Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula; $ x_{0}\equal{} 1$, $ y_{0}\equal{} 2007$ $ x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1})$, $ y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1})$ Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.

For a positive integer $n$ with prime factorization $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ let's define $\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}$. Define $L(n)$ as sum of $\lambda(x)$ over all integers from $1$ to $n$. Define $K(n)$ as sum of $\lambda(x)$ over all [b]composite[/b] integers from $1$ to $n$. For some $N>1$, we know, that for every $2\le n \le N$, $L(n)\le 0$. Prove that for this $N$, for every $2\le n \le N$, $K(n)\ge 0$. [i]Mykhailo Shtandenko[/i]

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters? A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

Find all positive integers $k$, so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$, $$f(n)=\text{lcm}(n+1, n+2, \ldots, n+k).$$

During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;... Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).

$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ . Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

\begin{quote} Ted quite likes haikus, \\ poems with five-seven-five, \\ but Ted knows few words. He knows $2n$ words \\ that contain $n$ syllables \\ for every int $n$. Ted can only write \\ $N$ distinct haikus. Find $N$. \\ Take mod one hundred. \end{quote} Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$. [i]Proposed by Lewis Chen[/i]

Prove that $$1\cdot 4 + 2\cdot 5 + 3\cdot 6 + \cdots + n(n+3) = \frac{n(n+1)(n+5)}{3}$$ for all positive integer $n$.