Almost Square Triangles
A perfect square is any integer in the form \(n^2\) where \(n\) is an integer. A triangle number is any integer in the form \(\frac{n(n+1)}{2}\) where \(n\) is an integer. An almost-square-triangle is any triangle number which is one less than a perfect square. For example: \(3\) is an almost-square-triangle because \(3\) is a triangle number and \(4\) is a perfect square. Let \(S(x)\) denote the sum of all almost-square-triangles less than \(x\). Since \(3\) and \(0\) are the only almost-square-triangles less than \(10\), we have \(S(10) = 3\).| Part A | What is \(S(1000)\)? | |
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| Part B | What is the remainder when \(S(10^{12})\) is divided by \((10^9+7)\)? | |
| Part C | What is the remainder when \(S(10^{1000})\) is divided by \((10^9+7)\)? |