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: 68

2005 Grigore Moisil Urziceni, 1
Tags: equations , system of equations , algebra
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

2004 Gheorghe Vranceanu, 4
Tags: complex numbers , equations , algebra
Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $

2010 Laurențiu Panaitopol, Tulcea, 1
Tags: quadratics , algebra , equations
Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

2023 China Northern MO, 3
Tags: equations , floor function , algebra
Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

2009 District Olympiad, 3
Tags: equations , algebra , logarithms
Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

2006 Grigore Moisil Urziceni, 3
Tags: monotony , algebra , system of equations , equations
Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

2007 Mathematics for Its Sake, 2
Tags: equations , system of equations , algebra
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones. [i]Mugur Acu[/i]

2018 Ramnicean Hope, 1
Tags: monotony , algebra , equations
Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

2004 Nicolae Coculescu, 3
Tags: equations , Hermite , algebra , fractional , Floor
Prove the identity $ \frac{n-1}{2}=\sum_{k=1}^n \left\{ \frac{m+k-1}{n} \right\} , $ where $ n\ge 2, m $ are natural numbers, and $ \{\} $ denotes the fractional part.

2007 Nicolae Coculescu, 2
Tags: trigonometry , equations , algebra
Solve in the real numbers the equation $ \cos \left( \pi\log_3 (x+6) \right)\cdot \cos \left( \pi \log_3 (x-2) \right) =1. $

1986 Traian Lălescu, 2.1
Tags: algebra , equations , parameterization , parametric equation
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.

2012 Centers of Excellency of Suceava, 4
Tags: monotony , equations , Cartesian circle , algebra
Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

2004 Gheorghe Vranceanu, 4
Tags: diophantine , equations
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $

2019 Ramnicean Hope, 1
Tags: monotony , equations , algebra
Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/i]

2009 Romania National Olympiad, 1
Tags: functions , unique solution , equations , algebra
[b]a)[/b] Show that two real numbers $ x,y>1 $ chosen so that $ x^y=y^x, $ are equal or there exists a positive real number $ m\neq 1 $ such that $ x=m^{\frac{1}{m-1}} $ and $ y=m^{\frac{m}{m-1}} . $ [b]b)[/b] Solve in $ \left( 1,\infty \right)^2 $ the equation: $ x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} . $

2006 Petru Moroșan-Trident, 2
Tags: equations , system of equations , algebra , monotony
Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

2007 Nicolae Coculescu, 1
Tags: algebra , equations , Floor , fractional part
Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $ [i]Ovidiu Pop[/i]

2015 IFYM, Sozopol, 8
Tags: algebra , equations
The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations: $a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$ are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.

1985 Traian Lălescu, 2.1
Tags: equations , algebra , Floor
Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

2011 District Olympiad, 1
Tags: Increasing function , algebra , equations , easy , district olympiad
Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

2006 Petru Moroșan-Trident, 1
Tags: equations , monotony , Floor , algebra
Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $ [i]Nedelcu Ion[/i]

2018 VJIMC, 1
Tags: equations , real numbers , powers
Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

2004 Nicolae Coculescu, 2
Tags: floor function , algebra , equations
Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

2004 Nicolae Coculescu, 2
Tags: quadratics , monotony , equations , algebra , trigonometry , logarithms
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]

2006 Mathematics for Its Sake, 1
Tags: fractional part , algebra , equations , Floor
Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.