The first few terms of the Fibonacci sequence are as follows: $\begin{aligned} F_1 = 1,&\, F_2 = 1,\, F_3 = 2,\, F_4 = 3,\,\\\\ {\color{#D61F06}F_5 = 5},&\, F_6 = 8,\, {\color{#D61F06}F_ 7 = 13},\, F_8 = 21, {\color{#D61F06}F_9 = 34},\, … \end{aligned}$ It just so happens that $5$, $13$, and $34$ are also the hypotenuses of right triangles with all integer sides: $(3, 4, {\color{#D61F06}5}),\ (5, 12, {\color{#D61F06}13}),\ (16, 30, {\color{#D61F06}34}).$ Is every Fibonacci number $F_{2n - 1}$ for all $n \geq 3$ the hypotenuse of a right triangle with integer sides?
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How many ways can you arrange the numbers 1 through 9 in a $3 \times 3$ grid such that the following conditions hold?
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$\begin{aligned} S&=\sqrt{x+2}+\sqrt{y+5}+\sqrt{z+10}\\\\ T&=\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1} \end{aligned}$
If the above is true for $x, y, z >0,$ what is the minimum value of $S^{2}-T^{2}?$
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Four congruent green circles and two congruent blue circles are tangential to one another, to the red chord, and to the large black circle, as shown.
If $r_1$ is the radius of the green circles and $r_2$ is the radius of the blue circles, find the ratio $\frac{r_1}{r_2},$ which can be expressed as $\frac{a+b\sqrt{d}} {c}$, where $a, b, c$ are coprime integers and $d$ is a square-free integer.
Give the answer as the product $a \times b \times c \times d$.
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A soap bubble stretches between two identical unit rings (with radius 1).
What is the maximum separation between the two rings before the bubble pops?
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