[b]a)[/b] Prove that for any two square matrices $ A,B $ of same order the equality $ \text{ord} (AB)=\text{ord} (BA) $ is true. [b]b)[/b] Show that $ \text{ord} (ab) =\text{ord} (ba) $ if $ a,b $ are elements of a monoid and one of them is an unit.

Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by \[ T_n(w) = w^{w^{\cdots^{w}}},\] with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $ T_2(3) = 3^3 = 27.$ Find the smallest tower of $ 3's$ that exceeds the tower of $ 1989$ $ 2's.$ In other words, find the smallest value of $ n$ such that $ T_n(3) > T_{1989}(2).$ Justify your answer.

Let $ABCD$ be a convex quadrilateral such that your diagonals $AC$ and $BD$ are perpendiculars. Let $P$ be the intersection of $AC$ and $BD$, let $M$ a midpoint of $AB$. Prove that the quadrilateral $ABCD$ is cyclic, if and only if, the lines $PM$ and $DC$ are perpendiculars.

For how many integers $a$ with $|a| \leq 2005$, does the system $x^2=y+a$ $y^2=x+a$ have integer solutions?

The sum of two natural numbers is $17,402.$ One of the two numbers is divisible by $10.$ If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? $\textbf{(A) }10,272 \qquad \textbf{(B) }11,700 \qquad \textbf{(C) }13,362 \qquad \textbf{(D) }14,238 \qquad \textbf{(E) }15,426$

There are $k$ piles of stones with $2020$ stones in each pile. Amber can choose any two non-empty piles of stones, and Barbara can take one stone from one of the two chosen piles and puts it into the other pile. Amber wins if she can eventually make an empty pile. What is the least $k$ such that Amber can always win?

Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$. [i]A. Kovaldji, V. Senderov[/i]

What is the largest number of regions into which a plane can be divided by drawing $n$ pairs of parallel lines?

Identical rectangular cardboard pieces are handed out to $30$ students, one to each. Each student cuts (parallel to the edges) his or her piece into equally large squares. Two different students’ squares do not necessarily have the same size. After all the cutting it turns out that the total number of squares is a prime. Prove that the original cardboard pieces must have been quadratic.

i) Given line segments $A,B,C,D$ with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible. ii) Given any acute-angled triangle $ABC$ and one altitude $AH$, select any point $D$ on $AH$, then draw $BD$ and extend until it intersects $AC$ in $E$, and draw $CD$ and extend until it intersects $AB$ in $F$. Prove that $\angle AHE = \angle AHF$.

Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$ Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.

In the nation of Onewaynia, certain pairs of cities are connected by roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges). Some roads have a traffic capacity of 1 unit and other roads have a traffic capacity of 2 units. However, on every road, traffic is only allowed to travel in one direction. It is known that for every city, the sum of the capacities of the roads connected to it is always odd. The transportation minister needs to assign a direction to every road. Prove that he can do it in such a way that for every city, the difference between the sum of the capacities of roads entering the city and the sum of the capacities of roads leaving the city is always exactly one. [i]Proposed by Zuming Feng and Yufei Zhao[/i]

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right. \]

Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.

In the figure, $[ABC]$ and $[DEC]$ are right triangles . Knowing that $EB = 1/2, EC = 1$ and $AD = 1$, calculate $DC$. [img]https://1.bp.blogspot.com/-nAOrVnK5JmI/X4UMb2CNTyI/AAAAAAAAMmk/TtaBESxYyJ0FsBoY2XaCGlCTc6mgmA5TQCLcBGAsYHQ/s0/2000%2Bportugal%2Bp5.png[/img]

Let $A_{1}A_{2}B_{1}B_{2}$ be a convex quadrilateral. At adjacent vertices $A_{1}$ and $A_{2}$ there are two Argentinian cities. At adjacent vertices $B_{1}$ and $B_{2}$ there are two Brazilian cities. There are $a$ Argentinian cities and $b$ Brazilian cities in the quadrilateral interior, no three of which collinear. Determine if it's possible, independently from the cities position, to build straight roads, each of which connects two Argentinian cities ou two Brazilian cities, such that: $\bullet$ Two roads does not intersect in a point which is not a city; $\bullet$ It's possible to reach any Argentinian city from any Argentinian city using the roads; and $\bullet$ It's possible to reach any Brazilian city from any Brazilian city using the roads. If it's always possible, construct an algorithm that builds a possible set of roads.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Denote by $\omega$ its incircle. A line $\ell$ tangent to $\omega$ intersects $\overline{AB}$ and $\overline{AC}$ at $X$ and $Y$ respectively. Suppose $XY=5$. Compute the positive difference between the lengths of $\overline{AX}$ and $\overline{AY}$.

Two sums, each consisting of $n$ addends , are shown below: $S = 1 + 2 + 3 + 4 + ...$ $T = 100 + 98 + 96 + 94 +...$ . For what value of $n$ is it true that $S = T$ ?

Show that the number $ 3^{{4}^{5}} \plus{} 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$

The sum of the numerical coefficients in the complete expansion of $ (x^2 \minus{} 2xy \plus{} y^2)^7$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 128 \qquad \textbf{(E)}\ 128^2$

Consider the sequence $(x_n)_{n\in\mathbb{N^*}}$ such that $$x_0=0,\quad x_1=2024,\quad x_n=x_{n-1}+x_{n-2}, \forall n\geq2.$$ Prove that there is an infinity of terms in this sequence that end with $2024.$