Found problems: 22
Find the real values of $p$ for which the equation
\[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\]
in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
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 which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$,
and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$.
[i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]
Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ :
$\begin{cases}x + y = a - 1 \\
x(y + 1) - z^2 = b \end{cases}$
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
Determine all real numbers $x$ which satisfy
\[ x = \sqrt{a - \sqrt{a+x}} \]
where $a > 0$ is a parameter.
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
Determine all real numbers $x$ which satisfy
\[ x = \sqrt{a - \sqrt{a+x}} \]
where $a > 0$ is a parameter.
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
Find the real values of $p$ for which the equation
\[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\]
in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).
Given is the equation:
\[x^2+mx+2022=0\]
a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers
b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation
$$ 1=2mx(2x-1)(2x-2)(2x-3) $$
are real.
Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$,
and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$.
[i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]