suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

For the numbers $ a$, $ b$, $ c$, $ d$, $ e$ define $ m$ to be the arithmetic mean of all five numbers; $ k$ to be the arithmetic mean of $ a$ and $ b$; $ l$ to be the arithmetic mean of $ c$, $ d$, and $ e$; and $ p$ to be the arithmetic mean of $ k$ and $ l$. Then, no matter how $ a$, $ b$, $ c$, $ d$, and $ e$ are chosen, we shall always have: $ \textbf{(A)}\ m \equal{} p \qquad \textbf{(B)}\ m \ge p \qquad \textbf{(C)}\ m > p \qquad \textbf{(D)}\ m < p \qquad \textbf{(E)}\ \text{none of these}$

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

Angle $C$ of an isosceles triangle $ABC$ equals $120^o$. Each of two rays emitting from vertex $C$ (inwards the triangle) meets $AB$ at some point ($P_i$) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle $ABC$ at point $Q_i$ ($i = 1,2$). Given that angle between the rays equals $60^o$, prove that area of triangle $P_1CP_2$ equals the sum of areas of triangles $AQ_1P_1$ and $BQ_2P_2$ ($AP_1 < AP_2$).

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$

Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that \[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\] [i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]

Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.

Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively. Find all possible values of the ratio $d_1:d_2$. [i](I. Voronovich)[/i]

After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that [b]exactly[/b] nine letters were inserted in the proper envelopes?

Robert has two stacks of five cards numbered 1--5, one of which is randomly shuffled while the other is in numerical order. They pick one of the stacks at random and turn over the first three cards, seeing that they are 1, 2, and 3 respectively. What is the probability the next card is a 4? [i]Proposed by Connor Gordon[/i]

The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$? $\textbf{(A)} \text{ 13} \qquad \textbf{(B)} \text{ 26} \qquad \textbf{(C)} \text{ 32} \qquad \textbf{(D)} \text{ 39} \qquad \textbf{(E)} \text{ 42}$

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

A $3\times3$ grid is to be painted with three colors (red, green, and blue) such that [list=i] [*] no two squares that share an edge are the same color and [*] no two corner squares on the same edge of the grid have the same color. [/list] As an example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? (Rotations and reflections are considered distinct colorings.)

Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over a) the side $AB$; b) the interior of $\triangle OAB$.

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations: 1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta. 2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up. 3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively. 4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair. 5. Remove four consecutive beads of one color. Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'', b) ``cyan, yellow, cyan, magenta, cyan'', c) ``magenta, magenta, cyan, cyan, cyan'', d) ``yellow, cyan, cyan, cyan''. [i]Proposed by Sammy Luo[/i]

In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points. Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true: $$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]

Suppose $x_0, x_1, \ldots , x_n$ are integers and $x_0 > x_1 > \cdots > x_n.$ Prove that at least one of the numbers $|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|,$ where \[F(x) = x^n + a_1x^{n-1} + \cdots+ a_n, \quad a_i \in \mathbb R, \quad i = 1, \ldots , n,\] is greater than $\frac{n!}{2^n}.$

Let $a, b$ be positive integers such that $a^2 \ge b$. Let $x = \sqrt{a+\sqrt{b}} - \sqrt{a-\sqrt{b}}$ (a) Prove that for all integers $a \ge 2$, there exists a positive integer $b$ such that $x$ is also a positive integer. (b) Prove that for all sufficiently large $a$, there are at least two $b$ such that $x$ is a positive integer. $\textbf{Note}$: We’ve received some questions about what is meant by “for all sufficiently large $a$.” To give a simple example of this phrasing, it is true that for all sufficiently large positive integers $n$, we have $n^2 \ge 100$. Specifically, this is true for all $n \ge 10$.

Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that : $(2^k)!=2^n*m$