Heart Area
Let \(H\) denote the area of a heart with equation \( x^2+y^2 = |x|y + 1\). Your task is to accurately compute the first few digits of \(H\). For example the \(1^{\mbox{st}}\) digit of \(H\) after the decimal point is \(6\).| Part A | What are the \(2^{\mbox{nd}}\) to \(7^{\mbox{th}}\) digits of \(H\) after the decimal point? | |
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| Part B | What are the \(2021^{\mbox{st}}\) to \(2029^{\mbox{th}}\) digits of \(H\) after the decimal point? |
Ross Says
The sentences Ross can say are comprised of any combination of the following phrases: "awesome", "very good", and "well done". For each positive integer \(n\) let \(R(n)\) denote the number of sentences that Ross can say, which have exactly \(n\) words. For example Ross can say \(5\) different sentences consisting of \(3\) words("awesome awesome awesome", "awesome very good", "awesome well done", "very good awesome", and "well done awesome") so \(R(3) = 5\).| Part A | What is \(R(10)\) | |
|---|---|---|
| Part B | What is the remainder when \(R(2021)\) is divided by \((10^9+7)\)? | |
| Part C | What is the remainder when \(R\left(2021^{2021}\right)\) is divided by \((10^9+7)\)? |