MathDB

1

Part of 2013 Iran MO (2nd Round)

Problems(2)

a/b = a.b

Source: Iran second round- 2013- P1

5/4/2013
Find all pairs (a,b)(a,b) of positive integers for which gcd⁡(a,b)=1\gcd(a,b)=1, and ab=b.a‾\frac{a}{b}=\overline{b.a}. (For example, if a=92a=92 and b=13b=13, then b/a=13.92b/a=13.92 )
number theory proposednumber theory
symmedian in two circles

Source: Iran second round- 2013- P4

5/4/2013
Let PP be a point out of circle CC. Let PAPA and PBPB be the tangents to the circle drawn from CC. Choose a point KK on ABAB . Suppose that the circumcircle of triangle PBKPBK intersects CC again at TT. Let P′{P}' be the reflection of PP with respect to AA. Prove that ∠PBT=∠P′KA \angle PBT = \angle {P}'KA
geometrycircumcirclegeometric transformationreflectiongeometry proposed