IMO qualification
To different numbers, a and b fulfills these to equations.
and
they have a common solution.
what are all possible values of a+b?
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1 solution
if x is a solution to both equations so is x to the equation we get for subtracting the last equation from the first. by doing this we get: a x − b x + 2 b − 2 a = 0 this equation is the same as ( a − b ) x = 2 ( a − b ) because a and b are different we can divide by a-b≠0. this gives us x=2 now we just plug this value into one of the equations. 2 2 0 1 9 + 2 a + 2 b = 0 where [a+b= − 2 2 2 0 1 9 ] =[ -2^{2018}] hence the possible values of a+b are [ -2^{2018}]