MathDB

8

Part of 2015 South East Mathematical Olympiad

Problems(2)

Product of two elements is an element

Source: China Southeast Math Olympiad 2015 Day 2 P8

6/6/2017
For any integers m,nm,n, we have the set A(m,n)={x2+mx+n∣x∈Z}A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}, where Z\mathbb{Z} is the integer set. Does there exist three distinct elements a,b,ca,b,c which belong to A(m,n)A(m,n) and satisfy the equality a=bca=bc?
number theory
2015 China South East MO Grade 11 P8

Source: 2015 China South East MO Grade 11 P8

1/16/2018
Find all prime number pp such that there exists an integer-coefficient polynomial f(x)=xp−1+ap−2xp−2+…+a1x+a0f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0 that has p−1p-1 consecutive positive integer roots and p2∣f(i)f(−i)p^2\mid f(i)f(-i), where ii is the imaginary unit.
algebranumber theory