MathDB

2

Part of 1997 Iran MO (3rd Round)

Problems(3)

Nice Geometric inequality on angles sines and side lengths

Source: Iran Third Round MO 1997, Exam 1, P2

2/17/2006
Show that for any arbitrary triangle ABCABC, we have sin⁡(A2)⋅sin⁡(B2)⋅sin⁡(C2)≤abc(a+b)(b+c)(c+a).\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.
inequalitiestrigonometrygeometrycircumcirclegeometry proposed
two traingles and 6 points of intersection

Source: Iran Third Round MO 1997, Exam 2, P2

3/23/2004
Let ABCABC and XYZXYZ be two triangles. Define A1=BC∩ZX,A2=BC∩XY,A_1=BC\cap ZX, A_2=BC\cap XY,B1=CA∩XY,B2=CA∩YZ,B_1=CA\cap XY, B_2=CA\cap YZ,C1=AB∩YZ,C2=AB∩ZX.C_1=AB\cap YZ, C_2=AB\cap ZX. Hereby, the abbreviation g∩hg\cap h means the point of intersection of two lines gg and hh.
Prove that C1C2AB=A1A2BC=B1B2CA\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA} holds if and only if A1C2XZ=C1B2ZY=B1A2YX\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}.
geometryparallelogramgeometry proposed
Circle pass through mid of bc

Source: Iran Third Round MO 1997, Exam 3, P2

10/18/2005
In an acute triangle ABCABC, points D,E,FD,E,F are the feet of the altitudes from A,B,CA,B,C, respectively. A line through DD parallel to EFEF meets ACAC at QQ and ABAB at RR. Lines BCBC and EFEF intersect at PP. Prove that the circumcircle of triangle PQRPQR passes through the midpoint of BCBC.
geometrycircumcirclegeometry proposed