MathDB

6

Part of 2019 Tuymaada Olympiad

Problems(2)

Fun easy number theory

Source: Tuymaada 2019 Senior P5 out of 8

7/15/2019
Let S\mathbb{S} is the set of prime numbers that less or equal to 26. Is there any a1,a2,a3,a4,a5,a6∈Na_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N} such that gcd(ai,aj)∈Sfor 1≤i≠j≤6 gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6 and for every element pp of S\mathbb{S} there exists a pair of 1≤k≠l≤6 1\leq k \ne l \leq 6 such that s=gcd(ak,al)?s=gcd(a_k,a_l)?
number theoryFunprime numbers
Prove that expression is not square of a natural number

Source: Tuymaada 2019 P6 out of 8

7/15/2019
Prove that the expression (14+12+1)(24+22+1)…(n4+n2+1) (1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1) is not square for all n∈Nn \in \mathbb{N}
number theorysquareTuymaada