Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$

An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called [i]central[/i] if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$. \\Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.

The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that \[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\] for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that \[\varphi(x,y,z) = h(x+y+z)\] for all real numbers $x,y$ and $z.$

Find the total number of $k$-tuples $(n_1,n_2,...,n_k)$ of positive integers so that $n_{i+1}\ge n_i$ for each $i$, and $k$ regular polygons with numbers of sides $n_1,n_2,...,n_k$ respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is $360^{\circ}$.

Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.

Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality \[ 1 < a < b+2 < 10. \] [i]Proposed by Lewis Chen [/i]

Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.

Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.

At CMIMC headquarters, there is a row of $n$ lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things: [list] [*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. [/list] Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most $n$ seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value $n$ such that he can complete his task on time? [i]Proposed by Adam Bertelli[/i]

Determine whether there exists a function $ f: \mathbb{N}\longrightarrow \mathbb{N}$ such that $ f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1))$ for all natural numbers $ n\ge 2$.