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
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
It is clear that each of a, b and c must be less than or equal to $10$. A brief inspection will show that the only combination of different square numbers which total $121$ is $81+36+4$.
More formally, the problem can be analysed by considering the remainders after dividing the square numbers less than $121$ $(1,4,9,16,25,36,49,64,81$ and $100)$ by three. The remainders are $(1,1,0,1,1,0,1,1,0$ and $1)$.
When $121$ is divided by $3$, the remainder is $1$. Therefore $a^2+b^2+c^2$ must also leave a remainder of $1$.
Now we can deduce that two of the three squares must leave a remainder of $0$ and so be multiples of $3$. There are three square numbers below $121$ which are multiples of three: $9, 36$ and $81$. Checking these, we see that $81$ and $36$ are the only pair to have a sum which differs from $121$ by a perfect square, namely $4$.
This problem is taken from the UKMT Mathematical Challenges.