MathDB

3

Part of 1996 Bulgaria National Olympiad

Problems(2)

Bulgaria National Olympiad 1996

Source:

6/10/2020
The quadratic polynomials ff and gg with real coefficients are such that if g(x)g(x) is an integer for some x>0x>0, then so is f(x)f(x). Prove that there exist integers m,nm,n such that f(x)=mg(x)+nf(x)=mg(x)+n for all xx.
algebrapolynomial
Bulgaria National Olympiad 1996

Source: Bulgaria National Olympiad 1996, Fourth round, P6

3/21/2021
A square table of size 7×77\times 7 with the four corner squares deleted is given.
[*] What is the smallest number of squares which need to be colored black so that a 5−5-square entirely uncolored Greek cross (Figure 1) cannot be found on the table? [*] Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive.
[asy] size(3.5cm); usepackage("amsmath"); MP("\text{Figure }1.", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy]
combinatorics