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Difference between revisions of "2011 UNCO Math Contest II Problems/Problem 4"

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== Solution ==
 
== Solution ==
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The smallest number in both sets is 101. This is because it is both 1^2+100 and 10^2+1. 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 15^2+100 and 18^2+1. There is only one left, which is 2501, because it is 49^2+100 and 50^2+1.
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The smallest number in both sets is 101. This is because it is both <math>1^2+100</math> and <math>10^2+1</math>. 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 <math>15^2+100</math> and <math>18^2+1</math>. There is only one left, which is 2501, because it is <math>49^2+100</math> and <math>50^2+1</math>.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 21:00, 28 August 2015

Problem

Let $A = \left\{ 2,5,10,17,\cdots,n^2+1,\cdots\right\}$ be the set of all positive squares plus $1$ and $B = \left\{101, 104, 109, 116,\cdots,m^2 + 100,\cdots\right\}$ be the set of all positive squares plus $100$.

(a) What is the smallest number in both $A$ and $B$?

(b) Find all numbers that are in both $A$ and $B$.


Solution

The smallest number in both sets is 101. This is because it is both $1^2+100$ and $10^2+1$. 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 $15^2+100$ and $18^2+1$. There is only one left, which is 2501, because it is $49^2+100$ and $50^2+1$.

See Also

2011 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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