For a positive integer \( n \), let \( S(n) \) be the sum of all decimal digits in \( n \), i.e., if \( n = n_1 n_2 \dots n_m \) is the decimal expansion of \( n \), then \( S(n) = n_1 + n_2 + \dots + n_m \). Find all positive integers \( n \) and \( r \) such that \( (S(n))^r = S(n^r) \).
Let \(a_1\), \(a_2\), \(\ldots\), \(a_m\) be distinct positive integers. Prove that if \(m>2\sqrt{N}\), then there exist \(i\), \(j\) such that the least common multiple of \(a_i\) and \(a_j\) is greater than \(N\).
Let \( S \) be an \( (n+1) \times (n+1) \) matrix defined by
\[
S_{ij} = \begin{cases}
(n+1)-i & \text{ if } j=i+1, \\
i-1 & \text{ if } j=i-1, \\
0 & \text{ otherwise. }
\end{cases}
\]
Find all eigenvalues of \( S \).
Let \(A_1,A_2,A_3,\ldots,A_n\) be the vertices of a regular \(n\)-gon on the unit circle. Evaluate \(\prod_{i=2}^n A_1A_i\). (Here, \(A_1A_i\) denotes the length of the line segment.)
A gambler is playing roulette and betting $1 on black each time. The probability of winning $1 is 18/38, and the probability of losing $1 is 20/38. Find the probability that starting with $20 the player reaches $40 before losing the money.
Let \(x_1,x_2,\ldots,x_n\) be reals such that \(x_1+x_2+\cdots+x_n=n\) and \(x_1^2+x_2^2+\cdots +x_n^2=n+1\). What is the maximum of \(x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1\)?
Find all integers \( n \) such that \( \sqrt{1} + \sqrt{2} + \dots + \sqrt{n} \) is an integer.
For \(n\ge 1\), let \(f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k \) be a polynomial with real coefficients. Prove that if \(f(x)>0\) for all \(x\in [-2,2]\), then \(f(x)\ge 4\) for some \(x\in [-2,2]\).
Define a sequence \( \{ a_n \} \) by \( a_1 = a \) and
\[
a_n = \frac{2n-1}{n-1} a_{n-1} -1
\]
for \( n \geq 2 \). Find all real values of \( a \) such that \( \lim_{n \to \infty} a_n \) exists.
Let \(p\), \(q\), \(r\) be positive integers such that \(p,q\ge r\). Ada and Betty independently read all source codes of their programming project. Ada found \(p\) bugs and Betty found \(q\) bugs, including \(r\) bugs that Ada found. What is the expected number of remaining bugs that neither Ada nor Betty found?