Recreational Problems (SM level)

[Ahmad]

Don't be discouraged, many of the problems here are difficult. Many are also very different to what is typically expected of you at school or university. The important thing is to give some of them a go and have fun doing so!

[bigtick]

made two arith errors, hope right this time.
max area 40.33
a3=5.7877+2.8916i
a4=1.8916+6.7877i

[dcc]

made two arith errors, hope right this time.
max area 40.33
a3=5.7877+2.8916i
a4=1.8916+6.7877i

What method did you use to attain this answer?

[bigtick]

let others try

[zzdfa]

q8

1.5


lim -> infinity = 1.5

p.s. if you could put solutions (or just the answers) in spoiler tags in the first post that'd be great, so we don't have to go through 15 pages+  to find out if we were correct

edit: learning LaTeX =D

[Ahmad]

I added a few links to some solutions by Coblin.

[Ahmad]

Here's a cute problem:

[enwiabe]

aww red x's are adorable

[kamil9876]

I saw on artofproblemsolving that problem 38 helps you solve this one, and that british flag theorem is of help here.
After reading those clues I came up with this method:

From the corner of each square connect a line to the centre of the circle. Connect a line from centre to point of contact between circle and tangent, note the 90 degree angle between tangent and this radius. Now express the lines that go from corner of square to centre in terms of radius, r, and the "known length". Example, when considering the top left corner, sqrt(2) is known length. Using the British flag theorem this yields:



Which gives B.

Never wouldve come up with this if i didn't know problem 38 nor the british flag theorem, hence this was more of a 'needle in a haystack' kind of problem which i don't find as 'cute' as some of your other problems. Or maybe I'm just not clever enough 

[Ahmad]

Well done. And yeah, hang out on AoPS long enough and you'll have seen power of a point and british flag theorem used dozens of times, although they aren't really necessary.

[kamil9876]

How did you do it? Did u merely derive the theorem as merely an intermediate step without explicitly knowing it before?

[bigtick]

[/0]

Solve:    , 


Hint:
Either use De Moivre's or google a nice formula

[Over9000]

By inspection is a solution.

[kamil9876]

Hint for another method: Try to look at the problem as finding a vector sum. Actually demoivre's theorem method is better