Let $S$ denote the set of words $W = w_1w_2\ldots w_n$ of any length $n\ge0$ (including the empty string $\lambda$), with each letter $w_i$ from the set $\{x,y,z\}$. Call two words $U,V$ [i]similar[/i] if we can insert a string $s\in\{xyz,yzx,zxy\}$ of three consecutive letters somewhere in $U$ (possibly at one of the ends) to obtain $V$ or somewhere in $V$ (again, possibly at one of the ends) to obtain $U$, and say a word $W$ is [i]trivial[/i] if for some nonnegative integer $m$, there exists a sequence $W_0,W_1,\ldots,W_m$ such that $W_0=\lambda$ is the empty string, $W_m=W$, and $W_i,W_{i+1}$ are similar for $i=0,1,\ldots,m-1$. Given that for two relatively prime positive integers $p,q$ we have \[\frac{p}{q} = \sum_{n\ge0} f(n)\left(\frac{225}{8192}\right)^n,\]where $f(n)$ denotes the number of trivial words in $S$ of length $3n$ (in particular, $f(0)=1$), find $p+q$. [i]Victor Wang[/i]

Determine the value of \[S=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\cdots+\sqrt{1+\dfrac{1}{1999^2}+\dfrac{1}{2000^2}}\]

In the quadrilateral $ABCD$, the diagonal $AC$ is the bisector $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X, \, \, Y$ are the feet of the perpendiculars drawn from the point $A$ on the lines $BC, \, \, CD$, respectively. Prove that the orthocenter $\Delta AXY$ lies on the line $BD$.

An eight-digit number is said to be 'robust' if it meets both of the following conditions: (i) None of its digits is $0$. (ii) The difference between two consecutive digits is $4$ or $5$. Answer the following questions: (a) How many are robust numbers? (b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all the super robust numbers.

Fix reals $x, y,z > 0$ such that $x + y + z = \sqrt[5]{x} + \sqrt[5]{y} +\sqrt[5]{z}$ . Prove that $x^x y^y z^z \ge 1$. (Paolo Leonetti)

Let $u_1,u_2,\ldots,u_n\in C([0,1]^n)$ be nonnegative and continuous functions, and let $u_j$ do not depend on the $j$-th variable for $j=1,\ldots,n$. Show that $$\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.$$

Let $a,b$, and $c$ be odd positive integers such that $a$ is not a perfect square and $$a^2+a+1 = 3(b^2+b+1)(c^2+c+1).$$ Prove that at least one of the numbers $b^2+b+1$ and $c^2+c+1$ is composite.

Construct a triangle given a side, the radius of the inscribed circle, and the radius of the exscribed circle tangent to that side. (Research is not required.)

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.

Find all f:N $\longrightarrow$ N that: [list][b]a)[/b] $f(m)=1 \Longleftrightarrow m=1 $ [b]b)[/b] $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ [b]c)[/b] $ f^{2000}(m)=f(m) $[/list]

Points $STABCD$ in space form a convex octahedron with faces $SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA$ such that there exists a sphere that is tangent to all of its edges. Prove that $A,B,C,D$ lie in one plane.

Consider the sequence $\{a_n\}$ such that $a_0=4$, $a_1=22$, and $a_n-6a_{n-1}+a_{n-2}=0$ for $n\ge2$. Prove that there exist sequences $\{x_n\}$ and $\{y_n\}$ of positive integers such that \[ a_n=\frac{y_n^2+7}{x_n-y_n} \] for any $n\ge0$.

Let triangle $ABC$ with $AB<AC$ has orthocenter $H$, and let the midpoint of $BC$ be $M$. The internal angle bisector of $\angle BAC$ meet $CH$ at $X$, and the external angle bisector of $\angle BAC$ meet $BH$ at $Y$. The circles $(BHX)$ and $(CHY)$ meet again at $Z$. Prove that $\angle HZM=90^{\circ}$. [i]Proposed by Ivan Chan Kai Chin[/i]

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.

We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, [u]can't[/u] be covered with 2x1 domino tiles?

In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that [b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i]; [b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i]. Find the least possible number of [i]central[/i] vertices in $G$.

How many permutations $p$ of $\{1, 2, ..., 8\}$ satisfy $|p(p(a)) - a| \leq 1$ for all $a$? [i]Team #9[/i]

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

For a positive integer $n$, let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$. Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > v(b)$?

We are given two triangles. Prove, that if $\angle{C}=\angle{C'}$ and $\frac{R}{r}=\frac{R'}{r'}$, then they are similar.

Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

Suppose $x, y, z, t$ are real numbers such that $\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}$ Prove that $x^2 + y^2 + z^2 + t^2 \le 1$.

Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers. (Day 1, 3rd problem author: Michal Rolínek)