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Maths Competition problem!

Hi guys I am new here. So I found this problem I can not solve, can someone help me out? :))

http://prntscr.com/27jqyk

or click here

Would like an answer for both questions and please provide valid proofs, thanks very much! :D

Note by Igor Filkovski
3 years, 11 months ago

No vote yet
7 votes

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

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Well, I can get the discussion started, but I don't have a full answer. It looks like \( f(x) = 1-x^2 \) satisfies the condition. (I rewrote \( f(y) \) as \( a \); the \( 2xa \) term suggested looking at squares.)

If \( f(y) = 0 \) for some \( y \), then \( f(0) = 1 \) (plug into the equation and solve). And in fact, if the range of \( f \) is all of \( \mathbb R \), it's easy to show that \( f(x) = 1-x^2 \): for all \( x \) find \( y \) such that \( f(y) = x \), then we get \( f(0) = f(x) + 2x^2 + f(x) - 1 \), and \( f(0) = 1 \), so \( f(x) = 1-x^2 \). (The same argument shows that regardless of the range of \( f \), \( f(f(y)) = C - f(y)^2, \), where \( C = (f(0)+1)/2. \) )

I'm not sure what to do without the assumption on the range of \( f \), though.

Patrick Corn - 3 years, 11 months ago

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Oh also just paste the link I put there, I could not get it work :((

Igor Filkovski - 3 years, 11 months ago

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Hi Igor,

Welcome to Brilliant! I just edited your discussion to add a direct link to your problem. You can see what I did by clicking the "edit this discussion" button on your post. Further info on using markdown formatting can be found here.

Peter Taylor Staff - 3 years, 11 months ago

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Thank you very much Mr. Taylor! I was directed here by a friend of mine, I can see what he meant by saying the people are friendly :))

Igor Filkovski - 3 years, 11 months ago

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Hi! I am new here .Can u plz teel me how to post a problem that we are not able to solve?

Anuva Agrawal - 3 years, 11 months ago

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