MathDB

Problem 2

Part of 2005 Slovenia National Olympiad

Problems(4)

p^2+11 less than 11 divisors

Source: Slovenia National MO 2005 1st Grade P2

4/5/2021
Find all prime numbers pp for which the number p2+11p^2+11 has less than 1111 divisors.
number theoryprime numbers
(p+1)^q square

Source: Slovenia National MO 2005 2nd Grade P2

4/5/2021
For which prime numbers pp and qq is (p+1)q(p+1)^q a perfect square?
number theoryprime numbers
p^3+2p^2+p has 42 divisors

Source: Slovenia National MO 2005 3rd Grade P2

4/5/2021
Find the smallest prime number pp for which the number p3+2p2+pp^3+2p^2+p has exactly 4242 divisors.
number theory
sequence defined on geometrical progression with log

Source: Slovenia National MO 2005 4th Grade P2

4/5/2021
Let (an)(a_n) be a geometrical progression with positive terms. Define Sn=log⁡a1+log⁡a2+…+log⁡anS_n=\log a_1+\log a_2+\ldots+\log a_n. Prove that if Sn=SmS_n=S_m for some m≠nm\ne n, then Sn+m=0S_{n+m}=0.
Sequencealgebra