NZ-NZMO 2023

For any positive integer $n$ let $n! = 1 \times 2 \times 3 \times \dots \times n$ . Do there exist infinitely many triples $(p, q, r)$ , of positive integers with $p > q > r > 1$ such that the product

$$p!\cdot q!\cdot r!$$

is a perfect square?

Let $a$ , $b$ and $c$ be positive real numbers such that $a + b + c = abc$ . Prove that at least one of $a$ , $b$ or $c$ is greater than $\frac{17}{10}$ .

Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation.

$$9^{x + 1} + 2187 = 3^{6x - x^{2}}$$

Let $p$ be a prime and let $f(x) = ax^{2} + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \leq p$ . Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$ .

Let $x$ , $y$ and $z$ be real numbers such that: $x^{2} = y + 2$ , and $y^{2} = z + 2$ , and $z^{2} = x + 2$ . Prove that $x + y + z$ is an integer.

Let triangle $ABC$ be right-angled at $A$ . Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$ . Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} = \frac{1}{2AB}$ .

Let $n,m$ be positive integers. Let $A_{1},A_{2},A_{3},\ldots ,A_{m}$ be sets such that $A_{i}\subseteq \{1,2,3,\ldots ,n\}$ and $|A_{i}| = 3$ for all $i$ (i.e. $A_{i}$ consists of three different positive integers each at most $n$ ). Suppose for all $i< j$ we have

$$|A_{i}\cap A_{j}|\leqslant 1$$

(i.e. $A_{i}$ and $A_{j}$ have at most one element in common).

(a) Prove that $m\leqslant \frac{n(n - 1)}{6}$ .

(b) Show that for all $n\geq 3$ it is possible to have $m\geqslant \frac{(n - 1)(n - 2)}{6}$ .

Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots.

$$P(x) = ax^{4} - 8ax^{3} + bx^{2} - 32cx + 16c$$