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Golden Thoughts
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 golden ratio.
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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.]
Square Mean
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Age 14 to 16
Challenge Level
How Old Am I? printable worksheet
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"
or:
"In 4 years' time, my age will be the square of my age 4 years ago"
or:
"In 5 years' time, my age will be the square of my age 5 years ago"
or:
"In 6 years' time, my age will be the square of my age 6 years ago"
or...?
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.