Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$a) Is $2023$ in the sequence? b) Show that there are no perfect squares in the sequence.

Let $a,b,c$ real numbers such that $a^n+b^n= c^n$ for three positive integers consecutive of $n$. Prove that $abc= 0$

Let $ABC$ an acute triangle and $D$ and $E$ the feet of heights by $A$ and $B$, respectively, and let $M$ be the midpoint of $AC$. The circle that passes through $D$ and $B$ and is tangent to $BE$ in $B$ intersects the line $BM$ in $F, F\neq B$. Show that $FM$ is the angle bisector of $\angle AFD$.

Determine all $n$ positive integers such that exists an $n\times n$ where we can write $n$ times each of the numbers from $1$ to $n$ (one number in each cell), such that the $n$ sums of numbers in each line leave $n$ distinct remainders in the division by $n$, and the $n$ sums of numbers in each column leave $n$ distinct remainders in the division by $n$.

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.

Let $S$ be a set not empty of positive integers and $AB$ a segment with, initially, only points $A$ and $B$ colored by red. An operation consists of choosing two distinct points $X, Y$ colored already by red and $n\in S$ an integer, and painting in red the $n$ points $A_1, A_2,..., A_n$ of segment $XY$ such that $XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY$ and $XA_1<XA_2<...<XA_n$. Find the least positive integer $m$ such exists a subset $S$ of $\{1,2,.., m\}$ such that, after a finite number of operations, we can paint in red the point $K$ in the segment $AB$ defined by $\frac{AK}{KB}= \frac{2709}{2022}$. Also, find the number of such subsets for such a value of $m$.

Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.