View source for Pre-Olympiad Level Tournament By Mathtime

=Problem 1=

[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=456734 <math>\boxed{1}</math>]

Suppose we have a sequence, with the first term equal to <math>a_1</math>, with <math>a_1>0</math>, an a second term of <math>3</math> and each term after that, <math>a_n</math> equal to <math>F_{a_{n-1}*a_{n-2}}</math>, which is the <math>a_{n-1}*a_{n-2}</math>'th Fibonacci number.  Assume that <math>a_k</math> is always an integer in this problem, and that <math>k</math> must always be an integer in this problem.

Find (with proof) all integers <math>a_1</math>, such that this sequence has the integer <math>832040</math> in it.

=Problem 2=
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=456747 <math>\boxed{2}</math>]

In a cyclic quadrilateral with sides <math>AB, BC, CD, AC</math> prove that:
<math>(AB+BC+CD+AD)^4\ge 64(-(AB)(CD)-(BC)(AD)+2(AC)(BD))^2-64((AB)^2(CD)^2-2(AB)(BC)(CD)(AD)+(BC)^2(AD)^2)</math>

=Problem 3=
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=456748 <math>\boxed{3}</math>]

Prove that there is no positive integers <math>k</math> such that <math>\frac{(x^2+y^2+xz)+x^m*y^z}{x^2-y^2}=3+kx+ky\ \forall x,y \in \mathbb{N}</math>, and this equation must satify for all <math>x</math> and <math>y</math>, and <math>m</math> and <math>z</math> are positive integers.