Question 38
Let $n$ be a positive integer. Real numbers $a_1, a_2, \dots, a_{2n}$ satisfy the following conditions:
(1) For every $i = 1, 2, \dots, 2n - 1$, one has $0 \lt a_{i+1} - a_i \le 1$;
(2) Rounding the numbers $a_1, a_2, \dots, a_{2n}$ to the closest integer (numbers equidistant from two closest integers are rounded up) gives pairwise distinct positive integers.
Numbers $a_1, a_2, \dots, a_{2n}$ are placed as the numerators and denominators of $n$ fractions. Prove that the sum of the obtained fractions is greater than $\frac{n}{4}$.
(1) For every $i = 1, 2, \dots, 2n - 1$, one has $0 \lt a_{i+1} - a_i \le 1$;
(2) Rounding the numbers $a_1, a_2, \dots, a_{2n}$ to the closest integer (numbers equidistant from two closest integers are rounded up) gives pairwise distinct positive integers.
Numbers $a_1, a_2, \dots, a_{2n}$ are placed as the numerators and denominators of $n$ fractions. Prove that the sum of the obtained fractions is greater than $\frac{n}{4}$.