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$0<b<a$

$a^{2}+b^{2}=6ab$

What is the value of $\dfrac{a-b}{a+b}$?

As shown above, three identical ellipses of the minor axis $2$ are positioned, such that each of them has four points of tangency with the regular hexagon at two opposite …

In the rectangular plane, let $A, B, C, D, E$ be points, such that

The sequence $\{a_1, a_2, a_3, \dots \}$ is defined by $a_{2n + 1} = (-1)^n$ and $a_{2n} = a_n$. A point $P$ moves on the coordinate plane as follows.

$2020, \lfloor \alpha \rfloor, \lfloor \beta \rfloor, 4040, \lfloor 2\alpha \rfloor, \lfloor 2\beta\rfloor, 6060, \lfloor 3\alpha \rfloor, \lfloor 3\beta\rfloor, \dots$

Do there exist two positive real numbers $\alpha$ and $\beta$, …

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