Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$must be a multiple of $12$.

Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$

Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

Let $P$ be a point inside triangle $ABC$ such that $\angle CPA = 90^o$ and $\angle CBP = \angle CAP$. Prove that $\angle P XY = 90^o$, where $X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively.

Let $a, b$ and $c$ be positive real numbers satisfying $$\frac{1}{a + 2019}+\frac{1}{b + 2019}+\frac{1}{c + 2019}=\frac{1}{2019}.$$Prove that $abc \ge 4038^3$.

The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

Let $N$ be the number of ways to colour each cell in a $2 \times 50$ rectangle either red or blue such that each $2 \times 2$ block contains at least one blue cell. Show that $N$ is a multiple of $3^{25}$, but not a multiple of $3^{26}$

Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.

Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$for all $x, y \in R$.