IMO 2024 — Complete Shortlist

Determine all real numbers $ \alpha $ such that the number

$$ \left\lfloor\alpha\right\rfloor+\left\lfloor2\alpha\right\rfloor+\cdots+\left\lfloor n\alpha\right\rfloor $$

is a multiple of n for every positive integer n. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to z.)

Let n be a positive integer. Find the minimum possible value of

$$ S=2^{0}x_{0}^{2}+2^{1}x_{1}^{2}+\cdots+2^{n}x_{n}^{2}, $$

where $x_{0}, x_{1}, \ldots, x_{n}$ are nonnegative integers such that $x_{0} + x_{1} + \cdots + x_{n} = n$.

Decide whether for every sequence $ (a_{n}) $ of positive real numbers,

$$ \frac{3^{a_{1}}+3^{a_{2}}+\cdots+3^{a_{n}}}{(2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}})^{2}} \lt \frac{1}{2024} $$

is true for at least one positive integer n.

Comment. The question can be asked in several forms, as follows:

(i) students could be asked, as above, to show the existence of such an n;

(ii) students could be asked to show that this happens for all sufficiently large n;

(iii) students could be given a concrete positive integer N and asked to show it for all $n\gt N$.

The solutions below provide varying bounds for N.

Let $\mathbb{Z}_{ \gt 0}$ be the set of all positive integers. Determine all subsets $\mathcal{S}$ of $\{2^0, 2^1, 2^2, \ldots\}$ for which there exists a function $f: \mathbb{Z}_{ \gt 0} \to \mathbb{Z}_{ \gt 0}$ such that

$$ \mathcal{S}=\{f(a+b)-f(a)-f(b)\mid a,b\in\mathbb{Z}_{ \gt 0}\}. $$

Find all periodic sequences $ a_{1}, a_{2}, \ldots $ of real numbers such that the following conditions hold for all $ n \geqslant 1 $:

$$ a_{n+2}+a_{n}^{2}=a_{n}+a_{n+1}^{2}\qquad\text{and}\qquad|a_{n+1}-a_{n}|\leqslant1. $$

Let $ a_0, a_1, a_2, \ldots $ be an infinite strictly increasing sequence of positive integers such that for each $ n \geq 1 $ we have

$$ a_{n}\in\left\{\frac{a_{n-1}+a_{n+1}}{2},\sqrt{a_{n-1}\cdot a_{n+1}}\right\}. $$

Let $b_{1}, b_{2}, \ldots$ be an infinite sequence of letters defined as

$$ b_{n}=\begin{cases}A,&\text{if} a_{n}=\frac{1}{2}(a_{n-1}+a_{n+1});\\G,&\text{otherwise}.\end{cases} $$

Prove that there exist positive integers $ n_{0} $ and d such that for all $ n \geq n_{0} $ we have $ b_{n+d} = b_{n} $.

Let Q be the set of rational numbers. Let $f: \mathbb{Q} \to \mathbb{Q}$ be a function such that the following property holds: for all $x, y \in \mathbb{Q}$,

$$ f(x+f(y))=f(x)+y\qquad \text{or}\qquad f(f(x)+y)=x+f(y). $$

Determine the maximum possible number of elements of $ \{f(x) + f(-x) \mid x \in \mathbb{Q}\} $.

Let $p \neq q$ be coprime positive integers. Determine all infinite sequences $a_1, a_2, \ldots$ of positive integers such that the following conditions hold for all $n \geq 1$:

$$ \begin{aligned}\max(a_{n},a_{n+1},\cdots,a_{n+p})&-\min(a_{n},a_{n+1},\cdots,a_{n+p})=p\qquad \text{and}\\\max(a_{n},a_{n+1},\cdots,a_{n+q})&-\min(a_{n},a_{n+1},\cdots,a_{n+q})=q.\end{aligned} $$

Let $n$ be a positive integer. A class of $n$ students run $n$ races, in each of which they are ranked with no draws. A student is eligible for a rating $(a,b)$ for positive integers $a$ and $b$ if they come in the top $b$ places in at least $a$ of the races. Their final score is the maximum possible value of $a-b$ across all ratings for which they are eligible.

