Found problems: 134
Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$
Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that
\[a_1^m+a_2^m+a_3^m+\cdots=m\]
for every positive integer $m?$
Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\
x_3 - x_4 = b \\
x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.
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 $p.$
=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that
\[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\}
\]
Are the following subsets of plane an algebraic sets?
1. A square
[img]http://i36.tinypic.com/28uiaep.png[/img]
2. A closed half-circle
[img]http://i37.tinypic.com/155m155.png[/img]
Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis.
(1) Find the value of $a.$
(2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$
Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter
Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.
Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?
For each positive integer $n$, find all triples $a,b,c$ of real numbers for which
\[\begin{cases}a=b^n+c^n\\
b=c^n+a^n\\
c=a^n+b^n\end{cases}\]
Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.
$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.
$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable
suppose $f(a)=f(-a)$
$ f'(-a)=f'(a)=a^2$
Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$
Study equality case ?
Radu Miculescu
Determine the values of the real parameter $a$, such that the solutions of the system of inequalities
$\begin{cases}
\log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\
\log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\
\end{cases}$
form an interval of length $\frac{1}{3}$.
Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points)
Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
Let $A,B,P$ positive reals with $P\le A+B$.
(a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that \[ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} \]
(b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$.
(c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$.
Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.
The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is
$\textbf{(A)}~m<\frac14$
$\textbf{(B)}~m\ge0$
$\textbf{(C)}~0\le m\le\frac14$
$\textbf{(D)}~0\le m<\frac14$
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions
Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$
[i]G. Tusnady[/i]
Consider the polynomials
\[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\]
Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$