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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Since no has not posted any problems ...............i am giving some
$1.$ what is the unit digit of $13^{2010}$ ?
$2.$ $m$ points are taken in each sides of a ragular $n$ gon .what is the total number of straight lines that can be drawn using all those lines ?
$1.$ what is the unit digit of $13^{2010}$ ?
$2.$ $m$ points are taken in each sides of a ragular $n$ gon .what is the total number of straight lines that can be drawn using all those lines ?
Re: Problems
What means unit digit?
Every logical solution to a problem has its own beauty.
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Re: Problems
একক স্থানীয় অংকZzzz wrote:What means unit digit?
Re: Problems
1. It should be 9!
2. I know thatthis prob has to be solved by combinatorix. But ami ekhono combinatorix sikhi nai, so....
2. I know thatthis prob has to be solved by combinatorix. But ami ekhono combinatorix sikhi nai, so....
r@k€€/|/
Re: Problems
its better to post the logic behind your solutionrakeen wrote:1. It should be 9!
2. I know thatthis prob has to be solved by combinatorix. But ami ekhono combinatorix sikhi nai, so....
Re: Problems
Probably Rakeen will give solution of problem 1.
Here is solution to problem 2:
If we choose any two points that are not on same side of the $n\ gon$ then we will find a straight line.
Lets number the sides of $n\ gon$ with $1,2,3,...,n$.
Start with side $1$. For each of the $m$ points of this side, we have $(n-1)m$ points. So in total $(n-1)m^2$ lines.
For each point of side #$2$, we have $(n-2)m$ points. So $(n-2)m^2$ lines.
Thus, total number of line is\[(n-1)m^2+(n-2)m^2+(n-3)m^2+...+1\cdot m^2\] \[=\frac{n(n-1)}{2}\cdot m^2\]
Here is solution to problem 2:
If we choose any two points that are not on same side of the $n\ gon$ then we will find a straight line.
Lets number the sides of $n\ gon$ with $1,2,3,...,n$.
Start with side $1$. For each of the $m$ points of this side, we have $(n-1)m$ points. So in total $(n-1)m^2$ lines.
For each point of side #$2$, we have $(n-2)m$ points. So $(n-2)m^2$ lines.
Thus, total number of line is\[(n-1)m^2+(n-2)m^2+(n-3)m^2+...+1\cdot m^2\] \[=\frac{n(n-1)}{2}\cdot m^2\]
Every logical solution to a problem has its own beauty.
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Re: Problems
1.ans is 9
logic 13^{fi(10)}_=1(mod10)
13^4 _=1(mod10)
13^2008 _=1(mod10)
13^2008+2_=13^2_=9(mod10)
ঠিক আছে?
2.solution টাও বুঝি নাই।কারণ আমি combinatorics অতো ভাল বুঝি না।
এখানে _= হচ্ছে is congruent to
logic 13^{fi(10)}_=1(mod10)
13^4 _=1(mod10)
13^2008 _=1(mod10)
13^2008+2_=13^2_=9(mod10)
ঠিক আছে?
2.solution টাও বুঝি নাই।কারণ আমি combinatorics অতো ভাল বুঝি না।
এখানে _= হচ্ছে is congruent to
Re: Problems
শাহরিয়ার, ফোরামে স্বাগতম।
আশা করি লক্ষ্য করেছ যে এই ফোরামে সমীকরণ লেখার চমৎকার সুবিধ আছে। তুমিও ইচ্ছা করলে কিভাবে LaTeX দিয়ে লিখতে হয় এই টপিকটা পড়ে সমীকরণ লিখতে পার।
আশা করি লক্ষ্য করেছ যে এই ফোরামে সমীকরণ লেখার চমৎকার সুবিধ আছে। তুমিও ইচ্ছা করলে কিভাবে LaTeX দিয়ে লিখতে হয় এই টপিকটা পড়ে সমীকরণ লিখতে পার।
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: Problems
Ami congurance ekhono sikhi nai. Ei jonno ami ebhabe ber korsi:
$3^1$ er ekok sthanio onko 3
$3^2$ er ekok sthanio onko 9
$3^3$ er ekok sthanio onko 7
$3^4$ er ekok sthanio onko 1
and 2010 is divided by 2, not by 4. So the answer is 9!
$3^1$ er ekok sthanio onko 3
$3^2$ er ekok sthanio onko 9
$3^3$ er ekok sthanio onko 7
$3^4$ er ekok sthanio onko 1
and 2010 is divided by 2, not by 4. So the answer is 9!
r@k€€/|/
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Re: Problems
ভাই exclamatory sign use কইরো না প্লিজ।এইটায় ৩৬২৮৮০ বুঝায়।@রাকীন।