Question:

Step 1: Express the sum in a general form. The sum $S$ is given by $$S = \left[\frac{1}{4}\right] + \left[\frac{1}{4} + \frac{1}{200}\right] + \left[\frac{1}{4} + \frac{1}{100}\right] + \left[\frac{1}{4} + \frac{3}{200}\right] + \cdots + \left[\frac{1}{4} + \frac{199}{200}\right]$$ This can be written as a summation: $$S = \sum_{k=0}^{199} \left[\frac{1}{4} + \frac{k}{200}\right]$$ There are $199 - 0 + 1 = 200$ terms in the sum. Step 2: Determine the value of each term in the sum. The integral part of $x$, denoted by $[x]$, is the greatest integer less than or equal to $x$. For the terms $\left[\frac{1}{4} + \frac{k}{200}\right]$: The smallest value occurs when $k=0$: $\left[\frac{1}{4} + \frac{0}{200}\right] = \left[\frac{1}{4}\right] = [0.25] = 0$. The largest value occurs when $k=199$: $\left[\frac{1}{4} + \frac{199}{200}\right] = [0.25 + 0.995] = [1.245] = 1$. Thus, each term in the sum can only be 0 or 1. A term $\left[\frac{1}{4} + \frac{k}{200}\right]$ equals 0 if $\frac{1}{4} + \frac{k}{200} < 1$. $$\frac{k}{200} < 1 - \frac{1}{4}$$ $$\frac{k}{200} < \frac{3}{4}$$ $$k < \frac{3}{4} \times 200$$ $$k < 150$$ So, for $k \in \{0, 1, \dots, 149\}$, the value of the term is 0. The number of such terms is $149 - 0 + 1 = 150$. A term $\left[\frac{1}{4} + \frac{k}{200}\right]$ equals 1 if $\frac{1}{4} + \frac{k}{200} \ge 1$. $$\frac{k}{200} \ge \frac{3}{4}$$ $$k \ge 150$$ So, for $k \in \{150, 151, \dots, 199\}$, the value of the term is 1. The number of such terms is $199 - 150 + 1 = 50$. Step 3: Calculate the total sum $S$. The sum $S$ is the sum of all these terms: $$S = \sum_{k=0}^{149} 0 + \sum_{k=150}^{199} 1$$ $$S = (150 \times 0) + (50 \times 1)$$ $$S = 0 + 50$$ $$S = 50$$ Step 4: Determine the properties of $S$. The value of $S$ is 50. 1. $S$ is a composite number. Its prime factorization is $50 = 2^1 \times 5^2$. Since $S$ has factors other than 1 and itself (e.g., 2, 5, 10, 25), it is a composite number. 2. The exponent of 2 in the prime factorization of $S$ is 1. 3. The number of factors of $S=50$ is determined by the exponents in its prime factorization. For $S = p_1^{e_1} p_2^{e_2} \cdots p_n^{e_n}$, the number of factors is $(e_1+1)(e_2+1)\cdots(e_n+1)$. For $S=2^1 \times 5^2$, the number of factors is $(1+1)(2+1) = 2 \times 3 = 6$. The factors are $\{1, 2, 5, 10, 25, 50\}$. 4. For the binomial coefficient $\binom{2S}{r}$, which is $\binom{100}{r}$, its maximum value occurs when $r = \frac{2S}{2} = \frac{100}{2} = 50$.