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IMO Shortlist 1983 problem 20

Find all solutions of the following system of $n$ equations in $n$ variables: $\begin{align*} x_{1}|x_{1}|-(x_{1}-a)&|x_{1}-a| = x_{2}|x_{2}|, \\ x_{2}|x_{2}|-(x_{2}-a)&|x_{2}-a| = x_{3}|x_{3}|, \\ &\vdots \\ x_{n}|x_{n}|-(x_{n}-a)&|x_{n}-a| = x_{1}|x_{1}| \\ \end{align*}$ where $a$ is a given number.

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IMO Shortlist 1967 problem 4

In what case does the system of equations

$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$

have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

IMO Shortlist 1967 problem 2

Find all real solutions of the system of equations:
$\sum^n_{k=1} x^i_k = a^i$ for $i = 1,2, \ldots, n.$

IMO Shortlist 1969 problem 41

$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$ , find an expression for the solution of the system $x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots$

IMO Shortlist 1971 problem 3

Knowing that the system
$x + y + z = 3,$ $x^3 + y^3 + z^3 = 15,$ $x^4 + y^4 + z^4 = 35,$
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$ , find the value of $x^5 + y^5 + z^5$ for that solution.

IMO Shortlist 1977 problem 6

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$ , does the following system of inequalities have:

$1 \leq -j + k + l \leq n$ $1 \leq i - k + l \leq n$ $1 \leq i - j + l \leq n$ $1 \leq i + j - k \leq n \ ?$

IMO Shortlist 1978 problem 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.