Pre-Olympiad Level Tournament By Mathtime

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Problem 1

$\boxed{1}$

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

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

Problem 2

$\boxed{2}$

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

Problem 3

$\boxed{3}$

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

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