Amy has a $2 \times 10$ puzzle grid which she can use $1 \times 1$ and $1 \times 2$ (1 vertical, 2 horizontal) tiles to cover. How many ways can she exactly cover the grid without any tiles overlapping and without rotating the tiles?

There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.

Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities $$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$ Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

According to the standard convention for exponentiation, \[2^{2^{2^2}} \equal{} 2^{\left(2^{\left(2^2\right)}\right)} \equal{} 2^{16} \equal{} 65,\!536.\] If the order in which the exponentiations are performed is changed, how many [u]other[/u] values are possible? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

In a triangle $ABC$ the bisectors through vertices $B$ and $C$ meet the sides $\left [ AC \right ]$ and $\left [ AB \right ]$ at $D$ and $E$ respectively. Let $I_{c}$ be the center of the excircle which is tangent to the side $\left [ AB \right ]$ and $F$ the midpoint of $\left [ BI_{c} \right ]$. If $\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2$, show that $ABC$ is an equilateral triangle.

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

Given is a convex quadrilateral $ABCD$ with $AB=CD$. Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$. Prove that the midpoints of $\overline{AD}$, $\overline{BC}$ and $\overline{EF}$ are collinear.

A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$ BNC$ and $DNA$ are equilateral. (I.Sharygin)

Let $ABCD$ be a rectangle with $\angle ABD = 15^o, BD = 6$ cm. Compute the area of the rectangle. A. $9$ cm$^2$ B. $9 \sqrt3$ cm$^2$ C. $18$ cm$^2$ D. $18 \sqrt3$ cm$^2$ E. $24 \sqrt3$ cm$^2$

A grasshopper is in the left above corner of a $10\times 10$ square. At each step he can jump a square below or a square to the right. Also, he can also fly from a cell of the bottom row to a cell of the above row, and from a cell of the rightmost column to a cell of the leftmost column. Prove that the grasshopper has to do at leat $9$ flies in order to visit each cell of the square at least once. [I]Proposed by N. Vlasova[/I]

The doctor instructed a person to take $48$ pills for next $30$ days. Every day he take at least $1$ pill and at most $6$ pills. Show that exist the numbers of conscutive days such that the total numbers of pills he take is equal with $11$.

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that \[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]

Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.

Find all solutions of the equation $y\partial u/\partial x-\sin x\partial u/\partial y=u^2$ in a neighbourhood of the point $0,0$.

Given positive integers $a,b,c$ such that $a^3$ is divisible by $b$, $b^3$ is divisible by $c$, $c^3$ is divisible by $a$. Prove that $(a+b+c)^{13}$ is divisible by $abc$.

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?

A [u]positive sequence[/u] is a finite sequence of positive integers. [u]Sum of a sequence[/u] is the sum of all the elements in the sequence. We say that a sequence $A$ can be [u]embedded[/u] into another sequence $B$, if there exists a strictly increasing function $$\phi : \{1,2, \ldots, |A|\} \rightarrow \{1,2, \ldots, |B|\},$$ such that $\forall i \in \{1, 2, \ldots ,|A|\}$, $$A[i] \leq B[\phi(i)],$$ where $|S|$ denotes the length of a sequence $S$. For example, $(1,1,2)$ can be embedded in $(1,2,3)$, but $(3,2,1)$ can not be in $(1,2,3)$\\ Given a positive integer $n$, construct a positive sequence $U$ with sum $O(n \, \log \, n)$, such that all the positive sequences with sum $n$, can be embedded into $U$.\\

We call a positive integer $n{}$ [i]peculiar[/i] if, for any positive divisor $d{}$ of $n{}$ the integer $d(d + 1)$ divides $n(n + 1).$ Prove that for any four different peculiar positive integers $A, B, C$ and $D{}$ the following holds: \[\gcd(A, B, C, D) = 1.\]

Let $x=\frac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}$. The value of $$H=(1+x^5-x^7)^{{2012}^{3^{11}}}$$ is (A) $1$ (B) $11$ (C) $21$ (D) $101$ (E) None of the above

Let be two complex numbers $ a,b $ chosen such that $ |a+b|\ge 2 $ and $ |a+b|\ge 1+|ab|. $ Prove that $$ \left| a^{n+1} +b^{n+1} \right|\ge \left| a^{n} +b^{n} \right| , $$ for any natural number $ n. $ [i]Alin Pop[/i]

Let $ABC$ be an acute triangle with incenter $I$; ray $AI$ meets the circumcircle $\Omega$ of $ABC$ at $M \neq A$. Suppose $T$ lies on line $BC$ such that $\angle MIT=90^{\circ}$. Let $K$ be the foot of the altitude from $I$ to $\overline{TM}$. Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$, and $\frac{BK}{CK} = \frac mn$ in lowest terms, compute $m+n$. [i]Proposed by Evan Chen[/i]

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its: $ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad \textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\ \textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad \textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\ \textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$