MathDB

4

Part of BIMO 2022

Problems(2)

Sum of P(x_i) are equal

Source: Own. IMO 2022 Malaysian Training Camp 1

1/29/2022
Given a polynomial P∈Z[X]P\in \mathbb{Z}[X] of degree kk, show that there always exist 2d2d distinct integers x1,x2,⋯x2dx_1, x_2, \cdots x_{2d} such that P(x1)+P(x2)+⋯P(xd)=P(xd+1)+P(xd+2)+⋯+P(x2d)P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d}) for some d≤k+1d\le k+1.
[Extra: Is this still true if d≤kd\le k? (Of course false for linear polynomials, but what about higher degree?)]
algebrapolynomial
Minimum degree in 2 variables

Source: Malaysian IMO TST 2022 P4

5/13/2022
Given a positive integer nn, suppose that P(x,y)P(x,y) is a real polynomial such that P(x,y)=11+x+yfor all x,y∈{0,1,2,…,n}P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} What is the minimum degree of PP?
Proposed by Loke Zhi Kin
algebra