A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

Let be a triangle $ ABC $ with $ \angle BAC = 90^{\circ } . $ On the perpendicular of $ BC $ through $ B, $ consider $ D $ such that $ AD=BC. $ Find $ \angle BAD. $

Let $k$ be a fixed positive integer. Prove that the sequence $\binom{2}{1},\binom{4}{2},\binom{8}{4},\ldots, \binom{2^{n+1}}{2^n},\ldots$ is eventually constant modulo $2^k$. [i]Proposed by V. Rastorguyev[/i]

Let $-1 < x < 1.$ Show that \[ \sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}. \] Deduce that \[ \csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8. \]

Show that there exist (at least) a rearrangement $a_0, a_1, a_2,..., a_{63}$ of the numbers $0,1, 2,..., 63$, such that $a_i - a_j \ne a_j - a_k$, for any $i < j < k \in \{0,1, 2,..., 63\}$.

Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.

Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \] [i]Dinu Serbanescu[/i]

In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP \cdot OQ$.

Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

Given two triangles with the same perimeter. Both triangles have integer side lengths. The first triangle is an equilateral triangle. The second triangle has a side with length 1 and a side with length $d$. Prove that when $d$ is divided by 3, the remainder is 1.

Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$

Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.

Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations \[ a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \] for every $1 \leq k \leq n$, the condition \[ r \mid a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \] also holds.

A bird flying in a straight line, initially at $\text{10 m/s}$, uniformly increases its speed to $\text{18 m/s}$ while covering a distance of $\text{40 m}$. What is the magnitude of the acceleration of the bird? (a) $\text{0.1 m/s}^2$ (b) $\text{0.2 m/s}^2$ (c) $\text{2.0 m/s}^2$ (d) $\text{2.8 m/s}^2$ (e) $\text{5.6 m/s}^2$