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Difference between revisions of "2019 USAJMO Problems/Problem 5"
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==Solution==
 
==Solution==
 
We claim the answer is <math>(2n)!\cdot 2^{n^2}</math>.
 
We claim the answer is <math>(2n)!\cdot 2^{n^2}</math>.
 +
 
Proof:
 
Proof:
 
Note that there are <math>(2n)!</math> ways to choose <math>S_{1, 0}, S_{2, 0}... S_{n, 0}, S_{n, 1}, S_{n, 2}... S{n, n}</math>, because there are <math>2n</math> ways to choose which number <math>S_{1, 0}</math> is, <math>2n-1</math> ways to choose which number to append to make <math>S_{2, 0}</math>, <math>2n-2</math> ways to choose which number to append to make <math>S_{3, 0}</math>... After that, note that <math>S_{n-1, 1}</math> contains the <math>n-1</math> in <math>S_{n-1. 0}</math> and 1 other element chosen from the 2 elements in <math>S_{n, 1}</math> not in <math>S_{n-1, 0}</math> so there are 2 ways for <math>S_{n-1, 1}</math>. By the same logic there are 2 ways for <math>S_{n-1, 2}</math> as well so <math>2^n</math> total ways for all <math>S_{n-1, j}</math>, so doing the same thing <math>n-1</math> more times yields a final answer of <math>(2n)!\cdot 2^{n^2}</math>, as desired.
 
Note that there are <math>(2n)!</math> ways to choose <math>S_{1, 0}, S_{2, 0}... S_{n, 0}, S_{n, 1}, S_{n, 2}... S{n, n}</math>, because there are <math>2n</math> ways to choose which number <math>S_{1, 0}</math> is, <math>2n-1</math> ways to choose which number to append to make <math>S_{2, 0}</math>, <math>2n-2</math> ways to choose which number to append to make <math>S_{3, 0}</math>... After that, note that <math>S_{n-1, 1}</math> contains the <math>n-1</math> in <math>S_{n-1. 0}</math> and 1 other element chosen from the 2 elements in <math>S_{n, 1}</math> not in <math>S_{n-1, 0}</math> so there are 2 ways for <math>S_{n-1, 1}</math>. By the same logic there are 2 ways for <math>S_{n-1, 2}</math> as well so <math>2^n</math> total ways for all <math>S_{n-1, j}</math>, so doing the same thing <math>n-1</math> more times yields a final answer of <math>(2n)!\cdot 2^{n^2}</math>, as desired.
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-Stormersyle

Revision as of 19:49, 19 April 2019

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:


1. for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and

2. $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.


Proposed by Ricky Liu

Solution

We claim the answer is $(2n)!\cdot 2^{n^2}$.

Proof: Note that there are $(2n)!$ ways to choose $S_{1, 0}, S_{2, 0}... S_{n, 0}, S_{n, 1}, S_{n, 2}... S{n, n}$, because there are $2n$ ways to choose which number $S_{1, 0}$ is, $2n-1$ ways to choose which number to append to make $S_{2, 0}$, $2n-2$ ways to choose which number to append to make $S_{3, 0}$... After that, note that $S_{n-1, 1}$ contains the $n-1$ in $S_{n-1. 0}$ and 1 other element chosen from the 2 elements in $S_{n, 1}$ not in $S_{n-1, 0}$ so there are 2 ways for $S_{n-1, 1}$. By the same logic there are 2 ways for $S_{n-1, 2}$ as well so $2^n$ total ways for all $S_{n-1, j}$, so doing the same thing $n-1$ more times yields a final answer of $(2n)!\cdot 2^{n^2}$, as desired.

-Stormersyle

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