Maths Olympiad Prep

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Combinatorics Difficulty 6.3 National Olympiad Prove it South Africa

One hundred glasses are arranged in a 10×1010 \times 10 array. Now we pick aa of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick bb of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves in a glass have the same colour, then that colour remains as it is.

a. Determine all possible combinations of values for aa and bb so that exactly half of the glasses contain green liquid at the end.

b. Is it possible that precisely one quarter of the glasses contain green liquid at the end?

Solution

The total number of glasses that are green at the end of the procedure is
a(10−b)+b(10−a)=10a+10b−2ab=2(5a+5b−ab). a(10 - b) + b(10 - a) = 10a + 10b - 2ab = 2(5a + 5b - ab).
We immediately observe that this number is always even, so the number of green glasses cannot be 2525 (i.e., one quarter). Hence the answer to the second question is no.

For the first question, we have to solve the equation
10a+10b−2ab=50, 10a + 10b - 2ab = 50,
which is equivalent to
2ab−10a−10b+50=2(a−5)(b−5)=0. 2ab - 10a - 10b + 50 = 2(a - 5)(b - 5) = 0.
Thus exactly half of the glasses contain green liquid if either a=5a = 5 or b=5b = 5 (and the other is arbitrary).

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.