Found problems: 91
2012 Argentina National Olympiad, 1
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
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!).
1933 Eotvos Mathematical Competition, 1
Let $a, b,c$ and $d$ be rea] numbers such that $a^2 + b^2 = c^2 + d^2 = 1$ and $ac + bd = 0$. Determine the value of $ab + cd$.
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
2003 Junior Balkan Team Selection Tests - Moldova, 6
The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$.
1964 Poland - Second Round, 4
Find the real numbers $ x, y, z $ satisfying the system of equations
$$(z - x)(x - y) = a $$
$$(x - y)(y - z) = b$$
$$(y - z)(z - x) = c$$
where $ a, b, c $ are given real numbers.
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 $b = y$
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
1983 Swedish Mathematical Competition, 6
Show that the only real solution to
\[\left\{ \begin{array}{l}
x(x+y)^2 = 9 \\
x(y^3 - x^3) = 7 \\
\end{array} \right.
\]
is $x = 1$, $y = 2$.
1992 Swedish Mathematical Competition, 3
Solve:
$$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\
2x_2 - 5x_3 + 3x4 \ge 0 \\
...\\
2x_{23} - 5x_{24} + 3x_{25} \ge 0\\
2x_{24} - 5x_{25} + 3x_1 \ge 0\\
2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$
2010 Saudi Arabia Pre-TST, 4.1
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
1999 Romania National Olympiad, 1
Solve the system $$\begin{cases} \displaystyle 4^{-x}+27^{-y}= \frac{5}{6} \\ \displaystyle 27^y-4^x \le 1 \\ \displaystyle \log_{27}y-\log_4 x \ge \frac{1}{6} \end{cases}.$$
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:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
2013 Hanoi Open Mathematics Competitions, 9
Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\
\frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$
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.
2010 Saudi Arabia BMO TST, 4
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$