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 $\Omega$.

For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2 $ such that (1) $l_1, l_2 $ are not parallel. (2) $l_1, l_2 $ do not pass through any of $a+b+c+d$ points. (3) There are $ a, b, c, d $ points on each region separated by two lines $l_1, l_2 $.

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Alice is stacking balls on the ground in three layers using two sizes of balls: small and large. All small balls are the same size, as are all large balls. For the first layer, she uses $6$ identical large balls $A, B, C, D, E$, and $F$ all touching the ground and so that $D, E, F$ touch each other, A touches $E$ and $F$, $B$ touches $D$ and $F$, and $C$ touches $D$ and $E$. For the second layer, she uses $3$ identical small balls, $G, H$, and $I$; $G$ touches $A, E$, and $F, H$ touches $B, D$, and $F$, and $I$ touches $C, D$, and $E$. Obviously, the small balls do not intersect the ground. Finally, for the top layer, she uses one large ball that touches $D, E, F, G, H$, and $I$. If the large balls have volume $2015$, the sum of the volumes of all the balls in the pyramid can be written in the form $a\sqrt{b}+c$ for integers $a, b, c$ where no integer square larger than $1$ divides $b$. What is $a + b + c$? (This diagram may not have the correct scaling, but just serves to clarify the layout of the problem.) [asy] size(6cm); pair A, B, C, D, E, F; A = (0,0); F = (1,0); B = (2,0); E = rotate(60, A)*F; D = F + E; C = rotate(60, A)*B; draw(Circle(A, 0.5), mediumblue); draw(Circle(B, 0.5), mediumblue); draw(Circle(C, 0.5), mediumblue); draw(Circle(D, 0.5), mediumblue); draw(Circle(E, 0.5), mediumblue); draw(Circle(F, 0.5), mediumblue); pair G = (E+A+F)/3; pair I = (C+E+D)/3; pair H = (D+B+F)/3; draw(Circle(G, 0.25), mediumblue); draw(Circle(I, 0.25), mediumblue); draw(Circle(H, 0.25), mediumblue); label("A", A, fontsize(10pt)); label("B", B, fontsize(10pt)); label("C", C, fontsize(10pt)); label("D", D, fontsize(10pt)); label("E", E, fontsize(10pt)); label("F", F, fontsize(10pt)); label("G", G, fontsize(5pt)); label("I", I, fontsize(5pt)); label("H", H, fontsize(5pt)); label("Figure 1: The projection of the balls onto the ground", (1,-1), fontsize(10pt)); [/asy]

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

There is a set of $2n$ chips of $n$ different colors, two chips of each color. The chips are randomly placed in a row. Prove that the probability that there are two adjacent chips of the same color in a row is greater than $1/2$. [i]From the folklore[/i]

Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number. [b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.

[b]9.[/b] Let $A_1, \dots , A_n$ and $B$ be ideals of an assoticative ring $R$ such that $B$ is contained in the set-union of the ideals $A_i$($i=1, \dots , n$) but not contained in the union of any $n-1$ of the ideals $A_i$. Show that, for some positive integer $k$, $B_k$ is contained in the intersection of the ideals $A_i$. [b](A. 19)[/b]

$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.

[asy] size(2.5inch); pair A, B, C, E, F, G; A = (0,3); B = (-1,0); C = (3,0); E = (0,0); F = (1,2); G = intersectionpoint(B--F,A--E); draw(A--B--C--cycle); draw(A--E); draw(B--F); label("$A$",A,N); label("$B$",B,W); label("$C$",C,dir(0)); label("$E$",E,S); label("$F$",F,NE); label("$G$",G,SE); //Credit to chezbgone2 for the diagram[/asy] In triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio $\textbf{(A) }1:4\qquad\textbf{(B) }1:3\qquad\textbf{(C) }2:5\qquad\textbf{(D) }4:11\qquad \textbf{(E) }3:8$

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

We're given the functions $f(x)=|x-1|-|x-2|$ and $g(x)=|x-3|$. a) Draw the graph of the function $f(x)$. b) Determine the area of the section enclosed by the functions $f(x)$ and $g(x)$.

Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties: $\bullet$ $a+b$ divides $ab+1$, $\bullet$ $a-b$ divides $ab -1$, $\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.

The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by \[ L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2. \] Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.

A rook stands in a corner of an $n$ by $n$ chess board. For what $n$, moving alternately along horizontals and verticals, can the rook visit all the cells of the board and return to the initial corner after $n^2$ moves? (A cell is visited only if the rook stops on it, those that the rook “flew over” during the move are not counted as visited.) Proposed by A. Spivak

We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. Lines $\overline{AC}$ and $\overline{BD}$ are drawn such that $C$ is on $O_1$ and $D$ is on $O_2$ and $\overline{AC} \perp \overline{AB}$ and $\overline{BD} \perp \overline{AB}$. If minor arc $AB= 45$ degrees relative to $O_1$ and minor arc $AB= 60$ degrees relative to $O_2$ and the radius of $O_2 = 10$, the area of quadrilateral $CADB$ can be expressed in simplest form as $a + b\sqrt{k} + c\sqrt{\ell}$. Compute $a + b + c + k +\ell$.

Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic. (ltf0501).

Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.

Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

There is a standard deck of $52$ cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “$2$” to “$A$” (i.e. $2, 3,\ldots , Q, K, A$). A pair of cards is called a “[i]straight flush[/i]” if these two cards belong to the same suit and their ranks are adjacent. Additionally, "$A$" and "$2$" are considered to be adjacent (i.e. "A" is also considered as "$1$"). For example, spade $A$ and spade $2$ form a “[i]straight flush[/i]”; diamond $10$ and diamond $Q$ are not a “[i]straight flush[/i]” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “[i]straight flush[/i]” exists in them.

Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$. [i]Liubomir Chiriac, Moldova[/i]