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.
This is another intriguing problem from Presh Talwalkar.
This interesting problem comes from Colin Hughes at the Maths Challenge website.
A glutton for punishment I considered another Sam Loyd puzzle:
This is from the UKMT Senior Challenge of 2004.
This statement showed up recently at 
In my search for new problems I came across this one from Martin Gardner:
This is a delightful little problem from Dick Hess that exercises one’s basic facility with logarithms:
I was reading yet another book on the Scientific Revolution when I came across a discussion of the mathematical significance of the invention of perspective for painting in the 15th century Italian Renaissance. The main player in the saga was Leon Battista Alberti (1404 – 1472) and his tome De Pictura (On Painting) (1435-6), which contained the first mathematical presentation of perspective. Even though mathematics was advertised, it was not at the level of trigonometry I used in my post “
How many tiles are there in the complete pattern?
