Maths Olympiad Prep

Library / /18 of 32

Algebra Difficulty 5.9 AIME, harder Prove it Netherlands

We have two integers consisting of two digits, and both numbers do not start with a 00. If you add these numbers, you get the number SS. If you interchange the two digits of both numbers and add the new numbers, you get 4S4S.
Determine all possible pairs of two-digit numbers satisfying these constraints. Make sure to clearly indicate in your answer which numbers form a pair.

Solution

{14,19}\{14, 19\}, {15,18}\{15, 18\}, and {16,17}\{16, 17\}

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.