Annabel, Belinda, Carmen and Davina went shopping during the new year sales.
"What did you buy?" Belinda asked Annabel at the end of the day.
"Handbags. They were a real bargain! Each one cost me ten times as many pounds as the total number of handbags I bought."
"That's a coincidence," said Belinda. "I bought shoes, and each pair cost me ten times as many pounds as the total number of pairs I bought."
"Me too," said Carmen. "I bought dresses, and each one cost me ten times as many pounds as the total number of dresses I bought."
"The same goes for me," said Davina. "I bought necklaces, and each one cost me ten times as many pounds as the total number of necklaces I bought."
Comparing prices, they discovered that Davina had spent the most, followed by Carmen, then Belinda, and finally Annabel.
"How much did we spend?" asked Davina.
"Well, two of us spent £2,210 between them," Belinda replied. "And, remarkably, so did the other two."
How much did Carmen spend?
Scroll down for the answer
The answer
Carmen spent £1,210.
Annabel bought a items at £10a each, a total of 10a2.
Similarly the others spent 10b2, 10c 2, and 10d2.
Since a < b < c < d, we must have 10a2+10d2 =2210
and 10b2+10c2 = 2210.
That is, a2+d2 = 221 and b2+c2 = 221.
There are two different ways to write 221 as a sum of two squares:
52+142 and 102+112.
Since a< b < c < d, we have a = 5, b = 10, c = 11, d = 14.
Carmen bought 11 dresses at £110 each, making £1,210.
The winner is Jolyon Kay from Cyprus
