For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]and determine all pairs of positive integers $(a,b)$ where we have equality case.

Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

Let $O$ be a fixed point in the plane. We have $2024$ red points, $2024$ yellow points and $2024$ green points in the plane, where there isn't any three colinear points and all points are distinct from $O$. It is known that for any two colors, the convex hull of the points that are from any of those two colors contains $O$ (it maybe contain it in it's interior or in a side of it). We say that a red point, a yellow point and a green point make a bolivian triangle if said triangle contains $O$ in its interior or in one of its sides. Determine the greatest positive integer $k$ such that, no matter how such points are located, there is always at least $k$ bolivian triangles.

We color some points in the plane with red, in such way that if $P,Q$ are red and $X$ is a point such that triangle $\triangle PQX$ has angles $30º, 60º, 90º$ in some order, then $X$ is also red. If we have vertices $A, B, C$ all red, prove that the barycenter of triangle $\triangle ABC$ is also red.

Let $n \ge 2$ be an integer and let $a_1, a_2, \cdots a_n$ be fixed positive integers (not necessarily all distinct) in such a way that $\gcd(a_1, a_2 \cdots a_n)=1$. In a board the numbers $a_1, a_2 \cdots a_n$ are all written along with a positive integer $x$. A move consists of choosing two numbers $a>b$ from the $n+1$ numbers in the board and replace them with $a-b,2b$. Find all possible values of $x$, with respect of the values of $a_1, a_2 \cdots a_n$, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.

Determine all infinite sets $A$ of positive integers with the following propety: If $a,b \in A$ and $a \ge b$ then $\left\lfloor \frac{a}{b} \right\rfloor \in A$