1. A chord of a circle has length of 3n, where n is a positive integer. The segment cutoff my the chord has height n, as shown below. What is the smallest value of n for which the radius of the circle is also a positive integer?

Answer: 8

Explanation:

From the diagram, O is the center of the circle and $r$ is the radius.
$\begin{align}
O{A^2} &= O{C^2} + A{C^2}\\[6px]
{r^2} &= {(r - n)^2} + {\left( {\dfrac{3}{2}n} \right)^2} \\[6px]
{r^2} &= {r^2} + {n^2} - 2nr + \frac{9}{4}{n^2} \\[6px]
2nr &= \dfrac{{13}}{4}{n^2} \\[6px]
8r &= 13n \\[6px]
r &= \dfrac{{13}}{8}n
\end{align}$
For, n = 8, r becomes integer. r = 13.

2. If n is a positive integer, let s(n) denote the integer obtained by removing the last digit of n and placing it in front. For example, s(731) = 173. What is the smallest positive integer n ending in 6 satisfying s(n) = 4(n)?

Answer: D

Explanation:
Let $n = 10a + 6$ and here $a$ has $k$ digits.
In effect n is a (k + 1) digits number.
If we shift 6 to the left side, It becomes $6 \times {10^k} + a$
Given, $S(n) = 4n$
$ \Rightarrow 6 \times {10^k} + a = 4n = 4(10a + 6)$
$ \Rightarrow 6 \times {10^k} - 24 = 39a$
$ \Rightarrow 2 \times {10^k} - 8 = 13a$
Now by trial and error, we need check for what value of k, $2 \times {10^k} - 8$ is divisible by 13.
For, $k = 5$,
$a = \dfrac{{2 \times {{10}^5} - 8}}{{13}} = 15384$
Therefore, $n = 15384 \times 10 + 6 = 153846$

3. Of all the nonempty subsets S of {1, 2, 3, 4, 5, 6, 7}, how many do not contain the number |S|, where |S| denotes the number of elements in S? For example, {3, 4} is one such subset, since it does not contain the number 2.

Answer: 63

Explanation:
Easy question. Number of subsets without element 2 are ${2^6} - 1 = 63$

4. A function f satisfies f(0) = 0, f(2n) = f(n), and f(2n+1) = f(n) + 1 for all positive integers n. What is the value of f(2018) ?