BALTIC-WAY 2025

Does there exist a sequence of integers $a_1, a_2, \ldots$ such that for each integer $d \neq 0$ there are exactly 2025 distinct pairs of indices $(i, j)$ for which $a_i - a_j = d$?

Let $a_1, a_2, \ldots$ be a sequence of real numbers such that

$$\{a_n\} \lfloor a_{n+1} \rfloor = \lfloor a_n \rfloor \{a_{n+1}\}$$

for each positive integer $n$. Prove that there is a real number $\lambda$ such that $\{a_m\} \lfloor a_m \rfloor = \lambda$ for infinitely many positive integers $m$.

Remark: For a real number $x$, $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$, and $\{x\} = x - \lfloor x \rfloor$.

Let $f: \mathbb{Q} \to \mathbb{Q}$ be a function that satisfies

$$f(x) + f(y) \geq f(x + y)$$

for all $x, y \in \mathbb{Q}$. Show that there exists an $\alpha \in \mathbb{R}$ such that $f(x) \geq \alpha x$ for all $x \in \mathbb{Q}$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that

$$(f(a - c) + f(b - d))(f(a) + f(b)) = f(ad - bc) + f(f(a) + f(b) - ac - bd)$$

for all real numbers $a, b, c, d$.

Let $n$ be a positive integer and let $x_1 \leq x_2 \leq \cdots \leq x_n$ be positive real numbers satisfying

$$\frac{x_1^3 + x_2^3 + \cdots + x_n^3}{x_1 + x_2 + \cdots + x_n} = \frac{x_1^2 + x_n^2}{2}.$$

Prove that

$$x_1^2 + x_2^2 + \cdots + x_n^2 \geq nx_1x_n.$$

Let $n \geq 3$ friends play dodgeball. Initially all players are free. When a free player $A$ knocks out another free player $B$, $A$ imprisons $B$ and frees all players currently imprisoned by $B$. Two knock-outs cannot happen at the same time.

Find the least number of knock-outs after which it is possible that among every two players at least one has freed the other one.

Let $n$ be an even positive integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. A set of $n$ unit cubes that forms a $1 \times 1 \times n$ box is called a needle, which can have three orientations. Find the largest integer $K$, such that it is possible to select $3K$ needles, $K$ in each orientation, such that no two of them share a unit cube.

A sequence of non-negative integers $a_1, \ldots, a_n$ is called self-descriptive if

$$a_k = |\{i \mid a_i \geq k\}|$$

for each $k \in \{1, \ldots, n\}$, i.e. $a_k$ equals the number of elements of the sequence that are greater than or equal to $k$, for each $k \in \{1, \ldots, n\}$.

For each positive integer $n$, determine the number of self-descriptive sequences of length $n$.

A Baltic polyiamond is an $n$-gon with side lengths $n, n-1, \ldots, 2, 1$ in exactly this order, and all internal angles $120^\circ$ or $240^\circ$. Prove that for every Baltic polyiamond, $n$ is divisible by 6.

A positive integer with 9 digits is called nice if it contains each digit from 1 to 9. A number is called wonderful if it is nice and for each $k = 1, \ldots, 9$, it holds that the $k$-th digit in the number is equal to the position of $k$ in the number. For example, 847296315 is wonderful:

• the first digit is 8, which is the position of 1 in the number,
• the second digit is 4, which is the position of 2 in the number,
• the third digit is 7, which is the position of 3 in the number, and so on.

Show that the number of wonderful numbers is even.

Let $A_1, \ldots, A_n$ be a sequence of points on the Euclidean plane. Some of the points may coincide, but the sequence contains at least two distinct points. A Baltic Way is a route defined by a permutation $B_1, \ldots, B_n$ of the points $A_1, \ldots, A_n$. The length of a Baltic Way is

$$|B_1B_2| + |B_2B_3| + \cdots + |B_{n-1}B_n| + |B_nB_1|,$$

where $|PQ|$ denotes the Euclidean distance between points $P$ and $Q$.

Let $K$ and $L$ be the lengths of some two Baltic Ways on the same sequence of $n$ points. For a given $n$, determine the maximum possible value of $K/L$ over all possible sequences $A_1, \ldots, A_n$.

In an acute triangle $ABC$ with $AB < AC$, the incentre is $I$ and the circumcircle is $\omega$. The line $AI$ intersects the side $BC$ at $D$. Let $T$ be the point on $\omega$ such that $AT \parallel BC$. The line $TI$ intersects $\omega$ again at $P$ and the circumcircle of triangle $APD$ again at $Q$. Prove that $AI = DQ$.

In a cyclic quadrilateral $ABCD$, the lines $AB$ and $CD$ intersect at $E$. A point $P$ lies inside $ABCD$ and satisfies

$$\angle BAP = \angle PCB \quad \text{and} \quad \angle CBP = \angle PDC.$$

Prove that $PE \perp BC$.

In an acute triangle $ABC$ with $AB < AC$, the circumcircle is $\omega$ and the orthocentre is $H$. The point $P$ on major arc $BC$ of $\omega$ satisfies $\angle PCB + \angle ACB = 90^\circ$. The point $R$ on segment $AC$ satisfies $BR = CR$. Prove that $\angle HPR = \angle ACB$.

In an acute, scalene triangle $ABC$, the perpendicular bisector of $BC$ intersects $AC$ and $AB$ in $A_b$ and $A_c$, respectively. Define $B_c, B_a, C_a$ and $C_b$ similarly. Prove that the circumcentres of $ABC, A_bB_cC_a$, and $A_cB_aC_b$ are collinear.

Let $a, b, c, n$ be positive integers satisfying $abc = 10^n - 1$. If $S(x)$ denotes the sum of digits of $x$ in decimal notation, prove that

$$S(a) + S(b^2) + S(c^4) \geq \sqrt[3]{243n}.$$

A positive integer is called brilliant if its positive divisors can be partitioned into two sets with an equal number of elements and an equal sum of elements.

Find all possible values of the number of positive divisors of a brilliant number.

Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(2) = 1$ and

$$\text{lcm}(f(a+b), f(b)) \mid \text{lcm}(a+f(b), b)$$

for all positive integers $a, b$.

There are 100 positive integers written on a blackboard. Anna plays a game. A move consists of choosing two integers $a$ and $b$ on the blackboard with the property that $a \mid b$, erasing them both and writing the integer $b/a$. She makes moves until there are no more valid moves, at which point the game ends.

Find the largest $N$ such that for some initial state Anna can both:

• finish the game with only one number remaining, and
• finish the game with $N$ numbers remaining.

Determine all positive integers $n$ such that there exists a permutation $a_1, \ldots, a_n$ of $1, \ldots, n$, where the numbers $a_1, 2a_2, \ldots, na_n$ are all pairwise distinct modulo $n$.