MathDB

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Part of 2014 India Regional Mathematical Olympiad

Problems(5)

Given a^2+5b^2+4c^2-4ab-4bc=0, can a,b,c form sides of Δ ?

Source: Mumbai Region RMO 2014 Problem 1

12/7/2014
Three positive real numbers a,b,ca,b,c are such that a2+5b2+4c2−4ab−4bc=0a^2+5b^2+4c^2-4ab-4bc=0. Can a,b,ca,b,c be the lengths of te sides of a triangle? Justify your answer.
algebra proposedalgebra
Dividing the altitude of a triangle into four equal parts

Source: RMO 2014 West Bengal Problem 1

12/7/2014
In acute △ABC,\triangle ABC, let DD be the foot of perpendicular from AA on BCBC. Consider points K,L,MK, L, M on segment ADAD such that AK=KL=LM=MDAK= KL= LM= MD. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that BD=DCBD= DC.
http://s27.postimg.org/a0d0plr4z/Untitled.png
geometrytrapezoidsimilar trianglesgeometry proposed
circumcentre of a triangle is the incentre of another

Source: CRMO 2014 region 2 p1

9/29/2018
In an acute-angled triangle ABC,∠ABCABC, \angle ABC is the largest angle. The perpendicular bisectors of BCBC and BABA intersect AC at XX and YY respectively. Prove that circumcentre of triangle ABCABC is incentre of triangle BXYBXY .
geometrycircumcircleincenterperpendicular bisector
Indian RMO 2014 P1

Source:

1/1/2015
Let ABCABC be a triangle with ∠ABC\angle ABC as the largest angle. Let RR be its circumcenter. Let the circumcircle of triangle ARBARB cut ACAC again at XX. Prove that RXRX is perpendicular to BCBC.
geometrycircumcircleAsymptoteperpendicular bisector
Indian RMO P1

Source:

1/1/2015
let ABCDABCD be a isosceles trapezium having an incircle with ABAB parallel to CDCD. let CECE be the perpendicular from CC on ABAB prove that CE2=AB.CD CE^2 = AB. CD
geometrytrapezoidsymmetryincenter