Found problems: 451
2012 Junior Balkan Team Selection Tests - Romania, 2
Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.
2013 IFYM, Sozopol, 6
For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?
1999 Swedish Mathematical Competition, 1
Solve $|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30$.
1979 Vietnam National Olympiad, 5
Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.
2015 German National Olympiad, 2
A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying
\[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\]
Find all smooth numbers.
2013 Bosnia And Herzegovina - Regional Olympiad, 2
Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$
1978 Vietnam National Olympiad, 1
Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.
2011 India IMO Training Camp, 2
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2023 Poland - Second Round, 3
Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
2013 India IMO Training Camp, 2
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.
[b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.
[b]b)[/b] Decide whether $a=2$ is friendly.
2016 District Olympiad, 1
Solve in the interval $ (2,\infty ) $ the following equation:
$$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers
2001 Estonia National Olympiad, 4
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.
2000 District Olympiad (Hunedoara), 2
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.
2010 VTRMC, Problem 3
Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$
2013 Bosnia And Herzegovina - Regional Olympiad, 1
Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$
2004 Nicolae Coculescu, 2
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
1992 IMO Longlists, 68
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy
\[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]
1966 IMO Shortlist, 10
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
1980 IMO Shortlist, 3
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
1992 Nordic, 1
Determine all real numbers $x > 1, y > 1$, and $z > 1$,satisfying the equation
$x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2})$
1987 ITAMO, 4
Given $I_0 = \{-1,1\}$, define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$,
where $y$ ranges over all elements of $I_{n-1}$. Determine the union of the sets $I_n$ over all nonnegative integers $n$.
1966 IMO Longlists, 12
Find digits $x, y, z$ such that the equality
\[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\]
holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
2005 Grigore Moisil Urziceni, 1
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system:
$$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$
1984 IMO Longlists, 12
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]