MathDB

5

Part of 2024 Malaysian IMO Training Camp

Problems(2)

Good residue sets

Source: Malaysian IMO TST 2024 P5

4/21/2024
Let nn be an odd integer and m=ϕ(n)m=\phi(n) be the Euler's totient function. Call a set of residues T={a1,⋯ ,ak}(modn)T=\{a_1, \cdots, a_k\} \pmod n to be good if gcd⁡(ai,n)>1\gcd(a_i, n) > 1 ∀i\forall i, and gcd⁡(ai,aj)=1,∀i≠j\gcd(a_i, a_j) = 1, \forall i \neq j. Define the set SnS_n consisting of the residues ∑i=1kaim(modn)\sum_{i=1}^k a_i ^m\pmod{n} over all possible residue sets T={a1,⋯ ,ak}T=\{a_1,\cdots,a_k\} that is good. Determine ∣Sn∣|S_n|.
Proposed by Anzo Teh Zhao Yang
number theory
(a^2+1)(b^2+1)((a+b)^2+1) being a square

Source: Malaysian SST 2024 P5

9/5/2024
Do there exist infinitely many positive integers a,ba, b such that (a2+1)(b2+1)((a+b)2+1)(a^2+1)(b^2+1)((a+b)^2+1) is a perfect square?
Proposed Ivan Chan Guan Yu
algebra