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2018-09-17 Advanced


The first few terms of the Fibonacci sequence are as follows: F1=1,F2=1,F3=2,F4=3,F5=5,F6=8,F7=13,F8=21,F9=34,\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 55, 1313, and 3434 are also the hypotenuses of right triangles with all integer sides: (3,4,5), (5,12,13), (16,30,34).(3, 4, {\color{#D61F06}5}),\ (5, 12, {\color{#D61F06}13}),\ (16, 30, {\color{#D61F06}34}). Is every Fibonacci number F2n1F_{2n - 1} for all n3n \geq 3 the hypotenuse of a right triangle with integer sides?

How many ways can you arrange the numbers 1 through 9 in a 3×33 \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.

S=x+2+y+5+z+10T=x+1+y+1+z+1\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,x, y, z >0, what is the minimum value of S2T2?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 r1r_1 is the radius of the green circles and r2r_2 is the radius of the blue circles, find the ratio r1r2,\frac{r_1}{r_2}, which can be expressed as a+bdc \frac{a+b\sqrt{d}} {c} , where a,b,ca, b, c are coprime integers and dd is a square-free integer.

Give the answer as the product a×b×c×d 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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