Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
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+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | |||
+ | draw(Circle((0,0),1)); | ||
+ | draw((0,0)--dir(75)); | ||
+ | draw((1.5,0)--(-1.5,0),grey); | ||
+ | draw((0,1.5)--(0,-1.5),grey); | ||
+ | markscalefactor=0.01; | ||
+ | draw(anglemark(dir(0),(0,0),dir(75))); | ||
+ | label("$\theta$",0.07*dir(37.5),dir(37.5)); | ||
+ | draw(dir(180)--dir(75)--dir(0)); | ||
+ | label("$P$",dir(75),dir(75)); | ||
+ | label("$A$",dir(0),dir(0)); | ||
+ | label("$B$",dir(180),dir(180));</asy> |
Revision as of 15:26, 4 May 2014
Contents
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1
Picture 2
physics problem
Solution
inscribed triangle
moar images
yay