MathDB

3

Part of 2007 Moldova Team Selection Test

Problems(3)

6 geometry problems out of 8 at our TST so far..

Source: Moldova 2007 IMO-BMO TST II problem 3

3/23/2007
Let M,NM, N be points inside the angle ∠BAC\angle BAC usch that ∠MAB≡∠NAC\angle MAB\equiv \angle NAC. If M1,M2M_{1}, M_{2} and N1,N2N_{1}, N_{2} are the projections of MM and NN on AB,ACAB, AC respectively then prove that M,NM, N and PP the intersection of M1N2M_{1}N_{2} with N1M2N_{1}M_{2} are collinear.
geometryratiotrigonometrycircumcircletrapezoidradical axisprojective geometry
an elegant identity in elements of a triangle

Source: Moldova 2007 IMO-BMO TST III problem 3

3/24/2007
Consider a triangle ABCABC, with corresponding sides a,b,ca,b,c, inradius rr and circumradius RR. If rA,rB,rCr_{A}, r_{B}, r_{C} are the radii of the respective excircles of the triangle, show that a2(2rA−rrBrC)+b2(2rB−rrArC)+c2(2rC−rrArB)=4(R+3r)a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r)
geometryinradiuscircumcircletrigonometrygeometry proposed
A circle tangent to the circumcircle and two sides

Source: Moldova 2007 IMO-BMO TST IV problem 3

3/25/2007
Let ABCABC be a triangle. A circle is tangent to sides AB,ACAB, AC and to the circumcircle of ABCABC (internally) at points P,Q,RP, Q, R respectively. Let SS be the point where ARAR meets PQPQ. Show that ∠SBA≡∠SCA\angle{SBA}\equiv \angle{SCA}
geometrycircumcircleincentergeometry proposed