MathDB

5

Part of 2012 India Regional Mathematical Olympiad

Problems(6)

RMO 2012 Kar Region

Source: RMO 2012

12/16/2012
Let ABCABC be a triangle. Let BEBE and CFCF be internal angle bisectors of ∠B\angle B and ∠C\angle C respectively with EE on ACAC and FF on ABAB. Suppose XX is a point on the segment CFCF such that AXAX perpendicular CFCF; and YY is a point on the segment BEBE such that AYAY perpendicular BEBE. Prove that XY=(b+c−a)/2XY = (b + c-a)/2 where BC=a,CA=bBC = a, CA = b and AB=cAB = c.
trigonometrygeometryincentergeometry proposed
ratio chasing inside a triangle, segment trisecting

Source: CRMO 2012 Region 2 p5

9/30/2018
Let ABCABC be a triangle. Let D,ED, E be a points on the segment BCBC such that BD=DE=ECBD =DE = EC. Let FF be the mid-point of ACAC. Let BFBF intersect ADAD in PP and AEAE in QQ respectively. Determine BP:PQBP:PQ.
ratiogeometrymidpoint
Ratio of areas in a familiar situation

Source: RMO 2012

12/2/2012
Let ABCABC be a triangle. Let D,ED,E be points on the segment BCBC such that BD=DE=ECBD=DE=EC. Let FF be the mid-point of ACAC. Let BFBF intersect ADAD in PP and AEAE in QQ respectively. Determine the ratio of the area of the triangle APQAPQ to that of the quadrilateral PDEQPDEQ.
ratiogeometrytrigonometryarea of a triangle
ratio chasing inside a triangle, double segment

Source: CRMO 2012 Region 4 p5

9/30/2018
Let ABCABC be a triangle. Let EE be a point on the segment BCBC such that BE=2ECBE = 2EC. Let FF be the mid-point of ACAC. Let BFBF intersect AEAE in QQ. Determine BQ:QFBQ:QF.
ratiogeometrymidpoint
\frac{1}{a}+ \frac{2}{b} +\frac{3}{c}= 1 find when a is prime and a \le b \le c

Source: CRMO 2012 region 5 p5 Mumbai

9/30/2018
Determine with proof all triples (a,b,c)(a, b, c) of positive integers satisfying 1a+2b+3c=1\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1, where aa is a prime number and a≤b≤ca \le b \le c.
number theoryDiophantine equationdiophantineprimepositive integers
Angle bisectors AL, BK; then LN=NA[Indian RMO 2012(b) Q5]

Source:

12/2/2012
Let ALAL and BKBK be the angle bisectors in a non-isosceles triangle ABC,ABC, where LL lies on BCBC and KK lies on AC.AC. The perpendicular bisector of BKBK intersects the line ALAL at MM. Point NN lies on the line BKBK such that LNLN is parallel to MK.MK. Prove that LN=NA.LN=NA.
geometrycircumcircleperpendicular bisectorangle bisectorgeometry proposed