JMO 2025

Let $n$ be an integer greater than or equal to $2$ . Consider real numbers $a_1, a_2, \dots, a_{2n}$ satisfying the condition $$|a_k - a_{n+k}| \geqslant 1 \text{for all } 1 \leqslant k \leqslant n.$$ Determine the minimum possible value of $$(a_1 - a_2)^2 + (a_2 - a_3)^2 + \dots + (a_{2n-1} - a_{2n})^2 + (a_{2n} - a_1)^2.$$

Let $ABC$ be an acute-angled triangle with circumcenter $O$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $ABO$ and $ACO$ , respectively. The circumcircle of $\triangle AO_1O_2$ intersects segment $BC$ at two distinct points $P$ and $Q$ , with the four points $B, P, Q, C$ appearing in this order along $BC$ . Let $O_3$ be the circumcenter of $\triangle OPQ$ . Prove that points $A, O, O_3$ are collinear.

Let $n$ be a positive integer. There exist $n$ ordered triples $$(x_1, y_1, z_1), (x_2, y_2, z_2), \dots, (x_n, y_n, z_n)$$ where each coordinate is an integer between $1$ and $100$ (inclusive), satisfying the following condition:

[center] For every infinite sequence $(a_1, a_2, a_3, \dots)$ of integers between $1$ and $100$ , there exist a positive integer $i$ and an index $j$ (with $1 \leqslant j \leqslant n$ ) such that $(a_i, a_{i+1}, a_{i+2}) = (x_j, y_j, z_j)$ . [/center]

Determine the minimum possible value of $n$ .

Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$ :

• $f(n) > 0$ .
• $f(n)$ divides $n^{f(n)} - 1$ .

Let $ABC$ be an acute-angled scalene triangle. Inside the triangle, distinct points $A_1, B_1, C_1$ are chosen such that $$\frac{AB_1}{CB_1} = \frac{AB}{CB} \text{and} \frac{AC_1}{BC_1} = \frac{AC}{BC}.$$ Let $A_2, B_2, C_2$ be the reflections of $A_1, B_1, C_1$ across lines $BC, AC, AB$ , respectively. These points satisfy the following conditions:

• The four points $A, A_2, B, C_2$ are concyclic.
• The four points $A, A_2, B_2, C$ are concyclic.
• The four points $B, B_2, C, C_2$ are concyclic.
• The three points $A_2, B_2, C_2$ do not lie on the circumcircle of $\triangle{ABC}$ .

Prove that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.