ldots, n\}$. Pokaži da vrijedi $$\frac{1}{p_1 + p_2} + \frac{1}{p_2 + p_3} + \ldots + \frac{1}{p_k + p_{k+1}} + \ldots + \frac{1}{p_{n-1} + p_n} > \frac{n - 1}{n + 2}.$$

U nogometnom klubu je $n$ igrača koji imaju dresove s međusobno različitim brojevima od 1 do $n$. Na kraju sezone igrač s brojem 1 završava karijeru. Uprava bira jednog od ostalih igrača kojeg prodaje nekom drugom klubu, dok svih preostalih $n - 2$ igrača dobiva dresove s međusobno različitim brojevima od 1 do $n$.

Na koliko načina uprava može odabrati igrača za prodaju i preostalima dati brojeve tako da nijedan igrač nema veći broj od onog koji je imao ove sezone?