NZ-NZMO 2025

Let $a$ and $b$ be positive integers with no common factor greater than 1. What are the possible values for the greatest common divisor of $(a + b)$ and $(a - b)$ ?

Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ , $\angle ABC = 70^{\circ}$ , and $AB = 1$ . Let $M$ be the midpoint of $BC$ . Let $D$ be the point on the extension of $AM$ beyond $M$ such that $\angle CDA = 110^{\circ}$ . Find the length of $CD$ .

Let $P(x) = x^{3} + ax^{2} + bx - 8$ be a polynomial with 3 real roots. Show that $a^{2} \geq 2b + 12$.

Find the largest integer $k$ such that any string of 2025 letters consisting only of A's and B's contains a palindromic substring of length $k$ or longer. A palindromic substring is a string of consecutive letters which reads the same backwards as forwards.

Alice plays a game with the Mad Hatter. The Mad Hatter will write two rows of numbers on a blackboard, each a distinct permutation of $\{1,2,\ldots ,n\}$ . On each move, Alice is allowed to swap the positions of the numbers $a$ and $a + 1$ in the first row, for some $1 \leq a < n$ . What is the minimum number of moves Alice needs in order to guarantee that she can turn the first row of numbers into the second, regardless of the permutations the Mad Hatter writes?

Determine the largest real number $M$ such that for each infinite sequence $x_{0}, x_{1}, x_{2}, \ldots$ of real numbers satisfying $x_{0} = 1$ , $x_{1} = 3$ and

$$x_{0} + x_{1} + \dots +x_{n - 1}\geq 3x_{n} - x_{n + 1} \mathrm{for~all~}n\geq 1,$$

the inequality

$$\frac{x_{n + 1}}{x_{n}} >M$$

holds for all $n \geq 0$ .

Let $ABC$ be a triangle and let $D$ be a point inside the triangle $ABC$ such that $AD$ bisects $\angle BAC$. Let line $BD$ meet side $AC$ at $E$. Let line $CD$ meet side $AB$ at $F$. Let $T$ be the intersection of the (internal) angle bisectors of $\angle AED$ and $\angle AFD$. Prove that if $T$ lies on segment $AD$, then triangle $ABC$ is isosceles.

Show that there are infinitely many triples $(a, b, c)$ of positive integers such that

$$a^{2} + b^{2} + c^{2} + (a + b + c)^{2} = abc.$$