An integer \(n \geq 2\) is said to be tuanis if, when you add the smallest prime divisor of \(n\) and the largest prime divisor of \(n\) (these divisors can be the same), you obtain an odd result. Calculate the sum of all tuanis numbers that are less or equal to \(2023\).

In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.

Let $ABC$ an acute triangle and $D,E$ and $F$ be the feet of altitudes from $A,B$ and $C$, respectively. The line $EF$ and the circumcircle of $ABC$ intersect at $P$, such that $F$ it´s between $E$ and $P$. Lines $BP$ and $DF$ intersect at $Q$. Prove that if $ED=EP$, then $CQ$ and $DP$ are parallel.

In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.

Find all pairs of primes $(p,q)$ such that $6pq$ divides $$p^3+q^2+38$$

Let $n \geq 2$ be an integer. Lucia chooses $n$ real numbers $x_1,x_2,\ldots,x_n$ such that $\left| x_i-x_j \right|\geq 1$ for all $i\neq j$. Then, in each cell of an $n \times n$ grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the $n^2$ numbers that Lucia calculated can take.