MathDB

4

Part of 2014 Danube Mathematical Competition

Problems(2)

\Sum a_1=0 from i=1,...,2n, among all pairs 2n-1 have sum >=0

Source: Danube junior 2014 p4

7/22/2019
Consider the real numbers a1,a2,...,a2na_1,a_2,...,a_{2n} whose sum is equal to 00. Prove that among pairs (ai,aj),i<j(a_i,a_j) , i<j where i,j∈{1,2,...,2n} i,j \in \{1,2,...,2n\} .there are at least 2n−12n-1 pairs with the property that ai+aj≥0a_i+a_j\ge 0.
algebracombinatoricsSum
max cardinality of set of lattice point with lines no parallel to given triangle

Source: Danube 2014 p4

7/22/2019
Let nn be a positive integer and let △\triangle be the closed triangular domain with vertices at the lattice points (0,0),(n,0)(0, 0), (n, 0) and (0,n)(0, n). Determine the maximal cardinality a set SS of lattice points in △\triangle may have, if the line through every pair of distinct points in SS is parallel to no side of △\triangle.
lattice pointsparallelogramMaximalcardinalitycombinatoricsgeometrycombinatorial geometry