Virginia Tech Regional Math Contest (2019)

Problem 1

For each positive integer $n$, define $f(n)$ to be the sum of the digits of $2771^n$ (so $f(1) = 17$). Find the minimum value of $f(n)$ (where $n \ge 1$). Justify your answer.

Problem 2

Let $X$ be the point on the side $AB$ of the triangle $ABC$ such that $BX/XA = 9$. Let $M$ be the midpoint of $BX$ and let $Y$ be the point on $BC$ such that $\angle BMY = 90^\circ$. Suppose $AC$ has length 20 and that the area of the triangle $XYC$ is $9/100$ of the area of the triangle $ABC$. Find the length of $BC$.

Problem 3

Let $n$ be a nonnegative integer and let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \in \mathbb{R}[x]$ be a polynomial with real coefficients $a_i$. Suppose that $$\frac{a_n}{(n+1)(n+2)} + \frac{a_{n-1}}{n(n+1)} + \dots + \frac{a_1}{6} + \frac{a_0}{2} = 0.$$ Prove that $f(x)$ has a real zero.

Problem 4

Compute $\displaystyle \int_0^1 \frac{x^2}{x + \sqrt{1 - x^2}} dx$ (the answer is a rational number).

Problem 5

Find the general solution of the differential equation $$x^4 \frac{d^2y}{dx^2} + 2x^2 \frac{dy}{dx} + (1 - 2x)y = 0$$ valid for $0 < x < \infty$.

Problem 6

Let $S$ be a subset of $\mathbb{R}$ with the property that for every $s \in S$, there exists $\varepsilon > 0$ such that $(s - \varepsilon, s + \varepsilon) \cap S = \lbrace s \rbrace$. Prove there exists a function $f : S \to \mathbb{N}$, the positive integers, such that for all $s, t \in S$, if $s \ne t$ then $f(s) \ne f(t)$.

Problem 7

Let $S$ denote the positive integers that have no 0 in their decimal expansion. Determine whether $\displaystyle \sum_{n \in S} n^{-99/100}$ is convergent.