MathDB

P10

Part of 2023 Bangladesh Mathematical Olympiad

Problems(2)

Colouring the grid for points

Source: BdMO 2023 Secondary National P10 Higher Secondary National P8

2/12/2023
Joy has a square board of size n×nn \times n. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has 00 points. Every step, after colouring a cell cc, he takes the largest possible set SS that creates a "++" sign where all cells are coloured and cc lies in the centre. Then, he gets the size of set SS as points. After colouring the whole n×nn \times n board, what is the maximum possible amount of points he can get?
combinatoricsgrid
Polynomial with 2023 roots.

Source: BdMO 2023 Higher Secondary National P10

2/12/2023
Let all possible 20232023-degree real polynomials: P(x)=x2023+a1x2022+a2x2021+⋯+a2022x+a2023P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}, where P(0)+P(1)=0P(0)+P(1)=0, and the polynomial has 2023 real roots r1,r2,⋯r2023r_1, r_2,\cdots r_{2023} [not necessarily distinct] so that 0≤r1,r2,⋯r2023≤10\leq r_1,r_2,\cdots r_{2023}\leq1. What is the maximum value of r1⋅r2⋯r2023?r_1 \cdot r_2 \cdots r_{2023}?
algebrapolynomialinequalities