MathDB

1

Part of 2010 Turkey Team Selection Test

Problems(2)

Turkey TST 2010 Q1

Source:

9/1/2010
D, E, FD, \: E , \: F are points on the sides AB, BC, CA,AB, \: BC, \: CA, respectively, of a triangle ABCABC such that AD=AF, BD=BE,AD=AF, \: BD=BE, and DE=DF.DE=DF. Let II be the incenter of the triangle ABC,ABC, and let KK be the point of intersection of the line BIBI and the tangent line through AA to the circumcircle of the triangle ABI.ABI. Show that AK=EKAK=EK if AK=AD.AK=AD.
geometryincentercircumcircletrigonometrytrig identitiesLaw of Sinesgeometry proposed
Turkey TST 2010 Q4

Source:

9/1/2010
Let 0≤k<n0 \leq k < n be integers and A={a : a≡k(modn)}.A=\{a \: : \: a \equiv k \pmod n \}. Find the smallest value of nn for which the expression am+3ma2−3a+1 \frac{a^m+3^m}{a^2-3a+1} does not take any integer values for (a,m)∈A×Z+.(a,m) \in A \times \mathbb{Z^+}.
modular arithmeticnumber theory proposednumber theory