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Part of 2024 OMpD

Problems(2)

kawaii subsets

Source: 2024 5th OMpD L3 P1 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
We say that a subset T T of {1,2,…,2024}\{1, 2, \dots, 2024\} is kawaii if T T has the following properties: 1. T T has at least two distinct elements; 2. For any two distinct elements x x and y y of T T , x−y x - y does not divide x+y x + y .
For example, the subset T={31,71,2024} T = \{31, 71, 2024\} is kawaii, but T={5,15,75} T = \{5, 15, 75\} is not kawaii because 15−5=10 15 - 5 = 10 divides 15+5=20 15 + 5 = 20 .
What is the largest possible number of elements that a kawaii subset can have?
Subsetscombinatoricsnumber theory
find all the possible values for ompd

Source: 2024 5th OMpD L2 P1 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
Let O,M,PO, M, P and DD be distinct digits from each other, and different from zero, such that O<M<P<DO < M < P < D, and the following equation is true: OMPD‾×(OM‾−D‾)=MDDMP‾−OM‾ \overline{\text{OMPD}} \times \left( \overline{\text{OM}} - \overline{\text{D}} \right) = \overline{\text{MDDMP}} - \overline{\text{OM}}
(a) Using estimates, explain why it is impossible for the value of OO to be greater than or equal to 33. (b) Explain why OO cannot be equal to 11. (c) Is it possible for MM to be greater than or equal to 55? Justify. (d) Determine the values of MM, PP, and DD.
algebranumber theory