
Pessimist Singularitarian
#1
February 13th, 2018,
22:34
Hello, MHB Community!
anemone has asked for me to fill in for her while she's away on holiday.
Here is this week's POTW:
Find all quadruples $(a,b,c,d)$ of real numbers that simultaneously satisfy the following equations:$$\left\{\begin{array}{rcl}a^3+c^3 & = & 2 \\ a^2b+c^2d & = & 0 \\ b^3+d^3 & = & 1 \\ ab^2+cd^2 & = & 6 \end{array}\right.$$
Remember to read the to find out how to !

February 13th, 2018 22:34
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Pessimist Singularitarian
#2
February 16th, 2018,
15:30
Thread Author
Hello, MHB Community!
It has been brought to my attention that the problem posted this week has been used in the past, found here:
Problem of the week #277 Aug 28th, 2017
I apologize for that...here is the solution with which I was provided:
So, I am going to post another problem, from my old physics homework.
A uniform rod of mass $M$ and length $d$ rotates in a horizontal plane about a fixed, vertical, frictionless pin through its center. Two small beads, each of mass $m$, are mounted on the rod such that they are able to slide without friction along its length. Initially the beads are held by catches st positions $x$ (where $x<d/2$) on each side of the center, at which time the system rotates with an angular speed $\omega$.
Suddenly the catches are released and the small beads slide outward along the rod. Find:
 (a) the angular speed of the system at the instant the beads reach the ends of the rod
 (b) the angular speed of the rod after the beads fly off the ends
Remember to read the to find out how to !

Pessimist Singularitarian
#3
February 21st, 2018,
23:21
Thread Author
No one answered this weeks POTW...my solution follows: