Search found 110 matches
- Sun Aug 21, 2016 10:39 pm
- Forum: Combinatorics
- Topic: USA TSTST 2015 - P1
- Replies: 1
- Views: 2351
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: 114643
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...
- Sat Aug 20, 2016 8:56 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114643
Re: IMO Marathon
$\boxed{\textbf{Problem 44}}:$Let $k$ be a nonzero natural number and $m$ an odd natural number.Prove that there exists a natural number $n$ such that the number $m^{n}+n^{m}$ has at least $k$ distinct prime factors. $\textbf{Source}:$ Romania TST $2014$ My solution uses a useful idea that is handy...
- Fri Aug 19, 2016 4:23 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114643
Re: IMO Marathon
Too easy. $A$ has a winning strategy.First,$A$ can write $pq$,where $p$ and $q$ are distinct prime divisors of $N$.The next numbers should be of the forms $pn$ ar $qn$ where $n > 1$ and $(n,pq)=1$.If $B$ writes $pn$,$A$ will write $qn$ and vice versa.Continuing like this,$A$ will eventually win. Yo...
- Fri Aug 19, 2016 2:02 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 114643
Re: IMO Marathon
Problem ${42}$ The number $N$ is the product of $k$ different primes ($k \ge 3$). A and B take turns writing composite divisors of $N$ on a board, according to the following rules. One may not write $N$. Also, there may never appear two coprime numbers or two numbers, one of which divides the other...
- Tue Aug 16, 2016 4:00 pm
- Forum: Number Theory
- Topic: ISL 2003 N1
- Replies: 1
- Views: 2291
- Tue Aug 16, 2016 12:53 am
- Forum: Combinatorics
- Topic: ISL 2007 C1
- Replies: 1
- Views: 2315
ISL 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 + n + 2} + \ldot...
- Tue Aug 16, 2016 12:52 am
- Forum: Algebra
- Topic: ISL 2003 A1
- Replies: 1
- Views: 2364
ISL 2003 A1
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
- Tue Aug 16, 2016 12:50 am
- Forum: Number Theory
- Topic: ISL 2003 N1
- Replies: 1
- Views: 2291
ISL 2003 N1
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
$x_i= 2^i$ if $0 \leq i\leq m-1$ and $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$.
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$
$x_i= 2^i$ if $0 \leq i\leq m-1$ and $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$.
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$
- Mon Aug 15, 2016 7:15 pm
- Forum: Geometry
- Topic: China TST $2011$ Day 2
- Replies: 2
- Views: 2992