26th IMC

IMC 2019

Blagoevgrad, Bulgaria · papers 30 July & 31 July 2019 · 10 problems across 2 papers

Top of the individual standings

Day 1

30 July 2019 · 5 problems

Problem 1
Evaluate the product
$\prod_{n=3}^{\infty} \frac{(n^3 + 3n)^2}{n^6 - 64}.$
Problem 2
A four-digit number $YEAR$ is called very good if the system
$\begin{aligned} Yx + Ey + Az + Rw &= Y \\ Rx + Yy + Ez + Aw &= E \\ Ax + Ry + Yz + Ew &= A \\ Ex + Ay + Rz + Yw &= R \end{aligned}$
of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good YEARs in the 21st century.

(The 21st century starts in 2001 and ends in 2100.)
Problem 3
Let $f \colon (-1, 1) \to \mathbb{R}$ be a twice differentiable function such that
$2f'(x) + xf''(x) \ge 1 \quad \text{for } x \in (-1, 1).$
Prove that
$\int_{-1}^{1} xf(x)\,\mathrm{d}x \ge \frac{1}{3}.$
Problem 4
Define the sequence $a_0, a_1, \ldots$ of numbers by the following recurrence:
$a_0 = 1, \quad a_1 = 2, \quad (n + 3)a_{n+2} = (6n + 9)a_{n+1} - na_n \quad \text{for } n \ge 0.$
Prove that all terms of this sequence are integers.
Problem 5
Determine whether there exist an odd positive integer $n$ and $n \times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:

(1) $\det(B) = 1$ ;

(2) $AB = BA$ ;

(3) $A^4 + 4A^2B^2 + 16B^4 = 2019I$ .

(Here $I$ denotes the $n \times n$ identity matrix.)

31 July 2019 · 5 problems

Problem 6
Let $f, g \colon \mathbb{R} \longrightarrow \mathbb{R}$ be continuous functions such that $g$ is differentiable. Assume that $\big(f(0) - g'(0)\big)\big(g'(1) - f(1)\big) > 0$ . Show that there exists a point $c \in (0, 1)$ such that $f(c) = g'(c)$ .
Problem 7
Let $C = \{4, 6, 8, 9, 10, \ldots\}$ be the set of composite positive integers. For each $n \in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$ . Determine whether the following series converges:
$\sum_{n \in C} \left(\frac{a_n}{n}\right)^n. \tag{1}$
Problem 8
Let $x_1, \ldots, x_n$ be real numbers. For any set $I \subset \{1, 2, \ldots, n\}$ let $s(I) = \sum_{i \in I} x_i$ . Assume that the function $I \mapsto s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1, 2, \ldots, n\}$ . Prove that the number of sets $I \subset \{1, 2, \ldots, n\}$ for which $s(I) = 2019$ does not exceed $1.7^n$ .
Problem 9
Determine all positive integers $n$ for which there exist $n \times n$ real invertible matrices $A$ and $B$ that satisfy $AB - BA = B^2A$ .
Problem 10
2019 points are chosen at random, independently, and distributed uniformly in the unit disc $\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \le 1\}$ . Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices?