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The Proof is Trivial!

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Original post by jack.hadamard
Problem 257 */**

How many subsets of {1,2,3,4,5,6,7,8,9,10}\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} are there such that the sum of the smallest and largest element is 1111?


Solution # 257

Spoiler

(edited 10 years ago)
Original post by Blutooth
...


Do you want to put "Solution #" on top. :tongue:
(edited 10 years ago)
Original post by jack.hadamard
Do you want to put "Solution #" on top. :tongue:


no problem :smile:
Reply 1823
Avatar for Jkn
Jkn
11
Original post by theterminator

LOL :lol: Blatantly gonna start sending people that link hahahahahaha :lol:

und
x

What did you call it "The Proof is Trivial!" btw? :tongue:
Original post by Jkn
What did you call it "The Proof is Trivial!" btw? :tongue:


The name of this thread comes from that link (it used to be in my signature) :wink:
Original post by Lord of the Flies
Here is a truly sensuous result.

Problem 192*

ff is a twice-differentiable function with continuous derivatives, and satisfies the following conditions over (a,b):(a,b):

(i)  f(x)>0(ii)  f′′(x)+f(x)>0(\text{i})\;f(x)>0\qquad (\text{ii})\; f''(x)+f(x)>0

Additionally,

(iii)  f(a)=f(b)=0(\text{iii})\; f(a)=f(b)=0

Show that b−a>πb-a>\pi



Has anybody got a solution they'd like to post? :smile:
Original post by bananarama2
Why does the first one seem so familiar? :pierre:


I remember it, too. I think it was the first problem I solved which you guys (current 6th formers) were interested in.
Original post by Lord of the Flies
The name of this thread comes from that link (it used to be in my signature) :wink:


You've got a good taste in music BTW :wink:
Reply 1828
Avatar for MW24595
MW24595
Original post by Felix Felicis

Spoiler



Gorgeous. :biggrin:
Shamelessly so.
Original post by theterminator
Has anybody got a solution they'd like to post? :smile:

There's a link to the solution in the OP
Original post by Felix Felicis
There's a link to the solution in the OP

I see. Thank you. I only just realised that there was a second post.. :P
Is there a function which takes prime numbers as values on all positive integers? :tongue:

Spoiler

Original post by jack.hadamard
Is there a function which takes prime numbers as values on all positive integers? :tongue:

Spoiler



Could you elaborate a bit more on the conditions? I can think of periodic functions where every integer input of x gives prime numbers but only a specific subset. However, I'm not sure if this is quite what you're asking.

(E.g. f(x)=5+2cos(πx) f(x) = 5+2cos( \pi x ) gives a prime output for every integer input, but only 2 specific ones.)
(edited 10 years ago)
Original post by DJMayes
However, I'm not sure if this is quite what you're asking.


What you gave is a good example. I am looking for a function that assumes infinitely many primes as values.
Original post by jack.hadamard
Is there a function which takes prime numbers as values on all positive integers? :tongue:

Spoiler



Surely there is no known function which gives different primes for each integer. If I find it do I get a millions pounds (and a job which GCHQ)?
Original post by bananarama2
Surely there is no known function which gives different primes for each integer. If I find it do I get a millions pounds (and a job which GCHQ)?


I claim there exists a constant m\mathfrak{m} such that ⌊m3n⌋\lfloor \mathfrak{m}^{3^n} \rfloor is a prime for all n∈Nn \in \mathbb{N}. :tongue:
Original post by jack.hadamard
Is there a function which takes prime numbers as values on all positive integers? :tongue:

Spoiler



5(n2 − n) + 1
Original post by jack.hadamard
I claim there exists a constant m\mathfrak{m} such that ⌊m3n⌋\lfloor \mathfrak{m}^{3^n} \rfloor is a prime for all n∈Nn \in \mathbb{N}. :tongue:



Original post by MAyman12
5(n2 − n) + 1


Interesting, it appears I'm very much mistaken. Pfft all this maths nonsense :colone:
Original post by jack.hadamard
Is there a function which takes prime numbers as values on all positive integers? :tongue:

Spoiler



Yes,

f(x)= x if x is prime, 0 otherwise

will work.
Original post by bananarama2
Interesting, it appears I'm very much mistaken. Pfft all this maths nonsense :colone:


I just think Mills' constant is under-appreciated and decided to promote it. :tongue: Actually, there are uncountably many such m\mathfrak{m}.

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