MathDB

5

Part of 2013 Czech-Polish-Slovak Junior Match

Problems(2)

midpoint lies on perpendicular bisectors, circumcenters related

Source: Czech-Polish-Slovak Junior Match 2013, Individual p5 CPSJ

3/14/2020
Point MM is the midpoint of the side ABAB of an acute triangle ABCABC. Point PP lies on the segment ABAB, and points S1S_1 and S2S_2 are the centers of the circumcircles of APCAPC and BPCBPC, respectively. Show that the midpoint of segment S1S2S_1S_2 lies on the perpendicular bisector of segment CMCM.
geometryperpendicular bisectormidpointCircumcenter
a+b+c>= 3 if ab+ac+bc >= a +b +c and a,b,c>0

Source: Czech-Polish-Slovak Junior Match 2013, Team p5 CPSJ

3/21/2020
Let a,b,ca, b, c be positive real numbers for which ab+ac+bc≥a+b+cab + ac + bc \ge a + b + c. Prove that a+b+c≥3a + b + c \ge 3.
inequalitiesalgebra