Proof contest

chess64

Week 4 Solution

The statement of Germany 1996 #2 is the contrapositive of this problem, and a nice solution to that is given here. Some people also used the contrapositive technique for this problem.

**Posted:** Sun May 21, 2006 5:43 am

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

I will take over the proof contest, but I want someone else to help me (as I am not always available.) Are there any volunteers?

Anyway:

Week 5

Prove that, for all values of $\theta$ , (provided that $\tan\theta$ and $\sec^{2}\theta$ are defined), $2\tan\theta \leq \sec^{2}\theta$

(PM me your proof)

**Posted:** Sun May 21, 2006 6:41 am

darkdevil

now you are the one who gets the solutions? not chess64 anymore?

**Posted:** Sun May 21, 2006 10:04 am

_________________
$\mathfrak{Radvan Marius}$ $aka$ $\mathfrak{darkdevil}$

darkdevil

and btw.....what is $sec \theta$ ??

**Posted:** Sun May 21, 2006 10:09 am

_________________
$\mathfrak{Radvan Marius}$ $aka$ $\mathfrak{darkdevil}$

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

**Posted:** Sun May 21, 2006 10:16 am

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

Scoreboard

drunner2007

any new updates/remarks?

**Posted:** Thu May 25, 2006 5:33 am

_________________
It is never too late to be what you could have been...

The only way to predict the future is to make it.

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

I might not be available for the next few days, so I am posting week 6 today.

Week 5 Solution

I would like to remind everybody to try to write proofs that are as clear as possible.

**Posted:** Fri May 26, 2006 11:38 am

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

I think that after Week 6 I may start again from week 1 and have proof contests in 6-week cycles. Any feedback on that idea?

Also, does anyone want to volunteer to help out with the proof contest or to be a backup?

Week 6

Let $a$ , $b$ , and $c$ be fixed natural numbers such that, for every positive integer $n$ , there is a triangle whose sides have length $a^n$ , $b^n$ , and $c^n$ . Prove that these triangles are not scalene.

**Posted:** Fri May 26, 2006 11:42 am

seesaw · Hodge Conjecture Offline Joined: 09 Mar 2006 Posts: 53

Week 6 Solution

I am sorry to say that due to certain circumstances, I am no longer able to run the Proof Contest. If there are any volunteers to take over the contest please PM me.

**Posted:** Sun Jun 04, 2006 2:54 pm

darkdevil

Week 7

$f=x^3-x-1$
Prove that $f$ has only one real root $r \in \left(1, \frac 32 \right)$
Prove that if $a$ is a root and $a \notin \mathbb{R}$ then $|a| \in \left( \sqrt{\left(\frac 23\right)}, 1\right)$

**Posted:** Tue Jun 06, 2006 5:48 am

_________________
$\mathfrak{Radvan Marius}$ $aka$ $\mathfrak{darkdevil}$

1=2

Since week 7 has no solution posted yet, and it is overdue, I would like to turn this into a proof marathon.

Here's an easy one.

Prove that
$3a_{1}+3a_{2}+3a_{3}+\cdot\cdot \cdot+3a_{n}\geq6\sqrt[n]{a_{1}a_{2}a_{3}\cdot \cdot \cdot a_{n}}$