[list=a] [*]Give an example of matrices $A$ and $B$ from $\mathcal{M}_{2}(\mathbb{R})$, such that $ A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) . $ [*]Let $A$ and $B$ be matrices from $\mathcal{M}_{2}(\mathbb{R})$, such that $\displaystyle A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) $. Prove that $AB\neq BA$.[/list]

Equilateral triangle $ABC$ has $AD = DB = FG = AE = EC = 4$ and $BF = GC = 2$. From $D$ and $G$ are drawn perpendiculars to $EF$ intersecting at $H$ and $I$, respectively. The three polygons $ECGI$, $FGI$, and $BFHD$ are rearranged to $EANL$, $MNK$, and $AMJD$ so that the rectangle $HLKJ$ is formed. Find its area. [img]https://cdn.artofproblemsolving.com/attachments/d/4/7e6667f0f0544b6fbc860f8d86c8ceaaf85cc1.png[/img]

Midori and Momoi are arguing over chores. Each of $5$ chores may either be done by Midori, done by Momoi, or put off for tomorrow. Today, each of them must complete at least one chore, and more than half of the chores must be completed. How many ways can they assign chores for today? (The order in which chores are completed does not matter.)

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

Does there exist a natural number that can be represented as the product of two numeric palindromes in more than $100{}$ ways?

Let $ABCDE$ be a convex pentagon inscribed in a circle $\Gamma$, such that $AB = BC = CD$. Let $F$ and $G$ be the intersections of $BE$ with $AC$ and of $CE$ with $BD$, respectively. Show that: a) $[ABC] = [FBCG]$ b) $\frac{[EFG]}{[EAD]} = \frac{BC}{AD}$ [b]Note: [/b] $[X]$ denotes the area of polygon $X$.

Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

On the sides $AB$ and $AC$ of a triangle $ABC$ select points $D$ and $E$ respectively, such that $AB = 6$, $AC = 9$, $AD = 4$ and $AE = 6$. It is known that the circumscribed circle of $\vartriangle ADE$ interects the side $BC$ at points $F, G$ , where $BF < BG$. Knowing that the point of intersection of lines $DF$ and $EG$ lies on the circumscribed circle of $\vartriangle ABC$ , find the ratio $BC:FG$.

As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all. Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?" "Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines. Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.

Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that \[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \] (Proposed by Gerhard Woeginger, Austria)

Find the remainder when $7^{7^7}$ is divided by $1000$.

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.

Determine all integral solutions of \[ a^2\plus{}b^2\plus{}c^2\equal{}a^2b^2.\]

In Mexico City, to limit traffic flow, each private car is set two days of the week on which it cannot go out onto the city streets. A family needs to have at least 10 cars at its disposal every day. What is the smallest number of cars can get by with the seven if its members can choose forbidden days for your cars?