MathDB

2

Part of 2016 International Zhautykov Olympiad

Problems(2)

Number Theory

Source: IZHO 2016 p2

1/15/2016
a1,a2,...,a100a_1,a_2,...,a_{100} are permutation of 1,2,...,1001,2,...,100. S1=a1,S2=a1+a2,...,S100=a1+a2+...+a100S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}Find the maximum number of perfect squares from SiS_i
number theory
Izho 2016P5

Source: Izho 2016 P5

1/16/2016
A convex hexagon ABCDEFABCDEF is given such that AB∣∣DEAB||DE, BC∣∣EFBC||EF, CD∣∣FACD||FA. The point M,N,KM, N, K are common points of the lines BDBD and AEAE, ACAC and DFDF, CECE and BFBF respectively. Prove that perpendiculars drawn from M,N,KM, N, K to lines AB,CD,EFAB, CD, EF respectively concurrent.