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IMO Shortlist 1969 problem 19

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$ ? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$ . Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$ . If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

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IMO Shortlist 1961 problem 1

Solve the system of equations: $x+y+z=a$ $x^2+y^2+z^2=b^2$ $xy=z^2$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

IMO Shortlist 1963 problem 4

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system $x_5+x_2=yx_1$ $x_1+x_3=yx_2$ $x_2+x_4=yx_3$ $x_3+x_5=yx_4$ $x_4+x_1=yx_5$ where $y$ is a parameter.

IMO Shortlist 1965 problem 2

Consider the sytem of equations
$a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0$ $a_{21}x_1+a_{22}x_2+a_{23}x_3 =0$ $a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0$ with unknowns $x_1, x_2, x_3$ . The coefficients satisfy the conditions:

a) $a_{11}, a_{22}, a_{33}$ are positive numbers;

b) the remaining coefficients are negative numbers;

c) in each equation, the sum ofthe coefficients is positive.

Prove that the given system has only the solution $x_1=x_2=x_3=0$ .

IMO Shortlist 1966 problem 31

Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$ . Which pairs $(x,m)$ of integers satisfy this equation ?

IMO Shortlist 1966 problem 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.

MEMO 2008 pojedinačno problem 1

Let $(a_n)^{\infty}_{n=1}$ be a sequence of integers with $a_{n} < a_{n+1}, \quad \forall n \geq 1.$ For all quadruple $(i,j,k,l)$ of indices such that $1 \leq i < j \leq k < l$ and $i + l = j + k$ we have the inequality $a_{i} + a_{l} > a_{j} + a_{k}.$ Determine the least possible value of $a_{2008}$ .