MathDB

2014 IMC

Part of IMC

Subcontests

(5)
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IMC 2014, Problem 4

Let n>6n>6 be a perfect number, and let n=p1e1⋅⋅⋅pkekn=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k} be its prime factorisation with 1<p1<⋯<pk1<p_1<\dots <p_k. Prove that e1e_1 is an even number. A number nn is perfect if s(n)=2ns(n)=2n, where s(n)s(n) is the sum of the divisors of nn.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)

IMC 2014, Problem 9

We say that a subset of Rn\mathbb{R}^n is kk-almost contained by a hyperplane if there are less than kk points in that set which do not belong to the hyperplane. We call a finite set of points kk-generic if there is no hyperplane that kk-almost contains the set. For each pair of positive integers (k,n)(k, n), find the minimal number of d(k,n)d(k, n) such that every finite kk-generic set in Rn\mathbb{R}^n contains a kk-generic subset with at most d(k,n)d(k, n) elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)
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IMC 2014, Problem 2

Consider the following sequence (an)n=1∞=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,… )(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) Find all pairs (α,β)(\alpha, \beta) of positive real numbers such that lim⁡n→∞∑k=1naknα=β\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta.
(Proposed by Tomas Barta, Charles University, Prague)

IMC 2014, Problem 7

Let A=(aij)i,j=1nA=(a_{ij})_{i, j=1}^n be a symmetric n×nn\times n matrix with real entries, and let λ1,λ2,…,λn\lambda _1, \lambda _2, \dots, \lambda _n denote its eigenvalues. Show that ∑1≤i<j≤naiiajj≥∑1≤i<j≤nλiλj\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j and determine all matrices for which equality holds.
(Proposed by Matrin Niepel, Comenius University, Bratislava)
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IMC 2014, Problem 6

For a positive integer xx, denote its nthn^{\mathrm{th}} decimal digit by dn(x)d_n(x), i.e. dn(x)∈{0,1,…,9}d_n(x)\in \{ 0,1, \dots, 9\} and x=∑n=1∞dn(x)10n−1x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}. Suppose that for some sequence (an)n=1∞(a_n)_{n=1}^{\infty}, there are only finitely many zeros in the sequence (dn(an))n=1∞(d_n(a_n))_{n=1}^{\infty}. Prove that there are infinitely many positive integers that do not occur in the sequence (an)n=1∞(a_n)_{n=1}^{\infty}.
(Proposed by Alexander Bolbot, State University, Novosibirsk)

IMC 2014, Problem 1

Determine all pairs (a,b)(a, b) of real numbers for which there exists a unique symmetric 2×22\times 2 matrix MM with real entries satisfying trace(M)=a\mathrm{trace}(M)=a and det(M)=b\mathrm{det}(M)=b.
(Proposed by Stephan Wagner, Stellenbosch University)
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IMC 2014, Problem 5 [Day 1]

Let A1A2…A3nA_{1}A_{2} \dots A_{3n} be a closed broken line consisting of 3n3n lines segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index i=1,2,…,3ni=1,2,\dots,3n, the triangle AiAi+1Ai+2A_{i}A_{i+1}A_{i+2} has counterclockwise orientation and ∠AiAi+1Ai+2=60º\angle A_{i}A_{i+1}A_{i+2} = 60º, using the notation A3n+1=A1A_{3n+1} = A_{1} and A3n+2=A2A_{3n+2} = A_{2}. Prove that the number of self-intersections of the broken line is at most 32n2−2n+1\frac{3}{2}n^{2} - 2n + 1

IMC 2014, Problem 10

For every positive integer nn, denote by DnD_n the number of permutations (x1,…,xn)(x_1, \dots, x_n) of (1,2,…,n)(1,2,\dots, n) such that xj≠jx_j\neq j for every 1≤j≤n1\le j\le n. For 1≤k≤n21\le k\le \frac{n}{2}, denote by Δ(n,k)\Delta (n,k) the number of permutations (x1,…,xn)(x_1,\dots, x_n) of (1,2,…,n)(1,2,\dots, n) such that xi=k+ix_i=k+i for every 1≤i≤k1\le i\le k and xj≠jx_j\neq j for every 1≤j≤n1\le j\le n. Prove that Δ(n,k)=∑i=0k=1(k−1i)D(n+1)−(k+i)n−(k+i)\Delta (n,k)=\sum_{i=0}^{k=1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}
(Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)