Authors: Martynnov P., Promynova N.

Six positive irreducible fractions are recorded, the sum of whose numerators is equal to the sum of their denominators. Pasha converted each of the improper fractions into a mixed number. Is it necessarily true that there will be two numbers with either the same whole parts or the same fractional parts?
For example, consider the fractions: $3 / 2=1 \frac{112}{2}, 7 / 3=2 \frac{1}{3}, 13 / 4=3 \frac{114}{4}, 21 / 5=4 \frac{1}{5}, { }^{31} / 6=51 / 6$ and $1 / 56$. The sum of their numerators, as well as the sum of their denominators, is 76. At the same time, both the integer parts and the fractional parts of these numbers are pairwise distinct.

## Answer

Not necessarily.