Found problems: 91
Determine all sets of real numbers $x,y,z$ which satisfy the system of equations
$$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$
Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$
Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\
\frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$
Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality
$$ p+q + r=1$$
$$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$
then for any numbers $ a $, $ b $, $ c $ equality holds
$$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$
Solve a set of inequalities in the domain of integer numbers:
$$3x^2 +2yz \le 1+y^2$$
$$3y^2 +2zx \le 1+z^2$$
$$3z^2 +2xy \le 1+x^2$$
Show that there are positive reals $a$, $b$, $c$ such that
\[\left\{ \begin{array}{l}
a^2 + b^2 + c^2 > 2 \\
a^3 + b^3 + c^3 <2 \\
a^4 + b^4 + c^4 > 2 \\
\end{array} \right.
\]
Find the values of the real numbers $x,y,z$ that correspond to the system of equations.
$$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$
$$xy + yz + zx=1$$
[i](Heir of Ramanujan)[/i]
Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$
Determine all solutions of the system
$$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$
in which the first three are equations and the last one is a linear inequality.
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
Solve the equations:
\[\left\{ \begin{array}{l}
x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\
2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\
3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\
\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\
(n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\
n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1
\end{array} \right.
\]
The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.
Determine all numbers $x, y$ and $z$ satisfying the system of equations
$$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$
Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied:
\begin{eqnarray*} x=a^2+2b^2\\
y=c^2+d^2\\
z=ac+2bd\\
\end{eqnarray*}
[i]Proposed by Isaac Jiménez[/i]
Show that the only integral solution to
\[\left\{ \begin{array}{l}
xy + yz + zx = 3n^2 - 1\\
x + y + z = 3n \\
\end{array} \right.
\]
with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.
Find all possible real values for $a, b$ and $c$ such that
(a) $a + b + c = 51$,
(b) $abc = 4000$,
(c) $0 < a \le 10$ and $c \ge 25$.
Given $x,y,z\in \mathbb{R^+}$. Find all sets of $x,y,z$ that correspond to $$x+y+z=x^2+y^2+z^2+18xyz=1$$
[i](Zhuge Liang)[/i]
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$.
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
Show that the only real solution to
\[\left\{ \begin{array}{l}
x(x+y)^2 = 9 \\
x(y^3 - x^3) = 7 \\
\end{array} \right.
\]
is $x = 1$, $y = 2$.
Let $a, b,c$ satis es the conditions $\begin{cases}
5 \ge a \ge b \ge c \ge 0 \\
a + b \le 8 \\
a + b + c = 10 \end{cases}$
Prove that $a^2 + b^2 + c^2 \le 38$
Determine all real solutions to the system of equations:
$$x_1 + x_2 +...+ x_{2004 }= 2004$$
$$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$