Subcontests
(6)CIIM 2019 Problem 5
Let {k1,k2,…,km} a set of m integers. Show that there exists a matrix m×m with integers entries A such that each of the matrices A+kjI,1≤j≤m are invertible and their entries have integer entries (here I denotes the identity matrix).
CIIM 2019 Problem 4
Let (G,∗) a group of n>1 elements, and let g∈G be an element distinct from the identity.
Ana and Bob play with the group G on the following way:
Starting with Ana and playing alternately, each player selects an element of G that has not been selected before, until each element of G have been selected or a player have selected the elements a and a∗g for some a∈G.
In that case it is said that the player loses and his opponent wins.a) If n is odd, show that, independent of element g, one of the two players has
a winning strategy and determines which player
possesses such a strategy.b) If n is even, show that there exists an element g∈G for which none of the players
has a winning strategy.Note: A group (G,∗) es a set G together with a binary operation ∗:G×G→G that satisfy the following properties
(i) ∗ is asociative: ∀a,b,c∈G(a∗b)∗c=a∗(b∗c);
(ii) there exists an identity element e∈G such that ∀a∈G,a∗e=e∗a=a;
(iii) there exists inverse elements: ∀a∈G∃a−1∈G such that a∗a−1=a−1∗a=e. CIIM 2019 Problem 3
Let {an}n∈N a sequence of non zero real numbers.
For m≥1, we define:
Xm={X⊆{0,1,…,m−1}:x∈X∑ax>m1}.
Show that
n→∞lim2n∣Xn∣=1. CIIM 2019 Problem 2
Consider the set
{0,1}n={X=(x1,x2,…,xn):xi∈{0,1},1≤i≤n}.
We say that X>Y if X=Y and the following n inequalities are satisfy
x1≥y1,x1+x2≥y1+y2,…,x1+x2+⋯+xn≥y1+y2+⋯+yn.
We define a chain of length k as a subset Z1,…,Zk⊆{0,1}n of distinct elements such that Z1>Z2>⋯>Zk.
Determine the lenght of longest chain in {0,1}n.