# Day \(1\)

July 28, 2012

Problem 1. 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)

Problem 2. 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)

Problem 3. Given an integer \(n>1\), let \(S_n\) be the group of permutations of the numbers \(1,2,3,\dotsc,n\). 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)

Problem 4. Let \(f:\;\Bbb R \to\Bbb R\) be a continuously differentiable function that satisfies \(f^\prime(t)>f(f(t))\) for all \(t\in\Bbb R\). Prove that \(f(f(f(t)))\leqslant0\) for all \(t\geqslant0\).

(Proposed by Tomas Barta, Charles University, Prague)

Problem 5. 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 \(\Bbb Q[X]\) of polynomials with rational coefficients.

(Proposed by Vincent Juge, Ecole Polytechnique, Paris)

# Day \(2\)

July 29, 2012

Problem 1. Consider a polynomial

\[f(x)=x^{2012}+a_{2011}x^{2011}+\dotsb+a_1x+a_0.\]

Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients \( a_0,a_1,\dotsc,a_{2011}\) 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)

Problem 2. Define the sequence \( a_0,a_1,\dotsc\) inductively by \(a_0=1,a_1=\frac 12\) and

\[ a_{n+1}=\frac{na_n^2}{1+(n+1)a_n}\quad\text{for}\: n\geqslant1.\]

Show that the series \( \sum\limits_{k=0}^\infty\frac{a_{k+1}}{a_k}\) converges and determine its value.

(Proposed by Christophe Debry, KU Leuven, Belgium)

Problem 3. Is the set of positive integers \(n\) such that \(n!+1\) divides \((2012n)!\) finite or infinite?

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

Problem 4. Let \(n\geqslant2\) be an integer. Find all real numbers \(a\) such that there exist real numbers \(x_1,\dotsc,x_n\) satisfying

\[x_1(1-x_2)=x_2(1-x_3)=\ldots=x_{n-1}(1-x_n)=x_n(1-x_1)=a.\]

(Proposed by Walther Janous and Gerhard Kirchner, Innsbruck)

Problem 5. Let \(c\geqslant1\) be a real number. Let \(G\) be an Abelian group and let \( A\subset G\) be a finite set satisfying \(|A+A|\leqslant c|A|\), where \( X+Y:=\{x+y| x\in X, y\in Y\}\) and \(|Z|\) denotes the cardinality of \(Z\). Prove that

\[|\underbrace{A+A+\dotsb +A}_{k \:\text{times}}|\leqslant c^k|A|\]

for every positive integer \(k\). (Plunnecke’s inequality)

(Proposed by Przemyslaw Mazur, Jagiellonian University)