Let $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers $$f(a_1),f(a_2),\ldots,f(a_n)$$ form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.

Right triangle $ABC$ has a right angle at $A.$ Points $D$ and $E$ respectively lie on $\overline{AC}$ and $\overline{BC}$ so that $\angle BDA \cong \angle CDE.$ If the lengths $DE,$ $DA,$ $DC,$ and $DB,$ in this order, form an arithmetic sequence of distinct positive integers, then the set of all possible areas of $\triangle ABC$ is a subset of the positive integers. Compute the smallest element in this set that is greater than $1000.$

Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Let $d(n)$ be the number of divisors of $n,$ where $n$ is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence $\left\{a_{1}, a_{2}, \cdots\right\}$ if $\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}$ is an nonconstant geometric series then $\left\{a_{1}, a_{2}, \cdots\right\}$ does not bear same colour.

Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that $a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$ Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$

Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$, $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$. Prove that for any integers $1\leqslant k\leqslant n$, $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$.

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.) $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$

The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$. [i]Proposed by Mikhail Ivanov[/i]

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

Colour a $20000\times 20000$ square grid using 2000 different colours with 1 colour in each square. Two squares are neighbours if they share a vertex. A path is a sequence of squares so that 2 successive squares are neighbours. Mark $k$ of the squares. For each unmarked square $x$, there is exactly 1 marked square $y$ of the same colour so that $x$ and $y$ are connected by a path of squares of the same colour. For any 2 marked squares of the same colour, any path connecting them must pass through squares of all the colours. Find the maximum value of $k$.