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Part of 2018 China Northern MO

Problems(2)

Inequality on Nonnegative Reals

Source: 2018 China North Mathematical Olympiad Grade 10 Test 1 P2

7/26/2018
Let aa,bb,cc be nonnegative reals such that a2+b2+c2+ab+23ac+43bc=1a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1 Find the maximum and minimum value of a+b+ca+b+c.
inequalitiesChinathree variable inequality
good prime

Source: 2018 China North Mathematical Olympiad Grade 11 Test 1 P2

6/24/2019
Let pp be a prime. We say pp is good if and only if for any positive integer a,b,a,b, such that a≡b(modp)⇔a3≡b3(modp).a\equiv b (\textup{mod}p)\Leftrightarrow a^3\equiv b^3 (\textup{mod}p).Prove that (1)There are infinite primes pp which are good; (2)There are infinite primes pp which are not good.
number theory