Search found 22 matches
- Wed Aug 10, 2016 4:56 pm
- Forum: Combinatorics
- Topic: Korea 1995
- Replies: 2
- Views: 3233
Re: Korea 1995
Full Solution: Let $M=\frac{1+\sqrt{2}}{m+2} \text{ and } N(a,b)=a+b\sqrt{2}, 0\leq a,b \leq m.$ So in total there are $(m+1)^2$ $N's.$ Now the highest value among the $N's$ is $m+m\sqrt{2}.$ So each $N$ will be in the interval $[0,m(1+\sqrt{2})]$. Divide this interval into $m(m+2)$ equal parts. The...
- Tue Aug 09, 2016 6:05 pm
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5876
Re: even odd even odd
Nice solu joss application
^ Little typo which might confuse others...
"Black squares are counted twice and all the white squares are counted once."
^ Little typo which might confuse others...
"Black squares are counted twice and all the white squares are counted once."
- Tue Aug 09, 2016 1:38 pm
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5876
Re: even odd even odd
Solution: I think my notation isn't clear. I meant when considering only values, $x_i = y_i = i$. But when I say $x_i$, I am refering to the row no. of the $i$'th row, not the column no. of the $i$'th column. Now let $\sum_{(x_i,y_j)\in S}x_i+y_j = N$ We'll show $N$ is even. First set $N = 0.$ Then ...
- Mon Aug 08, 2016 5:30 pm
- Forum: Combinatorics
- Topic: even odd even odd
- Replies: 7
- Views: 5876
Re: even odd even odd
Hint:
- Fri Aug 05, 2016 6:47 pm
- Forum: Combinatorics
- Topic: Guide the rook
- Replies: 1
- Views: 2517
Guide the rook
In a $n \times n$ board, there are some walls between adjacent squares, such that any two squares are still connected by a path not going through any wall. $A$ chooses a finite string of UP, DOWN, RIGHT, LEFT. A rook placed in some square of that board moves according to that string unless a wall or...
- Fri Apr 11, 2014 11:49 am
- Forum: Number Theory
- Topic: Swap those numbers
- Replies: 1
- Views: 2700
Swap those numbers
The numbers from $1$ to $n$ are written in a line in increasing order and at each time you are allowed to swap any two neighboring numbers. Now, is it possible to have this initial sequence of numbers after $2007$ such swappings?
- Tue Feb 11, 2014 9:15 pm
- Forum: Junior Level
- Topic: Need Of Difference
- Replies: 6
- Views: 4880
Re: Need Of Difference
Let the numbers be $10a+b$ and $10c+d,$ $10a+b>10c+d.$ Then their difference is $10(a-c)-(d-b).$ Now, the minimum possible value of $a-c$ is $1(a=6$ and $c=5)$ and the maximum possible value of $d-b is$ $7(d=8$ and $b=1).$ So the required two numbers are $61$ and $58$ and their difference is $3.$
- Tue Feb 11, 2014 9:01 pm
- Forum: Junior Level
- Topic: Acute angled circular
- Replies: 13
- Views: 10183
Re: Acute angled circular
As the distance between every two consecutive points are the same, any triange is acute-angled iff the difference between the points is $\leq 3$. Now, there are $7$ points. So the number of triangles having that property is one-third of the coefficient of the term $x^7$ multiplied by $7($as each poi...
- Tue Feb 11, 2014 8:48 pm
- Forum: Junior Level
- Topic: Need Of Difference
- Replies: 6
- Views: 4880
Re: Need Of Difference
Are $12$ and $21$ valid two numbers? I mean each digit can occur at most once. But is it in the sense of just one number?
- Tue Feb 11, 2014 8:25 pm
- Forum: Algebra
- Topic: simple equation
- Replies: 3
- Views: 7886
Re: simple equation
Letting $a = 2^{x}$ and $b = 3^{x}:$ $\frac{8^{x}+27^{x}}{8^{x}+12^{x}}=\frac{7}{6}$ $\Leftrightarrow \frac{(2^{x})^{3}+(3^{x})^{3}}{4^{x}(2^{x}+3^{x})}=\frac{7}{6}$ $\Leftrightarrow \frac{a^{3}+b^{3}}{a^{2}(a+b)}=\frac{7}{6}$ $\Leftrightarrow \frac{a^{2}-ab+b^{2}}{a^{2}}=\frac{7}{6}$ $\Leftrightarr...