The first few terms of the Fibonacci sequence are as follows: \[\begin{align} F_1 = 1,&\, F_2 = 1,\, F_3 = 2,\, F_4 = 3,\,\\\\ {\color{Red}F_5 = 5},&\, F_6 = 8,\, {\color{Red}F_ 7 = 13},\, F_8 = 21, {\color{Red}F_9 = 34},\, … \end{align}\] It just so happens that \(5\), \(13\), and \(34\) are also the hypotenuses of right triangles with all integer sides: \[(3, 4, {\color{Red}5}),\ (5, 12, {\color{Red}13}),\ (16, 30, {\color{Red}34}).\] 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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\[\begin{align} S&=\sqrt{x+2}+\sqrt{y+5}+\sqrt{z+10}\\\\ T&=\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1} \end{align}\]
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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