## Secondary and Higher Secondary Marathon

For students of class 11-12 (age 16+)
SANZEED
Posts: 550
Joined: Wed Dec 28, 2011 6:45 pm

### Re: Secondary and Higher Secondary Marathon

Forgot to mention the source. Source of my problem : www.brilliant.org
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

sourav das
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### Re: Secondary and Higher Secondary Marathon

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?
Solution:
Anyone can take my turn.
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### 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.
বড় ভালবাসি তোমায়,মা

sourav das
Posts: 461
Joined: Wed Dec 15, 2010 10:05 am
Location: Dhaka
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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......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

Reza_raj
Posts: 9
Joined: Thu Jan 17, 2013 7:31 pm

### Re: Secondary and Higher Secondary Marathon

I don't understand this!

Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### 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
বড় ভালবাসি তোমায়,মা

sourav das
Posts: 461
Joined: Wed Dec 15, 2010 10:05 am
Location: Dhaka
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### Re: Secondary and Higher Secondary Marathon

Reza_raj wrote:Can you please explain this!
I don't understand this!
Which part?
@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......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

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### Re: Secondary and Higher Secondary Marathon

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
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 $\angle NLA=\angle NAL=\angle B/2$.
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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.

*Mahi*
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### Re: Secondary and Higher Secondary Marathon

Find all functions $f:\mathbb R\to \mathbb R$ satisfying this equation:
$f(xy+f(x))=xf(y)+f(x)$
Use $L^AT_EX$, It makes our work a lot easier!