Found problems: 744
Determine all three integers $(x, y, z)$ that are solutions of the system
$$x + y -z = 6$$
$$x^3 + y^3 -z^3 = 414$$
Find all real numbers $x, y, z$ that satisfy the following system
$$\sqrt{x^3 - y} = z - 1$$
$$\sqrt{y^3 - z} = x - 1$$
$$\sqrt{z^3 - x} = y - 1$$
Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l}
x+y^2 +z^3 = 3\\
y+z^2 +x^3 = 3\\
z+x^2 +y^3 = 3\\
\end{array} \right.
\]
Does
\[\left\{ \begin{array}{l}
x^y = z \\
y^z = x \\
z^x = y \\
\end{array} \right.
\]
have any solutions in positive reals apart from $x = y = z= 1$?
Find all values of the real parameter $a$, for which the system
$(|x| + |y| - 2)^2 = 1$
$y = ax + 5$
has exactly three solutions
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
Let $a,b,c$ be reals satisfying\\
$\hspace{2cm} 3ab+2=6b, \hspace{0.5cm} 3bc+2=5c, \hspace{0.5cm} 3ca+2=4a.$\\
\\
Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s}\in \mathbb{Q}$ and $\frac{t}{u}\in \mathbb{Q}$ , in lowest form, find $r+s+t+u$.
Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*}
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
Find all real quadruples $(a,b,c,d)$ satisfying the system of equations
$$
\left\{ \begin{array}{ll}
ab+cd = 6 \\
ac + bd = 3 \\
ad + bc = 2 \\
a + b + c + d = 6.
\end{array} \right.
$$
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]
Find all integer solutions to
\begin{align*}
x^2+2y^2+3z^2&=36,\\
3x^2+2y^2+z^2&=84,\\
xy+xz+yz&=-7.
\end{align*}
Solve the system of equations in reals:
\[
\begin{cases}
y = x^2 + 2x \\
z = y^2 + 2y \\
x = z^2 + 2z
\end{cases}
\]
[i]Proposed by Mykhailo Shtandenko[/i]
Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n,
x^2 + y^2 = a^m$$ for some natural $a, n, m$.
Let $m, n$ be odd prime numbers.
Find all pairs of integers numbers $a, b$ for which the system of equations:
$x^m+y^m+z^m=a$,
$x^n+y^n+z^n=b$
has many solutions in integers $x, y, z$.
Find real solutions of the system :
$$\begin{cases} \sin x + 2 \sin (x+y+z) = 0 \\
\sin y + 3 \sin (x+y+z) = 0\\
\sin z + 4 \sin (x+y+z) = 0\end{cases}$$
Solve the system of equations for positive real numbers:
$$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$.
Find all real solutions to the following system of equations:
\[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\
x + y + 2z = 4(a^2 + 1) \\
z^2 - xy = a^2
\end{cases}$
Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]
Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1,\] \[\sqrt{z-b}+\sqrt{x-b}=1,\] \[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x, y, z)$ in real numbers.