Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
(→Solution 2) |
(→Picture 1) |
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</asy> | </asy> | ||
<cmath>\text{Find the probability that }b>a \text{.}</cmath> | <cmath>\text{Find the probability that }b>a \text{.}</cmath> | ||
+ | ==Picture 2== | ||
+ | <asy> | ||
+ | for (int i=0;i<6;i=i+1){ | ||
+ | draw(dir(60*i)--dir(60*i+60)); | ||
+ | } | ||
+ | draw(dir(120)--(dir(0)+dir(-60))/2); | ||
+ | draw(dir(180)--(dir(60)+dir(0))/2); | ||
+ | fill(dir(120)--dir(180)--intersectionpoint(dir(120)--(dir(0)+dir(-60))/2,dir(180)--(dir(60)+dir(0))/2)--cycle,grey); | ||
+ | fill((dir(0)+dir(-60))/2--dir(0)--(dir(60)+dir(0))/2--intersectionpoint(dir(120)--(dir(0)+dir(-60))/2,dir(180)--(dir(60)+dir(0))/2)--cycle,grey); | ||
+ | </asy> | ||
+ | <cmath>\text{Prove the shaded areas are equal.}</cmath> |
Revision as of 21:25, 19 October 2013
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