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斯坦福大学数学夏令营保录取Sumac代写2023

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斯坦福大学数学夏令营保录取Sumac代写2023

Consider an $m \times n$ grid, that is, a grid with $m$ rows and $n$ columns, where $m$ and $n$ are relatively prime (that is, where $m$ and $n$ have no prime factors in common). For example, here is a $9 \times 10$ grid:
Each one-by-one square in the grid represents a hole, and we fill in some of these holes as follows. For each integer $d>1$ that shares a prime factor with $m$, we fill in all holes in row $d$, and for each integer $d>1$ that shares a prime factor with $n$, we fill in all holes in column $d$ (that haven’t already been filled in). For example, in the $9 \times 10$ grid above, we fill in the holes as follows:

We filled in the third, sixth, and ninth rows since 3,6 , and 9 are integers greater than 1 that share the prime factor 3 with 9, and we filled in second, fourth, fifth, sixth, eighth and tenth columns since 2,4 , $5,6,8$ and 10 all share a prime factor with 10. In this example, there are 24 holes (white squares) left over. Define the hole number of a grid to be the number of holes (white squares) that remain after the rest of the grid is filled in according to the above procedure, so the hole number of the above grid is 24 . The following two grids have hole number 8 .

斯坦福大学数学夏令营保录取Sumac代写2023

We say that two grids are hole equivalent if they have the same hole numbers. So, the above two grids $\left(4 \times 5\right.$ and $3 \times 5$ ) are hole equivalent. Let $h_k$ be the number of $m \times n$ grids with $m0$ such that $h_k=0$. That is, what is the smallest positive $k$ such that there are no $m \times n$ grids with $k$ holes.
(ii) What is $h_8$ ?
(iii) What is the smallest value of $k$ such that $h_k>h_8$ ?

(a) Find all non-empty finite sets of integers $A$ and $B$ with the following properties:
(i) Whenever $x$ is in $A, x+1$ is in $B$.
(ii) Whenever $x$ is in $B, x^2-4$ is in $A$.
(b) Find all positive integers $a$ and $b$ such that there are non-empty finite sets $A$ and $B$ with the property that whenever $x$ is in $A, x+a$ is in $B$, and whenever $x$ is in $B, x^2-b$ is in $A$.

斯坦福大学数学夏令营保录取Sumac代写2023

(ii) 什么是 $h_8$ ?
(iii) 的最小值是多少 $k$ 这样 $h_k>h_8$ ?
(a) 找出所有非空的有限整数集 $A$ 和 $B$ 具有以下特性:
(i) 每当 $x$ 在 $A, x+1$ 在 $B$.
(ii) 每当 $x$ 在 $B, x^2-4$ 在 $A$.
(b) 找出所有正整数 $a$ 和 $b$ 使得存在非空有限集 $A$ 和 $B$ 具有每当 $x$ 在 $A, x+a$ 在 $B$, 并且每当 $x$ 在 $B, x^2-b$ 在 $A$.