IL-GROSSMAN 2022

For each positive integer $n$ denote: $$n!=1\cdot 2\cdot 3\dots n$$ Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

We call a sequence of length $n$ of zeros and ones a "string of length $n$ " and the elements of the same sequence "bits". Let $m,n$ be two positive integers so that $m<2^n$ . Arik holds $m$ strings of length $n$ . Giora wants to find a new string of length $n$ different from all those Arik holds. For this Giora may ask Arik questions of the form:

"What is the value of bit number $i$ in string number $j$ ?"

where $1\leq i\leq n$ and $1\leq j\leq m$ . What is the smallest number of questions needed for Giora to complete his task when:a) $m=n$ ?b) $m=n+1$ ?

An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$ ; the width is considered to be $0$ ). Prove that there is a circle of radius $\frac{1}{4}$ containing the path.

Illustration of an example path:

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