Determine whether there exist real numbers $x$, $y$, $z$, such that \[x+\frac{1}{y}=z,\quad y+\frac{1}{z}=x,\quad z+\frac{1}{x}=y.\]

There are integers $a$ and $b$, such that $a>b>1$ and $b$ is the largest divisor of $a$ different from $a$. Prove that the number $a+b$ is not a power of $2$ with integer exponent.

Triangle $ABC$ is given, where $AC<BC$ and $\angle ACB=60^\circ\!\!.$ Point $D$, distinct from $A$, lies on the segment $AC$ such that $AB=BD$, and point $E$, distinct from $B$, lies on the line $BC$ such that $AB=AE$. Prove that $\angle DEC=30^\circ$.

On the sides $AB$ and $BC$ of triangle $ABC$, there are points $D$ and $E$, respectively, such that \[\angle ADC=\angle BDE\quad\text{and}\quad \angle BCD=\angle AED.\]Prove that $AE=BE$.

Initially, the numbers $2$ and $5$ are written on the board. A \emph{move} consists of replacing one of the two numbers on the board with their sum. Is it possible to obtain (in a finite numer of moves) a situation in which the two integers written on the board are consecutive? Justify your answer.

A natural number $n$ is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number $n$. Find all numbers $n$ with this property.

Given is a rectangle with perimeter $x$ cm and side lengths in a $1:2$ ratio. Suppose that the area of the rectangle is also $x$ $\text{cm}^2$. Determine all possible values of $x$.

Kamil wrote on a board an expression consisting of alternating addition and subtraction of natural numbers from $1$ to $100$: \[1-2+3-4+5-6+\ldots-98+99-100.\]Then, Kamil erased one of the plus or minus signs and replaced it with an equals sign, obtaining a true equality. Which number preceded the erased sign? Find all possibilities and justify your answer.

Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.