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/sci/ - Science & Math

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5858621 No.5858621[DELETED]  [Reply] [Original]

hey /sci/

today I thought we could use some problems at different levels, problems which you could try your hand on.
If you like it, I'll post more topics like this in the future stating:
-an approximate level at which you can hope to find a solution
-the problem itself

First problem, which you can probably do after highschool:
let <span class="math">f : \mathbb{R} \rightarrow \mathbb{R}[/spoiler] be a strictly decreasing function.
Show that the system <span class="math"> \left\{ x=f(y), y=f(z), z=f(x) \right\} [/spoiler] has a unique solution.

Second problem: freshmen can try it!
let <span class="math">a<b \in \mathbb{R}[/spoiler]. Show that there exists a unique <span class="math">c \in ]a,b[ [/spoiler] such as <span class="math"> \int_{a}^{b} P(x)dx[/spoiler] is a linear combination of <span class="math">P(a), P(b)[/spoiler] and <span class="math">P(c)[/spoiler].

Third problem is for people with more experience I think:
Give an equivalent of <span class="math"> \displaystyle \sum_{k=1}^nk \lfloor \frac{n}{k} \rfloor [/spoiler]

There you go, good luck and ask anything!

>> No.5858628

>>5858621

Since I already did Linear Algebra
and Advanced Calculus...I just want to say:

Hulk would kick both their asses.

>> No.5858634

>>5858628
superman would too!

>> No.5858661

Hi. First: keep going on! Those are a good alternative for us plebs who can't tack on putnam problems.

Second: I don't know what your notation exactly means on the second problem (the part with the interval containing c).

>> No.5858673

You didn't define P in your second problem. Your interval notation is confusing, too.

Your third problem is vague and has infinitely many solutions, most of which are unenlightening.

Good idea, though. You just need to be more rigorous in phrasing your problems.

>> No.5858681

>>5858673
P is a polynomial, sorry
I was so affraid the latex wouldn't work I didn't pay attention to those details!

>> No.5858674

>>5858661
first, thank you for your appreciation, I thought this could be a good idea!
Second:
The notation means <span class="math"> a < c < b [/spoiler] (I don't even know why I didn't write it this way in the first place)

>> No.5858684

>>5858673
the third problem is asking for an equivalent as n goes to infinity!

>> No.5858715

The statement in the first problem is fucking wrong.
Trivially <span class="math">x,y,z[/spoiler] have to be fixed points of <span class="math">f \circ f \circ f[/spoiler] and since <span class="math">f[/spoiler] is strictly decreasing, they also have to be fixed points of <span class="math">f[/spoiler].
But there are strictly decreasing functions without a fixed point like
<span class="math">f(x)=-x + 1[/spoiler] for <span class="math">x \leq 0[/spoiler] and <span class="math">f(x)=-x - 1[/spoiler] for <span class="math">x > 0[/spoiler].

>> No.5858724

>>5858715
yet if you choose x=y=z=1/2 it works for f(x)=-x+1!

I should add that f is continuous.
This thread is fucked isn't it?

>> No.5858807
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5858807

>>5858715
>>5858724
should I give hints?

for example, in the first problem, you have to show that the line of equation y=x always intersects the line of equation y=f(x) at only one point.
You could study an intermediary function...
pic related