Found problems: 744
2017-IMOC, A3
Solve the following system of equations:
$$\begin{cases}
x^3+y+z=1\\
x+y^3+z=1\\
x+y+z^3=1\end{cases}$$
2021 Czech-Polish-Slovak Junior Match, 5
Find all three real numbers $(x, y, z)$ satisfying the system of equations $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}$$ $$x^2 + y^2 + z^2 = xy + yz + zx + 4$$
II Soros Olympiad 1995 - 96 (Russia), 10.4
Solve system of equations
$$\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1
\\ x+\dfrac{x-y}{x^2+y^2}=2
\end{cases}$$
2016 Dutch BxMO TST, 2
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations
$\begin{cases} x^2 - y = (z - 1)^2\\
y^2 - z = (x - 1)^2 \\
z^2 - x = (y -1)^2 \end{cases}$.
1966 German National Olympiad, 4
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:
$$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$
$$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$
$$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$
$$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$
2008 Tournament Of Towns, 2
Solve the system of equations $(n > 2)$
\[\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, \] \[x_1-x_2=1.\]
1966 IMO Longlists, 62
Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.
2011 IFYM, Sozopol, 6
Solve the following system of equations in integers:
$\begin{cases}
x^2+2xy+8z=4z^2+4y+8\\
x^2+y+2z=156 \\
\end{cases}$
2005 Austrian-Polish Competition, 7
For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$:
$$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$
where
$$S=x_1^2+x_2^2+\dots+x_n^2.$$
2017 Hanoi Open Mathematics Competitions, 8
Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases}
x^3 - 3x = 4 - y \\
2y^3 - 6y = 6 - z \\
3z^3 - 9z = 8 - x\end{cases}$
1976 IMO Shortlist, 5
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\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
2001 Singapore Senior Math Olympiad, 1
Let $n$ be a positive integer. Suppose that the following simultaneous equations
$$\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}$$
has a solution, where $x_1 x_2,.., x_n$ are the unknowns. Find the smallest possible positive integer $n$. Justify your answer.
2011 Swedish Mathematical Competition, 3
Find all positive real numbers $x, y, z$, such that
$$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$
2022 Latvia Baltic Way TST, P1
Find all triplets of positive real numbers $(x,y,z)$ that satisfy the following system of equations:
$$ \begin{cases}
x+y^2+z^3=3\\
y+z^2+x^3=3\\
z+x^2+y^3=3.
\end{cases}$$
1994 Poland - First Round, 2
Given a positive integer $n \geq 2$. Solve the following system of equations:
$
\begin{cases}
\ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\
\ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\
&\dots \\
\ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\
\end{cases}
$
2018 Dutch Mathematical Olympiad, 3
Determine all triples $(x, y,z)$ consisting of three distinct real numbers, that satisfy the following system of equations:
$\begin {cases}x^2 + y^2 = -x + 3y + z \\
y^2 + z^2 = x + 3y - z \\
x^2 + z^2 = 2x + 2y - z \end {cases}$
2024 Indonesia MO, 1
Determine all positive real solutions $(a,b)$ to the following system of equations.
\begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}
1993 Denmark MO - Mohr Contest, 3
Determine all real solutions $x,y$ to the system of equations
$$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$
1983 All Soviet Union Mathematical Olympiad, 352
Find all the solutions of the system
$$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$
2022 Israel TST, 1
Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$:
\begin{align*}
&(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\
=&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\
&\qquad \qquad \qquad \qquad \vdots \\
=&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1})
\end{align*}
has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.
2011 Morocco National Olympiad, 1
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
1998 German National Olympiad, 6a
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}
2019 German National Olympiad, 1
Determine all real solutions $(x,y)$ of the following system of equations:
\begin{align*}
x&=3x^2y-y^3,\\
y &= x^3-3xy^2
\end{align*}
2024 AMC 12/AHSME, 17
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
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.