This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 28

2004 Gheorghe Vranceanu, 2
Tags: equations , inequalities , monotony
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]

2018 Ramnicean Hope, 2
Tags: trigonometry , equations , algebra , monotony
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

2007 Nicolae Păun, 4
Tags: function , topology , IVT , darboux s property , real analysis , monotony , Pathological
Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/i]