Found problems: 744
Solve the system of equations
$$\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}$$
Find all possible pairs of real numbers $ (x, y) $ that satisfy the equalities $ y ^ 2- [x] ^ 2 = 2001 $ and $ x ^ 2 + [y] ^ 2 = 2001 $.
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 all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that
$$\int_{0}^{1} f(x)\; dx =1, $$
$$\int_{0}^{1} xf(x)\; dx =\alpha,$$
$$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$
where $\alpha$ is a given real number.
Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases}
x^3 - 3x = 4 - y \\
2y^3 - 6y = 6 - z \\
3z^3 - 9z = 8 - x\end{cases}$
Determine all positive real solutions $(a,b)$ to the following system of equations.
\begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}
Real numbers $a, b$ and $c$ satisfy $$\begin{cases} a^2 + b^2 + c^2 = 1 \\ a^3 + b^3 + c^3 = 1. \end{cases}$$ Find $a + b + c$.
Solve the system of equations in integers
$$x + y + z = 3$$
$$x^3 + y^3 + z^3 = 3$$
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:
\[a_1^2 + a_1 - 1 = a_2\]
\[ a_2^2 + a_2 - 1 = a_3\]
\[\hspace*{3.3em} \vdots \]
\[a_{n}^2 + a_n - 1 = a_1\]
Determine all real solutions to the following system of equations:
$$
\begin{cases}
y = 4x^3 + 12x^2 + 12x + 3\\
x = 4y^3 + 12y^2 + 12y + 3.
\end{cases}
$$
For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$:
$$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$
where
$$S=x_1^2+x_2^2+\dots+x_n^2.$$
Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
Determine all real numbers $a, b, c, d$ for which
$ab+cd=6$
$ac+bd=3$
$ad+bc=2$
$a+b+c+d=6$
Find all pairs of real numbers $x, y$ that satisfy the following system of equations $$\begin{cases} x^2 + 3y = 10 \\ 3 + y = \frac{10}{ x} \end{cases}$$
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.
Find all pairs of real numbers $(x,y)$ for which
\[
\begin{aligned}
x^2+y^2+xy&=133 \\
x+y+\sqrt{xy}&=19
\end{aligned}
\]
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$.
If $a$ and $b$ are integers and if the solutions of the system of equations
$$y - 2x - a = 0$$
$$y^2 - xy + x^2 - b = 0$$
are rational, prove that the solutions are integers.
Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations
$$\begin{cases} x^2 +y^2 = a^2 +b^2 \\
x^3 +y^3 = a^3 +b^3 \end{cases}$$
where $a$ and $b$ are given positive numbers.
Let $x, y, z \in R$, find all triples $(x, y, z)$ that satisfy the following system of equations:
$2x^2 - 3xy + 2y^2 = 1$
$y^2 - 3yz + 4z^2 = 2$
$z^2 + 3zx - x^2 = 3$
Find all pairs $(x, y)$ of real numbers satisfying the system :
$\begin{cases} x + y = 3 \\
x^4 - y^4 = 8x - y \end{cases}$
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.
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
$$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$
Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.