Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define \[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.

Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$. Let $P$ a point on $\Gamma$. Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$. Prove that when $P$ varies on $\Gamma$, the line $AB$ is tangent to a fixed circunference.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

Let $\omega$ be the circumcircle of an actue triangle $ABC$ and let $H$ be the feet of aliitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$. The lines $BM$ and $CN$ intersect each other at $G$ and intersect $\omega$ at $P$ and $Q$ respectively. The circles $(HMG)$ and $(HNG)$ intersect the segments $HP$ and $HQ$ again at $R$ and $S$ respectively. Prove that $PQ\parallel RS$.

Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos({\angle CMD})$? $\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{1}{3} \qquad\textbf{(C)} ~\frac{2}{5} \qquad\textbf{(D)} ~\frac{1}{2} \qquad\textbf{(E)} ~\frac{\sqrt{3}}{2} $

We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $200$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid. [asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("$5$",(0,5/2),W); label("$8$",(4,0),S); [/asy]

High schoolers chew a lot of gum. At the supermarket, $15$ packs of $14$ sticks of gum costs $\$10$. If $1400$ high schoolers chew $3$ sticks of gum per day, find the total number of dollars spent by these high schoolers on gum per week.

Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.

Let $a_1, a_2, ..., a_n$ be positive real numbers. Furthermore, let $S_n$ denote the set of all permutations of set $\{1, 2, ..., n\}$. Prove that $$\sum_{\pi \in S_n} \frac{1}{a_{\pi(1)}(a_{\pi(1)}+a_{\pi(2)})...(a_{\pi(1)}+a_{\pi(2)}+...+a_{\pi(n)})}=\frac{1}{a_1 a_2 ... a_n}$$

Alice and the Mad Hatter are playing a game. At the start of the game, three $2024$’s are written on the blackboard. Then, Alice and the Mad Hatter alternate turns, with the Mad Hatter starting. On the Mad Hatter’s turn, he must pick one of the numbers on the blackboard and increase it by $1$. On Alice’s turn, she must: - pick one of the numbers on the blackboard and decrease it by 1, and then - replace the two numbers $a$ and $b$ on the blackboard which were not chosen by the Mad Hatter on the previous turn with $\sqrt{ab}$. Alice wins if, on the start of her turn, any of the three numbers are less than $1$. Can the Mad Hatter prevent Alice from winning?

A dart is thrown at a square dartboard of side length $2$ so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance $1$ of a corner?

The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that \[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\] for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that \[\varphi(x,y,z) = h(x+y+z)\] for all real numbers $x,y$ and $z.$

4) find all polynom with coeffs a permutation of $[1,...,n]$ and all roots rational

We consider the set \begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \nu \geq 2 \end{align*} and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the $\nu-$tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$th roots of $-1$, where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*} Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$, we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine [list=i] [*] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) $ have measures less than or equal to $1$ [*] for $\nu =4$, the minimal value of $\kappa \in \mathbb{N}$, for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}

Let $x_1, x_2, x_3, y_1, y_2, y_3$ be real numbers in $[-1, 1]$. Find the maximum value of \[(x_1y_2-x_2y_1)(x_2y_3-x_3y_2)(x_3y_1-x_1y_3).\]

Let $ABCD$ be a trapezoid, where $AD\parallel BC$, $BC<AD$, and $AB\cap DC=T$. A circle $k_1$ is inscribed in $\Delta BCT$ and a circle $k_2$ is an excircle for $\Delta ADT$ which is tangent to $AD$ (opposite to $T$). Prove that the tangent line to $k_1$ through $D$, different than $DC$, is parallel to the tangent line to $k_2$ through $B$, different than $BA$.

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$. Is it a must that $ n \equal{} 2^{m} \plus{} 1$?

Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent: a) $f$ is differentiable, with continuous first derivative. b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.