Find the maximum possible sum of all the scores of the $n$ students.

Let $n$ be a positive integer. The integers 1, 2, 3, ..., $n^{2}$ are to be written in the cells of an $n \times n$ board such that each integer is written in exactly one cell and each cell contains exactly one integer. For every integer $d$ with $d \mid n$, the $d$-division of the board is the division of the board into $(n/d)^{2}$ nonoverlapping sub-boards, each of size $d \times d$, such that each cell is contained in exactly one $d \times d$ sub-board.

We say that $n$ is a cool number if the integers can be written on the $n\times n$ board such that, for each integer $d$ with $d\mid n$ and $1 \lt d \lt n$, in the $d$-division of the board, the sum of the integers written in each $d\times d$ sub-board is not a multiple of $d$.

Determine all even cool numbers.

Let n be a positive integer. There are 2n knights sitting at a round table. They consist of n pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places.

Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some time.

On a board with 2024 rows and 2023 columns, Turbo the snail tries to move from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then moves one step at a time to an adjacent cell sharing a common side. He wins if he reaches any cell in the last row. However, there are 2022 predetermined, hidden monsters in 2022 of the cells, one in each row except the first and last rows, such that no two monsters share the same column. If Turbo unfortunately 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.

Suppose Turbo is allowed to take n attempts. Determine the minimum value of n for which he has a strategy that guarantees reaching the last row, regardless of the locations of the monsters.

Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers 1, 2, ..., $N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$, where $k \geq 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k \mid n - s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of N for which Geoff can ensure that he wins, no matter how Ceri plays.

Let $n$ and $T$ be positive integers. James has $4n$ marbles with weights 1, 2, ..., $4n$. He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most $T$.

Find, in terms of n, the minimum positive integer T such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.

Let $N$ be a positive integer and let $a_1, a_2, \ldots$ be an infinite sequence of positive integers. Suppose that, for each $n \gt N$, $a_n$ 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} $, ... and $ a_{2} $, $ a_{4} $, $ a_{6} $, ... is eventually periodic.

Let $n$ be a positive integer. Given an $n \times n$ board, the unit cell in the top left corner is initially coloured black, and the other cells are coloured white. We then apply a series of colouring operations to the board. In each operation, we choose a $2 \times 2$ square with exactly one cell coloured black and we colour the remaining three cells of that $2 \times 2$ square black.

Determine all values of n such that we can colour the whole board black.

Let $ABCD$ be a cyclic quadrilateral such that $AC \lt BD \lt AD$ and $\angle DBA \lt 90^{\circ}$. Point $E$ lies on the line through $D$ parallel to $AB$ such that $E$ and $C$ lie on opposite sides of line $AD$, and $AC = DE$. Point $F$ lies on the line through $A$ parallel to $CD$ such that $F$ and $C$ lie on opposite sides of line $AD$, and $BD = AF$.

Prove that the perpendicular bisectors of segments BC and EF intersect on the circumcircle of ABCD.

Let $ABC$ be a triangle with $AB \lt AC \lt BC$, incentre $I$ and incircle $\omega$. Let $X$ be the point in the interior of side $BC$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point in the interior of side $BC$ 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 $AB$ and $AC$, respectively.

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

Let ABCDE be a convex pentagon and let M be the midpoint of AB. Suppose that segment AB is tangent to the circumcircle of triangle CME at M and that D lies on the circumcircles of triangles AME and BMC. Lines AD and ME intersect at K, and lines BD and MC intersect at L. Points P and Q lie on line EC so that $ \angle PDC = \angle EDQ = \angle ADB $.

Prove that lines $ KP, LQ $, and MD are concurrent.

Let ABCD be a quadrilateral with AB parallel to CD and AB < CD. Lines AD and BC intersect at a point P. Point $ X \neq C $ on the circumcircle of triangle ABC is such that $ PC = PX $. Point $ Y \neq D $ on the circumcircle of triangle ABD is such that $ PD = PY $. Lines AX and BY intersect at Q.

Prove that PQ is parallel to AB.