## Secondary and Higher Secondary Marathon

For students of class 11-12 (age 16+)
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### Re: Secondary and Higher Secondary Marathon

Problem $21$: Determine all positive rationals $x,y,z$ such that $x+y+z$,$xyz$,$\frac{1} {x}+\frac{1} {y}+\frac{1} {z}$ are all integers.

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### Re: Secondary and Higher Secondary Marathon

It seems you folks lack stamina for a marathon.But resting is enough already i suppose.Let the marathon commence again with an easier problem.
Problem $22$:Let $CH$ be the altitude of triangle $ABC$ with $∠ACB = 90°$. The bisector of $∠BAC$ intersects $CH$, $CB$ at $P$, $M$ respectively. The bisector of $∠ABC$ intersects $CH$, $CA$ at $Q$, $N$, respectively. Prove that the line passing through the midpoints of $PM$ and $QN$ is parallel to line $AB$.

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### Re: Secondary and Higher Secondary Marathon

zadid xcalibured wrote:Problem $22$:Let $CH$ be the altitude of triangle $ABC$ with $∠ACB = 90°$. The bisector of $∠BAC$ intersects $CH$, $CB$ at $P$, $M$ respectively. The bisector of $∠ABC$ intersects $CH$, $CA$ at $Q$, $N$, respectively. Prove that the line passing through the midpoints of $PM$ and $QN$ is parallel to line $AB$.
Proof:
Let $I$ be the incenter of $ABC$ and $D,E$ be the mid-points of $QN$ and $PM$, respectively.
$\angle PMC=\angle AMC=90^{\circ}-\angle A/2=\angle APH=\angle CPM\Longrightarrow CP=CM$.
Similarly $CQ=CN$. Now it's easy to see $\measuredangle DCE=45^{\circ}=\measuredangle AIN$.
Therefore $C,D,I,E$ concyclic.
So $\angle IDE=\angle ICE=\angle ICM-\angle ECM=45^{\circ}-\angle A/2=\angle B/2=\angle IBA$.
Hence $DE||AB$.
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### Re: Secondary and Higher Secondary Marathon

Here goes another easy one.
Problem $23$:Find all continuous functions from the set of real numbers to itself satisfying $f(x + y) = f(x) + f(y) + f(x)f(y)$.

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### Re: Secondary and Higher Secondary Marathon

zadid xcalibured wrote:Here goes another easy one.
Problem $23$:Find all continuous functions from the set of real numbers to itself satisfying $f(x + y) = f(x) + f(y) + f(x)f(y)$.
Constant solutions $f(x)=0,-1$
Define $g(x)=f(x)+1$.
The equation can be re-written as $g(x+y)=g(x)g(y)$
By continuity, this famous equation has only solution $g(x)=a^x$ where $a$ is an arbitrary real. Therefore, $f(x)=a^x-1$.
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*Mahi*
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### Re: Secondary and Higher Secondary Marathon

zadid xcalibured wrote:Here goes another easy one.
Problem $23$:Find all continuous functions from the set of real numbers to itself satisfying $f(x + y) = f(x) + f(y) + f(x)f(y)$.
By continuity, this famous equation has only solution $g(x)=a^x$ where $a$ is an arbitrary real. Therefore, $f(x)=a^x-1$.
It should be "arbitrary positive real".

Use $L^AT_EX$, It makes our work a lot easier!

SANZEED
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### Re: Secondary and Higher Secondary Marathon

$\boxed {24}$
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?
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

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### Re: Secondary and Higher Secondary Marathon

Some moderator edit this.

SANZEED
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### Re: Secondary and Higher Secondary Marathon

edited now.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$