Found problems: 744
1989 Greece National Olympiad, 1
Find all real solutions of $$ \begin{matrix}
\sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\
\sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15}
\end{matrix}$$
2011 Dutch Mathematical Olympiad, 4
Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations:
$a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.
2018 Hanoi Open Mathematics Competitions, 6
Write down all real numbers $(x, y)$ satisfying two conditions: $x^{2018} + y^2 = 2$, and $x^2 + y^{2018} = 2$.
1969 IMO Shortlist, 41
$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]
1969 German National Olympiad, 4
Solve the system of equations:
$$|\log_2(x + y)| + | \log_2(x - y)| = 3$$
$$xy = 3$$
2010 Saudi Arabia Pre-TST, 4.4
Find all pairs $(x, y)$ of real numbers that satisfy the system of equations
$$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\
y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$
1978 Vietnam National Olympiad, 4
Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.
1986 IMO Shortlist, 7
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:
\[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\]
Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
1996 Denmark MO - Mohr Contest, 2
Determine all sets of real numbers $x,y,z$ which satisfy the system of equations
$$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$
1976 IMO Longlists, 25
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
2020 MMATHS, 3
Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.
2012 Middle European Mathematical Olympiad, 1
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]
2023 Bosnia and Herzegovina Junior BMO TST, 1.
Determine all real numbers $a, b, c, d$ for which
$ab+cd=6$
$ac+bd=3$
$ad+bc=2$
$a+b+c+d=6$
2004 Canada National Olympiad, 1
Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations:
\[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\
xz & \equal{} & y \minus{} x \minus{} z \\
yz & \equal{} & x \minus{} y \minus{} z \end{array} \right.
\]
2006 Federal Competition For Advanced Students, Part 2, 3
Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$.
$ x \plus{} y^2 \plus{} z^3 \equal{} A$
$ \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A}$
$ xy^2z^3 \equal{} A^2$
2011 Canadian Mathematical Olympiad Qualification Repechage, 3
Determine all solutions to the system of equations:
\[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\]
[This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]
2016 Irish Math Olympiad, 8
Suppose $a, b, c$ are real numbers such that $abc \ne 0$.
Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$.
Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.
2024 Kyiv City MO Round 2, Problem 1
Solve the following system of equations in real numbers:
$$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$
[i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]
2009 Tuymaada Olympiad, 1
Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational.
[i]Proposed by A. Golovanov[/i]
2011 Hanoi Open Mathematics Competitions, 6
Find all pairs $(x, y)$ of real numbers satisfying the system :
$\begin{cases} x + y = 2 \\
x^4 - y^4 = 5x - 3y \end{cases}$
2015 India Regional MathematicaI Olympiad, 3
Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).
2004 Federal Competition For Advanced Students, P2, 5
Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\
b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\
c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\
d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$
2024 Belarusian National Olympiad, 9.1
Find all triples $(x,y,z)$ of positive real numbers such that
$$
\begin{cases}
2x^2+y^3=3 \\
3y^2+z^3=4 \\
4z^2+x^3=5 \\
\end{cases}
$$
[i]M. Zorka[/i]
1988 All Soviet Union Mathematical Olympiad, 484
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?
2016 JBMO TST - Turkey, 1
Find all pairs $(x, y)$ of real numbers satisfying the equations
\begin{align*} x^2+y&=xy^2 \\
2x^2y+y^2&=x+y+3xy.
\end{align*}