数量关系
加减同质:两个数相加相减的结果 奇偶是一定的
判断一个数是否能被x整除:3.9看各位和,和能整除就可以整除;
4看末两位,8看末三位
2.5看末位
拆分:减去x的倍数 然后判断
因式分解:分解成互质的因子
不定方程 m*x+n*y=t 如果n*y和t都能被一个数整除,那么m*x也能被该数整除
公式 来回走相遇问题 (2n-1)*S=(v1+v2)*t
涨了两次 a b 现价=原价*(1+a)(1+b) 如果a+b和一定 每次涨总数的一半最大
这个是隔板法呀
n 个相同的物品分给 m 人,每人至少分 1 个,有 C(n-1,m-1)种不同的分法
若每人至少分 a 个:先给每人分 a-1 个。比如至少分 5 个,则先给 4 个,转化为至少 1 个。
![](https://img2020.cnblogs.com/blog/326461/202104/326461-20210423172927319-132393752.png)
6个元素错排 265
容斥原理三个:![](data:image/png;base64,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)
这里的满足两项是只满足两项
典型三角形:1-2-根号3;1-1根号2;5-12-13
球面面积 4πR平方
![](https://img2020.cnblogs.com/blog/326461/202104/326461-20210423173047297-1511542850.png)