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BdMO Online Forum • Search

## Search found 135 matches

### Re: Combi Marathon

Solution $\boxed{15}$ Suppose $a_1,...,a_n$ are the number of gems of each type. If they are arranged serially like, all $a_1$ gems of type $1$ at first, all $a_2$ gems of type $2$ next, and so on till gems of type $n$, then clearly we'll need $n$ cuts to split evenly (each cut at the middle of eac...
Thu Mar 02, 2017 9:20 pm

Forum: Combinatorics
Topic: Combi Marathon
Replies: 41
Views: 1272

### Re: Favorite mathematician?

John Gabriel for the win. :3

Those who haven't heard of this great mathemagician, come to the path of light.
Sun Feb 26, 2017 11:57 pm

Forum: Social Lounge
Topic: Favorite mathematician?
Replies: 30
Views: 2701

### Re: IMO Marathon

Solution $\boxed{52}$ Let $P(x,y)$ denote $g(f(x+y))=f(x)+(2x+y)g(y)$. If $g$ is constant then setting $g\equiv c$ we see that $f(x)=(1-2x-y)c$ for all $(x,y)\in\mathbb R^2$, implying $c=0$. So $f\equiv 0$ and $g\equiv 0$. This...
Mon Feb 06, 2017 7:45 pm

Topic: IMO Marathon
Replies: 184
Views: 19052

### Re: IMO Marathon

Solution $\boxed{51}$ Screams probabilistic approach. Take a random set $B$. Clearly $|A+B|=N^2$, we let $X$ to be the number of distinct elements in $A+B$. For some fixed residue $r$ we have $N$ possible elements to complement with an element from $A$, precisely $r-A_i$ for $1\le i\le N$. So the p...
Sat Feb 04, 2017 2:48 am

Topic: IMO Marathon
Replies: 184
Views: 19052

Let $\triangle (a,b,c)$ denote that lengths $a,b,c$ form a valid triangle. Let $P(a,b)$ denote $\triangle\left(a,f(b),f\left(b+f(a)-1\right)\right)$. Easy to prove: $\triangle (1,a,b)\Rightarrow a=b$ and $\triangle (2,a,b)\Rightarrow \left|a-b\right|\le 1$. Now $P(1,b)\Rightarrow f\left(b+f(1)-1\rig... Wed Feb 01, 2017 9:04 pm Forum: Algebra Topic: 2009 IMO SL A3 Replies: 2 Views: 130 ### Re: IMO Marathon Solution$\boxed{49}$The key fact is that for any positive integer$d$the set of edges divisible by$d$forms a clique. Indeed, for any positive integer$d$if there's only one edge divisible by$d$then it's an induced$K_1$. Otherwise take two edges$\alpha d$and$\beta d$. For any edg... Sun Sep 04, 2016 3:29 am Forum: International Mathematical Olympiad (IMO) Topic: IMO Marathon Replies: 184 Views: 19052 ### Re: IMO Marathon The$\dbinom n 2$consecutive natural numbers don't necessarily have to be the first$\dbinom n 2$natural numbers. Sat Sep 03, 2016 3:56 pm Forum: International Mathematical Olympiad (IMO) Topic: IMO Marathon Replies: 184 Views: 19052 ### Re: IMO Marathon Solution$\boxed{48}$Suppose$\displaystyle a_i = \prod_{j=1}^k p_j^{e_{i_j}}$for each$1\le i\le n.$Take$n$distinct primes$d_1,d_2,...,d_n.$Then we can take$b=\displaystyle\prod_{j=1}^k p_j^{\alpha_j}$where$\alpha_j+e_{i_j}\equiv 0~(\bmod~d_i)~$for each$1\le j \le k$and$1\le ...
Tue Aug 30, 2016 8:03 pm

Topic: IMO Marathon
Replies: 184
Views: 19052

### Re: IMO Marathon

Post a problem that's not lame.
Thu Aug 11, 2016 10:04 pm

Topic: IMO Marathon
Replies: 184
Views: 19052

### Re: UK 1996/2

Let $f(1)=C$ and $P(n)~\Rightarrow ~f(1)+\cdots+f(n)=n^2 f(n)$. Then rearranging $P(n)-P(n-1)$ gives $\dfrac{f(n)}{f(n-1)}=\dfrac{n-1}{n+1}$. Multiplying similar expressions leads to \[\prod_{k=2}^n \dfrac{f(k)}{f(k...
Mon Sep 21, 2015 11:00 pm

Forum: Algebra
Topic: UK 1996/2
Replies: 2
Views: 436
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