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Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

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Relative Powers

Weekly Problem 10 - 2007
The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?

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Summing Squares

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Square Sum

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2
The mean of the nine consecutive integers is the fifth number in the series, so their sum is nine times the fifth number. As nine is itself a perfect square, the sum will be a perfect square if and only if the fifth number is also a perfect square. For the options given, the fifth numbers would be 114, 124, 134, 144, 154 respectively. Of these, only 144 is a perfect square, so the answer is D.

This problem is taken from the UKMT Mathematical Challenges.
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