Some weeks ago the idea of making a daily math challenge thread gained popularity and then a guy started doing it, first with a problem about circles and then a problem about an integral with circles. Unfortunately he stopped, but I am here to save it.
Here is Today's problem:
Suppose that a sequence [math] a_1 , a_2 , a_3 , ... [/math] of positive real numbers satisfies:
[math] a_{k+1} \geq \frac{k a_k}{ (a_k)^2 + k - 1} [/math] for every positive integer [math] k [/math].
Prove that [math] a_1 + a_2 + ... + a_n \geq n [/math] for every [math] n \geq 2 [/math]
RULES: 1) If you have a proof, show full work. This means that: 2) The method of proof of leaving it as an exercise for the reader is invalid, for even the most trivial detail of your proof. 3) Everyone welcome, which means 4) You may tackle this problem in any way you see fit. The most elementary and the most complex methods will be praised equally. 5) When you post your solution, also tell us your major. Just for fun. 6) If you do not understand any detail of the problem, feel free to ask questions 7)Latex is encouraged but pictures of handwriting are also okay.
You didnt give a0? Or am I such a brainlet that I can't see that it's unnecessary to know?
Kevin Reyes
clever homework thread, but do it yourself
Brayden Morales
Great first equation.
First, there is no a0, our first term is a1. But that is not important.
Regarding the heart of your question, to solve this problem you do not need to know the first term. The "hard" part of the statement is that you need to prove that for ANY arbitrary first term, the theorem holds.
So for your proof you will have to assume an arbitrary a1 and prove it like that.
Hudson Wood
And obviously an induction proof is best applicable here.
Owen Hughes
Not a homework thread. This is a problem right from the international math olympiad. Not gonna tell you what year tho. I don't want anyone going to copy paste the solution given on the website.
Benjamin Diaz
Shit nigga IMOs pretty hard, maybe start with USAMO
Ethan Perry
But here we supposedly have university students from freshman to senior. And even some masters and PhDs students.
And the point is for it to be a challenge... and I've never heard of USAMO before.
Brody Foster
A sketchy answer [eqn] \frac{k}{a_k+ \frac{k-1}{a_k}} > \frac{ka_k}{k-1} > a_k [/eqn]If [math]a_1 \geq 1 [/math], we're done.[eqn] a_2 \geq \frac{1}{a_1}[/eqn]By AM-GM, [math]a_2+a_1 \geq 2[/math]. If [math]a_1 < 1[/math], [math]a_2 >1[/math], so [math] a_k > 1 [/math] for k >1 and so we're done
James Russell
We have retarded university students, that's the problem. To be fair though, I suppose IMOs do get harder every year
