Found problems: 744
Solve the system $\begin{cases} x+ y +z = a \\
x^2 + y^2 + z^2 = a^2 \\
x^3 + y^3 +z^3 = a^3
\end{cases}$
Solve the system of equations:
$$\begin{cases} xy+zu=14
\\ xz+yu=11
\\ xu+yz=10
\\ x+y+z+u=10
\end{cases}$$
Determine all real numbers $a$ for which the system of equations
\begin{align*}
3x^2+2y^2+2z^2&=a\\
4x^2+4y^2+5z^2&=1-a
\end{align*}
has at least one solution $(x,y,z)$ in the real numbers.
Find all of the positive real numbers like $ x,y,z,$ such that :
1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$
2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$
Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.
Find all pairs $(x, y)$ of real numbers satisfying the equations
\begin{align*} x^2+y&=xy^2 \\
2x^2y+y^2&=x+y+3xy.
\end{align*}
Find all real solutions of the system of equations $\begin{cases}
(x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases} $
(Based on an idea by A . Aho , J. Hop croft , J. Ullman )
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.
Find the "?"
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
Find all pairs of real numbers $(x,y)$ which satisfy the system
$$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$
Find all the natural numbers $x, y, z$ that satisfy simultaneously
$$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
If $x$, $y$, $z$ are real numbers satisfying
\begin{align*}
(x + 1)(y + 1)(z + 1) & = 3 \\
(x + 2)(y + 2)(z + 2) & = -2 \\
(x + 3)(y + 3)(z + 3) & = -1,
\end{align*}
find the value of
$$ (x + 20)(y + 20)(z + 20). $$
Determine all real numbers $x$ and $y$ such that
$x^2 + x = y^3 - y$,
$y^2 + y = x^3 - x$
Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations:
$x^3-4x^2-16x+60=y$;
$y^3-4y^2-16y+60=z$;
$z^3-4z^2-16z+60=x$.
If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?
Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations
$$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$
Find all real solutions to the system of equations
$$\begin{cases} x^2 +y^2 = 6z \\
y^2 +z^2 = 6x \\
z^2 +x^2 = 6y \end{cases}$$
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Find all pairs $(x, y)$ of real numbers satisfying the system :
$\begin{cases} x + y = 2 \\
x^4 - y^4 = 5x - 3y \end{cases}$
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*}
abcd &= 2007,\\
a &= \sqrt{55 + \sqrt{k+a}},\\
b &= \sqrt{55 - \sqrt{k+b}},\\
c &= \sqrt{55 + \sqrt{k-c}},\\
d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*}
a) Solve the system of equations $\begin{cases}
1 - x_1x_2 = 0 \\
1 - x_2x_3 = 0 \\
...\\
1 - x_{14}x_{15} = 0 \\
1 - x_{15}x_1 = 0 \end{cases}$
b) Solve the system of equations $\begin{cases}
1 - x_1x_2 = 0 \\
1 - x_2x_3 = 0 \\
...\\
1 - x_{n-1}x_{n} = 0 \\
1 - x_{n}x_1 = 0 \end{cases}$
How does the solution vary for distinct values of $n$?
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. $
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
Find all real solutions of the system of equations
$$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$