{"problem":{"name":"Ex - Sum of Prod of Min","description":{"content":"You are given positive integers $N$ and $M$. Here, it is guaranteed that $N\\leq M \\leq 2N$. Print the sum, modulo $200\\ 003$ (a prime), of the following value over all sequences of positive integers $","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc279_h"},"statements":[{"statement_type":"Markdown","content":"You are given positive integers $N$ and $M$. Here, it is guaranteed that $N\\leq M \\leq 2N$.\nPrint the sum, modulo $200\\ 003$ (a prime), of the following value over all sequences of positive integers $S=(S_1,S_2,\\dots,S_N)$ such that $\\displaystyle \\sum_{i=1}^{N} S_i = M$ (notice the unusual modulo):\n\n*   $\\displaystyle \\prod_{k=1}^{N} \\min(k,S_k)$.\n\n## Constraints\n\n*   $1 \\leq N \\leq 10^{12}$\n*   $N \\leq M \\leq 2N$\n*   All values in the input are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$ $M$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc279_h","tags":[],"sample_group":[["3 5","14\n\nThere are six sequences $S$ that satisfy the condition: $S=(1,1,3), S=(1,2,2), S=(1,3,1), S=(2,1,2), S=(2,2,1), S=(3,1,1)$.\nThe value $\\displaystyle \\prod_{k=1}^{N} \\min(k,S_k)$ for each of those $S$ is as follows.\n\n*   $S=(1,1,3)$ : $1\\times 1 \\times 3 = 3$\n*   $S=(1,2,2)$ : $1\\times 2 \\times 2 = 4$\n*   $S=(1,3,1)$ : $1\\times 2 \\times 1 = 2$\n*   $S=(2,1,2)$ : $1\\times 1 \\times 2 = 2$\n*   $S=(2,2,1)$ : $1\\times 2 \\times 1 = 2$\n*   $S=(3,1,1)$ : $1\\times 1 \\times 1 = 1$\n\nThus, you should print their sum: $14$."],["1126 2022","40166\n\nPrint the sum modulo $200\\ 003$."],["1000000000000 1500000000000","180030"]],"created_at":"2026-03-03 11:01:13"}}