2018 IMO Problems/Problem 2
Find all numbers for which there exists real numbers
satisfying
and
for
Solution
We find at least one series of real numbers for for each
and we prove that if
then the series does not exist.
Case 1
Let We get system of equations
We subtract the first equation from the second and get:
So
Case 1'
Let
Real numbers
satisfying
and
.
Case 2
Let We get system of equations
We multiply each equation by the number on the right-hand side and get:
We multiply each equation by a number that precedes a pair of product numbers in a given sequence
So we multiply the equation with product
by
, we multiply the equation with product
by
etc. We get: