What is the sum of the digits of the square of $ 111,111,111$? $ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 63 \qquad \textbf{(E)}\ 81$

Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$.

An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$). The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow. a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple. b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple ( i.e. everybody is in a triple).

On a board there are $2022$ numbers: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots,\frac{1}{2022}$. During a $move$ two numbers are chosen, $a$ and $b$, they are erased and $a+b+ab$ is written in their place. The moves take place until only one number is left on the board. What are the possible values of this number?

Arne has $2n + 1$ tickets. Each card has one number on it. One card has the number $0$ on it. The natural numbers $1, 2, . . . , n$ occur on exactly two cards each. Prove that Arne can arrange cards in a row so that there are exactly $m$ cards between the two cards with the number $m$, for every $m \in \{1, 2, . . . , n\}$.

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

Let $ABC$ be a triangle with right angle $ACB$. Denote by $F$ the projection of $C$ on $AB$. A circle $\omega$ touches $FB$ at point $P$, touches $CF$ at point $Q$, and the circumcircle of $ABC$ at point $R$. Prove that the points $A, Q, R$ all lie on the same line and $AP=AC$.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.

find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$

In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point. Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.

Prove that for any four points in the plane, no three of which are collinear, there exists a circle such that three of the four points are on the circumference and the fourth point is either on the circumference or inside the circle.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]

Let $ABCD$ be a trapezoid with bases $ AB$ and $CD$ $(AB>CD)$. Diagonals $AC$ and $BD$ intersect in point $ N$ and lines $AD$ and $BC$ intersect in point $ M$. The circumscribed circles of $ADN$ and $BCN$ intersect in point $ P$, different from point $ N$. Prove that the angles $AMP$ and $BMN$ are equal.

For each integer sequence $a_1, a_2, a_3, \dots, a_n$, a [i]single parity swapping[/i] is to choose $2$ terms in this sequence, say $a_i$ and $a_j$, such that $a_i + a_j$ is odd, then switch their placement, while the other terms stay in place. This creates a new sequence. Find the minimal number of single parity swapping to transform the sequence $1,2,3, \dots, 2025$ to $2025, \dots, 3, 2, 1$, using only single parity swapping.

Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that \[(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0\]

How to hang a picture? What a strange question? It's simple. We take a piece of rope, attach its ends to the picture frame on the back side, then drive it into the wall. nail and throw a rope over the nail. The picture is hanging. If you pull out the nail, then, of course, it will fall. But Professor No wonder acted differently. At first, he attached the rope to the painting in the same way, only he took it a little longer. Then he hammered two nails into the wall nearby and threw a rope over these nails in a special way. The painting hangs on these nails, but if you pull out any nail, the painting will fall. Moreover, the professor claims that he can hang a painting on three nails so that the painting hangs on all three, but if any nail is pulled out, the painting will fall. You have two tasks: indicate how you can hang the picture in the right way on a) two nails; b) three nails.

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Let $a,b,k$ be positive integers such that $gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$ Prove that $k$ is an even number.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?

In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.

Determine if there exist positive integer pairs $(m,n)$, such that (i) the greatest common divisor of m and $n$ is $1$, and $m \le 2007$, (ii) for any $k=1,2,..., 2007$, $\big[\frac{nk}{m}\big]=\big[\sqrt2 k\big]$ . (Here $[x]$ stands for the greatest integer less than or equal to $x$.)