Found problems: 134
Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \]
has a unique solution.
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.
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$?
$\textbf{(A) }38\qquad
\textbf{(B) }40\qquad
\textbf{(C) }42\qquad
\textbf{(D) }44\qquad
\textbf{(E) }46\qquad$
Find values of the parameter $u$ for which the expression
\[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\]
does not depend on $x.$
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x).
\]
Determine all real $a$, such that the solutions to the system of equations
$\begin{cases}
\frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\
(2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a
\end{cases}$
form an interval with length $\frac{32}{225}$.
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?
For all valid values of $a$ and $b$, solve the equation
$$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$