MathDB

3

Part of 2016 Czech-Polish-Slovak Match

Problems(2)

Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P3,day 1

7/12/2016
Let nn be a positive integer. For a fi nite set MM of positive integers and each i∈{0,1,...,n−1}i \in \{0,1,..., n-1\}, we denote sis_i the number of non-empty subsets of MM whose sum of elements gives remainder ii after division by nn. We say that MM is "nn-balanced" if s0=s1=....=sn−1s_0 = s_1 =....= s_{n-1}. Prove that for every odd number nn there exists a non-empty nn-balanced subset of {0,1,...,n}\{0,1,..., n\}. For example if n=5n = 5 and M={1,3,4}M = \{1,3,4\}, we have s0=s1=s2=1,s3=s4=2s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2 so MM is not 55-balanced.(Czech Republic)
combinatorics
Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P3,day 2

7/12/2016
Let ABCABC be an acute-angled triangle with AB<ACAB < AC. Tangent to its circumcircle Ω\Omega at AA intersects the line BCBC at DD. Let GG be the centroid of △ABC\triangle ABC and let AGAG meet Ω\Omega again at H≠AH \neq A. Suppose the line DGDG intersects the lines ABAB and ACAC at EE and FF, respectively. Prove that ∠EHG=∠GHF\angle EHG = \angle GHF.(Slovakia)
geometry proposedgeometrycircumcircleCentroid