Found problems: 91
Determine all real solutions $x,y$ to the system of equations
$$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$
Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.
Find all triples $(x,y,z)$ of positive integers such that
$$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$
Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$
The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$.
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled:
$$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$
$$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$
$$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$
$$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$
Solve the system $$\begin{cases} \displaystyle 4^{-x}+27^{-y}= \frac{5}{6} \\ \displaystyle 27^y-4^x \le 1 \\ \displaystyle \log_{27}y-\log_4 x \ge \frac{1}{6} \end{cases}.$$
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?
A system of equations
$$\begin{cases} x^2 \,\, ? \,\, z^2 = -8 \\ y^2 \,\, ? \,\, z^2 = 7 \end{cases}$$
is written on a piece of paper, but unfortunately two of the symbols are a little blurred. However, it is known that the system has at least one solution, and that each of the two question marks stands for either $+$ or $-$. What are the two symbols?
Solve the system of equations
$$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
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 $x = y = z= 1$?
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 $x, y,z,t$.
Solve the system of equations
$$\begin{cases} x^2+x+y=8\\
y^2+2xy+z=168\\
z^2+2yz+2xz=12480 \end{cases}$$
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
$$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$
If $a$ and $b$ are distinct real numbers, solve the systems
(a) $\begin{cases} x+y = 1 \\ (ax+by)^2 \le a^2x+b^2y \end{cases}$ and (b) $\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}$
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 real solutions $x, y$ of the system of equations
$$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$
Solve in $ \mathbb{Z} $ the system:
$$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$