Search found 108 matches

by rah4927
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

$N$ terms of an AP are divisible by product of first $N$

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

$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
by rah4927
Thu Nov 10, 2016 1:06 pm
Forum: Combinatorics
Topic: USAMO 2005/4 (Stable table)
Replies: 1
Views: 1428

Re: USAMO 2005/4 (Stable table)

Try to balance any two opposite legs first and then the other ones. Try similarity.
by rah4927
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...
by rah4927
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...
by rah4927
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...
by rah4927
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 +...
by rah4927
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 ...
by rah4927
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...
by rah4927
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...
by rah4927
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...