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Part of 2021 South East Mathematical Olympiad

Problems(2)

China South East Mathematical Olympiad 2021 Grade10 P7

Source:

7/30/2021
Let a,b,ca,b,c be pairwise distinct positive real, Prove thatab+bc+ca(a+b)(b+c)(c+a)<17(1∣a−b∣+1∣b−c∣+1∣c−a∣).\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).
inequalitiesalgebra
China South East Mathematical Olympiad 2021 Grade11 P7

Source:

7/30/2021
Determine all the pairs of positive odd integers (a,b),(a,b), such that a,b>1a,b>1 and 7φ2(a)−φ(ab)+11φ2(b)=2(a2+b2),7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2), where φ(n)\varphi(n) is Euler's totient function.
number theoryEuler s totient function