Found problems: 5
2013 USAJMO, 4
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
Mathematical Minds 2024, P3
On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column?
[i]Proposed by Vlad Spătaru[/i]
2019 USAJMO, 5
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]
Gheorghe Țițeica 2024, P4
For a set $M$ of $n\geq 3$ points in the plane we define a [i]path[/i] to be a polyline $A_1A_2\dots A_n$, where $M=\{A_1,A_2,\dots ,A_n\}$ and define its length to be $A_1A_2+A_2A_3+\dots +A_{n-1}A_n$. We call $M$ [i]path-unique[/i] if any two distinct paths have different lengths and [i]segment-unique[/i] if any two nondegenerate segments with their ends among the points in $M$ have different lengths. Determine the positive integers $n\geq 3$ such that:
a) any segment-unique set $M$ of $n$ points in the plane is path-unique;
b) any path-unique set $M$ of $n$ points in the plane is segment-unique.
[i]Cristi Săvescu[/i]
2019 USAMO, 4
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]