Question 1
Peter starts with a positive integer and multiplies it by 5, then multiplies the result by 5, and continues for a total of $k$ multiplications. Neither the initial integer nor any result contains the decimal digit 7. Prove that there is a positive integer which can be multiplied successively by 2 a total of $k$ times with no initial or resulting number containing 7. (4 points)
Neutral editorial illustration.
Practice scoring adds all whole-question scores (maximum 24). Official ITOT scoring adds only the three highest whole-question scores (maximum 16), after summing subparts within each question. The importer does not enforce best-three scoring. Duration: 300 minutes from historical organizer general rules, not an individually printed duration on this paper.
Neutral editorial illustration.
Practice scoring adds all whole-question scores (maximum 24). Official ITOT scoring adds only the three highest whole-question scores (maximum 16), after summing subparts within each question. The importer does not enforce best-three scoring. Duration: 300 minutes from historical organizer general rules, not an individually printed duration on this paper.