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

August 1, 2014 Leave a comment

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<j\leq n} a_{ii}a_{jj} \geq \sum_{1 \leq i<j\leq n}\lambda_i \lambda_j

and determine all matrices for which equality holds.

IMC 2014 Day 2 Problem 2

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IMC 2014 Day 2 Problem 1

August 1, 2014 Leave a comment

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

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Categories: Algebra Tags: , ,

IMC 2014 Day 1 Problem 5

July 31, 2014 Leave a comment

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<p_1<...<p_k}. 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

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