[math-fun] Divisors of divisors
Given a multiset S, how can we determine whether there is an integer n such that each s in S corresponds to (is in bijection with) a divisor t of n such that d(t) = s? This came up recently when considering multisets with the property s_1^3 + s_2^3 + ... + s_k^3 = (s_1 + s_2 + s_3 + ... + s_k)^2. One infinite family are the multisets with this divisor-of-divisors property. Another is {n, n, ..., n} with n elements. Charles Greathouse Analyst/Programmer Case Western Reserve University
This was a problem posed to me by a friend: Let circle C have center O. Let point P satisfy OP = 4. Draw tangent line L to C at point Q. Place point R on L with QR = 12. Let segment PQ cut C at point S between P and Q, so that PS = 4 and QS = 8. What is the radius of C? I give up on this one. An algebraic solution would be nice, a geometric one better.
Hello David, do we need L and R ?! Best, É. Envoyé d'un aPhone Le 26 janv. 2013 à 16:30, "davidwwilson@comcast.net" <davidwwilson@comcast.net> a écrit :
This was a problem posed to me by a friend:
Let circle C have center O.
Let point P satisfy OP = 4.
Draw tangent line L to C at point Q. Place point R on L with QR = 12.
Let segment PQ cut C at point S between P and Q, so that PS = 4 and QS = 8.
What is the radius of C?
I give up on this one.
An algebraic solution would be nice, a geometric one better. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Furthermore, if PQ cuts the circle at S between P and Q, then the radius must inceed OP = 4. Therefore QS must inceed 8, and the data which are not irrelevant are contradictory. Still, never mind --- it must have been a good party! WFL On 1/26/13, Eric Angelini <Eric.Angelini@kntv.be> wrote:
Hello David, do we need L and R ?! Best, É.
Envoyé d'un aPhone
Le 26 janv. 2013 à 16:30, "davidwwilson@comcast.net" <davidwwilson@comcast.net> a écrit :
This was a problem posed to me by a friend:
Let circle C have center O.
Let point P satisfy OP = 4.
Draw tangent line L to C at point Q. Place point R on L with QR = 12.
Let segment PQ cut C at point S between P and Q, so that PS = 4 and QS = 8.
What is the radius of C?
I give up on this one.
An algebraic solution would be nice, a geometric one better. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Sorry, let's try again Let segment PR cut C at point S bete This was a problem posed to me by a friend: Let circle C have center O. Let point P satisfy OP = 4. Draw tangent line L to C at point Q. Place point R on L with QR = 12. Let segment PR cut C at point S so that RS = 8 and RP = 12. What is the radius of C? Perhaps I got it right this time.
participants (4)
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Charles Greathouse -
davidwwilson＠comcast.net -
Eric Angelini -
Fred lunnon