Found problems: 38
Find all values of the parameter $a$ for which the inequality
\[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\]
has a unique solution.
Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.
Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.
Find all values of $a$ such that absolute value of one of the roots of the equation $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$ is twice of absolute value of the other root.
Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$
has three distinct real roots which are sides of a right triangle.
Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\
y^2 - 3z + p = x, \\
z^2 - 3x + p = y \end{cases}$ with real parameter $p$.
a) For $p \ge 4$, solve the considered system in the field of real numbers.
b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$.
(Jaroslav Svrcek)
For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.
Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\
x + y+ z = 2b \\
x^2 + y^2-z^2 = b^2
\end{cases}$ in $C$
Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$
Find all possible values of $a$.
For what real values of $a$ the equation $\frac{2^{2x}}{2^{2x}+2^{x+1}+1}+a \frac{2^x}{2^x+1}+(a-1) = 0$ has a single root ?
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
Determine the number of real roots of the equation
$$16x^5 - 20x^3 + 5x + m = 0.$$
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?
$C$ 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?
Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\
x + y + 2z = 4(a^2 + 1) \\
z^2 - xy = a^2
\end{cases}$
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
For which values of the parameter $a$ does the equation
\[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\]
has three different real roots.
Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.
The system $\begin{cases} x^2 - y^2 = 0 \\
(x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?
$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$
$b)$ Does inequality $a)$ holds for
$1)$ arbitrary real numbers $a$, $b$ and $c$
$2)$ any integer $m$
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.
Solve the system $\begin{cases} x^3 - y^3 = 26 \\
x^2y - xy^2 = 6
\end{cases}$ in $C$
[hide=other version]solved below
Solve the system $\begin{cases} x^3 - y^3 = 2b \\
x^2y - xy^2 = b
\end{cases}$[/hide]
Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\
x + y = b \end{cases}$ in $C$
Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.