Secondary and Higher Secondary Marathon
Forgot to mention the source. Source of my problem : www.brilliant.org
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Re: Secondary and Higher Secondary Marathon
Solution:SANZEED wrote:$\boxed {24}$
Let $N={1,2,3,...,2012}$. A $3$ element subset $S$ of $N$ is called $4$-splittable if there is an $n\in (N-S)$ such that $S\bigcup {n}$ can be partitioned into two sets such that the sum of each set is the same. How many $3$ element subsets of $N$ is non-$4$-splittable?
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- Tahmid Hasan
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Re: Secondary and Higher Secondary Marathon
Problem $25$: Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC$, $AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
Source: Centroamerican 2012-2.
Source: Centroamerican 2012-2.
বড় ভালবাসি তোমায়,মা
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Re: Secondary and Higher Secondary Marathon
Solution:
Anyone can take my turn
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: Secondary and Higher Secondary Marathon
Can you please explain this!
I don't understand this!
I don't understand this!
- Tahmid Hasan
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Re: Secondary and Higher Secondary Marathon
Problem $26$: Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
Source: JBMO-2010-3.
বিঃদ্রঃ বেশি অভিজ্ঞদের(বিশেষ করে সৌরভ ভাই) এই ম্যারাথনে এত তাড়াতাড়ি সমাধান না দেওয়ার অনুরোধ জানাচ্ছি
Source: JBMO-2010-3.
বিঃদ্রঃ বেশি অভিজ্ঞদের(বিশেষ করে সৌরভ ভাই) এই ম্যারাথনে এত তাড়াতাড়ি সমাধান না দেওয়ার অনুরোধ জানাচ্ছি
বড় ভালবাসি তোমায়,মা
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Re: Secondary and Higher Secondary Marathon
Which part?Reza_raj wrote:Can you please explain this!
I don't understand this!
@Tahmid, মোটামুটি ৪ মাস পর এই বছরের সমস্যা সমাধান করা শুরু করলাম। নেশার মত লাগতাসে। নেশা...... চেষ্টা করব পরে পোস্ট করার (সমাধান করতে পারলে)
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon
In $\triangle ABK,$ the angle bisector of $\angle A$ and perpendicular bisector of opposite side, $BK$, meet at point $M$. So $M$ must lie on $\bigcirc ABK$. After some angle chasing, we get $\angle NLA=\angle NAL=\angle B/2$.Tahmid Hasan wrote:Problem $26$: Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
Source: JBMO-2010-3.
বিঃদ্রঃ বেশি অভিজ্ঞদের(বিশেষ করে সৌরভ ভাই) এই ম্যারাথনে এত তাড়াতাড়ি সমাধান না দেওয়ার অনুরোধ জানাচ্ছি
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- Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon
Problem 27
Find all functions $f:\mathbb R\to \mathbb R$ satisfying this equation:
\[f(xy+f(x))=xf(y)+f(x)\]
Source: An excalibur problem. I solved it in the last month, so can't remember the particular source.
Find all functions $f:\mathbb R\to \mathbb R$ satisfying this equation:
\[f(xy+f(x))=xf(y)+f(x)\]
Source: An excalibur problem. I solved it in the last month, so can't remember the particular source.
Re: Secondary and Higher Secondary Marathon
Phlembac Adib Hasan wrote:Problem 27
Find all functions $f:\mathbb R\to \mathbb R$ satisfying this equation:
\[f(xy+f(x))=xf(y)+f(x)\]
Source: An excalibur problem. I solved it in the last month, so can't remember the particular source.
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi