VTRMC 2018

It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$

Let $A, B \in M_6 (\mathbb{Z} )$ such that $A \equiv I \equiv B \text{ mod }3$ and $A^3 B^3 A^3 = B^3$ . Prove that $A = I$ . Here $M_6 (\mathbb{Z} )$ indicates the $6$ by $6$ matrices with integer entries, $I$ is the identity matrix, and $X \equiv Y \text{ mod }3$ means all entries of $X-Y$ are divisible by $3$ .

Prove that there is no function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(f(n))=n+1.$ Here $\mathbb{N}$ is the positive integers $\{1,2,3,\dots\}.$

Let $m, n$ be integers such that $n \geq m \geq 1$ . Prove that $\frac{\text{gcd} (m,n)}{n} \binom{n}{m}$ is an integer. Here $\text{gcd}$ denotes greatest common divisor and $\binom{n}{m} = \frac{n!}{m!(n-m)!}$ denotes the binomial coefficient.

For $n \in \mathbb{N}$ , let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$ . Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.

For $n \in \mathbb{N}$ , define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$ . Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$ .

A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals $$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$ such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$ . We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$ , and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$ . If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$ -times, so $f^2 (x) = f(f(x))$ etc.