2022年普通高等学校招生全国统一考试数学(北京卷)T21
题目:
待补充(求发个$\LaTeX$的, 懒得码)
昨晚上刷到这题, 看了一眼就摸鱼去了反正不是京爷考不到我, 今早(12:00)起来做了一下
但还是感谢一下2022年北京高考数学压轴题赏析, 提到了富比尼原理, 指明了方向
(1)(2)自己做
(3):
实际上本题有个条件是无用的自己猜是哪个
只证$k=6$不成立, $k<6$的情况补$0$即为$k=6$的情况, 同时得知$k=6$时$\forall i\in{\{1,2,3,4,5,6\}}, a_{i}\neq 0$
$k=6$时
设$S$为所有连续和组成的可重集合
由
$|S|=21$
$\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}\subseteq S$
知
$\exists t\in{S}, \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}=S/\{t\}$
算两次
$\sum \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}=210=6\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}-t=\sum S/\{t\}$
$\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}=\frac{210+t}{6}$
若$\exists j\in{\{1,2,3,4,5,6\}}, a_{j}<0$
则$t=a_{j}<0$
$\sum_{i\in{\{1,2,3,4,5,6\}}, i\neq j}a_{i}=\frac{210-5t}{6}\geq\frac{210-5(-6)}{6}=40$
又$a_{j}$至多将$\{a_{n}\}$分为两部, 或有一部连续和大于$20$, 或有两部连续和均大于等于$20$
均矛盾
若$\forall i\in{\{1,2,3,4,5,6\}}, a_{i}>0$
则$t>0$
$\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}=\frac{210+t}{6}\geq\frac{210+6}{6}=36$
考察等于$20$的连续和
$l,r\in{\mathbb{Z}^{+}},l,r-1\in{\{1,2,3,4,5,6\}},\sum_{i\in{\mathbb{Z}^{+}},i\in{[l,r-1]}}a_{i}=20$
若$r-l<5$
则由$(r-l)+1<6$知$\exists L,R\in{\mathbb{Z}^{+}},L,R-1\in{\{1,2,3,4,5,6\}},[l,r-1]\subset[L,R-1],R-L=5$
$\sum_{i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}}a_{i}>\sum_{i\in{\mathbb{Z}^{+}},i\in{[l,r-1]}}a_{i}=20$
又
$\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}\geq 36>20$
即
$\sum_{i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}}a_{i}\notin \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$
$\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}\notin \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$
$\{i|i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}\}\neq \{i|i\in{\{1,2,3,4,5,6\}}\}$
矛盾
且由
$\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}\notin \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$
知$r-l\neq 6$又$r-l\leq 6$故$r-l=5$
考察$j\notin{[l,r-1]}$
$a_{j}=\sum_{i\in{\{1,2,3,4,5,6\}}}a_{i}-\sum_{i\in{\mathbb{Z}^{+}},i\in{[l,r-1]}}a_{i}\geq 36-20=16$
$\exists L,R\in{\mathbb{Z}^{+}},L,R-1\in{\{1,2,3,4,5,6\}},j\in{[L,R-1]},R-L=5$
$\sum_{i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}}a_{i}\geq 16+1+1+1+1=20$
$\{i|i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}\}\neq \{i|i\in{\{1,2,3,4,5,6\}}\}$
$\{i|i\in{\mathbb{Z}^{+}},i\in{[L,R-1]}\}\neq \{i|i\in{\mathbb{Z}^{+}},i\in{[l,r-1]}\}$
或有两个连续和等于$20$, 或有两个连续和不属于$\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$
均矛盾
$Q.E.D.$