MathDB

2

Part of 2014 IberoAmerican

Problems(2)

Finding polynomials

Source:

9/24/2014
Find all polynomials P(x)P(x) with real coefficients such that P(2014)=1P(2014) = 1 and, for some integer cc:
xP(x−c)=(x−2014)P(x)xP(x-c) = (x - 2014)P(x)
algebrapolynomialalgebra unsolved
Prove that 4 points lie on a circumference

Source: Iberoamerican Olympiad 2014, Problem 5

9/24/2014
Let ABCABC be an acute triangle and HH its orthocenter. Let DD be the intersection of the altitude from AA to BCBC. Let MM and NN be the midpoints of BHBH and CHCH, respectively. Let the lines DMDM and DNDN intersect ABAB and ACAC at points XX and YY respectively. If PP is the intersection of XYXY with BHBH and QQ the intersection of XYXY with CHCH, show that H,P,D,QH, P, D, Q lie on a circumference.
geometrygeometric transformationreflectiontrapezoidcircumcirclecyclic quadrilateralgeometry unsolved