Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?

We consider $5 \times 5$ tables containing a real number in each of the $25$ cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is balanced if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell is called small if the number in that cell is strictly smaller than the number in the cell in the very middle of the table. What is the least number of small cells that a balanced table can have?

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.

A quadruple $(p, a, b, c)$ of positive integers is a karaka quadruple if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

Find all polynomials $P(x)$ with real coefficients such that the polynomial $$Q(x) = (x + 1)P(x-1) -(x-1)P(x)$$is constant.

Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $P \ne E$ be the point of tangency of the circle with radius $HE$ centred at $H$ with its tangent line going through point $C$, and let $Q \ne E$ be the point of tangency of the circle with radius $BE$ centred at $B$ with its tangent line going through $C$. Prove that the points $D, P$ and $Q$ are collinear.

Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$has positive integer solutions.

Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.

An $n$-tuple $(a_1, a_2 . . . , a_n)$ is occasionally periodic if there exist a non-negative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+j+p}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, a_2 . . . , a_n)$ with elements from the set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, a_2, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.