MathDB

2

Part of 2016 Azerbaijan Team Selection Test

Problems(2)

Polynomial Equation

Source: Azerbaijan IMO TST 2016,D1 P2

5/26/2018
Find all polynomials P(x)P(x) with real coefficents, such that for all x,y,zx,y,z satisfying x+y+z=0x+y+z=0, the equation below is true: P(x+y)3+P(y+z)3+P(z+x)3=3P((x+y)(y+z)(z+x))P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))
algebrapolynomial
Interesting Polynomial

Source: Azerbaijan IMO TST 2016, D3 P2

5/26/2018
A positive interger nn is called rising if its decimal representation akak−1⋯a0a_ka_{k-1}\cdots a_0 satisfies the condition ak≤ak−1≤⋯≤a0a_k\le a_{k-1}\le\cdots \le a_0. Polynomial PP with real coefficents is called interger-valued if for all integer numbers nn, P(n)P(n) takes interger values. P(n)P(n) is called rising-valued if for all rising numbers nn, P(n)P(n) takes integer values. Does it necessarily mean that, "every rising-valued PP is also interger-valued PP"?
algebrapolynomial