Found problems: 68
2019 Teodor Topan, 1
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
2004 Nicolae Coculescu, 3
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $
[i]Florian Dumitrel[/i]
2006 Petru Moroșan-Trident, 1
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $
[i]Constantin Nicolau[/i]
2015 German National Olympiad, 1
Determine all pairs of real numbers $(x,y)$ satisfying
\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.
\end{align*}
2019 Danube Mathematical Competition, 1
Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $
[i]Lucian Petrescu[/i]
2004 Gheorghe Vranceanu, 2
Solve in $ \mathbb{R}^2 $ the following equation.
$$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$
[i]O. Trofin[/i]
1995 IMO Shortlist, 2
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
\[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]
2006 Cezar Ivănescu, 2
[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $
[b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that
$$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$
for any nonnegative integer $ n. $
2018 Ramnicean Hope, 2
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $
[i]Ovidiu Țâțan[/i]
2004 Unirea, 1
Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $
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.
1998 Junior Balkan Team Selection Tests - Romania, 1
Solve in $ \mathbb{Z}^2 $ the following equation:
$$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$
[i]Adrian Zanoschi[/i]
2006 Petru Moroșan-Trident, 2
Solve the following Diophantines.
[b]a)[/b] $ x^2+y^2=6z^2 $
[b]b)[/b] $ x^2+y^2-2x+4y-1=0 $
[i]Dan Negulescu[/i]
2007 Alexandru Myller, 1
Solve $ x^3-y^3=2xy+7 $ in integers.
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.
1960 Putnam, A1
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation
$$\frac{xy}{x+y}=n?$$
1978 Romania Team Selection Test, 4
Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $
2016 District Olympiad, 1
Solve in $ \mathbb{N}^2: $
$$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$