Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.

Define two sets: $M=\{ (x,y)|y\geq x^2\} ,N=\{ (x,y)|x^2+(y-a)^2\leq 1\}$. If $M\cup N=N$, then the range value of $a$ is $\text{(A)}a\geq 1\frac{1}{4}\qquad\text{(B)}a=1\frac{1}{4}\qquad\text{(C)}a\geq 1\qquad\text{(D)}0<a<1$

Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$, and let $M$ be a midpoint of the side $BC$. Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$. Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$. $\omega$, the circumcircle of triangle $ADE$, meets $EM$ again at $F$. Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$.

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable. [i]A. Hajnal, I. Juhasz[/i]

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?

[b](a)[/b] A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different. [i]($2$ points)[/i] [b](b)[/b] A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different? [i]($6$ points)[/i]

Show that there are positive reals $a$, $b$, $c$ such that \[\left\{ \begin{array}{l} a^2 + b^2 + c^2 > 2 \\ a^3 + b^3 + c^3 <2 \\ a^4 + b^4 + c^4 > 2 \\ \end{array} \right. \]

Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$?

The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles. (A Shapovalov)

Compute the number of ways to select $99$ cells in a $19 \times 19$ square grid such that no two selected cells share an edge or a vertex.

Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A. a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido" b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?

The perimeter, that is, the sum of the lengths of all sides of a convex quadrilateral $ ABCD $, is equal to $2004$ meters; while the length of its diagonal $ AC $ is equal to $1001$ meters. Find out if the length of the other diagonal $ BD $ can: a) To be equal to only one meter. b) Be equal to the length of the diagonal $ AC $.

On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two. Determine the smallest number of crickets such that this is possible.

A line passes through the points $\left(-1,3\right)$ and $\left(7,-2\right)$. At what value of $x$ does this line intercept the $x$-axis? $\text{(A) }\frac{7}{5}\qquad\text{(B) }\frac{19}{8}\qquad\text{(C) }\frac{19}{5}\qquad\text{(D) }\frac{27}{5}\qquad\text{(E) }\frac{23}{4}$

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$.

Let $\Gamma$ a circumference. $T$ a point in $\Gamma$, $P$ and $A$ two points outside $\Gamma$ such that $PT$ is tangent to $\Gamma$ and $PA=PT$. Let $C$ a point in $\Gamma (C\neq T)$, $AC$ and $PC$ intersect again $\Gamma$ in $D$ and $B$, respectively. $AB$ intersect $\Gamma$ in $E$. Prove that $DE$ it´s parallel to $AP$

Bilbo, Gandalf, and Nitzan play the following game. First, Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo now compiles a string of length $4044$ built from the three letters $a,b,c$. Nitzan looks at the string, chooses one of the three letters $a,b,c$, and removes from the string all instances of the chosen letter. Only then is the string revealed to Gandalf. He must now guess the number Nitzan chose. Can Bilbo and Gandalf work together and come up with a strategy beforehand that will always allow Gandalf to guess Nitzan's number correctly, no matter how he acts?

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection. Can this statement happen to be true? [i](М. Антипов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus. [i]Authored by Nikola Velov[/i]