7/29/2019 Rmo 2011 Solutions Previous year Question Papers of
Regional Mathematical Olympiad with solutions
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Problems and Solutions: CRMO-2011
1. LetABC be a triangle. LetD, E, F be points respectively on
the segments BC,CA, AB such that AD, BE, CF concur at the point K.
Suppose BD/DC =BF/FA and ADB = AF C. Prove thatABE = CAD.
Solution: Since BD/DC = BF/FA,the lines DF and CA are parallel.
Wealso have BDK = ADB = AF C =180 BF K, so thatBDKF is a
cyclicquadrilateral. Hence F BK = F DK.Finally, we get
ABE = F BK = F DK
= F DA = DAC,
since F D AC.
2. Let(a1, a2, a3, . . . , a2011) be a permutation (that is a
rearrangement) of the num-bers 1, 2, 3, . . . , 2011. Show that
there exist two numbers j, k such that 1 j
