10.10

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

Source: IVSoros Olympiad 1997-98 Corr. 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

A circle touches the extensions of sides

CA

CB

ABC

, and also touches side

AB

of this triangle at point

P

. Prove that the radius of the circle tangent to segments

AP

CP

and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.

geometryequal circles

15 teams in football tournament (IV Soros Olympiad 1997-98 R3 10.10)

Source:

The football tournament, held in one round, involved

16

teams, each two of which scored a different number of points. (

3

points were given for a victory,

1

point for a draw,

0

points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?

number theoryalgebra