Found problems: 134
1970 Czech and Slovak Olympiad III A, 3
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
1994 Balkan MO, 1
An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$.
[i]Greece[/i]
1978 Vietnam National Olympiad, 2
Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.
1991 Arnold's Trivium, 20
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 $A = 0$.
2005 IMC, 1
1. Let $f(x)=x^2+bx+c$, M = {x | |f(x)|<1}. Prove $|M|\leq 2\sqrt{2}$ (|...| = length of interval(s))
2013 ELMO Shortlist, 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]
2012 Macedonia National Olympiad, 1
Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.
1992 India Regional Mathematical Olympiad, 7
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 $A$.
2000 French Mathematical Olympiad, Exercise 2
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.
2009 Indonesia TST, 2
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]
2004 Bundeswettbewerb Mathematik, 2
Let $k$ be a positive integer. In a circle with radius $1$, finitely many chords are drawn. You know that every diameter of the circle intersects at most $k$ of these chords.
Prove that the sum of the lengths of all these chords is less than $k \cdot \pi$.
2019 LIMIT Category B, Problem 5
The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is
$\textbf{(A)}~m<\frac14$
$\textbf{(B)}~m\ge0$
$\textbf{(C)}~0\le m\le\frac14$
$\textbf{(D)}~0\le m<\frac14$
2024 Moldova EGMO TST, 1
Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$ where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.
2003 Flanders Math Olympiad, 4
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.
2008 Bulgarian Autumn Math Competition, Problem 12.1
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 $\frac{1}{3}$.
2024 Austrian MO National Competition, 1
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]
2024 Polish MO Finals, 4
Do there exist real numbers $a,b,c$ such that the system of equations
\begin{align*}
x+y+z&=a\\
x^2+y^2+z^2&=b\\
x^4+y^4+z^4&=c
\end{align*}
has infinitely many real solutions $(x,y,z)$?
1988 Spain Mathematical Olympiad, 6
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation
$$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .
2008 Greece Team Selection Test, 4
Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter.
$i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation.
$ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.
1991 Arnold's Trivium, 28
Sketch the phase portrait and investigate its variation under variation of the small complex parameter $\epsilon$:
\[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]
2002 AIME Problems, 8
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 $x.$)
1981 Czech and Slovak Olympiad III A, 1
Determine all $a\in\mathbb R$ such that the inequality \[x^4+x^3-2(a+1)x^2-ax+a^2<0\] has at least one real solution $x.$
2013 Greece Team Selection Test, 2
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]
2001 Turkey Team Selection Test, 3
For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$
$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
1979 Bulgaria National Olympiad, Problem 4
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.