If $S$ is a sequence of positive integers let $p(S)$ be the product of the members of $S$. Let $m(S)$ be the arithmetic mean of $p(T)$ for all non-empty subsets $T$ of $S$. Suppose that $S'$ is formed from $S$ by appending an additional positive integer. If $m(S) = 13$ and $m(S') = 49$, find $S'$.

[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.

The increasing sequence of natural numbers $a_1,a_2,\ldots$ is such that for every $n>100$ the number $a_n$ is equal to the smallest natural number greater than $a_{n-1}$ and not divisible by any of the numbers $a_1,\ldots,a_{n-1}$. Prove that there is only a finite number of composite numbers in such a sequence. [i]Proposed by P. Kozhevnikov[/i]

Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and \[ a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots. \] for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.

here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

You have an $n\times n$ grid and want to remove all edges of the grid by the sequence of the following moves. In each move, you can select a cell and remove exactly three edges surrounding that cell; in particular, that cell must have at least three remaining edges for the operation to be valid. For which positive integers $n$ is this possible?

Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?

Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots , a_{2r}$ of $2r \ge 4$ pairwise distinct red squares is called a [i]bishop circuit[/i] if for every $k \in \{1, \ldots , 2r \}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1}=a_1$ and $a_{2r+2}=a_2$). In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit. ([i]Remark.[/i] Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^\circ$.)

The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$ $n \ge 1$. Prove that all the terms in the sequence are integers.

How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ? (A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.

Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relatively prime, $\ldots$, and $a_7$ and $a_8$ are relatively prime. How many cool sequences are there?

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

Consider an infinite sequence of positive integers $a_1, a_2, a_3, \dots$ such that $a_1 > 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible for two terms of such a sequence to be equal? [i]Proposed by Pavel Kozlov, Russia[/i]

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.