This is a fairly challenging problem from BL Math Games.
“Find the volume of the solid obtained by rotating the region enclosed by y = x and y = x2 about the line y = x.”
See Slanted Volume of Revolution for solutions.
This is a Valentine’s Day puzzle from BL’s (Barry Leung) Math Games.
“Happy Valentine’s Day everyone, I hope you are having a euphoric moment, but if not, you can try this algebra puzzle about maximizing the expression LUV + LU + UV + LV given L + U + V = 12, where L, U, V are non-negative integers.”
See Maximizing Love for a solution.
This is another challenging sum from the 2024 Math Calendar.
“Find x where x = et and
”
As before, recall that all the answers are integer days of the month.
See Yet Another Sum for a solution.
Here is another problem from BL’s Weekly Math Games.
“For every point P on y = 2x2, areas A and B are equal. Find the equation for curve C.”
See Mystery Curve Puzzle for a solution
A while ago James Tanton provided a series of puzzles:
Puzzle #1 At what value between 0 and 1 does a horizontal line at that height produce two regions of equal area as shown on the graph of y = x2?
Puzzle #2 A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = x2 into two regions as shown. At what height should that line be drawn so that the sum of the areas of these two regions is minimal?
Puzzle #3 A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = xn into two regions as shown (n > 0). At what height should that line be drawn so that the sum of the areas of these two regions is minimal? Does that height depend on the value of n?
Puzzle #4 What horizontal line drawn between y = 0 and y = 1 on the graph of y = 2√x – 1 minimizes the sum of the two shaded areas shown?
See Double Areas Puzzles for solutions.
This is another puzzle from BL’s Weekly Math Games.
“a + b + c = 2, and
a2 + b2 + c2 = 12
where a, b, and c are real numbers. What is the difference between the maximum and minimum possible values of c?”
The original problem statement mentioned a fourth real number d, but I considered it a typo, since it was not involved in the problem.
See Sphere and Plane Puzzle for a solution.
I came across this remarkable result in Futility Closet:
“On each of these two black lines is a trio of red points marked by the same distances. The midpoints of segments drawn between corresponding points are collinear.
(Discovered by Danish mathematician Johannes Hjelmslev.)”
This result seems amazing and mysterious. I wondered if I could think of a proof. I found a simple approach that did not use plane geometry. And suddenly, like a magic trick exposed, the result seemed obvious.
Well, I discovered that the 2024 Math Calendar has some interesting problems, so I guess things will limp along for a while. This is a challenging but imaginative problem from the calendar.
_______________
As before, recall that all the answers are integer days of the month.
See the Amazing Root Problem for a solution.
This problem is from Colin Hughes’s Maths Challenge website (mathschallenge.net).
“Four corners measuring x by x are removed from a sheet of material that measures a by a to make a square based open-top box. Prove that the volume of the box is maximised iff the area of the base is equal to the area of the four sides.”
See the Maximized Box Problem
Since the changes in Twitter (now X), I have not been able to see the posts, not being a subscriber. But I noticed poking around that some twitter accounts were still viewable. However, like some demented aging octogenarian they had lost track of time, that is, instead of being sorted with the most recent post first, they showed a random scattering of posts from different times. So a current post could be right next to one several years ago. That is what I discovered with the now defunct MathsMonday site. I found a post from 10 May 2021 that I had not seen before, namely,
“The points A and B are on the curve y = x2 such that AOB is a right angle. What points A and B will give the smallest possible area for the triangle AOB?”
See the Pythagorean Parabola Puzzle for solution.
(Update 9/1/2023) Elegant Alternative Solution by Oscar Rojas
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