FE Marathon!

For discussing Olympiad Level Algebra (and Inequality) problems
Dustan
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FE Marathon!

Unread post by Dustan » Mon Oct 05, 2020 4:32 pm

আমি যতদূর সম্ভব সার্চ দিয়ে কোনো FE ম্যারাথন খুঁজে পাই নি। তাই এই ম্যারাথন চালু করতেসি।সবাই এগিয়ে আসলে হয়তো এটা চালু হবে। হ্যাপি প্রব্লেম সলভিং!

Problem 1:
Find all functions $\ f$ such that
$\ f: \mathbb{R} \rightarrow \mathbb{R}$ and $\ f(xf(x)+f(y))=f(x)^2+y $ for all real $x,\ y$.

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Mehrab4226
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Location: Dhaka, Bangladesh

Re: FE Marathon!

Unread post by Mehrab4226 » Thu Dec 10, 2020 9:49 pm

Is $f(x)^2=f(x) \times f(x)$?
or $ f(x)^2=f(f(x))$?
Last edited by Mehrab4226 on Fri Dec 11, 2020 12:02 am, edited 1 time in total.

tanmoy
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Location: Rangpur,Bangladesh

Re: FE Marathon!

Unread post by tanmoy » Thu Dec 10, 2020 10:24 pm

Dustan wrote:
Mon Oct 05, 2020 4:32 pm
আমি যতদূর সম্ভব সার্চ দিয়ে কোনো FE ম্যারাথন খুঁজে পাই নি। তাই এই ম্যারাথন চালু করতেসি।সবাই এগিয়ে আসলে হয়তো এটা চালু হবে। হ্যাপি প্রব্লেম সলভিং!

Problem 1: Find all functions $\ f$ such that
$\ f: \mathbb{R} \rightarrow \mathbb{R}$ and $\ f(xf(x)+f(y))=f(x)^2+y $ for all real $x,\ y$.
Good initiative. Use ${\large LaTeX}$ properly. I have edited this one.
"Questions we can't answer are far better than answers we can't question"

tanmoy
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Location: Rangpur,Bangladesh

Re: FE Marathon!

Unread post by tanmoy » Thu Dec 10, 2020 10:28 pm

Mehrab4226 wrote:
Thu Dec 10, 2020 9:49 pm
Is $f(x)^2=f(x) \times f(x)$
or $ f(x)^2=f(f(x))$
$\ f(x)^2 = f(x) \times f(x)$
"Questions we can't answer are far better than answers we can't question"

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Mehrab4226
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Location: Dhaka, Bangladesh

Re: FE Marathon!

Unread post by Mehrab4226 » Fri Dec 11, 2020 9:06 pm

If we can prove $f(f(x))$ is onto can we prove $f(x)$ is onto too? If we can than how?
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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Mehrab4226
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Location: Dhaka, Bangladesh

Re: FE Marathon!

Unread post by Mehrab4226 » Fri Dec 11, 2020 11:26 pm

[N.B I am not sure the solution is correct. If somebody would kindly check it, I will be grateful.]
Given that,
$f(xf(x)+f(y))=f(x)^2+y \cdots (1)$
Let, $x = 0$ then,
$f(0f(0)+f(y))=f(0)^2+y$
Or, $f(f(y))=f(0)^2+y$
Or, $f(f(y))=k^2+y [\text{Let, f(0)=k}] $
Now proving $f^2(y)$ is onto function,
proof: $f(f((x))) = k^2 + y$
For all $y \in \mathbb{R}$, $f(f((x))) \in \mathbb{R}$
Range $f = \mathbb{R}$ $=$ codomain
$\therefore f(f((x)))$ is onto function
$\therefore f(x)$ is a onto function Proof is in link( https://www.youtube.com/channel/UCyzsvF ... MRQ/videos )
This video proves if $g(f(x))$ is onto so is $g$ , we can use it in my proof by replacing $g$ by $f$.
Now back to the main solution,
Let, for some $t, f(t)=0$, $t=x$ in (1) we get,
$f(tf(t)+f(y))=f(t)^2+y$
Or, $f(0+f(y))=0+y$
Or, $f(f(y))=y \cdots (2)$
Finally , Let, $x=f(q)$ for some $q$ which is real
$f(f(q)f(f(q))+f(y))=f(f(q))^2+y$
Or, $f(qf(q)+f(y))=q^2+y$ [Using (2)]
Or, $f(q)^2+y=q^2+y$ [$f(qf(q)+f(y))=f(q)^2+y$ using (1)]
Or $f(q)^2=q^2$
Or, $f(q)= \pm q$
$\therefore$ the required function $f(x)= \pm x$
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

