When two red amoebas join, the result in one blue amoeba; when a red amoeba and a blue amoeba join, they turn into three red amoeba; and when two blue amoeba join, they become four red amoeba. Fernando observes a test tube with initially 201 blue amoebas and 112 red amoebas.
a. Is it possible that after some amoebas transform the test tube contains 100 blue amoebas and 314 red amoebas?
b. Is it possible that after some amoebas transform the test tube contains 99 blue amoebas and 314 red amoebas?
Solution
If the number of blue amoebas is b and the number of red amoebas is r then 2b+r is invariant: indeed, whenever one blue amoeba appears/disappears, two red amoebas disappear/appear. In the problem, such number is 2⋅201+112=514.
a. Since 2⋅100+314=514, it can be possible. Indeed, if 50 pairs of blue amoebas turn into 200 red amoebas and a pair of one amoeba of each color turn into 3 red amoebas, we have 201−2⋅50−1=100 blue amoebas and 112+200−1+3=314 red amoebas.
b. Since 2⋅99+314=512, it is not possible.
Want a route through all this instead of an archive? The track
puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.
Source: MathNet,
licensed CC-BY-4.0.
Statement reproduced verbatim; metadata (topic, difficulty) added by this project.