Found problems: 744
Solve in $ \mathbb{Z} $ the following system of equations:
$$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$
Solve the following system of equations in real numbers:
$$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$
[i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
$\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\
x+y+z+w= 48
\end{cases}$
Given a positive integer $n \geq 2$. Solve the following system of equations:
$
\begin{cases}
\ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\
\ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\
&\dots \\
\ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\
\end{cases}
$
Let $x$ and $y$ are real numbers such that $$\begin{cases} x + y = 2 \\ x^3 + y^3 = 3\end{cases} $$ What is $x^2+y^2$?
Solve the system of equations
$$\begin{cases} x^2+x+y=8\\
y^2+2xy+z=168\\
z^2+2yz+2xz=12480 \end{cases}$$
Find all triples $(a, b, c)$ of positive integers such that:
$$a + b + c = 24$$
$$a^2 + b^2 + c^2 = 210$$
$$abc = 440$$
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations:
$$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$
Find all triples of complex numbers $(x, y, z)$ for which
$$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$
For real numbers $x$, $y$ and $z$, solve the system of equations:
$$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations:
$x+y+z=7; x^2+y^2+z^2=27; xyz=5$.
Given is a natural number $n \geq 3$. Solve the system of equations:
$\[
\begin{cases}
\tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\
\tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\
& \dots \\
\tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1) \\
\end{cases}
\]$
Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$
Determine all pairs of real numbers $(k, d)$ such that the system of equations
$$\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases}$$ has no solutions $(x, y)$ with $x$ and $y$ real numbers.
Find all solutions $(x,y,z)$ to the system
$$\begin{cases}(x^2 - 6x + 13)y = 20 \\
(y^2 - 6y + 13)z = 20 \\
(z^2 - 6z + 13)x = 20 \end{cases}$$
Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations
$$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\
y^2 + 4z^2 = 4xy \\
z^2 + 4x^2 = 4yz \end{cases}$
Find all real numbers $a$ and $b$ that satisfy the system of equations:
$$\begin{cases}
a &= \frac{2}{a+b} \\
\\
b &= \frac{2}{3a-b} \\
\end{cases}$$
Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$)
a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$
b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)?
D. Bazylev