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    Solution 223

    Clearly, \begin{aligned} \displaystyle f(2x)+1 = \sum_{k\ge 1} 2p_{k}\cos^{2} x\beta_{k} = 2 \sum_{k \ge 1} \frac{p^{2}_{k}\cos^{2} \beta_{k} x}{p_{k}} \ge 2\left(\sum_{k \ge 1} p_{k}\cos \beta_{k} x \right)^{2}\left(\sum_{k \ge 1} p_{k} \right)^{-1} = 2f^{2}(x) \end{aligned}.
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    (Original post by Mladenov)
    Solution 222

    Clearly, \begin{aligned} \displaystyle f(2x)+1 = \sum_{k\ge 1} 2p_{k}\cos^{2} x\beta_{k} = 2 \sum_{k \ge 1} \frac{p^{2}_{k}\cos^{2} \beta_{k} x}{p_{k}} \ge 2\left(\sum_{k \ge 1} p_{k}\cos \beta_{k} x \right)^{2}\left(\sum_{k \ge 1} p_{k} \right)^{-1} = 2f^{2}(x) \end{aligned}.
    ****s sake, you're too fast! :lol: I'm wondering if there is a way to do it without Cauchy Schwartz...

    Oh and, btw, do you know how the Engel form is derived?

    Edit: Btw henpen, I didn't think this was too boring or easy, the result is rather nice!
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    (Original post by Jkn)
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    ****s sake, you're too fast! :lol: I'm wondering if there is a way to do it without Cauchy Schwartz...

    Oh and, btw, do you know how the Engel form is derived?

    Edit: Btw henpen, I didn't think this was too boring or easy, the result is rather nice!
    From C-S, \displaystyle \left(b_{1} + \cdots + b_{k} \right)\left(\sum_{i \ge 1} \frac{a^{2}_{i}}{b_{i}} \right) \ge \left(\sqrt{b_{1}}\frac{a_{1}}{ \sqrt{b_{1}}} + \cdots + \sqrt{b_{k}}\frac{a_{k}}{\sqrt{b  _{k}}} \right)^{2}.
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    (Original post by Mladenov)
    From C-S, \displaystyle \left(b_{1} + \cdots + b_{k} \right)\left(\sum_{i \ge 1} \frac{a^{2}_{i}}{b_{i}} \right) \ge \left(\sqrt{b_{1}}\frac{a_{1}}{ \sqrt{b_{1}}} + \cdots + \sqrt{b_{k}}\frac{a_{k}}{\sqrt{b  _{k}}} \right)^{2}.
    You have a nice way of making things seem so easy :lol:

    I'm such a fan of it now! It trivialises so many things! :awesome:
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    How about the form in Theorem 4.3? Engel form's easy enough to derive, but that form seems very far removed from the vanilla CS.
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    (Original post by henpen)
    How about the form in Theorem 4.3? Engel form's easy enough to derive, but that form seems very far removed from the vanilla CS.
    Induction.
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    (Original post by Mladenov)
    Problem 221**

    Find all f: \mathbb{R} \to \mathbb{R} which satisfy f(x^{2})+f(xy)= f(x)f(y)+yf(x)+xf(x+y).
    Solution 221

    When x=0, \displaystyle f(0)(y+f(y)-2)=0.

    Hence, for f(0) \not= 0, f(x)=2-x

    For f(0)=0, when y=0, \displaystyle f(x^2)=xf(x). Letting x \to -x and equating the two equations gives \displystyle xf(x)=-xf(-x) \Rightarrow f(x)=-f(-x) which we know also works when x=0, by assumption.

    Letting y=-x gives \displaystyle f(x^2)+f(-x^2)=f(x)f(-x)-xf(x)+xf(0). In the case where f(0)=0, and using the equations just derived, this reduces to \displaystyle f(x)f(-x)=xf(x) \Rightarrow f(-x)=x \Rightarrow f(x)=-x which also works for f(x)=0.