tanmoy
Posts: 305
Joined: Fri Oct 18, 2013 11:56 pm
Location: Rangpur,Bangladesh

Re: FE Marathon!

Unread post by tanmoy » Sun Dec 13, 2020 11:39 pm

Mehrab4226 wrote:
Fri Dec 11, 2020 11:26 pm
[N.B I am not sure the solution is correct. If somebody would kindly check it, I will be grateful.]
Given that,
$f(xf(x)+f(y))=f(x)^2+y \cdots (1)$
Let, $x = 0$ then,
$f(0f(0)+f(y))=f(0)^2+y$
Or, $f(f(y))=f(0)^2+y$
Or, $f(f(y))=k^2+y [\text{Let, f(0)=k}] $
Now proving $f^2(y)$ is onto function,
proof: $f(f((x))) = k^2 + y$
For all $y \in \mathbb{R}$, $f(f((x))) \in \mathbb{R}$
Range $f = \mathbb{R}$ $=$ codomain
$\therefore f(f((x)))$ is onto function
$\therefore f(x)$ is a onto function Proof is in link( https://www.youtube.com/channel/UCyzsvF ... MRQ/videos )
This video proves if $g(f(x))$ is onto so is $g$ , we can use it in my proof by replacing $g$ by $f$.
Now back to the main solution,
Let, for some $t, f(t)=0$, $t=x$ in (1) we get,
$f(tf(t)+f(y))=f(t)^2+y$
Or, $f(0+f(y))=0+y$
Or, $f(f(y))=y \cdots (2)$
Finally , Let, $x=f(q)$ for some $q$ which is real
$f(f(q)f(f(q))+f(y))=f(f(q))^2+y$
Or, $f(qf(q)+f(y))=q^2+y$ [Using (2)]
Or, $f(q)^2+y=q^2+y$ [$f(qf(q)+f(y))=f(q)^2+y$ using (1)]
Or $f(q)^2=q^2$
Or, $f(q)= \pm q$
$\therefore$ the required function $f(x)= \pm x$
The solution is okay, well done! Just a few remarks:
(1) Hide your solution. I have hidden it; check the latex code.
(2) Use proper spaces between your solutions so that they become reader friendly.
(3) You have latexed your solution, that's great. Though there were some typos. I have edited them; check.
(4) After solving a problem in a marathon, try to post a new problem.
Good luck! :)
"Questions we can't answer are far better than answers we can't question"

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Mehrab4226
Posts: 212
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Location: Dhaka, Bangladesh

Re: FE Marathon!

