MathDB

3

Part of 2007 Iran Team Selection Test

Problems(4)

Incircle, and a tangency of circle and a line

Source: Iran TST 2007, Day 2

5/7/2007
Let ω\omega be incircle of ABCABC. PP and QQ are on ABAB and ACAC, such that PQPQ is parallel to BCBC and is tangent to ω\omega. AB,ACAB,AC touch ω\omega at F,EF,E. Prove that if MM is midpoint of PQPQ, and TT is intersection point of EFEF and BCBC, then TMTM is tangent to ω\omega. By Ali Khezeli
geometryincentergeometric transformationhomothetytrapezoidanalytic geometryreflection
A routine Functional Equation

Source: Iran TST 2007, Day 1

5/5/2007
Find all solutions of the following functional equation: f(x2+y+f(y))=2y+f(x)2.f(x^{2}+y+f(y))=2y+f(x)^{2}.
algebra proposedalgebra
Tangency of circles

Source: Iran TST 2007, Day 3

5/23/2007
OO is a point inside triangle ABCABC such that OA=OB+OCOA=OB+OC. Suppose B′,C′B',C' be midpoints of arcs \overarc{AOC} and AOBAOB. Prove that circumcircles COC′COC' and BOB′BOB' are tangent to each other.
geometrycircumcirclegeometry proposed
Sequence and light ray

Source: Iran TST 2007, Day 4

5/28/2007
Let PP be a point in a square whose side are mirror. A ray of light comes from PP and with slope α\alpha. We know that this ray of light never arrives to a vertex. We make an infinite sequence of 0,10,1. After each contact of light ray with a horizontal side, we put 00, and after each contact with a vertical side, we put 11. For each n≥1n\geq 1, let BnB_{n} be set of all blocks of length nn, in this sequence. a) Prove that BnB_{n} does not depend on location of PP. b) Prove that if απ\frac{\alpha}{\pi} is irrational, then ∣Bn∣=n+1|B_{n}|=n+1.
analytic geometrygraphing linesslopetrigonometrygeometrygeometric transformationreflection