VTRMC 2022

Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.

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 $P$ .

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.

Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.

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 $1$ .

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.