2011 UNCO Math Contest II Problems/Problem 4
Let be the set of all positive squares plus and be the set of all positive squares plus .
(a) What is the smallest number in both and ?
(b) Find all numbers that are in both and .
The smallest number in both sets is 101. This is because it is both and . Subtracting 10-1 gives us 9. This means that all other numbers in both sets must be less than 9 apart, because squares get further apart the bigger they get. Following a geometric sequence gives us 9,3,1 so the squares must be that far apart. This means the next term is 325, because it is and . There is only one left, which is 2501, because it is and .
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