A pair $(a,b)$ of positive integers is good if $\gcd(a,b)=1$ and for each pair of sets $A,B$ of positive integers such that $A,B$ are, respectively, complete residues system modulo $a,b$, there are $x \in A, y \in B$ such that $\gcd(x+y,ab)=1$. For each pair of positive integers $a,k$, let $f(N)$ the number of $b \leq N$ such $b$ has $k$ distinct prime factors and $(a,b)$ is good. Prove that \[\liminf_{n \to \infty} f(n)/\frac{n}{(\log n)^k}\ge e^{k}\]

Let $ABC$ be a triangle and $\Omega$ its circumcircle. Let the internal angle bisectors of $\angle BAC, \angle ABC, \angle BCA$ intersect $BC,CA,AB$ on $D,E,F$, respectively. The perpedincular line to $EF$ through $D$ intersects $EF$ on $X$ and $AD$ intersects $EF$ on $Z$. The circle internally tangent to $\Omega$ and tangent to $AB,AC$ touches $\Omega$ on $Y$. Prove that $(XYZ)$ is tangent to $\Omega$.

positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least $Cn$ cells have the same color. a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$. b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.

Prove that there exists a positive integer $x<5^{2022}$ such that \[\{\varphi\sqrt[3]{x}\}<\varphi^{-2022}.\]