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
This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.
“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.
Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”
I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.
See the Fireworks Rocket for solutions.
This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) hearkens back to basic physics.
“A boy drops a stone down a well and hears the splash from the bottom after three seconds. Given that sound travels at a constant speed of 300 m/s and the acceleration of the stone due to gravity is 10 m/s2, how deep is the well?”
See the Falling Sound Problem for solutions.
This math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) is a bit more challenging.
“In the diagram, AB represents the diameter, C lies on the circumference of the circle, and you are given that
(Area of Circle) / (Area of Triangle) = 2π.
Prove that the two smaller angles in the triangle are exactly 15° and 75° respectively.”
See the 15 Degree Triangle Puzzle
The craziness of manipulating radicals strikes again. This 2006 four-star problem from Colin Hughes at Maths Challenge is really astonishing, though it takes the right key to unlock it.
“Problem Consider the following sequence:
For which values of [positive integer] n is S(n) rational?”
See Amazing Radical Sum for a solution.
This 2005 four-star problem from Colin Hughes at Maths Challenge is also a bit challenging.
“Problem
For any set of real numbers, R = {x, y, z}, let sum of pairwise products,
________________S = xy + xz + yz.
Given that x + y + z = 1, prove that S ≤ 1/3.”
Again, I took a different approach from Maths Challenge, whose solution began with an unexplained premise.
See the Pairwise Products
This is another stimulating math problem from Colin Hughes’s Maths Challenge website (mathschallenge.net).
“Problem
Find the exact value of the following infinite series:
1/2! + 2/3! + 3/4! + 4/5! + …”
See the Unexpected Sum for solutions.
Here is yet another surprising result from Colin Hughes at Maths Challenge.
“Problem
It can be shown that a unique circle passes through three given points. In triangle ABC three points A’, B’, and C’ lie on the edges opposite A, B, and C respectively. Given that the circle AB’C’ intersects circle BA’C’ inside the triangle at point P, prove that circle CA’B’ will be concurrent with P.”
I have to admit it took me a while to arrive at the final version of my proof. My original approach had some complicated expressions using various angles, and then I realized I had not used one of my assumptions. Once I did, all the complications faded away and the result became clear.
This problem from Colin Hughes at Maths Challenge is a most surprising result that takes a bit of tinkering to solve.
“Problem
We can see that 3 x 4 x 5 x 6 = 360 = 19² – 1. Prove that the product of four consecutive integers is always one less than a perfect square.”
The result is so mysterious at first that you begin to understand why the ancient Pythagoreans had a mystical relationship with mathematics.
See the Consecutive Product Square.
(Update 11/12/2020) Generalization and Visual Proof
Continue reading
This 2007 four-star problem from Colin Hughes at Maths Challenge is definitely a bit challenging.
“Problem
For any positive integer, k, let Sk = {x1, x2, … , xn} be the set of [non-negative] real numbers for which x1 + x2 + … + xn = k and P = x1 x2 … xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.
Prove that P is maximised when all the elements of S are equal in value and rational.”
I took a different approach from Maths Challenge, but for me, it did not rely on remembering a somewhat obscure formula. (I don’t remember formulas well at my age—only procedures, processes, or proofs, which is ironic, since at a younger age it was just the opposite.) It is also clear from the Maths Challenge solution that the numbers were assumed to be non-negative.
See Maximum Product.
