{"problem":{"name":"A. Okabe and Future Gadget Laboratory","description":{"content":"Okabe needs to renovate the _Future Gadget Laboratory_ after he tried doing some crazy experiments! The lab is represented as an _n_ by _n_ square grid of integers. A _good_ lab is defined as a lab in","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF821A"},"statements":[{"statement_type":"Markdown","content":"Okabe needs to renovate the _Future Gadget Laboratory_ after he tried doing some crazy experiments! The lab is represented as an _n_ by _n_ square grid of integers. A _good_ lab is defined as a lab in which every number not equal to 1 can be expressed as the sum of a number in the same row and a number in the same column. In other words, for every _x_, _y_ such that 1 ≤ _x_, _y_ ≤ _n_ and _a__x_, _y_ ≠ 1, there should exist two indices _s_ and _t_ so that _a__x_, _y_ = _a__x_, _s_ + _a__t_, _y_, where _a__i_, _j_ denotes the integer in _i_\\-th row and _j_\\-th column.\n\nHelp Okabe determine whether a given lab is _good_!\n\n## Input\n\nThe first line of input contains the integer _n_ (1 ≤ _n_ ≤ 50) — the size of the lab.\n\nThe next _n_ lines contain _n_ space-separated integers denoting a row of the grid. The _j_\\-th integer in the _i_\\-th row is _a__i_, _j_ (1 ≤ _a__i_, _j_ ≤ 105).\n\n## Output\n\nPrint \"_Yes_\" if the given lab is _good_ and \"_No_\" otherwise.\n\nYou can output each letter in upper or lower case.\n\n[samples]\n\n## Note\n\nIn the first sample test, the 6 in the bottom left corner is valid because it is the sum of the 2 above it and the 4 on the right. The same holds for every number not equal to 1 in this table, so the answer is \"_Yes_\".\n\nIn the second sample test, the 5 cannot be formed as the sum of an integer in the same row and an integer in the same column. Thus the answer is \"_No_\".","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"Okabe 需要在进行了一些疯狂的实验后翻新 _Future Gadget 实验室_！该实验室由一个 #cf_span[n] × #cf_span[n] 的整数方格表示。一个 _好_ 的实验室定义为：每个不等于 #cf_span[1] 的数都可以表示为同一行中的某个数与同一列中的某个数之和。换句话说，对于每个满足 #cf_span[1 ≤ x, y ≤ n] 且 #cf_span[ax, y ≠ 1] 的位置 #cf_span[x, y]，都应存在两个下标 #cf_span[s] 和 #cf_span[t]，使得 #cf_span[ax, y = ax, s + at, y]，其中 #cf_span[ai, j] 表示第 #cf_span[i] 行第 #cf_span[j] 列的整数。\n\n帮助 Okabe 判断给定的实验室是否为 _好_ 的！\n\n输入的第一行包含整数 #cf_span[n]（#cf_span[1 ≤ n ≤ 50]）——实验室的大小。\n\n接下来的 #cf_span[n] 行每行包含 #cf_span[n] 个用空格分隔的整数，表示网格的一行。第 #cf_span[i] 行的第 #cf_span[j] 个整数是 #cf_span[ai, j]（#cf_span[1 ≤ ai, j ≤ 105]）。\n\n如果给定的实验室是 _好_ 的，输出 \"_Yes_\"；否则输出 \"_No_\"。\n\n你可以以任意大小写输出每个字母。\n\n在第一个样例中，左下角的 #cf_span[6] 是合法的，因为它等于其上方的 #cf_span[2] 与右侧的 #cf_span[4] 之和。表中所有不等于 #cf_span[1] 的数都满足这一性质，因此答案为 \"_Yes_\"。\n\n在第二个样例中，#cf_span[5] 无法表示为同一行和同一列中任意两个整数的和，因此答案为 \"_No_\"。\n\n## Input\n\n输入的第一行包含整数 #cf_span[n]（#cf_span[1 ≤ n ≤ 50]）——实验室的大小。接下来的 #cf_span[n] 行每行包含 #cf_span[n] 个用空格分隔的整数，表示网格的一行。第 #cf_span[i] 行的第 #cf_span[j] 个整数是 #cf_span[ai, j]（#cf_span[1 ≤ ai, j ≤ 105]）。\n\n## Output\n\n如果给定的实验室是 _好_ 的，输出 \"_Yes_\"；否则输出 \"_No_\"。你可以以任意大小写输出每个字母。\n\n[samples]\n\n## Note\n\n在第一个样例中，左下角的 #cf_span[6] 是合法的，因为它等于其上方的 #cf_span[2] 与右侧的 #cf_span[4] 之和。表中所有不等于 #cf_span[1] 的数都满足这一性质，因此答案为 \"_Yes_\"。在第二个样例中，#cf_span[5] 无法表示为同一行和同一列中任意两个整数的和，因此答案为 \"_No_\"。","is_translate":true,"language":"Chinese"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ n \\in \\mathbb{Z} $ be the size of the grid.  \nLet $ A = (a_{i,j}) \\in \\mathbb{Z}^{n \\times n} $ be the grid, where $ a_{i,j} $ denotes the integer in row $ i $, column $ j $.\n\n**Constraints**  \n1. $ 1 \\leq n \\leq 50 $  \n2. $ 1 \\leq a_{i,j} \\leq 10^5 $ for all $ i, j \\in \\{1, \\dots, n\\} $\n\n**Objective**  \nDetermine whether for every cell $ (x, y) $ such that $ a_{x,y} \\neq 1 $, there exist indices $ s, t \\in \\{1, \\dots, n\\} $ satisfying:  \n$$\na_{x,y} = a_{x,s} + a_{t,y}\n$$  \nIf this holds for all such $ (x,y) $, output \"Yes\"; otherwise, output \"No\".","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF821A","tags":["implementation"],"sample_group":[["3\n1 1 2\n2 3 1\n6 4 1","Yes"],["3\n1 5 2\n1 1 1\n1 2 3","No"]],"created_at":"2026-03-03 11:00:39"}}