MathDB

5

Part of 2023 Kazakhstan National Olympiad

Problems(2)

Equation in primes

Source: Kazakhstan Junior Olympiad 2023 / Problem 5

3/22/2023
Solve the given equation in prime numbers p3+q3+r3=p2qrp^3+q^3+r^3=p^2qr
number theoryprime numbers
Two weird numbers are coprime

Source: Kazakhstan National Olympiad (10-11).5

3/22/2023
Given are positive integers a,b,m,ka, b, m, k with k≥2k \geq 2. Prove that there exist infinitely many nn, such that gcd⁡(φm(n),⌊an+bk⌋)=1\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1, where φm(n)\varphi_m(n) is the mm-th iteration of φ(n)\varphi(n).
number theory