MathDB

1

Part of 2013 Turkey MO (2nd round)

Problems(2)

Show that four points are concyclic

Source: Turkey National Olympiad Second Round 2013 P1

11/28/2013
The circle ω1\omega_1 with diameter [AB][AB] and the circle ω2\omega_2 with center AA intersects at points CC and DD. Let EE be a point on the circle ω2\omega_2, which is outside ω1\omega_1 and at the same side as CC with respect to the line ABAB. Let the second point of intersection of the line BEBE with ω2\omega_2 be FF. For a point KK on the circle ω1\omega_1 which is on the same side as AA with respect to the diameter of ω1\omega_1 passing through CC we have 2⋅CK⋅AC=CE⋅AB2\cdot CK \cdot AC = CE \cdot AB. Let the second point of intersection of the line KFKF with ω1\omega_1 be LL. Show that the symmetric of the point DD with respect to the line BEBE is on the circumcircle of the triangle LFCLFC.
geometrycircumcirclegeometric transformationreflectionsymmetrycyclic quadrilateralsimilar triangles
2^n+n=m!

Source: Turkey National Olympiad Second Round 2013 P4

11/28/2013
Find all positive integers mm and nn satisfying 2n+n=m!2^n+n=m!.
number theoryDiophantine equationfactorial