Source: CSMO-2004-P4
Category: Extremal Problems

Problem: Given an integer $n > 2004$. Fill the numbers $1, 2, \\cdots, n^2$ into an $n \\times n$ grid, with one number in each cell. If a cell contains a number that is greater than at least $2004$ numbers in its row and greater than at least $2004$ numbers in its column, then the cell is called an \"excellent cell\". Find the maximum possible number of excellent cells.


Answer: n(n-2004)

Reference solution:
The maximum possible number of excellent cells is $n(n-2004)$.

By evaluating row constraints, the $2004$ smallest numbers in any given row cannot form excellent cells, restricting each row to at most $n-2004$ excellent cells and establishing the total upper bound of $n(n-2004)$.

Applying the optimal assignment directly: for $1 \le i \le n$, mark cells in columns $i, i+1, \dots, i+2003 \pmod n$ of row $i$ with $\ast$. Allocating the numbers $1, 2, \dots, 2004n$ to the $\ast$ cells and the remaining larger numbers to the unmarked cells guarantees that all unmarked cells strictly exceed the required threshold of $2004$ elements in both their respective rows and columns. This configuration achieves exactly $n(n-2004)$ excellent cells.

Grading guidelines:
(Partial) 1. Proved the per-row upper bound: in any row, the 2004 smallest entries cannot be excellent, hence each row has at most n-2004 excellent cells (or an equivalent counting argument giving the same bound). 2. Proved the per-column upper bound analogously (each column has at most n-2004 excellent cells), and used it to justify an O(n^2) global upper bound (even if not simplified to n(n-2004)). 3. Gave a plausible construction scheme aiming for n(n-2004) excellent cells (e.g. selecting 2004 “small” positions in each row/column and placing small numbers there), but did not fully verify the column condition or the strict “>2004” requirement. (Almost) 1. Correctly proved the upper bound n(n-2004), but the construction is only sketched or has a small gap (e.g. not fully justifying that every non-star cell has at least 2004 smaller numbers in its column, or mishandling the cyclic shift argument). 2. Gave the cyclic-star construction (2004 marked cells per row, shifted so each column has exactly 2004 marked cells) and the intended number placement, but made a localized mistake such as using 2004 vs 2003 shifts, or failing to check the boundary case n=2005 / the strictness of inequalities; overall method would work after minor fixes. 3. Achieved the value n(n-2004) with a nearly complete proof, but left one nontrivial verification step insufficiently justified (typically: why all unmarked entries are simultaneously among the largest n-2004 in their row and among the largest n-2004 in their column).

Construction task:
For the specific case $n = 35$ and using a threshold of $24$ instead of $2004$ (i.e., a cell is "excellent" if its value is strictly greater than at least $24$ numbers in its row and strictly greater than at least $24$ numbers in its column), provide an explicit construction of the $35 \times 35$ grid that maximizes the number of excellent cells.

Construction requirements:
- Output a single Python list of lists representing the $35 \times 35$ grid.
- The grid must contain every integer from $1$ to $1225$ exactly once.
- The number of excellent cells in your grid must be the maximum possible for $n=35$ and threshold $24$.
- Put ONLY the Python literal (the list of lists) inside the construction block.

Reference construction:
[
  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851],
  [852, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862],
  [863, 864, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 865, 866, 867, 868, 869, 870, 871, 872, 873],
  [874, 875, 876, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 877, 878, 879, 880, 881, 882, 883, 884],
  [885, 886, 887, 888, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 889, 890, 891, 892, 893, 894, 895],
  [896, 897, 898, 899, 900, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 901, 902, 903, 904, 905, 906],
  [907, 908, 909, 910, 911, 912, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 913, 914, 915, 916, 917],
  [918, 919, 920, 921, 922, 923, 924, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 925, 926, 927, 928],
  [929, 930, 931, 932, 933, 934, 935, 936, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 937, 938, 939],
  [940, 941, 942, 943, 944, 945, 946, 947, 948, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 949, 950],
  [951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 961],
  [962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288],
  [289, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312],
  [313, 314, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336],
  [337, 338, 339, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360],
  [361, 362, 363, 364, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384],
  [385, 386, 387, 388, 389, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408],
  [409, 410, 411, 412, 413, 414, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432],
  [433, 434, 435, 436, 437, 438, 439, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456],
  [457, 458, 459, 460, 461, 462, 463, 464, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480],
  [481, 482, 483, 484, 485, 486, 487, 488, 489, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504],
  [505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 1072, 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528],
  [529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1090, 1091, 1092, 1093, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552],
  [553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576],
  [577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600],
  [601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624],
  [625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 640, 641, 642, 643, 644, 645, 646, 647, 648],
  [649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 665, 666, 667, 668, 669, 670, 671, 672],
  [673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 690, 691, 692, 693, 694, 695, 696],
  [697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1170, 715, 716, 717, 718, 719, 720],
  [721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 740, 741, 742, 743, 744],
  [745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 1182, 1183, 1184, 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 765, 766, 767, 768],
  [769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200, 1201, 1202, 1203, 790, 791, 792],
  [793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 1204, 1205, 1206, 1207, 1208, 1209, 1210, 1211, 1212, 1213, 1214, 815, 816],
  [817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 1215, 1216, 1217, 1218, 1219, 1220, 1221, 1222, 1223, 1224, 1225, 840]
]

Verifier code:
import sys
import ast

def verify():
    input_text = sys.stdin.read().strip()
    if not input_text:
        print("Empty input.")
        return
        
    try:
        grid = ast.literal_eval(input_text)
    except Exception as e:
        print(f"Parse error: {e}")
        return
        
    if not isinstance(grid, list):
        print("The grid must be a list of lists.")
        return
        
    if len(grid) != 35:
        print(f"Expected exactly 35 rows, got {len(grid)}.")
        return
        
    elements = []
    for i, row in enumerate(grid):
        if not isinstance(row, list):
            print(f"Row {i} is not a list.")
            return
        if len(row) != 35:
            print(f"Row {i} must have exactly 35 elements, got {len(row)}.")
            return
        for j, val in enumerate(row):
            if type(val) is not int:
                print(f"Element at row {i}, col {j} is not an integer.")
                return
            elements.append(val)
            
    if sorted(elements) != list(range(1, 1226)):
        print("The grid must contain every integer from 1 to 1225 exactly once.")
        return
        
    excellent_count = 0
    for i in range(35):
        for j in range(35):
            val = grid[i][j]
            # Count strictly smaller numbers in the same row
            row_smaller = sum(1 for c in range(35) if grid[i][c] < val)
            # Count strictly smaller numbers in the same column
            col_smaller = sum(1 for r in range(35) if grid[r][j] < val)
            
            if row_smaller >= 24 and col_smaller >= 24:
                excellent_count += 1
                
    if excellent_count != 385:
        print(f"Expected exactly 385 excellent cells, but found {excellent_count}.")
        return
        
    print("True")

if __name__ == "__main__":
    verify()