2014 UNCO Math Contest II Problems/Problem 1

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Problem

The Duchess had a child on May 1st every two years until she had five children. This year the youngest is $1$ and the ages of the children are $1, 3, 5, 7$, and $9$. Alice notices that the sum of the ages is a perfect square: $1 + 3 + 5 + 7 + 9 = 25$. How old will the youngest be the next time the sum of the ages of the five children is a perfect square, and what is that perfect square?


Solution

We know that the sum of the ages is 25. To get another perfect square, we need to multiply it by a perfect square, so we get 100 for the next perfect square. We know their ages are (N-4)+(N-2)+(N)+(N+2)+(N+4)=100. Solving for N gives us 20, so the youngest is 20-4=16.

See also

2014 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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