SIMON-MARAIS 2021

Let $a, b, c$ be real numbers such that $a \neq 0$ . Consider the parabola with equation $$y = ax^2 + bx + c,$$ and the lines defined by the six equations \begin{align*} &y = ax + b, & y = bx + c, & y = cx + a, &y = bx + a, & y = cx + b, & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$ .

Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ , and $$a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n$$ for all integers $n \ge 1$ . Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$ . Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$ .

Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$ . Determine the size of the set $\{ \det A : A \in M \}$ . Here $\det A$ denotes the determinant of the matrix $A$ .

For each positive real number $r$ , define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$ . (a) Prove that for each positive real number $r$ , the limit $$L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n}$$ exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .

Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$ . Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$ . Call the resulting matrix $A$ . Determine the probability that $A^2 = O$ , as a function of $n$ .

Let $n$ be a positive integer. There are $n$ lamps, each with a switch that changes the lamp from on to off, or from off to on, each time it is pressed. The lamps are initially all off. You are going to press the switches in a series of rounds. In the first round, you are going to press exactly $1$ switch; in the second round, you are going to press exactly $2$ switches; and so on, so that in the $k$ th round you are going to press exactly $k$ switches. In each round you will press each switch at most once. Your goal is to finish a round with all of the lamps switched on. Determine for which $n$ you can achieve this goal.

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties. (i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$ . (ii) For every real number $x$ and every integer $n \ge 1$ we have $$f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt.$$

The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).

Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$ . Suppose that, for each $1 \le k \le d-1$ , $p(x)$ and its $k$ th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$ ; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$ . (a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that $$p(x) = c(ax + b)^d.$$ (b) For which integers $d \ge 2$ does the conclusion of part (a) hold?