MathDB

4

Part of 2023 Malaysian IMO Training Camp

Problems(2)

a^2+b^2+c^2 divides a!+b!+c!

Source: Malaysian IMO TST 2023 P4

4/30/2023
Do there exist infinitely many triples of positive integers (a,b,c)(a, b, c) such that aa, bb, cc are pairwise coprime, and a!+b!+c!a! + b! + c! is divisible by a2+b2+c2a^2 + b^2 + c^2?
Proposed by Anzo Teh Zhao Yang
number theory
Bounds on number of divisors

Source: Malaysian SST 2023 P4

8/30/2023
Find the largest constant c>0c>0 such that for every positive integer n≥2n\ge 2, there always exist a positive divisor dd of nn such that d≤nandτ(d)≥cτ(n)d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)} where τ(n)\tau(n) is the number of divisors of nn.
Proposed by Mohd. Suhaimi Ramly
number theory