## IMC 2014 Day 2 Problem 5

August 1, 2014 2 comments

For every positive integer ${n}$, denote by ${D_n}$ the number of permutations ${(x_1,...,x_n)}$ of ${(1,2,...,n)}$ such that ${x_j \neq j}$ for every ${1 \leq j \leq n}$. For ${1 \leq k \leq \frac{n}{2}}$, denote by ${\Delta(n,k)}$ the number of permutations ${(x_1,...,x_n)}$ of ${(1,2,...,n)}$ such that ${x_i = k+i}$ for every ${1 \leq i \leq k}$ and ${x_j \neq j}$ for every ${1 \leq j \leq n}$. Prove that

$\displaystyle \Delta(n,k) = \sum_{i = 0}^{k-1} {k-1 \choose i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}.$

IMC 2014 Day 2 Problem 5

## IMC 2014 Day 2 Problem 4

We say that a subset of ${\Bbb{R}^n}$ is ${k}$-almost contained by a hyperplane if there are less than ${k}$ points in that set which do not belong to the hyperplane. We call a finite set of points ${k}$-generic if there is no hyperplane that ${k}$-almost contains the set. For each pair of positive integers ${k}$ and ${n}$, find the minimal number ${d(k,n)}$ such that every finite ${k}$-generic set in ${\Bbb{R}^n}$ contains a ${k}$-generic subset with at most ${d(k,n)}$ elements.

IMC 2014 Day 2 Problem 4

## IMC 2014 Day 2 Problem 3

August 1, 2014 2 comments

Let ${f(x) = \displaystyle \frac{\sin x}{x}}$, for ${x >0}$ and let ${n}$ be a positive integer. Prove that ${\displaystyle | f^{(n)}(x) | <\frac{1}{n+1}}$, where ${f^{(n)}}$ denotes the ${n}$-th derivative of ${f}$.

IMC 2014 Day 2 Problem 3

Categories: Analysis, Olympiad Tags: ,

## IMC 2014 Day 2 Problem 2

August 1, 2014 5 comments

Let ${A=(a_{ij})_{i,j=1}^n}$ be a symmetric ${n \times n}$ matrix with real entries, and let ${\lambda_1,...,\lambda_n}$ denote its eigenvalues. Show that

$\displaystyle \sum_{1 \leq i

and determine all matrices for which equality holds.

IMC 2014 Day 2 Problem 2

## IMC 2014 Day 2 Problem 1

For a positive integer ${x}$ denote its ${n}$-th decimal digit by ${d_n(x)}$, i.e. ${d_n(x) \in \{0,1,..,9\}}$ and ${x = \displaystyle \sum_{n=1}^\infty d_n(x) 10^{n-1}}$. Suppose that for some sequence ${(a_n)_{n=1}^\infty}$ there are only finitely many zeros in the sequence ${(d_n(a_n))_{n=1}^\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence ${(a_n)_{n=1}^\infty}$.

IMC 2014 Day 2 Problem 1

Categories: Algebra Tags: , ,

## IMC 2014 Day 1 Problem 5

Let ${A_1A_2...A_{3n}}$ be a close broken line consisting of ${3n}$ line segments in the Euclidean plane. Suppose that no three of its vertices are collinear and for each index ${i=1,2,...,3n}$, the triangle ${A_iA_{i+1}A_{i+2}}$ has counterclockwise orientation and ${\angle A_iA_{i+1}A_{i+2}=60^\circ}$, using the notation modulo ${3n}$. Prove that the number of self-intersections of the broken line is at most ${\displaystyle \frac{3}{2}n^2 -2n+1}$.

IMC 2014 Day 1 Problem 5

## IMC 2014 Day 1 Problem 4

July 31, 2014 2 comments

Let ${n > 6}$ be a perfect number and ${n = p_1^{e_1}...p_k^{e_k}}$ be its prime factorization with ${1. Prove that ${e_1}$ is an even number.

A number ${n}$ is perfect if ${s(n)=2n}$, where ${s(n)}$ is the sum of the divisors of ${n}$.

IMC 2014 Day 1 Problem 4