Determine all squarefree positive integers $n\geq 2$ such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where $d_1,d_2,\ldots,d_k$ are all the positive divisors of $n$.

Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.

Find how many positive integers $k\in\{1,2,\ldots,2022\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal.

Let $ABC$ be a right triangle $(AB<AC)$ with heights $AD, BE,$ and $CF$ and orthocenter $H$. Let $M$ denote the midpoint of $BC$ and let $X$ be the second intersection of the circle with diameter $HM$ and line $AM.$ Given that lines $HX$ and $BC$ intersect at $T,$ prove that the circumcircles of $\triangle TFD$ and $\triangle AEF$ are tangent.

We call a set $A\subset \mathbb{R}$ free of arithmetic progressions if for all distinct $a,b,c\in A$ we have $a+b\neq 2c.$ Prove that the set $\{0,1,2,\ldots 3^8-1\}$ has a subset $A$ which is free of arithmetic progressions and has at least $256$ elements.

Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\] Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$ Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$

Find the largest positive integer $n$ such that the following is true: There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.

Decompose a $6\times 6$ square into unit squares and consider the $49$ vertices of these unit squares. We call a square good if its vertices are among the $49$ points and if its sides and diagonals do not lie on the gridlines of the $6\times 6$ square. Find the total number of good squares. Prove that there exist two good disjoint squares such that the smallest distance between their vertices is $1/\sqrt{5}.$

Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$

Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$ If $a+b+c+d=6,$ prove that $d<0,36.$ If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold?

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circles, internally tangent at $P$ ($\mathcal{C}_2$ lies inside of $\mathcal{C}_1$). A chord $AB$ of $\mathcal{C}_1$ is tangent to $\mathcal{C}_2$ at $C.$ Let $D$ be the second point of intersection between the line $CP$ and $\mathcal{C}_1.$ A tangent from $D$ to $\mathcal{C}_2$ intersects $\mathcal{C}_1$ for the second time at $E$ and it intersects $\mathcal{C}_2$ at $F.$ Prove that $F$ is the incenter of triangle $ABE.$

Let $p_i$ denote the $i^{\text{th}}$ prime number. For any positive integer $k$ let $a_k$ denote the number of positive integers $t$ such that $p_tp_{t+1}$ divides $k.$ Let $n$ be an arbitrary positive integer. Prove that \[a_1+a_2+\cdots+a_n<\frac{n}{3}.\]

For any $n$-tuple $a=(a_1,a_2,\ldots,a_n)\in\mathbb{N}_0^n$ of nonnegative integers, let $d_a$ denote the number of pairs of indices $(i,j)$ such that $a_i-a_j=1.$ Determine the maximum possible value of $d_a$ as $a$ ranges over all elements of $\mathbb{N}_0^n.$

Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]

Let $ABC$ be an acute scalene triangle. Let $D$ be the foot of the $A$-bisectrix and $E$ be the foot of the $A$-altitude. The perpendicular bisector of the segment $AD$ intersects the semicircles of diameter $AB$ and $AC$ which lie on the outside of triangle $ABC$ at $X$ and $Y$ respectively. Prove that the points $X,Y,D$ and $E$ lie on a circle.

Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]

Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$). A configuration of the $d\times d$ board is called feasible if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.