MathDB

5

Part of 2019 India Regional Mathematical Olympiad

Problems(2)

Perpendicular lines

Source: RMO 2019 P5

10/20/2019
In an acute angled triangle ABCABC, let HH be the orthocenter, and let D,E,FD,E,F be the feet of altitudes from A,B,CA,B,C to the opposite sides, respectively. Let L,M,NL,M,N be the midpoints of the segments AH,EF,BCAH, EF, BC respectively. Let X,YX,Y be the feet of altitudes from L,NL,N on to the line DFDF respectively. Prove that XMXM is perpendicular to MYMY.
geometryRMO
Counting problem with three types of objects

Source: RMO Maharashtra and Goa 2019 P5

11/10/2019
There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from {Δ,□,⊙}\{\Delta,\square,\odot\}; the second value is a letter from {A,B,C}\{A,B,C\}; and the third value is a number from {1,2,3}\{1,2,3\}. In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values. For example we can chose {ΔA1,ΔB2,⊙C3}\{\Delta A1,\Delta B2,\odot C3\} But we cannot choose {ΔA1,□B2,ΔC1}\{\Delta A1,\square B2,\Delta C1\}
combinatorics