Difference between revisions of "2018 OIM Problems/Problem 1"
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Both solutions work. Thus the solution sets are either <math>x_i=0</math> for all <math>i</math> or <math>x_i=(n-1)^{-\frac{2018}{2017}}</math> for all <math>i</math>. | Both solutions work. Thus the solution sets are either <math>x_i=0</math> for all <math>i</math> or <math>x_i=(n-1)^{-\frac{2018}{2017}}</math> for all <math>i</math>. | ||
− | ~ | + | ~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406] |
== See also == | == See also == | ||
[[OIM Problems and Solutions]] | [[OIM Problems and Solutions]] |
Revision as of 17:39, 23 March 2025
Problem
For each natural number , find the integer solutions to the following system of equations:
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Since the right-hand side (RHS) of each equation is nonnegative, then the left-hand side (LHS) is too; this implies that each is nonnegative. As a result, we can take the
root of both sides of the first and second equations, resulting in:
Subtracting yields:
If
, then the LHS is positive but the RHS is negative, which is not possible, and the same goes for
. Thus
, and if we repeat this process for each consecutive pair of equations we eventually find that all
are equal.
Using this property on the first equation results in:
Both solutions work. Thus the solution sets are either
for all
or
for all
.