MathDB

3

Part of 1997 Federal Competition For Advanced Students, Part 2

Problems(2)

PQRS is a parallelogram if and only if ACBD is cyclic

Source: Austrian Mathematical Olympiad 1997, Part 2, D1, P3

7/4/2011
Let be given a triangle ABCABC. Points PP on side ACAC and YY on the production of CBCB beyond BB are chosen so that YY subtends equal angles with APAP and PCPC. Similarly, QQ on side BCBC and XX on the production of ACAC beyond CC are such that XX subtends equal angles with BQBQ and QCQC. Lines YPYP and XBXB meet at RR, XQXQ and YAYA meet at SS, and XBXB and YAYA meet at DD. Prove that PQRSPQRS is a parallelogram if and only if ACBDACBD is a cyclic quadrilateral.
geometryparallelogramgeometry proposed
Find all p(x) = x^2+ax+b such that p(x) | x^{2n} + ax^n + b

Source: Austrian Mathematical Olympiad 1997, Part 2, D2, P3

7/4/2011
For every natural number nn, find all polynomials x2+ax+bx^2+ax+b, where a2≥4ba^2 \geq 4b, that divide x2n+axn+bx^{2n} + ax^n + b.
algebrapolynomialalgebra proposed