MathDB

3

Part of 2011 International Zhautykov Olympiad

Problems(2)

International Zhautykov Olympiad 2011 - Problem 3

Source:

1/16/2011
Let N\mathbb{N} denote the set of all positive integers. An ordered pair (a;b)(a;b) of numbers a,b∈Na,b\in\mathbb{N} is called interesting, if for any n∈Nn\in\mathbb{N} there exists k∈Nk\in\mathbb{N} such that the number ak+ba^k+b is divisible by 2n2^n. Find all interesting ordered pairs of numbers.
modular arithmeticlogarithmsinductionnumber theory unsolvednumber theory
International Zhautykov Olympiad 2011 - Problem 6

Source:

1/17/2011
Diagonals of a cyclic quadrilateral ABCDABCD intersect at point K.K. The midpoints of diagonals ACAC and BDBD are MM and N,N, respectively. The circumscribed circles ADMADM and BCMBCM intersect at points MM and L.L. Prove that the points K,L,M,K ,L ,M, and N N lie on a circle. (all points are supposed to be different.)
geometrycircumcirclesymmetryratiocyclic quadrilateralpower of a pointradical axis