MathDB

1

Part of 2010 Turkey MO (2nd round)

Problems(2)

Turkey NMO 2010 P1

Source:

12/15/2010
In a country, there are some two-way roads between the cities. There are 20102010 roads connected to the capital city. For all cities different from the capital city, there are less than 20102010 roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. kk roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of k.k.
ceiling functioncombinatorics proposedcombinatorics
Turkey NMO 2010 P4

Source:

12/15/2010
Let AA and BB be two points on the circle with diameter [CD][CD] and on the different sides of the line CD.CD. A circle Γ\Gamma passing through CC and DD intersects [AC][AC] different from the endpoints at EE and intersects BCBC at F.F. The line tangent to Γ\Gamma at EE intersects BCBC at PP and QQ is a point on the circumcircle of the triangle CEPCEP different from EE and satisfying ∣QP∣=∣EP∣. AB∩EF={R}|QP|=|EP|. \: AB \cap EF =\{R\} and SS is the midpoint of [EQ].[EQ]. Prove that DRDR is parallel to PS.PS.
geometrycircumcirclegeometry proposed