MathDB

23

Part of 1979 IMO Longlists

Problems(1)

All elements of the set E of pairs of positive integers

Source: IMO LongList 1979 - P23

6/1/2011
Consider the set EE consisting of pairs of integers (a,b)(a, b), with a≥1a \geq 1 and b≥1b \geq 1, that satisfy in the decimal system the following properties:
(i) bb is written with three digits, as α2α1α0‾\overline{\alpha_2\alpha_1\alpha_0}, α2≠0\alpha_2 \neq 0;
(ii) aa is written as βp…β1β0‾\overline{\beta_p \ldots \beta_1\beta_0} for some pp;
(iii) (a+b)2(a + b)^2 is written as βp…β1β0α2α1α0‾.\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.
Find the elements of EE.
modular arithmeticnumber theory proposednumber theory