Fractions: Types, Comparison and Arithmetic

Given two proper fractions, $a/b$ and $c/d$, where $a, c < b, d$ and $b \neq d$, under what precise condition is it impossible to determine which fraction is larger without converting them to a common denominator or decimal form?

Consider the infinite series of fractions, $\sum_{n=1}^{\infty} f_n$, where each $f_n$ is a unique proper fraction with a denominator that is a power of 2 (i.e., $f_n = a_n / 2^{k_n}$, $a_n < 2^{k_n}$). What is the highest upper bound this series can converge to, given that no two fractions share the same value?

Given a set $S$ of $n$ distinct unit fractions ($1/a_i$, where $a_i$ are distinct positive integers), what is the tightest upper bound on the smallest possible value of the least common multiple (LCM) of the denominators $a_i$, such that the sum of these unit fractions equals 1?

In the realm of abstract algebra, consider the set of all proper fractions $a/b$ where $a$ and $b$ are coprime integers and $b$ is less than some integer $n$. If we define an operation $\star$ such that $(a/b) \star (c/d) = (ad + bc) / (bd)$, under what conditions does this set, along with the operation $\star$, form a group?

Let $F$ be a field and consider the field of rational functions $F(x)$. If we take a proper fraction $p(x) / q(x)$ in $F(x)$ (where the degree of $p(x)$ is less than the degree of $q(x)$), and $q(x)$ has a complete factorization into distinct linear terms, what method guarantees the unique decomposition of $p(x) / q(x)$ into a sum of partial fractions?

Given a continuous uniform distribution over the interval $[0, 1]$, consider the expected value of the reciprocal of a randomly chosen number $x$ from this interval. If this reciprocal is then expressed as an improper fraction $a/b$, what can be inferred regarding the properties of $a$ and $b$?

Consider an iterative process where a proper fraction is repeatedly transformed by adding its numerator to its denominator, creating a new fraction. Formally, $f_{n+1} = a_n / (a_n + b_n)$, where $f_n = a_n / b_n$ is the fraction at the $n$-th step. What is the limit of this fraction as $n$ approaches infinity, assuming $a_0 / b_0$ is a positive proper fraction?

In the context of Diophantine equations, consider the equation $1/x + 1/y = a/b$ where $x$, $y$, $a$, and $b$ are positive integers. What conditions on $a$ and $b$ guarantee that there are infinitely many integer solutions for $x$ and $y$?

Consider a game where you start with a proper fraction $x_0$ between 0 and 1. In each turn, you are allowed to either add $1/n$ to the numerator or subtract $1/n$ from the denominator (where $n$ is any positive integer), but the fraction must remain proper after each move. What is the optimal strategy to reach a fraction arbitrarily close to 1 in the fewest number of steps?

Suppose a real number $r$ can be expressed as an infinite continued fraction $[a_0; a_1, a_2, a_3, ...]$ where the $a_i$ are positive integers. If we truncate this continued fraction at the $n$-th term to obtain a rational approximation $p_n/q_n$, and we know that $r$ is an irrational number between 0 and 1, what can be definitively said about the sequence of fractions $p_n/q_n$ as $n$ approaches infinity?