# [Solutions] William Lowell Putnam Mathematical Competition 2018

1. Find all ordered pairs $(a, b)$ of positive integers for which $\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.$
2. Let $S_1, S_2, \dots, S_{2^n - 1}$ be the nonempty subsets of $\{1, 2, \dots, n\}$ in some order, and let $M$ be the $(2^n - 1) \times (2^n - 1)$ matrix whose $(i, j)$ entry is $m_{ij} = \begin{cases} 0 & \text{ if } S_i \cap S_j = \emptyset, \\ 1 & \text{ otherwise}.\end{cases}$ Calculate the determinant of $M$.
3. Determine the greatest possible value of $\sum_{i = 1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \dots, x_{10}$ satisfying $\displaystyle\sum_{i = 1}^{10} \cos(x_i) = 0$.
4. Let $m$ and $n$ be positive integers with $\gcd(m, n) = 1$, and let $a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor$for $k = 1, 2, \dots, n$. Suppose that $g$ and $h$ are elements in a group $G$ and that $gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,$ where $e$ is the identity element. Show that $gh = hg$. (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
5. Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
6. Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient $\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}$is a rational number.
7. Let $\mathcal{P}$ be the set of vectors defined by $\mathcal{P} = \left\{\begin{pmatrix} a \\ b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.$Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P}\setminus\{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.
8. Let $n$ be a positive integer, and let $$f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}.$$ Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$.
9. Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.
10. Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_nx_{n-1} - x_{n-2}$ for $n \ge 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.
11. Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\tfrac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that $\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0$everywhere. Prove that $f$ is one-to-one.
12. Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most $2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.$
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