Found problems: 744
Find all real numbers $x,y$ such that $x^3-y^3=7(x-y)$ and $x^3+y^3=5(x+y)$.
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.
Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.
The real numbers $x, y, z$, distinct in pairs satisfy $$\begin{cases} x^2=2 + y \\ y^2=2 + z \\ z^2=2 + x.\end{cases}$$
Find the possible values of $x^2 + y^2 + z^2$.
Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\
\frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$
Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.
Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.)
\[ x_i^2 + x_i - 1 = x_{i+1} \]
Determine all solutions to the system of equations:
\[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\]
[This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]
Solve the system of equations:
\[x>y>z\]
\[x+y+z=1\]
\[x^2+y^2+z^2=69\]
\[x^3+y^3+z^3=271\]
[hide=Answer]x=7, y=-2, z=-4[/hide]
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.
Solve the system of equations
\[\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}\]
Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$?
Find all non-negative real numbers $x, y, z$ satisfying $x^2y^2 + 1 = x^2 + xy$, $y^2z^2 + 1 = y^2 + yz$ and $z^2x^2 + 1 = z^2 + xz$.
For each positive integer $n$, find all triples $a,b,c$ of real numbers for which
\[\begin{cases}a=b^n+c^n\\
b=c^n+a^n\\
c=a^n+b^n\end{cases}\]
$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start?
$\textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 26\qquad
\textbf{(C)}\ 28 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 32$
Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\
y(x^3-y^3) = 7 \\
\end{array} \right.
\]
In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of:
a) a glass of lemonade, a sandwich and a cake;
b) two glasses of lemonade, three sandwiches and five biscuits.
($1$ shilling = $12$ pence).
Find all real solutions of the system $\begin{cases} x^3 + y^3 = 1 \\
x^4 + y^4 = 1 \end{cases}$
Find all triples of real numbers $(x, y,z)$ such that
$$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$
Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
Solve in the real numbers the system:
$$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$
[i]Eduard Buzdugan[/i]
Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$