MathDB

2

Part of 2006 Vietnam Team Selection Test

Problems(2)

Problem 2 vietnamese tst 2006

Source: Vietnamese TST 2006

4/17/2006
Find all pair of integer numbers (n,k)(n,k) such that nn is not negative and kk is greater than 11, and satisfying that the number: A=172006n+4.172n+7.195n A=17^{2006n}+4.17^{2n}+7.19^{5n} can be represented as the product of kk consecutive positive integers.
number theory proposednumber theory
Problem 5 vietnamese tst 2006

Source: Vietnamese TST 2006

4/18/2006
Given a non-isoceles triangle ABCABC inscribes a circle (O,R)(O,R) (center OO, radius RR). Consider a varying line ll such that l⊥OAl\perp OA and ll always intersects the rays AB,ACAB,AC and these intersectional points are called M,NM,N. Suppose that the lines BNBN and CMCM intersect, and if the intersectional point is called KK then the lines AKAK and BCBC intersect. 11, Assume that PP is the intersectional point of AKAK and BCBC. Show that the circumcircle of the triangle MNPMNP is always through a fixed point. 22, Assume that HH is the orthocentre of the triangle AMNAMN. Denote BC=aBC=a, and dd is the distance between AA and the line HKHK. Prove that d≤4R2−a2d\leq\sqrt{4R^2-a^2} and the equality occurs iff the line ll is through the intersectional point of two lines AOAO and BCBC.
geometrycircumcircleparallelogramratioquadraticspower of a pointradical axis