Found problems: 134
In terms of real parameter $a$ solve inequality:
$\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$
in set of real numbers
Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter $A$.
Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.
Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations
$$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$
where $x$ and $y$ are real numbers.
Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]
Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$:
\begin{align*}
&(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\
=&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\
&\qquad \qquad \qquad \qquad \vdots \\
=&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1})
\end{align*}
has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation
$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$.
a) Prove that $ f_{\pi}$ is not periodic.
b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic.
[b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.
Say that a (nondegenerate) triangle is [i]funny[/i] if it satisfies the following condition: the altitude, median, and angle bisector drawn from one of the vertices divide the triangle into 4 non-overlapping triangles whose areas form (in some order) a 4-term arithmetic sequence. (One of these 4 triangles is allowed to be degenerate.) Find with proof all funny triangles.
Determine the values of the real parameter $a$, such that the equation
\[\sin 2x\sin 4x-\sin x\sin 3x=a\]
has a unique solution in the interval $[0,\pi)$.
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]
Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation
$$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
Find all ordered triples of non-negative integers $(a,b,c)$ such that $a^2+2b+c$, $b^2+2c+a$, and $c^2+2a+b$ are all perfect squares.
[i]Proposed by Matthew Babbitt[/i]
Prove that the system of equations
\[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \]
$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.
Find all real numbers $a$ for which the following equation has a unique real solution:
$$|x-1|+|x-2|+\ldots+|x-99|=a.$$
Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$:
\begin{align*}
&(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\
=&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\
&\qquad \qquad \qquad \qquad \vdots \\
=&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1})
\end{align*}
has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.
Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter.
Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function defined as the maximum of a finite number of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ of the form $g(x)=C\cdot10^{-|x-d|}$ (with different values of parameters $d{}$ and $C>0$). For real numbers $a<b$ we have $f(a)=f(b)$. Prove that on the segment $[a;b]$ the sum of legnths of segments on which $f$ is increasing is equal to the sum of legnths of segments on which $f$ is decreasing.