MathDB

2

Part of 2015 Czech-Polish-Slovak Match

Problems(2)

Maximal size of a perfect family of sets

Source: Czech-Polish-Slovak Match 2015, Problem 2

6/19/2015
A family of sets FF is called perfect if the following condition holds: For every triple of sets X1,X2,X3∈FX_1, X_2, X_3\in F, at least one of the sets (X1∖X2)∩X3, (X_1\setminus X_2)\cap X_3, (X2∖X1)∩X3(X_2\setminus X_1)\cap X_3 is empty. Show that if FF is a perfect family consisting of some subsets of a given finite set UU, then ∣F∣≤∣U∣+1\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1.
Proposed by Michał Pilipczuk
combinatoricsSets
Circles through A touching sides of the triangle

Source: Czech-Polish-Slovak Match 2015, Problem 5

6/19/2015
Let ABCABC be an acute triangle, which is not equilateral. Denote by OO and HH its circumcenter and orthocenter, respectively. The circle kk passes through BB and touches the line ACAC at AA. The circle ll with center on the ray BHBH touhes the line ABAB at AA. The circles kk and ll meet in XX (X≠AX\ne A). Show that ∠HXO=180∘−∠BAC\angle HXO=180^\circ-\angle BAC.
Proposed by Josef Tkadlec
geometry