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Part of 2023 Puerto Rico Team Selection Test

Problems(2)

2023-digit palindromes - Puerto Rico TST 2023.1.1

Source:

3/23/2024
A number is capicua if it is read equally from left to right as it is from right to the left. For example, 2343223432 and 111111111111 are capicua numbers. (a) How many 20232023-digit capicua numbers can be formed if you want them to have at least 20222022 equal digits? (b) How many 20232023-digit capicua numbers can be formed if you want them to have at least 20212021 equal digits?
number theory
a! +b! = 2^{c!}

Source: Puerto Rico TST 2023.2.1

3/24/2024
Determine all triples (a,b,c)(a, b, c) of positive integers such that a!+b!=2c!.a! +b! = 2^{c!} .
number theoryDiophantine equationdiophantine