2018 VTRMC

Part of VTRMC

Subcontests

(7)

1

Convergence of a Piecewise-Monotone Function Sequence

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 ]

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

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

f

composed with itself

n

f^2 (x) = f(f(x))

1

Extrema of Difference of Sequences

n \in \mathbb{N}

a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}

b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}

. Find the maximum and minimum of

a_n - b_n

1 \leq n \leq 999

1

Bounding an Integral Sequence

n \in \mathbb{N}

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

1

GCD and Binomial Divisibility

m, n

be integers such that

n \geq m \geq 1

\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.

1

Modular Matrices

A, B \in M_6 (\mathbb{Z} )

A \equiv I \equiv B \text{ mod }3

A^3 B^3 A^3 = B^3

A = I

M_6 (\mathbb{Z} )

6

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

1

2018 VTRMC #3

Prove that there is no function

f:\mathbb{N}\rightarrow \mathbb{N}

f(f(n))=n+1.

\mathbb{N}

is the positive integers

\{1,2,3,\dots\}.

1

2018 VTRMC #1

It is known that

\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)

for some rational number

q.

q.

0\leq\arctan(x)<\frac{\pi}{2}

0\leq x <\infty.