    \displaystyle \therefore f(x)=2-x, f(x)=0 and \displaystyle f(x)=-x are the only functions that satisfy the given functional equation. \square

    (maybe I should start doing functional equations questions. I've always avoided them for some reason...)
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    I'm waiting for the mechanics
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    (Original post by bananarama2)
    I'm waiting for the mechanics
    Mechanics? Can't give you something that easy... here's a problem that takes place in a non-mathematical world:

    Problem 223*

    A spherical ball is dropped from the Eiffel Tower from rest. Give a convincing argument as to why Olber's Paradox gives strong evidence to support the idea that the observable universe has existed for a finite period of time. Note that, for such an argument to be convincing, it must rely solely on assumptions that are supported by strong evidence. In this question you are required to quote primary data sources to support all assumptions made.
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    Solution 221

    a,b,c>0:\;x=a+b,\;y=b+c,\; z=a+c

    The inequality reduces to:

    \displaystyle \sum_{a,b,c}\frac{8a^4}{b(a+b)} \geq \sum_{a,b,c} a^2+3ab

    This is easily shown using CS:

    \displaystyle\begin{aligned} \frac{8a^4}{b(a+b)}+\frac{8b^4}{  c(b+c)}+\frac{8c^4}{a(c+a)} &\geq \frac{8(a^2+b^2+c^2)^2}{a^2+b^2+  c^2+ab+ac+bc}\\& \geq\frac{8(a^2+b^2+c^2)^2}{2(a^  2+b^2+c^2)}\\ &= 4(a^2+b^2+c^2)\\ & \geq a^2+b^2+c^2+3(ab+bc+ac)\end{alig  ned}
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    (Original post by Jkn)

    Letting y=-x gives \displaystyle f(x^2)+f(-x^2)=f(x)f(-x)-xf(x)+xf(0). In the case where f(0)=0, and using the equations just derived, this reduces to \displaystyle f(x)f(-x)=xf(-x) \Rightarrow f(x)=-x which also works when f(x)=0, by assumption.
    I do not understand this. From f(x)(f(-x)-x)=0 it does not follow that f(-x)=x for all x or f(x)=0 for all x. Can I not take f to be zero over the rationals and -x over the irrationals?
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    (Original post by Jkn)
    ...
    I've decided to give you a taste of what I do

    Problem 224*/**:

    Whenever we say "function" we mean total function.

    Denote for every n \in \mathbb N _0 the set \underline n = \{ m \in \mathbb N _0 | 0 < m \leq n\}.

    The mystery concept of a strange product \underline m \otimes \underline n for two such sets \underline m and \underline n satisfies the following property:



    There exist functions \pi_1 : \underline m \otimes \underline n \to \underline m and \pi_2 : \underline m \otimes \underline n \to \underline n such that:

    For each k \in \mathbb N_0 and each pair of functions f: \underline k \to \underline m and g: \underline k \to \underline n there exists a unique function h : \underline k \to \underline m \otimes \underline n such that for all x\in \underline k we have f(x) = \pi_1(h(x)) and g(x) = \pi_2(h(x))



    Show that regardless of what the exact definition for \otimes is, if it satisfies the above property then for all sets X and (non-zero) natural n, |X \otimes \underline n| = n iff |X| = 1 (in other words X = \underline 1).

    EDIT1: made it clearer, but question is still the same.

    EDIT2: Hmmm I just realised that the elegant solution is far beyond *-level, and the inelegant one may be quite long.

    EDIT3: Actually for the above reason I'm going to withdraw it as a formal problem for this thread, but it's open to anyone who wants to try it.
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    (Original post by ukdragon37)
    ]

    EDIT1: made it clearer, but question is still the same.
    Yeh. You just elucidated the whole problem
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    (Original post by ukdragon37)
    I've decided to give you a taste of what I do


    EDIT1: made it clearer, but question is still the same.

    EDIT2: Hmmm I just realised that the elegant solution is far beyond *-level, and the inelegant one may be quite long.

    EDIT3: Actually for the above reason I'm going to withdraw it as a formal problem for this thread, but it's open to anyone who wants to try it.
    I must say I was shamelessly looking up most of the notation, and was wondering how I was meant to solve it :lol:
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    (Original post by bananarama2)
    Yeh. You just elucidated the whole problem
    It's a royal pain writing out the question with a concrete example and in words. Normally you would express the property as a diagram:

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    (Original post by joostan)
    I must say I was shamelessly looking up most of the notation, and was wondering how I was meant to solve it :lol:
    Nah don't worry about it. Once again I posed a problem that requires too much definitions to understand.
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    (Original post by ukdragon37)
    It's a royal pain writing out the question with a concrete example and in words. Normally you would express the property as a diagram:

    My mind just can't plough through that question I'm sure someone else will have a go. I think I'll choose Matsci. over Compsci.
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    (Original post by ukdragon37)
    Nah don't worry about it. Once again I posed a problem that requires too much definitions to understand.
    Haha, it looks pretty
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    (Original post by joostan)
    Haha, it looks pretty
    My thoughts exactly for most of the solutions...
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    (Original post by ukdragon37)
    ...
    So how many kids do you have?
 
 
 
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