Unread post by Mehrab4226 » Tue Dec 15, 2020 9:32 pm

tanmoy wrote:
Sun Dec 13, 2020 11:39 pm
Mehrab4226 wrote:
Fri Dec 11, 2020 11:26 pm
[N.B I am not sure the solution is correct. If somebody would kindly check it, I will be grateful.]
Given that,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmr10">(</span></span><span class="icmr10">1)</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Let, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span></span></nobr></span> then,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(0</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(0</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">k</span><span class="spacer" style="margin-left:0.031em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="icmr10">[</span>Let, f(0)=k<span class="icmr10">]</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Now proving <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> is onto function,
proof: <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">k</span><span class="spacer" style="margin-left:0.031em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
For all <span class="typeset"><nobr><span class="scale"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span>, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char32.png" style=" width:9px; vertical-align:-1px; margin-right:0.019em;"></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span>
Range <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/msbm10/alpha/100/char52.png" style=" width:10px;"></span></span></nobr></span> <span class="typeset"><nobr><span class="scale"><span class="icmr10">=</span></span></nobr></span> codomain
<span class="typeset"><nobr><span class="scale"><img src="jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">((</span><span class="icmmi10">x</span><span class="icmr10">)))</span></span></nobr></span> is onto function
<span class="typeset"><nobr><span class="scale"><img src="jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span></span></nobr></span> is a onto function Proof is in link( https://www.youtube.com/channel/UCyzsvF ... MRQ/videos )
This video proves if <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">))</span></span></nobr></span> is onto so is <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> , we can use it in my proof by replacing <span class="typeset"><nobr><span class="scale"><span class="icmmi10">g</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> by <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span></nobr></span>.
Now back to the main solution,
Let, for some <span class="typeset"><nobr><span class="scale"><span class="icmmi10">t</span><img src="jsMath/fonts/cmmi10/alpha/100/char3B.png" style=" width:3px; vertical-align:-3px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span></span></nobr></span>, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">t</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">x</span></span></span></nobr></span> in (1) we get,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">t</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(0</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">0</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span><span style="position: relative; margin-left:0.166em;"><img src="jsMath/fonts/cmsy10/alpha/100/char01.png" style=" width:3px; vertical-align:2px; margin-right:0.062em;"></span></span><span style="position: relative; margin-left:0.166em;"><span class="icmr10">(</span></span><span class="icmr10">2)</span></span></nobr></span>
Finally , Let, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">x</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span></span></nobr></span> for some <span class="typeset"><nobr><span class="scale"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span> which is real
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> [Using (2)]
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> [<span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">))</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">y</span><span class="spacer" style="margin-left:0.035em"></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span> using (1)]
Or <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span class="blank" style="height:1.190em;vertical-align:-0.25em"></span></span></nobr></span>
Or, <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char06.png" style=" width:11px; margin-right:-0.013em;"></span><span class="icmmi10">q</span><span class="spacer" style="margin-left:0.035em"></span></span></nobr></span>
<span class="typeset"><nobr><span class="scale"><img src="jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"></span></nobr></span> the required function <span class="typeset"><nobr><span class="scale"><span class="icmmi10">f</span><span class="spacer" style="margin-left:0.108em"></span><span class="icmr10">(</span><span class="icmmi10">x</span><span class="icmr10">)</span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char06.png" style=" width:11px; margin-right:-0.013em;"></span><span class="icmmi10">x</span></span></nobr></span>
The solution is okay, well done! Just a few remarks:
(1) Hide your solution. I have hidden it; check the latex code.
(2) Use proper spaces between your solutions so that they become reader friendly.
(3) You have latexed your solution, that's great. Though there were some typos. I have edited them; check.
(4) After solving a problem in a marathon, try to post a new problem.
Good luck! :)
Thank you Bhaya. I will keep those in mind. I am very new in LaTeX and in the forum so sorry for this time.
$ \text{Problem 2:}$

Find all the function of $f : {\Bbb R} \to {\Bbb R}, satisfying$

$$f(x^2+y)=f(f(x)-y)+4f(x)y$$

for all real numbers x and y.Good Luck! :D
Source:
Some Iran TST
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

Dustan
Posts: 65
Joined: Mon Aug 17, 2020 10:02 pm

Re: FE Marathon!

Unread post by Dustan » Wed Dec 16, 2020 5:34 pm

$Solution 3$:
Let $P(x,y)$ be the assertion of the following equation.
From,$P(0,x)$ and $P(0,f(0)-x)$
$f(0)=0$
$P(x,-x^2)$:
$0$=$f(f(x)+x^2)+4\cdot f(x)\cdot (-x^2)$
Or,$4x^2f(x)$=$f(f(x)+x^2$...(1)

Again from, $P(x,f(x))$:
$f(f(x)+x^2)$=$4f(x)^2$....(2)
From this two, we get $f(x)=0$ and $f(x)=x^2$
which indeed fits the equation.
It took 2.30 hours :(

Dustan
Posts: 65
Joined: Mon Aug 17, 2020 10:02 pm

Re: FE Marathon!

Unread post by Dustan » Wed Dec 16, 2020 5:55 pm

$Problem 3$:

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
Source:IMO

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