# Problems of the Week

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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?

How many ways can you arrange the numbers 1 through 9 in a $$3 \times 3$$ grid such that the following conditions hold?

• Every number is greater than the number directly above it.
• Every number is greater than the number immediately to the left of it.

Inspired by this problem.

\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 $$S^{2}-T^{2}?$$

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 $$a \times b \times c \times d$$.

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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