Found problems: 744
2006 Grigore Moisil Urziceni, 3
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]
2016 Chile National Olympiad, 5
Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations:
$$x^2 - y = (z -1)^2 $$
$$y^2 - z = (x -1)^2$$
$$z^2 - x = (y - 1)^2$$
2018 Canadian Mathematical Olympiad Qualification, 1
Determine all real solutions to the following system of equations:
$$
\begin{cases}
y = 4x^3 + 12x^2 + 12x + 3\\
x = 4y^3 + 12y^2 + 12y + 3.
\end{cases}
$$
2005 MOP Homework, 6
Solve the system of equations:
$x^2=\frac{1}{y}+\frac{1}{z}$,
$y^2=\frac{1}{z}+\frac{1}{x}$,
$z^2=\frac{1}{x}+\frac{1}{y}$.
in the real numbers.
1953 Poland - Second Round, 4
Solve the system of equations $$ \qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}$$
2025 Kosovo National Mathematical Olympiad`, P1
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations:
$$\begin{cases}
ab-c = 3 \\
a+bc = 4 \\
a^2+c^2 = 5\end{cases}$$
2015 Kosovo Team Selection Test, 3
It's given system of equations
$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$
..........
$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$
such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
2003 Czech-Polish-Slovak Match, 1
Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*}
\max\{1, x_1\} &= x_2 \\
\max\{2, x_2\} &= 2x_3 \\
&\cdots \\
\max\{n, x_n\} &= nx_1. \\
\end{align*}
V Soros Olympiad 1998 - 99 (Russia), 9.3
Solve the system of equations:
$$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$
2009 Hanoi Open Mathematics Competitions, 6
Suppose that $4$ real numbers $a, b,c,d$ satisfy the conditions $\begin{cases} a^2 + b^2 = 4\\
c^2 + d^2 = 4 \\
ac + bd = 2 \end{cases}$
Find the set of all possible values the number $M = ab + cd$ can take.
2008 JBMO Shortlist, 4
Find all triples $(x,y,z)$ of real numbers that satisfy the system
$\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$
2021 Dutch BxMO TST, 2
Find all triplets $(x, y, z)$ of real numbers for which
$$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$
2004 Austrian-Polish Competition, 3
Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:
$
\begin{matrix}
a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\
b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\
c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\
d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\
\end{matrix}
$
1990 Austrian-Polish Competition, 4
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$
II Soros Olympiad 1995 - 96 (Russia), 10.4
Solve the system of equations
$$\begin{cases} x^2+ [y]=10
\\ y^2+[x]=13
\end{cases}$$
($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).
1978 IMO Shortlist, 16
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.
2008 ITest, 8
The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \begin{align*}2x+3y+3z&=8,\\3x+2y+3z&=808,\\3x+3y+2z&=80808.\end{align*} Their goal is not to find a solution $(x,y,z)$ to the system, but instead to compute the sum of the variables. Find the value of $x+y+z$.
1966 IMO Shortlist, 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.
1999 Junior Balkan Team Selection Tests - Moldova, 1
Solve in $R$ the system:
$$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\
\dfrac{xyz}{y + z - 1}= 1998000 \\ \\
\dfrac{xyz}{z+x}= 1998000 \end{cases}$$
III Soros Olympiad 1996 - 97 (Russia), 9.7
Solve the system of equations:
$$\begin{cases} xy+zu=14
\\ xz+yu=11
\\ xu+yz=10
\\ x+y+z+u=10
\end{cases}$$
2014 Singapore Junior Math Olympiad, 4
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
$$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$
2020 BMT Fall, 22
Suppose that $x, y$, and $z$ are positive real numbers satisfying
$$\begin{cases} x^2 + xy + y^2 = 64 \\
y^2 + yz + z^2 = 49 \\
z^2 + zx + x^2 = 57 \end{cases}$$
Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
1980 AMC 12/AHSME, 14
If the function $f$ is defined by
\[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is
$\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
1983 Austrian-Polish Competition, 5
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system
$$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\
a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\
a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$
has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
2006 Flanders Math Olympiad, 4
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that
\[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]