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The first few terms of the Fibonacci sequence are as follows: F1=1,F5=5,F2=1,F3=2,F4=3,F6=8,F7=13,F8=21,F9=34,… It just so happens that 5, 13, and 34 are also the hypotenuses of right triangles with all integer sides: (3,4,5), (5,12,13), (16,30,34). Is every Fibonacci number F2n−1 for all n≥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 3×3 grid such that the following conditions hold?
Inspired by this problem.
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ST=x+2+y+5+z+10=x+1+y+1+z+1
If the above is true for x,y,z>0, what is the minimum value of S2−T2?
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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 r1 is the radius of the green circles and r2 is the radius of the blue circles, find the ratio r2r1, which can be expressed as ca+bd, where a,b,c are coprime integers and d is a square-free integer.
Give the answer as the product a×b×c×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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