Olympiad problems

1936 Moscow Mathematical Olympiad, 022

Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.

1946 Moscow Mathematical Olympiad, 108

Tags: number theory , 4-digit , remainder

Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.

1992 All Soviet Union Mathematical Olympiad, 562

Tags: number theory , divisible , 4-digit

Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?

1954 Moscow Mathematical Olympiad, 261

Tags: number theory , 4-digit

Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines: [img]https://cdn.artofproblemsolving.com/attachments/2/a/e1d3c68ec52e11ad59de755c3dbdc2cf54a81f.png[/img]

1945 Moscow Mathematical Olympiad, 099

Tags: combinatorics , algebra , sum , number theory , 4-digit

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

1998 Tournament Of Towns, 2

Tags: sum , product , digit , 4-digit , number theory

For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)