IMO 1995 SL N2

Let Z denote the set of all integers. Prove that for any integers

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IMO 1995 SL N2

Origin: RUS | Category: Number Theory

Problem

Let Z denote the set of all integers. Prove that for any integers A and B, one can find an integer C for which M1 = {x2 +Ax+B : x \inZ} and M2 = {2x2 + 2x + C : x \inZ} do not intersect.

Solution

If A is odd, then every number in M1 is of the form x(x + A) + B \equivB (mod 2), while numbers in M2 are congruent to C modulo 2. Thus it is enough to take C \equivB + 1 (mod 2).

If A is even, then all numbers in M1 have the form  X + A 2 + B −A2 and are congruent to B −A2 4 or B −A2 4 + 1 modulo 4, while numbers in M2 are congruent to C modulo 4. So one can choose any C \equivB −A2 4 + 2 (mod 4).