Forget the condition thatHow can we have the example of? Didn't you say
needed to be integer?
does that mean that m must be an integer?Forget the condition that
What do you mean by the highest power of p that divides k? Why wouldn't we just have that beNo, I'm saying
I didn't worry too much about the fine details since it's a 4U thread, but I'll clarify:What do you mean by the highest power of p that divides k? Why wouldn't we just have that be, for
?
And what do we mean by 'divides k' if k isn't an integer?
wait, now youre saying that m is an integer?I didn't worry too much about the fine details since it's a 4U thread, but I'll clarify:
The domain of the function is the positive rationals.
The function can only be calculated directly if
Alsois the highest integer power of
that divides
Also, what was the meaning ofetc., since k may not be a prime number, and we defined the function for a prime number in the subscript?
No, I mean in that property of the function, you gave likeas the highest power of 2 that goes into 12 is 2 (2^2=4|12)
You only need prime values ofNo, I mean in that property of the function, you gave like, what is
's meaning here, if
given that i have interpreted the question correctly, d1= {1,3,6,8} results in dn = 2 if n even or 6 if n odd. d1{2 and 7} results in dn=6 for even n and 2 for odd n. d1{4,5,9} have dn =0 for all n>1.Not too difficult (but better suited for this thread than the other):
Ah sorry, mistyped the question, it should be, Around a spherical planet, there are 37 satellites. Prove that, there exists a point on the surface such that at most 17 satellites are visible
Wouldn't it matter how close the satellites are orbiting this planet? Or am I just misunderstanding the question
.yeah that was the basic idea behind my method during the exam (proving that there had to be at least 38 points if the claim was false).Cool question, it's a shame there isn't more solid geometry or combinatorial geometry in the HS syllabus
Take the origin 0 to be the centre of the sphere.
If all of the satellites are co-linear, then the problem is trivial (any point on the sphere that also lies on the plane through 0 which is orthogonal to this line does not have vision of any of the satellites).
Suppose then, that x,y are two linearly independent satellites (ie 0,x,y form a non-degenerate triangle, and hence define a plane E.)
This plane intersects the sphere in a great circle that after rotation we can view as the equator of the sphere (just for ease of nomenclature).
Then we draw tangent planes N and S at the "north" and "south" poles, relative to to this equator.
An observer at the north pole can only see everything lying above N, and an observer at S can only see everything lying below S.
So if the claim were not true there would have to be at least 18 points above N and at least 18 points below S.
As these sets are disjoint, there can be at most 1 point between the planes N and S.
But x and y both lie between N and S...contradiction! (note that the planes E, S and N are all parallel.)