Initially, the numbers $1$ and $2$ are written on the board. A move consists of choosing a positive real number $x$ and replacing $(a,b)$ on the board by $\left(a+\frac{x}{b},b+\frac{x}{a}\right)$. Is it possible to create in finitely many moves a situation where the numbers on the board are $2$ and $3$?

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

Let $ABCD$ be a convex quadrilateral with $AB=BD=DC$ and $AB \perp BD \perp DC$. Let $M$ be the midpoint of segment $BC$. Show that $\angle BAM+\angle DCA=45^\circ$.

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n)=3 \cdot S(2n)$.

Let $G$ be the barycenter of triangle $ABC$. Let $D$ be a point such that $AGDB$ is a parallelogram. Show that $BG \parallel CD$.

Among all triples $(a,b,c)$ of natural numbers satisfying \[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\]determine the one with the maximal value of $a$.

Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.

How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?

We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.