{"problem":{"name":"Integer Convex Hull","description":{"content":"We have $N$ points $P_1, P_2, \\dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line. For a subset $S$ of ${ P_1, P_2, \\dots, P_N }$ with a","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc202_f"},"statements":[{"statement_type":"Markdown","content":"We have $N$ points $P_1, P_2, \\dots, P_N$ on a plane. The coordinates of $P_i$ is $(X_i, Y_i)$. We know that no three points lie on the same line.\nFor a subset $S$ of ${ P_1, P_2, \\dots, P_N }$ with at least three elements, let us define the **convex hull** of $S$ as follows:\n\n*   The convex hull is the convex polygon with the minimum area such that every point of $S$ is within or on the circumference of that polygon.\n\nFind the number, modulo $(10^9 + 7)$, of subsets $S$ such that the area of the convex hull is an integer.\n\n## Constraints\n\n*   $3 \\leq N \\leq 80$\n*   $0 \\leq |X_i|, |Y_i| \\leq 10^4$\n*   No three points lie on the same line.\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$X_1$ $Y_1$\n$X_2$ $Y_2$\n$\\vdots$\n$X_N$ $Y_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc202_f","tags":[],"sample_group":[["4\n0 0\n1 2\n0 1\n1 0","2\n\n${ P_1, P_2, P_4 }$ and ${ P_2, P_3, P_4 }$ satisfy the condition."],["23\n-5255 7890\n5823 7526\n5485 -113\n7302 5708\n9149 2722\n4904 -3918\n8566 -3267\n-3759 2474\n-7286 -1043\n-1230 1780\n3377 -7044\n-2596 -6003\n5813 -9452\n-9889 -7423\n2377 1811\n5351 4551\n-1354 -9611\n4244 1958\n8864 -9889\n507 -8923\n6948 -5016\n-6139 2769\n4103 9241","4060436"]],"created_at":"2026-03-03 11:01:14"}}