MathDB

2

Part of 2024 Malaysian IMO Training Camp

Problems(2)

Sum of certain images of P is still an image

Source: Own. Malaysian IMO TST 2024 P2

4/21/2024
Let kk be a positive integer. Find all collection of integers (a1,a2,⋯ ,ak)(a_1, a_2,\cdots, a_k) such that there exist a non-linear polynomial PP with integer coefficients, so that for all positive integers nn there exist a positive integer mm satisfying: P(n+a1)+P(n+a2)+...+P(n+ak)=P(m)P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)
Proposed by Ivan Chan Kai Chin
algebra
Common Sequences

Source: Malaysian SST 2024 P2

9/5/2024
A finite sequence of decimal digits from {0,1,⋯ ,9}\{0,1,\cdots, 9\} is said to be common if for each sufficiently large positive integer nn, there exists a positive integer mm such that the expansion of nn in base mm ends with this sequence of digits.
For example, 00 is common because for any large nn, the expansion of nn in base nn is 1010, whereas 0000 is not common because for any squarefree nn, the expansion of nn in any base cannot end with 0000.
Determine all common sequences.
Proposed by Wong Jer Ren
number theoryDigits