Circumscribed Quadrilateral
A Quadrilateral is called "good" if it is convex and it's four sidelengths are \(3\), \(4\), \(5\) and \(6\) (but not necessarily in that order). Let \(A\) be the diameter of the smallest circle such that we can fit a good quadrilateral completely inside the circle. Let \(B\) be the diameter of the largest circle such that we can fit a good quadrilateral completely around the circle.
| Part A | What are the first \(2\) digits after the decimal point of \(A\)? | |
|---|---|---|
| Part B | What are the first \(8\) digits after the decimal point of \(B\)? |
House Numbers
Ross live's on a road with \(N\) houses numbered from \(1\) to \(N\). Amazingly, the sum of all the house numbers before his house is equal to the sum of all the house numbers after his house. So for example, it is possible that \(N=8\) if Ross lived in house number \(6\) because \(1+2+3+4+5 = 7+8\).
| Part A | If \(N>10\), what is the smallest possible value of \(N\)? | |
|---|---|---|
| Part B | If \(N>10000\) what is the smallest possible value of \(N\)? | |
| Part C | If \(N>10^9\) what is the smallest possible value of \(N\)? |