MathDB

5

Part of 2014 IMC

Problems(2)

IMC 2014, Problem 5 [Day 1]

Source:

7/31/2015
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
IMCgeometrygeometric transformationcollege contests
IMC 2014, Problem 10

Source: IMC 2014

7/27/2016
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)
IMCcollege contestspermutationscombinatorics