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2 two-digit nos in base z, one equal to the other one read in reverse order

Source: Czech and Slovak MO, III A, 2003 p5

January 12, 2020
number theorynumber basetrinomial

Problem Statement

Show that, for each integer z≥3z \ge 3, there exist two two-digit numbers AA and BB in base zz, one equal to the other one read in reverse order, such that the equation x2−Ax+Bx^2 -Ax+B has one double root. Prove that this pair is unique for a given zz. For instance, in base 1010 these numbers are A=18,B=81A = 18, B = 81.
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