International Competitions 2024

Determine all real numbers $\alpha$ such that, for every positive integer $n$, the integer $$\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \cdots + \lfloor n\alpha \rfloor$$ is a multiple of $n$. (Note that $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$. For example, $\lfloor -\pi \rfloor = -4$ and $\lfloor 2 \rfloor = \lfloor 2.9 \rfloor = 2$.)

Determine all pairs $(a, b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd(a^n + b, b^n + a) = g$$ holds for all integers $n \geq N$. (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y$.)

Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \ldots, a_{n-1}$.

Prove that at least one of the sequences $a_1, a_3, a_5, \ldots$ and $a_2, a_4, a_6, \ldots$ is eventually periodic.

(An infinite sequence $b_1, b_2, b_3, \ldots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \geq M$.)

Let $ABC$ be a triangle with $AB < AC < BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ again at $P \neq A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.

Prove that $\angle KIL + \angle YPX = 180^{\circ}$.

Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.

Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.

Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n^{\text{th}}$ attempt or earlier, regardless of the locations of the monsters.

Let $\mathbb{Q}$ be the set of rational numbers. A function $f\colon \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x, y \in \mathbb{Q}$, $$f(x + f(y)) = f(x) + y \quad \text{or} \quad f(f(x) + y) = x + f(y).$$

Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.

Determine all $k \in \mathbb{N}_0$ for which there exists a function $f \colon \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and

$$f(f(n)) \leq f(n + 1) - f(n)$$

for all $n \in \mathbb{N}_0$.

Remark. Here $\mathbb{N}_0$ denotes the set of nonnegative integers.

There is a sheet of paper (like this one) on an infinite blackboard. Marvin secretly chooses a convex 2024-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$ such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects segments $AB$ and $AC$ again in points $D \neq A$ and $E \neq A$, respectively. Let $F$ be the intersection of $BE$ and $CD$. Let $G$ be the point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$. Similarly, let $H$ be the point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T \neq A$, independent of the choice of $\omega$, such that the circumcircle of $AGH$ passes through $T$.

For any positive integer $n$, let $\sigma(n)$ denote the sum of positive divisors of $n$. Determine all polynomials $P$ with integer coefficients such that $P(k)$ is divisible by $\sigma(k)$ for all positive integers $k$.

Consider the two infinite sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ of real numbers such that $a_0 = 0$, $b_0 = 0$ and $$a_{k+1} = b_k, \qquad b_{k+1} = \frac{a_k b_k + a_k + 1}{b_k + 1}$$ for each integer $k \geq 0$. Prove that $a_{2024} + b_{2024} \geq 88$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$yf(x + 1) = f(x + y - f(x)) + f(x)f(f(y))$$ for all $x, y \in \mathbb{R}$.

There are 2024 mathematicians sitting in a row next to the river Tisza. Each of them is working on exactly one research topic, and if two mathematicians are working on the same topic, everyone sitting between them is also working on it.

Marvin is trying to figure out for each pair of mathematicians whether they are working on the same topic. He is allowed to ask each mathematician the following question: "How many of these 2024 mathematicians are working on your topic?" He asks the questions one by one, so he knows all previous answers before he asks the next one.

Determine the smallest positive integer $k$ such that Marvin can always accomplish his goal with at most $k$ questions.

A finite sequence $x_1, x_2, \ldots, x_r$ of positive integers is a palindrome if $x_i = x_{r+1-i}$ for all integers $1 \leq i \leq r$.

Let $a_1, a_2, \ldots$ be an infinite sequence of positive integers. For a positive integer $j \geq 2$, denote by $a[j]$ the finite subsequence $a_1, a_2, \ldots, a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1, b_2, \ldots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \leq b_{n+1} + b_n$. Prove that there exists a positive integer $T$ such that $a_i = a_{i+T}$ for every positive integer $i$.

Let $ABC$ be a triangle with $\angle BAC = 60°$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \neq F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC, ABM, ACM$, respectively. Let $P, Q$ be points on the lines $MK, MJ$, respectively, such that $\angle AJP = \angle ABC$ and $\angle AKQ = \angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

Define glueing of positive integers as writing their base ten representations one after another and interpreting the result as the base ten representation of a single positive integer.

Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \geq N_k$, we can glue the numbers $1, 2, \ldots, n$ in some order so that the result is a number divisible by $k$.

Remark. The base ten representation of a positive integer never starts with zero.

Example. Glueing 15, 14, 7 in this order makes 15147.

Let $k$ be a positive integer and $a_1, a_2, \ldots$ be an infinite sequence of positive integers such that $$a_i a_{i+1} \mid k - a_i^2$$ for all integers $i \geq 1$. Prove that there exists a positive integer $M$ such that $a_n = a_{n+1}$ for all integers $n \geq M$.