Find all integers $n$ such that $\frac{4n}{n^2 +3 }$is an integer.

Let $P(x)$ a grade 3 polynomial such that: $$P(1)=1, P(2)=4, P(3)=9$$Find the value of $P(10)+P(-6)$

Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively. Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$

In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop. Note: A bishops moves diagonally.

Find an acutangle triangle such that its sides and altitudes have integer length.

Find all positive integers $a, b, c$ such that $ab+1$ and $c$ are coprimes and: $$a(ba+1)(ca^2+ba+1)=2021^{2021}$$