Found problems: 11
2018 VTRMC, 4
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.
2022 VTRMC, 4
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.
2022 VTRMC, 2
Let $A$ and $B$ be the two foci of an ellipse and let $P$ be a point on this ellipse. Prove that the focal radii of $P$ (that is, the segments $\overline{AP}$ and $\overline{BP}$ ) form equal angles with the tangent to the ellipse at $P$.
2018 VTRMC, 6
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 $1 \leq n \leq 999$.
2018 VTRMC, 7
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.
2022 VTRMC, 5
Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.
2022 VTRMC, 6
Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.
2022 VTRMC, 1
Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.
2018 VTRMC, 5
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.
2018 VTRMC, 2
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 $3$.
2022 VTRMC, 3
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.