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:

Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

$n$ lines are given in the plane so that no three of them concur and no two are parallel. Show that there is a non-self-intersecting path consisting of $n$ straight segments so that each of the given lines contains exactly one of the segments of the path.

In the following image is a beehive lattice of hexagons. Each cell is colored in one of three colors Red, Blue, or Green (denoted by the letters $R, B, G$). The frame is colored according to the instructions in the image, and the rest of the hexagons are colored however one wants. Is there necessarily a point where three hexagons of different colors meet?

Let $k\leq n$ be two positive integers. $n$ points are marked on a line. It is known that for each marked point, the number of marked points at a distance $\leq 1$ from it (including the point itself) is divisible by $k$. Show that $k$ divides $n$ (without remainder).