Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$:

$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

## Search found 108 matches

- Thu Nov 10, 2016 1:08 pm
- Forum: Number Theory
- Topic: $N$ terms of an AP are divisible by product of first $N$
- Replies:
**0** - Views:
**1143**

- Thu Nov 10, 2016 1:06 pm
- Forum: Combinatorics
- Topic: USAMO 2005/4 (Stable table)
- Replies:
**1** - Views:
**1428**

- Thu Nov 10, 2016 1:03 pm
- Forum: Geometry
- Topic: Circle passing through the centers of two others'
- Replies:
**1** - Views:
**1381**

### Circle passing through the centers of two others'

$1.$ Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX...

- Thu Nov 10, 2016 12:59 pm
- Forum: Number Theory
- Topic: Functional divisibility
- Replies:
**2** - Views:
**2375**

### Re: Functional divisibility

Let $P(m,n)$ be the statement $f(m)+f(n) | (m+n)^k$ . We prove that the only solution is $f(n)=n$. Note that this indeed works. Claim 0: For two polynomials $P$ and $Q$ such that $P \nmid Q$, there are only finitely many integers $n$ such that $P(n)|Q(n)$ . Proof: Apply division algorithom on $P$ a...

- Tue Oct 11, 2016 4:04 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies:
**10** - Views:
**9409**

### Re: Combi Solution Writing Threadie

The problem is due to Rahul Saha, from an unknown source. 2016 nonnegative integers are written on a board. In each step, you can erase two of the numbers and replace them with their sum and their difference. For any given set of 2016 nonnegative integers on the blackboard, is it possible, in a fin...

- Tue Oct 11, 2016 4:01 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies:
**10** - Views:
**9409**

### Re: Combi Solution Writing Threadie

IMOSL 2007 C1 Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 + n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n; \] \[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i +...

- Tue Oct 11, 2016 3:56 am
- Forum: Combinatorics
- Topic: Combi Solution Writing Threadie
- Replies:
**10** - Views:
**9409**

### Re: Combi Solution Writing Threadie

There is a handout by Evan Chen used at MOP this year to teach students about solution writing. Should be relevant to this thread. You can find it in the first section titled "English" in the following link. http://www.mit.edu/~evanchen/olympiad.html I guess the big takeaways from the note that are ...

- Sun Aug 28, 2016 12:56 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**63501**

### Re: IMO Marathon

I will apply induction on the length of the string, and leave some of the finer details to the reader. Let's just do one case. Take $n=6$ (i.e. the string $123456$). Now let's randomly put some arrow signs inside, and try to apply the algorithm. The essence of this solution lies in knowing precisel...

- Sun Aug 21, 2016 10:39 pm
- Forum: Combinatorics
- Topic: USA TSTST 2015 - P1
- Replies:
**1** - Views:
**1377**

### USA TSTST 2015 - P1

Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let...

- Sun Aug 21, 2016 3:10 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies:
**184** - Views:
**63501**

### Re: IMO Marathon

$\boxed{\textbf{Problem 46}}$ Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is reachable from a string $A$ if i...