Search found 185 matches
- Sun Jan 04, 2015 7:10 pm
- Forum: Combinatorics
- Topic: n+1 rows and columns
- Replies: 10
- Views: 8055
n+1 rows and columns
Some cells of a $n\times n$ board are coloured black so that any set of $n$ cells, no $2$ are on the same line or column contains at least one black cell. Prove there exists $i$ rows and $j$ columns with $i+j\geq n+1$,all of whose intersections are black.
- Sun Jan 04, 2015 7:00 pm
- Forum: Combinatorics
- Topic: Sequence count
- Replies: 2
- Views: 4288
Re: Sequence count
We consider a $n\times n$ board and color the upper $a_{i}$ cells black of column $i-1$ for $i=2,3......n$. Now the board is divided into $2$ parts. This makes a path from the upper left vertex to the lower right vertex of length $2n-2$.The $3rd$ condition implies that the black squares don't inters...
- Tue Sep 30, 2014 11:26 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
Problem $39$: Let $ABC$ be a triangle and $\omega $ be it's incircle.$\omega _{A}$ is the circle that passes thgrough $B,C$ and touches $\omega$. Similarly define $\omega _{B},\omega _{C}$. Let $\omega _{B}\cap \omega _{C}=A'$. Similarly define $B',C'$. Prove $AA',BB'$ and $CC'$ concur on $OI$. [$O$...
- Tue Sep 30, 2014 11:16 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
The answer is $n+1$.We proceed by induction. The base case is trivial.Let this is true for $n=k-1$.Now we prove this for $n=k$. Tow countries are called friend if one has a difensive agreement with the other. We choose a country $D$ from the n countries.Let $P$ and $C$ be it's player and coach resp....
- Mon Sep 29, 2014 7:15 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
Is it right?
Country $A$ has a difensive agreement with country $B$ $\rightarrow$ Country $B$ has a difensive agreement with country $A$
Country $A$ has a difensive agreement with country $B$ $\rightarrow$ Country $B$ has a difensive agreement with country $A$
- Mon Sep 29, 2014 7:04 pm
- Forum: Combinatorics
- Topic: USAMO 2007
- Replies: 2
- Views: 3200
Re: USAMO 2007
Please check someone this solution. We say that a cell/animal is joined with another cell/animal if they share at least one common edge. An animal is called 'good' if there exists no cell that is joined with a cell of it. Let $D$ be the greatest primitive dinosaur. Now if another cell $x$ is joined ...
- Mon Sep 22, 2014 6:26 pm
- Forum: Number Theory
- Topic: Turkish TST 2009
- Replies: 1
- Views: 2139
Turkish TST 2009
Find all $f:Q^+\rightarrow Z $ functions that satisfy $f(\dfrac{1}{x})=f(x)$ and $(x+1)f(x-1)=xf(x)$ for all rational numbers that are bigger than 1
- Wed Sep 10, 2014 2:22 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
Mahi vai is right.The square is $100 \times 100$.
- Tue Sep 09, 2014 6:20 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
Problem $35$
Every unit square of a $100 \times 100$ is coloured by one of $4$ colours red,green,blue or yellow so that in every line and column,there are $25$ squares of every colour.Prove that thre exists $2$ rows and columns so that their $4$ intersection points have $4$ different colours.
Every unit square of a $100 \times 100$ is coloured by one of $4$ colours red,green,blue or yellow so that in every line and column,there are $25$ squares of every colour.Prove that thre exists $2$ rows and columns so that their $4$ intersection points have $4$ different colours.
- Fri Sep 05, 2014 12:23 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 115440
Re: IMO Marathon
At first,we prove the following lemma. Lemma : For all $n \in N$ $f(1)<f(2)<..........<f(n)<f(i)$ for all $i>n$ Proof : We induct on n.For the proof of base case,let $f(a)$ be the smallest element of the range of $f$.If $a>1$,then $a-1$ is positive. $f(a)>\dfrac{f(a-1)+f(f(a-1))}{2} \geq min(f(a-1),...