2018-09-17 Advanced Practice Problems Online

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?

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.

$\begin{aligned} S&=\sqrt{x+2}+\sqrt{y+5}+\sqrt{z+10}\\\\ T&=\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1} \end{aligned}$

If the above is true for $x, y, z >0,$ what is the minimum value of $S^{2}-T^{2}?$

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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