Let $n$ be a positive integer. The following $35$ multiplication are performed: $$1 \cdot n, 2 \cdot n, \dots, 35 \cdot n.$$Show that in at least one of these results the digit $7$ appears at least once.

Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that: $$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$for each integer $x$.

Ann and Beto play with a two pan balance scale. They have $2023$ dumbbells labeled with their weights, which are the numbers $1, 2, \dots, 2023$, with none of them repeating themselves. Each player, in turn, chooses a dumbbell that was not yet placed on the balance scale and places it on the pan with the least weight at the moment. If the scale is balanced, the player places it on any pan. Ana starts the game, and they continue in this way alternately until all the dumbbells are placed. Ana wins if at the end the scale is balanced, otherwise Beto win. Determine which of the players has a winning strategy and describe the strategy.

Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$. Note: The locus is the set of all points of the plane that satisfies the property.

A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is carioca if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments $$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$are all of the same length. Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is carioca.

Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-representable if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-representable, then all the even integers are $P$-representable or all the odd integers are $P$-representable.