Given is a square $ABCD$ with side length $1$. Points $K$, $L$, $M$, and $N$, distinct from the vertices of the square, lie on segments $AB$, $BC$, $CD$, and $DA$, respectively. Prove that the perimeter of at least one of the triangles $ANK$, $BKL$, $CLM$, $DMN$ is less than $2$.

Find all positive integers $n$ for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\]are prime.

Given a parallelogram $ABCD$ in which $\angle ABD=90^\circ$ and $\angle CBD=45^\circ$. Point $E$ lies on segment $AD$ such that $BC=CE$. Determine the measure of angle $BCE$.

The line segments $AB$ and $CD$ are perpendicular and intersect at point $X$. Additionally, the following equalities hold: $AC=BD$, $AD=BX$, and $DX=1$. Determine the length of segment $CX$.

Let $n\geq 1$ be an integer and let $a$ and $b$ be its positive divisors satisfying $a+b+ab=n$. Prove that $a=b$.

Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.

There are $17$ students in Marek's class, and all of them took a test. Marek's score was $17$ points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer.

In the rectangle $ABCD$, the ratio of the lengths of sides $BC$ and $AB$ is equal to $\sqrt{2}$. Point $X$ is marked inside this rectangle so that $AB=BX=XD$. Determine the measure of angle $BXD$.

Let $n\geq 1$ be an integer. Show that there exists an integer between $\sqrt{2n}$ and $\sqrt{5n}$, exclusive.