IBM Ponder This #289 · 2022 年 5 月

一个整数集合 $A$ 有 $2n$ 个元素，已知总存在大小为 $n$ 的 $B\subseteq A$，使 $\sum _{b\in B}b$ 被 $n$ 整除。

例如 $n=7$、$A=[70,18,15,74,71,40,92,54,23,82,24,6,73,62]$ 时，可选 $B=[71,82,24,62,54,23,6]$，和为 $322=7\times 46$。

现在对 $B$ 加入限制：若 $A=[a_1,a_2,\dots ,a_{2n}]$，则 $B$ 必须从每对 $(a_{2i-1},a_{2i})$ 中恰选一个，$1\le i\le n$。原例中的 $B$ 不合法，因为第九、十个元素 23 与 82，即 $i=5$ 这一对，都在 $B$ 中；另一组合法解为 $B=[70,74,71,92,23,24,73]$。

任务：对下列 A，找出满足大小、整除条件以及每对选一限制的子集：

[194, 236, 291, 430, 1, 721, 486, 819, 2, 44, 196, 238, 293, 529, 584, 45, 197, 433, 101, 46, 586, 337, 5, 725, 102, 822, 781, 144, 6, 920, 394, 242, 976, 533, 492, 244, 493, 632, 784, 729, 106, 536, 591, 149, 10, 537, 204, 634, 689, 150, 399, 538, 690, 926, 884, 54, 206, 539, 400, 831, 207, 735, 401, 832, 498, 929, 789, 251, 111, 542, 208, 639, 305, 736, 112, 930, 500, 349, 791, 931, 888, 156, 113, 253, 501, 933, 695, 62, 503, 159, 891, 838, 116, 935, 989, 742, 505, 937, 699, 841, 893, 551, 894, 842, 22, 67, 119, 358, 313, 843, 604, 359, 895, 553, 217, 747, 702, 941, 993, 457, 121, 70, 218, 555, 800, 168, 25, 362, 26, 653, 414, 363, 125, 753, 29, 949, 514, 950, 805, 78, 999, 757, 418, 467, 31, 80, 128, 468, 613, 857, 807, 276, 323, 761, 227, 858, 712, 277, 325, 374, 616, 665, 810, 763, 132, 86, 36, 474, 424, 862, 522, 961, 135, 89, 426, 186, 39, 672, 233, 866, 330, 770, 815, 287, 912, 384, 137, 191, 525, 676, 41, 192, 623, 581]

答案以同样的数字列表格式给出。

附加问题：对下面的 A 完成同样任务：

[900, 523, 1, 823, 151, 74, 301, 75, 152, 76, 752, 826, 902, 677, 4, 827, 304, 978, 604, 229, 455, 379, 755, 831, 905, 82, 6, 532, 456, 383, 606, 833, 906, 983, 7, 84, 308, 534, 758, 834, 908, 984, 9, 235, 609, 985, 909, 86, 610, 236, 760, 386, 311, 836, 461, 687, 611, 538, 761, 239, 612, 989, 762, 90, 313, 540, 463, 690, 613, 840, 164, 91, 464, 241, 764, 391, 914, 991, 465, 392, 615, 842, 915, 94, 166, 694, 466, 994, 916, 695, 317, 96, 618, 996, 768, 97, 169, 397, 320, 697, 620, 98, 770, 700, 921, 101, 323, 851, 623, 702, 923, 852, 174, 253, 625, 554, 26, 854, 926, 105, 177, 255, 478, 405, 778, 705, 179, 556, 779, 706, 180, 707, 630, 108, 930, 558, 31, 708, 631, 858, 781, 559, 482, 411, 932, 561, 483, 711, 933, 262, 34, 862, 184, 113, 784, 413, 934, 713, 186, 414, 337, 265, 637, 565, 787, 266, 339, 117, 190, 267, 490, 567, 790, 717, 940, 869, 491, 120, 641, 270, 642, 720, 792, 121, 193, 271, 643, 872, 344, 123, 944, 124, 195, 274, 495, 424, 795, 724, 945, 275, 196, 575, 796, 725, 47, 875, 947, 426, 798, 277, 948, 877, 49, 428, 199, 129, 799, 879, 200, 130, 350, 580, 501, 880, 801, 131, 951, 281, 53, 731, 203, 432, 54, 133, 205, 134, 805, 284, 806, 734, 507, 884, 359, 435, 60, 885, 960, 436, 61, 137, 361, 287, 511, 587, 811, 288, 212, 588, 662, 888, 363, 290, 963, 590, 364, 740, 964, 141, 365, 291, 815, 592, 816, 892, 817, 143, 68, 593, 518, 744, 69, 595, 519, 895, 669, 146, 819, 897, 71, 148, 72, 298, 372, 598, 672, 149, 73, 299, 373, 599]

解答

认真尝试后再打开

待补充。