Olympiad problems

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Found problems: 2

2019 USAMO, 4

Tags: AMC , USA(J)MO , USAMO , Olympiad Combinatorics , recursion , 2019 USAJMO , 2019 USAMO

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: [list] [*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and [*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. [/list] [i]Proposed by Ricky Liu[/i]

2019 USAJMO, 5

Tags: AMC , USA(J)MO , USAMO , Olympiad Combinatorics , recursion , 2019 USAJMO , 2019 USAMO

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: [list] [*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and [*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. [/list] [i]Proposed by Ricky Liu[/i]