Let $n \ge 3$ be an integer number. We say that a matrix $A \in \mathcal{M}_n(\mathbb{C})$ has the property $(\mathcal{P})$ if $\det(A + X_{ij}) = \det(A + X_{ji})$, for any $i, j \in \{1, 2, \dots, n\}$, where $X_{ij} \in \mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ at the position $(i, j)$ and $0$ elsewhere.
a) Assume a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, such that $\det(A) \neq 0$. Prove that $A = A^T$.
b) Give an example of a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, but $A \neq A^T$.
Let $n$, $m$ be positive integers such that $n + m$ is odd. Assume that the edges of a complete bipartite graph $K_{n,m}$ are labeled by $1$ and $-1$ such that the sum of the numbers written on all edges is zero. Show that this graph has a spanning tree such that the sum of the numbers on its edges is $0$.
*A spanning tree is a sub-tree of the graph that contains all the vertices.*
Let $n \ge 3$ be a natural number, and let $\mathcal{F}$ be a family consisting of at most $n$ distinct subsets of the set $\{1, 2, \dots, n\}$ with the property that one can consider $n$ distinct points in the plane, labeled with the numbers $1, 2, \dots, n$, then draw segments between some of these points such that, for any distinct numbers $i, j \in \{1, 2, \dots, n\}$, the points labeled $i$ and $j$ are connected by a segment if and only if the number $i$ belongs to exactly $j$ subsets in $\mathcal{F}$. Find the maximum possible value of the sum of the number of elements of the sets in $\mathcal{F}$.
Let $n \ge 2$ be an integer and $a$ and $b$ complex numbers such that $a \neq 0$ and $b^k \neq 1$, for any $k \in \{1, 2, \dots, n\}$. Suppose that the matrices $A, B \in \mathcal{M}_n(\mathbb{C})$ are such that $BA = aI_n + bAB$. Prove that $A$ and $B$ are invertible.
Sorin Rădulescu and Mihai Piticari
Digits $A$, $B$, $C$ are given distinct values from $1$ to $9$ to make the value of the expression $2024 \cdot AB \cdot CC \cdot BA$ a perfect square. How many distinct values can the expression $A+B+C$ obtain?
Find all the integers $n \ge 2$ with the property: the cells of a $n \times n$ board can be colored with several colors, so that each cell $C$ has exactly two neighbouring cells with the same color as $C$. Here neighbouring cells means cells with a common side.
Let $A_n$ denote the number of subsets of $\{1, 2, \dots, n\}$ that contain no two elements differing by $1$, and let $B_n$ denote the number of subsets that contain no two elements differing by $2$. Prove that $A_{2025} \lt B_{1013}^2$.
(Batbayasgalan Balkhuu and Nursoltan Khavalbolot)
A point $E$ is chosen on the side $AB$ of a rectangle $ABCD$ ($E \neq A, E \neq B$). The line segments $BD$ and $CE$ intersect at point $F$. Among the triangles $ADE$, $DEF$, $DCF$, $BCF$ and $BEF$, there are exactly two pairs of triangles with equal area (the order of components in a pair is not taken into account). Find the ratio of the lengths of the line segments $EB$ and $AB$.
Consider a sequence of integers $a_1, a_2, a_3, \dots$ such that $a_1 \gt 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible that two terms of such a sequence be equal?
Alexia has several marbles and her friend, Cristina, has none. Each day of one week, starting on Monday, Alexia gave Cristina some of her marbles, so that each day Alexia gave more marbles than the day before. On Monday Alexia gave five times less marbles than on Friday, on Tuesday she gave six times less marbles than on Saturday, and on Wednesday she gave seven times less marbles than on Sunday. At the end of that week, Cristina got 72 marbles. Find how many marbles got Cristina Thursday.
Lucian Dragomir
a) Prove that, if a ring $(A, +, \cdot)$ has property (P) and $a, b$ are distinct elements of $A$ such that $a$ and $a+b$ are invertible, then $b$ is not invertible, but $1+ab$ is invertible.
b) Give an example of a unitary ring possessing (P).
where property (P) is:
$$
(P) \quad \left\{ \begin{array}{l} \text{the set } A \text{ has at least 4 elements,} \\ \text{the element } 1+1 \text{ is invertible in } A, \\ x+x^4 = x^2+x^3, \text{ for any } x \in A. \end{array} \right.
$$
Let $\mathbb{Z}$ denote the set of integers and let $S \subset \mathbb{Z}$ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \to \mathbb{Z}$ satisfying
$$
f(xy + c) = f(x) + f(y) \quad \text{for all } x, y \in S.
$$
Let $ABCDEF$ be a convex hexagon with $\angle A \equiv \angle C \equiv \angle E$ and $\angle B \equiv \angle D \equiv \angle F$, such that there is a point $P$ in its interior that is equidistant from the sides $AB$, $CD$ and $EF$. If $G_1 \neq G_2$ are the centroids of triangles $ACE$ and $BDF$, respectively, prove that $\angle G_1 P G_2 = 60^\circ$.
We say that a natural number $n \ge 3$ is almost square-free if there exists a prime number $p$, with $p \equiv 1 \pmod 3$, such that $n$ is divisible by $p^2$, and the number $\frac{n}{p}$ is square-free (i.e., not divisible by the square of any prime number).
Show that, for any natural number $n$ that is almost square-free, the ratio between twice the sum of the divisors of $n$ and the number of divisors of $n$ is a natural number.
Let $f : [0, 1] \to \mathbb{R}$ be a differentiable function, with integrable derivative on $[0, 1]$, such that $f(1) = 0$. Prove that
$$
\int_{0}^{1} (x f'(x))^2 dx \geq 12 \cdot \left( \int_{0}^{1} x f(x) dx \right)^2 .
$$
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that for every positive integer $n$
$$
a_{n+1} = (n + 1)(a_n - n + 1).
$$
In terms of $a_1$, determine the greatest positive integer $k$ such that $\gcd(a_i, a_{i+1}) = k$
for some positive integer $i \ge 2$. (Note that $\gcd(x, y)$ denotes the greatest common
divisor of integers $x$ and $y$.)
Consider the sets
$$
A = \{(x, y) \mid x, y \in \mathbb{R} \text{ and } x + y + 1 = 0\}
$$
and
$$
B = \{(x, y) \mid x, y \in \mathbb{R} \text{ and } x^3 + y^3 + 1 = 3xy\}.
$$
a) Show that $A \subset B$.
b) Prove that the set $B \setminus A$ has exactly one element.