For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4) = 5$ because $$4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1.$$ Also define $p(0) = 1$.

Prove that $p(n) - p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1.

(Proposed by Fedor Duzhin, Nanyang Technological University)

Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n \times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.

(Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow)

Given an integer $n > 1$, let $S_n$ be the group of permutations of the numbers $1, 2, \dots, n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?

(Proposed by Fedor Petrov, St. Petersburg State University)

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function that satisfies $f'(t) > f(f(t))$ for all $t \in \mathbb{R}$. Prove that $f(f(f(t))) \le 0$ for all $t \ge 0$.

(Proposed by Tomáš Bárta, Charles University, Prague)

Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n} (X + a)^{2^n} + 1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.

(Proposed by Vincent Jugé, École Polytechnique, Paris)

Consider a polynomial $$f(x) = x^{2012} + a_{2011} x^{2011} + \dots + a_1 x + a_0.$$ Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients $a_0, \dots, a_{2011}$ and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values.

Homer's goal is to make $f(x)$ divisible by a fixed polynomial $m(x)$ and Albert's goal is to prevent this.

(a) Which of the players has a winning strategy if $m(x) = x - 2012$?

(b) Which of the players has a winning strategy if $m(x) = x^2 + 1$?

(Proposed by Fedor Duzhin, Nanyang Technological University)

Define the sequence $a_0, a_1, \dots$ inductively by $a_0 = 1$, $a_1 = \frac{1}{2}$ and $$a_{n+1} = \frac{n a_n^2}{1 + (n+1) a_n} \quad \text{for } n \ge 1.$$ Show that the series $\sum\limits_{k=0}^{\infty} \dfrac{a_{k+1}}{a_k}$ converges and determine its value.

(Proposed by Christophe Debry, KU Leuven, Belgium)

Is the set of positive integers $n$ such that $n! + 1$ divides $(2012n)!$ finite or infinite?

(Proposed by Fedor Petrov, St. Petersburg State University)

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1$, …, $x_n$ satisfying $$\tag{1} x_1 (1 - x_2) = x_2 (1 - x_3) = \dots = x_{n-1} (1 - x_n) = x_n (1 - x_1) = a.$$

(Proposed by Walther Janous and Gerhard Kirchner, Innsbruck)

Let $c \ge 1$ be a real number. Let $G$ be an abelian group and let $A \subset G$ be a finite set satisfying $|A + A| \le c |A|$, where $X + Y := \{ x + y \mid x \in X,\ y \in Y \}$ and $|Z|$ denotes the cardinality of $Z$. Prove that $$|\underbrace{A + A + \dots + A}_{k\ \text{times}}| \le c^k |A|$$ for every positive integer $k$. (Plünnecke's inequality)

(Proposed by Przemyslaw Mazur, Jagiellonian University)