MathDB

3

Part of 1990 Romania Team Selection Test

Problems(2)

the lcm and prime

Source: Laurentiu Panaitopol,Romania TST 1990

8/10/2005
Prove that for any positive integer nn, the least common multiple of the numbers 1,2,…,n1,2,\ldots,n and the least common multiple of the numbers: (n1),(n2),…,(nn)\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n} are equal if and only if n+1n+1 is a prime number. Laurentiu Panaitopol
number theoryleast common multiplecombinatorics proposedcombinatorics
2P(2x^2 -1) = P(x)^2 -1

Source: Romania IMO TST 1990 p3

2/19/2020
Find all polynomials P(x)P(x) such that 2P(2x2−1)=P(x)2−12P(2x^2 -1) = P(x)^2 -1 for all xx.
polynomialalgebrafunctionalfunctional equation