MathDB

Problem 3

Part of 2003 Federal Math Competition of S&M

Problems(3)

side equality in triangle gives angles

Source: Serbia 2003 1st Grade P3

5/12/2021
Let a,ba,b and cc be the lengths of the edges of a triangle whose angles are α=40∘,β=60∘\alpha=40^\circ,\beta=60^\circ and γ=80∘\gamma=80^\circ. Prove that a(a+b+c)=b(b+c).a(a+b+c)=b(b+c).
geometryTriangles
Determine the set of all points P....very easy

Source: Serbia and Montanagro 2003

7/31/2005
Let ABCDABCD be a rectangle. Determine the set of all points PP from the region between the parallel lines ABAB and CDCD such that ∠APB=∠CPD\angle APB=\angle CPD.
geometryrectanglesymmetrycircumcircleperpendicular bisectorgeometry solved
easy and nice

Source: Serbia and Montanagro Team Selection Test 2003

8/7/2005
Given a circle kk and the point PP outside it, an arbitrary line ss passing through PP intersects kk at the points AA and BB . Let MM and NN be the midpoints of the arcs determined by the points AA and BB and let CC be the point on ABAB such that PC2=PA⋅PBPC^2=PA\cdot PB . Prove that ∠MCN\angle MCN doesn't depend on the choice of ss. [Moderator edit: This problem has also been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=56295 .]
geometry unsolvedgeometry