Found problems: 91
1969 German National Olympiad, 4
Solve the system of equations:
$$|\log_2(x + y)| + | \log_2(x - y)| = 3$$
$$xy = 3$$
1926 Eotvos Mathematical Competition, 1
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
1972 Poland - Second Round, 1
Prove that there are no real numbers $ a, b, c $, $ x_1, x_2, x_3 $ such that for every real number $ x $
$$ ax^2 + bx + c = a(x - x_2)(x - x_3) $$
$$bx^2 + cx + a = b(x - x_3) (x - x_1)$$
$$cx^2 + ax + b = c(x - x_1) (x - x_2)$$
and $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.
1968 German National Olympiad, 1
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled:
$$x_1 + ax_2 + x_3 = b $$
$$x_2 + ax_3 + x_4 = b $$
$$x_3 + ax_4 + x_1 = b $$
$$x_4 + ax_1 + x_2 = b$$
Here $a$ and $b$ are real numbers (case distinction!).
2022 IFYM, Sozopol, 1
Find all triples of complex numbers $(x, y, z)$ for which
$$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$
1979 Swedish Mathematical Competition, 1
Solve the equations:
\[\left\{ \begin{array}{l}
x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\
2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\
3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\
\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\
(n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\
n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1
\end{array} \right.
\]
2021 Dutch BxMO TST, 2
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
2017 OMMock - Mexico National Olympiad Mock Exam, 3
Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied:
\begin{eqnarray*} x=a^2+2b^2\\
y=c^2+d^2\\
z=ac+2bd\\
\end{eqnarray*}
[i]Proposed by Isaac Jiménez[/i]
2006 Denmark MO - Mohr Contest, 2
Determine all sets of real numbers $(x,y,z)$ which fulfills
$$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$
1981 Swedish Mathematical Competition, 2
Does
\[\left\{ \begin{array}{l}
x^y = z \\
y^z = x \\
z^x = y \\
\end{array} \right.
\]
have any solutions in positive reals apart from $x = y = z= 1$?
2009 Bosnia And Herzegovina - Regional Olympiad, 2
Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers
2010 Saudi Arabia Pre-TST, 4.1
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
1950 Poland - Second Round, 1
Solve the system of equations
$$\begin{cases} x^2+x+y=8\\
y^2+2xy+z=168\\
z^2+2yz+2xz=12480 \end{cases}$$
1995 Swedish Mathematical Competition, 3
Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$
2009 Denmark MO - Mohr Contest, 2
Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$
2004 Peru MO (ONEM), 3
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
2007 Cuba MO, 1
Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$
2016 Hanoi Open Mathematics Competitions, 9
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$.
2021 Dutch IMO TST, 2
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2017 District Olympiad, 2
Solve in $ \mathbb{Z} $ the system:
$$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$
1983 Swedish Mathematical Competition, 3
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
2011 Mathcenter Contest + Longlist, 7
Given $k_1,k_2,...,k_n\in R^+$, find all the naturals $n$ such that
$$k_1+k_2+...+k_n=2n-3$$
$$\frac{1}{k_1}+\frac{1}{k_2}+...+\frac{1}{k_n}=3$$
[i](Zhuge Liang)[/i]
2022 Azerbaijan National Mathematical Olympiad, 4
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2005 Austria Beginners' Competition, 3
Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations:
$$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$