有回味的题目

王守恩

2025中国科技大学强基计划数学试题 ——第7题——

\(将1,2,3,...,10十个数重新排列,得到新序列{a_{1},a_{2},a_{3},...,a_{10}},若|a_{i} - i| ≤1。求满足条件的排列数量。\)

王守恩

在△ABC中, 内角A,B,C的对边分别为a,b,c。若A = 2B,  则a^2 = b^2 + b*c。

王守恩

Table[N[(10^(9 a) - 1)/(10^9 - 1)^a, 100], {a, 15}] —— 每一行都还是有理数吗？
1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,
1.000000002000000002000000002000000002000000002000000002000000002000000002000000002000000002000000002,
1.000000003000000006000000009000000012000000015000000018000000021000000024000000027000000030000000033,
1.000000004000000010000000020000000034000000052000000074000000100000000130000000164000000202000000244,
1.000000005000000015000000035000000070000000125000000205000000315000000460000000645000000875000001155,
1.000000006000000021000000056000000126000000252000000461000000786000001266000001946000002877000004116,
1.000000007000000028000000084000000210000000462000000924000001715000002996000004977000007924000012166,
1.000000008000000036000000120000000330000000792000001716000003432000006434000011432000019412000031704,
1.000000009000000045000000165000000495000001287000003003000006435000012870000024309000043749000075537,
1.000000010000000055000000220000000715000002002000005005000011440000024310000048620000092377000167950,
1.000000011000000066000000286000001001000003003000008008000019448000043758000092378000184756000352715,
1.000000012000000078000000364000001365000004368000012376000031824000075582000167960000352716000705432,
1.000000013000000091000000455000001820000006188000018564000050388000125970000293930000646646001352078,
1.000000014000000105000000560000002380000008568000027132000077520000203490000497420001144066002496144,
1.000000015000000120000000680000003060000011628000038760000116280000319770000817190001961256004457400,

每一行都还是有理数吗？——譬如第3行。
1.000000003000000006000000009000000012000000015000000018000000021000000024000000027000000030000000033000000036000000039000000042000000045000000048000000051000000054000000057000000060000000063
000000066000000069000000072000000075000000078000000081000000084000000087000000090000000093000000096000000099000000102000000105000000108000000111000000114000000117000000120000000123000000126
000000129000000132000000135000000138000000141000000144000000147000000150000000153000000156000000159000000162000000165000000168000000171000000174000000177000000180000000183000000186000000189
000000192000000195000000198000000201000000204000000207000000210000000213000000216000000219000000222000000225000000228000000231000000234000000237000000240000000243000000246000000249000000252
000000255000000258000000261000000264000000267000000270000000273000000276000000279000000282000000285000000288000000291000000294000000297000000300000000303000000306000000309000000312000000315
000000318000000321000000324000000327000000330000000333000000336000000339000000342000000345000000348000000351000000354000000357000000360000000363000000366000000369000000372000000375000000378
000000381000000384000000387000000390000000393000000396000000399000000402000000405000000408000000411000000414000000417000000420000000423000000426000000429000000432000000435000000438000000......

aimisiyou

王守恩 发表于 2025-7-10 06:09
2025中国科技大学强基计划数学试题 ——第7题——

\(将1,2,3,...,10十个数重新排列,得到新序列{a_{1},a_{2 ...

大概递归思路差不多。举个简单的，$$i-1\leq a(i) \leq  i+2$$,如图。
dp(n)=2*dp(n-2)+2*dp(n-3)+dp(n-4).

王守恩

正整数 a, b, c,   a < b < c 且满足:  a 是 b + c 的因子;  b 是 c + a 的因子;  c 是 a + b 的因子。

称这样的 a, b, c 为"三元好组"。若 a + b + c ≤ 2025,  有多少组这样的"三元好组"？