BDMO National Junior 2018/6
- nahin munkar
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Given $8$ lines on a plane and no two of them are parallel. Prove that, at least two of them form an angle less than 23°.
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Re: BDMO National Junior 2018/6
Using $PHP$
Anyone please clarify my short solution.
- Enthurelxyz
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Re: BDMO National Junior 2018/6
Let the $8$ lines are $l_1,l_2,\cdots ,l_8$. Take a point $O$ on the same plane. Now draw $8$ lines $L_i$ on point $O%$ such that $L_i \parallel l_i$.
Now, on point $O$, there are $8$ different angles created by $L_1,L_2,\cdots ,L_8$. So, there should be at least an angle less than $23$.
Let $\angle L_iOL_j < 23$ then $l_i , l_j$ create an angle less than $23$ as they create an equal angle with $\angle L_iOL_j$.
Now, on point $O$, there are $8$ different angles created by $L_1,L_2,\cdots ,L_8$. So, there should be at least an angle less than $23$.
Let $\angle L_iOL_j < 23$ then $l_i , l_j$ create an angle less than $23$ as they create an equal angle with $\angle L_iOL_j$.
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- Anindya Biswas
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Re: BDMO National Junior 2018/6
I have almost same argument, since those angles add up to $180^{\circ}$, then on average they are $\frac{180^{\circ}}{8}=22.5^{\circ}$, so in order to get such average, we must need at least one angle $\leq 22.5^{\circ}$Enthurelxyz wrote: ↑Wed Jan 20, 2021 10:54 amLet the $8$ lines are $l_1,l_2,\cdots ,l_8$. Take a point $O$ on the same plane. Now draw $8$ lines $L_i$ on point $O%$ such that $L_i \parallel l_i$.
Now, on point $O$, there are $8$ different angles created by $L_1,L_2,\cdots ,L_8$. So, there should be at least an angle less than $23$.
Let $\angle L_iOL_j < 23$ then $l_i , l_j$ create an angle less than $23$ as they create an equal angle with $\angle L_iOL_j$.
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- Mehrab4226
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Re: BDMO National Junior 2018/6
This was a clever one! And easy to express too!Anindya Biswas wrote: ↑Sat Feb 06, 2021 10:41 pmsince those angles add up to $180^{\circ}$, then on average they are $\frac{180^{\circ}}{8}=22.5^{\circ}$, so in order to get such average, we must need at least one angle $\leq 22.5^{\circ}$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
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