MathDB

9

Part of 2018 Hanoi Open Mathematics Competitions

Problems(4)

3 polygons of area 3 inside square of area 6 (HOMC 2018 Ind. p9)

Source:

1/31/2020
There are three polygons and the area of each one is 33. They are drawn inside a square of area 66. Find the greatest value of mm such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than mm.
geometrycombinatorial geometrygeometric inequalityarea
Hanoi diagrams, fill numbers 0-9 with 3 of same sum (HOMC 2018 Team j9)

Source:

1/31/2020
Each of the thirty squares in the diagram below contains a number 0,1,2,3,4,5,6,7,8,90, 1, 2, 3, 4, 5, 6, 7, 8, 9 of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to SS. https://cdn.artofproblemsolving.com/attachments/8/0/3e056ebc252aee9ade1f45fd337cc6a2f84302.png
Sumalgebra
ways of choosing 4 edges in a cube such that (HOMC 2018 ind. sen9)

Source:

2/2/2020
How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point.
combinatoricscube
circumcenter is equidistant from B,C (HOMC 2018 S Team 9)

Source:

2/17/2020
Let ABCABC be acute, non-isosceles triangle, inscribed in the circle (O)(O). Let DD be perpendicular projection of AA onto BCBC, and E,FE, F be perpendicular projections of DD onto CA,ABCA,AB respectively. (a) Prove that AO⊥EFAO \perp EF. (b) The line AOAO intersects DE,DFDE,DF at I,JI,J respectively. Prove that △DIJ\vartriangle DIJ and △ABC\vartriangle ABC are similar. (c) Prove that circumcenter of △DIJ\vartriangle DIJ is equidistant from BB and CC
geometryperpendicularCircumcentersimilar triangles