Found problems: 91
Solve:
$$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\
2x_2 - 5x_3 + 3x4 \ge 0 \\
...\\
2x_{23} - 5x_{24} + 3x_{25} \ge 0\\
2x_{24} - 5x_{25} + 3x_1 \ge 0\\
2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$
Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that
\begin{align*}
cy-bz &\ge 0, \\
az-cx &\ge 0, \\
bx-ay &\ge 0.
\end{align*}
Find all positive real numbers $x, y, z$, such that
$$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations
$$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$
where $x$ and $y$ are real numbers.
Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers
Let $a, b,c$ and $d$ be rea] numbers such that $a^2 + b^2 = c^2 + d^2 = 1$ and $ac + bd = 0$. Determine the value of $ab + cd$.
On the puzzle page of a newspaper this problem is proposed:
“Two children, Antonio and José, have $160$ comics. Antonio counts his by $7$ by $7$ and there are $4$ left over. José counts his $ 8$ by $8$ and he also has $4$ left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has $60$ comics and José has $100$.”
Analyze this solution and indicate what a mathematician would do with this problem.
For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$
Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations :
$$x^2+xy+y^2=57$$
$$y^2+yz+z^2=84$$
$$z^2+zx+x^2=111$$
What is the value of $xy+3yz+5zx$?
[i](maphybich)[/i]
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
Find all the natural numbers $x, y, z$ that satisfy simultaneously
$$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$
Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations
$$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$
Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities
$$a + b + c> 0,$$
$$ ab + bc + ca > 0$$
$$ abc > 0$$
then $a > 0, b > 0, c > 0$.
Solve the equations :
$$\begin{cases} a + b + c = 0 \\ a^2 + b^2 + c^2 = 1\\a^3 + b^3 +c^3 = 4abc \end{cases}$$ for $ a,b,$ and $c. $
Prove that there are no real numbers $ a, b, c $, $ x_1, x_2, x_3 $ such that for every real number $ x $
$$ ax^2 + bx + c = a(x - x_2)(x - x_3) $$
$$bx^2 + cx + a = b(x - x_3) (x - x_1)$$
$$cx^2 + ax + b = c(x - x_1) (x - x_2)$$
and $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.