MathDB

2

Part of 2023 China Second Round

Problems(2)

exponents and logarithms

Source: 2023 China Second Round Test 1 P2

6/1/2024
if a,b∈R+,alog⁡b=2a^{\log b}=2,alog⁡ablog⁡b=5a^{\log a}b^{\log b}=5,find out (ab)log⁡ab(ab)^{\log ab}
algebralogarithms
$n^2/2^n \neq m^2/2^m$ for Every $m$

Source: Problem 2, China National High School Mathematics League 2023

9/10/2023
For some positive integer nn, nn is considered a <spanclass=′latex−bold′>unique</span><span class='latex-bold'>unique</span> number if for any other positive integer m≠nm\neq n, {2nn2}≠{2mm2}\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\} holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.
number theorychang 09