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| $[\underline{\text { Evenness and Oddness }}]$ |  |  |

Authors: Akopyan E., Kalinin D. $\underline{\text {. }}$.

Malvina wrote down 2016 common proper fractions in order: $1 / 2,1 / 3,2 / 3,1 / 4,2 / 4,3 / 4, \ldots$ (including reducible ones). Fractions whose value is less than $1 / 2$ she painted red, and the rest of the fractions - blue. By how much is the number of red fractions less than the number of blue fractions?

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Note that only one fraction with a denominator of 2 is recorded, exactly two fractions with a denominator of 3, exactly three with a denominator of 4, and so on. Since

$2016=64 \cdot 63: 2$, the number 2016 is equal to the sum of all natural numbers from 1 to 63. Therefore, Malvina recorded all fractions with denominators $2,3, \ldots, 64$.

Consider the fractions with the same denominator, recorded in ascending order. If the denominator is odd, then each of the fractions in the first half is less than $1 / 2$, and each of the fractions in the second half is greater than $1 / 2$. If the denominator is even, then there is a fraction equal to $1 / 2$, and of the remaining fractions, half are less than $1 / 2$, and the other half are greater than $1 / 2$.

Thus, the number of red fractions is less than the number of blue fractions by exactly the number of fractions equal to $1 / 2$. These fractions are $1 / 2,2 / 4,{ }^{3} / 6, \ldots,{ }^{32} / 64$, their number is 32.

## Answer

By 32.