MathDB

5

Part of 2014 Dutch IMO TST

Problems(2)

Lamps on a 2014x2014 board

Source: Dutch IMO TST I Problem 5

7/17/2014
On each of the 201422014^2 squares of a 2014×20142014 \times 2014-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least 10071007 light bulbs are on and changing the state of all 20142014 light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer kk such that from each starting situation there is a finite sequence of moves to a situation in which at most kk light bulbs are on.
combinatorics proposedcombinatorics
Polynomial such that |P(10)-P(0)|<1000

Source: Dutch IMO TST II Problem 5

7/17/2014
Let P(x)P(x) be a polynomial of degree n≤10n \le 10 with integral coefficients such that for every k∈{1,2,…,10}k \in \{1, 2, \dots, 10\} there is an integer mm with P(m)=kP(m) = k. Furthermore, it is given that ∣P(10)−P(0)∣<1000|P(10) - P(0)| < 1000. Prove that for every integer kk there is an integer mm such that P(m)=k.P(m) = k.
algebrapolynomialcalculusintegrationratioarithmetic sequencealgebra unsolved