[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]

(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$ (b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$ [i]Proposed by Hungary.[/i]

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

Let the sequence $(a_n)$ be constructed in the following way: $$ a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. $$ Prove that $a_{180}>19$. [i](Folklore)[/i]

Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent. [hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?

Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}$. Suppose \[ a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1 \] Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.

Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. [i](Proposed by Danylo Khilko)[/i]

For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that \[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\ b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\ c^2 &= \alpha\beta\gamma. \end{cases} \] Also, let $ \lambda$ be a real number that satisfies the condition \[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\] Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.

In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$