NZ-NZMO 2021

A school offers three subjects: Mathematics, Art and Science. At least $80\%$ of students study both Mathematics and Art. At least $80\%$ of students study both Mathematics and Science. Prove that at least $80\%$ of students who study both Art and Science, also study Mathematics.

Prove that

$$x^{2} + \frac{8}{xy} +y^{2}\geq 8.$$

for all positive real numbers $x$ and $y$ .

In a sequence of numbers, a term is called golden if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1,2,3,\ldots ,2021?$

Find all triples $(x,p,n)$ of non-negative integers such that $p$ is prime and

$$2x(x + 5) = p^{n} + 3(x - 1).$$

Find all pairs of integers $x, y$ such that

$$y^{5} + 2x y = x^{2} + 2y^{4}.$$

Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied?

• Each number (not in the top row) is a proper divisor of the number immediately below.

• Each row consists of 10 consecutive positive integers (but not necessarily in order).

Let $a, b, c, d$ be integers such that $a > b > c > d \geq -2021$ and

$$\frac{a + b}{b + c} = \frac{c + d}{d + a}$$

(and $b + c \neq 0 \neq d + a$ ). What is the maximum possible value of $ac$ ?