Školjka

%V0 Consider the two square matrices $$A=\begin{bmatrix}+1 &+1 &+1&+1 &+1\\+1 &+1 &+1&-1 &-1\\ +1 &-1&-1 &+1&+1\\ +1 &-1 &-1 &-1 &+1\\ +1 &+1&-1 &+1&-1\end{bmatrix}\quad\text{ and }\quad B=\begin{bmatrix}+1 &+1 &+1&+1 &+1\\+1 &+1 &+1&-1 &-1\\ +1 &+1&-1&+1&-1\\ +1 &-1&-1&+1&+1\\ +1 &-1&+1&-1 &+1\end{bmatrix}$$ with entries $+1$ and $-1$. The following operations will be called elementary: (1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns. Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.

%V0 Neka je $M$ podskup skupa $\{1, 2, ..., 15\}$ koji ne sadrži $3$ elementa čiji je umnožak potpun kvadrat. Odredi maksimalan broj elemenata skupa $M$.

%V0 On a blackboard there are $n \geq 2, n \in \mathbb{Z}^{+}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $n$ for which it is possible to yield $n$ identical number after a finite number of steps.

%V0 Let $a$, $b$, $c$ be real numbers such that for every two of the equations $$x^2+ax+b=0, \quad x^2+bx+c=0, \quad x^2+cx+a=0$$ there is exactly one real number satisfying both of them. Determine all possible values of $a^2+b^2+c^2$.

%V0 For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane. What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?

%V0 Skup $S$ sastoji se od $14$ prirodnih brojeva. Pokažite da postoji $k\in\{1, \ldots, 7\}$ za koji je moguće naci $k$-člane disjunktne podskupove $\{a_1, \ldots, a_k\}$ i $\{b_1, \ldots, b_k\}$ skupa $S$ tako da se sume $$ A = \frac{1}{a_1} + \cdots + \frac{1}{a_k}, \quad B = \frac{1}{b_1} + \cdots + \frac{1}{b_k}$$ razlikuju za manje od $0.001$.