Solve the system of equations: $x^2=\frac{1}{y}+\frac{1}{z}$, $y^2=\frac{1}{z}+\frac{1}{x}$, $z^2=\frac{1}{x}+\frac{1}{y}$. in the real numbers.

In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes. On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?" "Shut up. I'm $\textit{not}$ sniffing the carpet. I'm $\textit{doing something}$." "Sure, if $\textit{sniffing the carpet}$ counts as $\textit{doing something}.$" At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel. Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm $\textit{designing}$ something." Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?" "Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome." "And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project. "I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be $9$ inches from one of the corners, $15$ inches from another, $16$ inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days. Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

Solve the system of equation $$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$

Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$

Solve in $ \mathbb{C}^3 $ the following chain of equalities: $$ x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3. $$

Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.

Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$

Let $n$ be a positive integer. Suppose that the following simultaneous equations $$\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\ \sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases}$$ has a solution, where $x_1 x_2,.., x_n$ are the unknowns. Find the smallest possible positive integer $n$. Justify your answer.

Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}

Find all real solutions of the system of equations $$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$

When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

Determine all real numbers $x$ and $y$ such that $x^2 + x = y^3 - y$, $y^2 + y = x^3 - x$

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system: \[a_1^2 + a_1 - 1 = a_2\] \[ a_2^2 + a_2 - 1 = a_3\] \[\hspace*{3.3em} \vdots \] \[a_{n}^2 + a_n - 1 = a_1\]

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

Find all real solutions of the system : (a) $$\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}$$ (b)* $$\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}$$

Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?

Determine all real solutions $(x,y)$ of the following system of equations: \begin{align*} x&=3x^2y-y^3,\\ y &= x^3-3xy^2 \end{align*}

Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]

Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\ bc + d + a = 5 \\ cd + a + b = 2 \\ da + b + c = 6 \end{cases}$

Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$