Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
On my last birthday, my friend said to me:
"In 15 years' time, your age will be the square of your age 15 years ago!"
Can you work out how old I am?
This got me thinking...
Was there ever a time in my life when I had other birthdays that were special in this way?
Could I have said:
"In 3 years' time, my age will be the square of my age 3 years ago"
"In 4 years' time, my age will be the square of my age 4 years ago"
"In 5 years' time, my age will be the square of my age 5 years ago"
"In 6 years' time, my age will be the square of my age 6 years ago"
Can you make any generalisations about which birthdays are special in this way?
Can you prove your findings?
Click here for a poster of this problem.