| You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his yard, which can be represented on a Cartesian plane. | |
| Boss Rob has a primary well at point \((A_x, A_y)\) and a backup well at a different point \((B_x, B_y)\), each able to water trees within an \(R\) unit radius. Using \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\) (unknown integers to you), Rob plants \(N\) happy little trees at real number points obtained by \(N\) calls to the function: | |
| ``` | |
| def gen_one_tree(A_x, A_y, B_x, B_y, R): | |
| while True: | |
| r = random.uniform(0, R) | |
| theta = random.uniform(0, 2*math.pi) | |
| x = A_x + r*math.cos(theta) | |
| y = A_y + r*math.sin(theta) | |
| if (x - B_x)**2 + (y - B_y)**2 <= R*R: | |
| return (x, y) | |
| ``` | |
| Here, `random.uniform(L, H)` returns a real number in \([L, H)\) uniformly at random. | |
| In other words, he picks a point \((x, y)\) in the circular range of the primary well using the special method above. If \((x, y)\) happens to be in range of the backup well, he plants a tree there (else he discards it and tries again with a new \((x, y)\)). This repeats until Rob has planted \(N\) trees. | |
| Given only the planted tree coordinates \((X_1, Y_1), \ldots, (X_N, Y_N)\), you are tasked to predict the exact values of \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\). As you are new, Boss Rob will accept your solution if it correctly predicts at least \(80\%\) of the test cases. | |
| # Constraints | |
| \(1 \le T \le 1{,}000\) | |
| \(500 \le N \le 1{,}000{,}000\) | |
| \(0 \le A_x, A_y, B_x, B_y \le 50\) | |
| \((A_x, A_y) \ne (B_x, B_y)\) | |
| \(1 \le R \le 50\) | |
| The sum of \(N\) across all test cases is at most \(2{,}000{,}000\). | |
| The intersection area of the two circular regions is strictly positive. | |
| Tree coordinates in the data were truly generated using the randomized algorithm as described above. The secret parameters \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\) have also been chosen uniformly at random for each case (rejecting cases where the circles are identical or do not have positive overlap). | |
| # Input Format | |
| Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\), the number of planted trees. Then, \(N\) lines follow, the \(i\)th of which contains two space-separated real numbers \(X_i\) and \(Y_i\), each given to \(6\) decimal places. | |
| # Output Format | |
| For the \(i\)th test case, print a line containing `"Case #i: "`, followed by the five space-separated integers \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\), in that order. | |
| # Sample Explanation | |
| The first sample case is pictured below, with the primary well's range in red, the backup well's range in blue, and the \(500\) randomly-generated trees in green: | |
| {{PHOTO_ID:6502772429739994|WIDTH:700}} | |