MathDB

1

Part of 2012 Serbia National Math Olympiad

Problems(2)

Parallelogram and equality of angles

Source: Serbian National Olympiad 2012, Problem 1

4/4/2012
Let ABCDABCD be a parallelogram and PP be a point on diagonal BDBD such that ∠PCB=∠ACD\angle PCB=\angle ACD. Circumcircle of triangle ABDABD intersects line ACAC at points AA and EE. Prove that ∠AED=∠PEB.\angle AED=\angle PEB.
geometryparallelogramcircumcircletrigonometrygeometry proposed
Permutation of 1,2,...,n and complete residue system

Source: Serbian National Olympiad 2012, Problem 4

4/4/2012
Find all natural numbers nn for which there is a permutation (p1,p2,...,pn)(p_1,p_2,...,p_n) of numbers (1,2,...,n)(1,2,...,n) such that sets {p1+1,p2+2,...,pn+n}\{p_1 +1, p_2 + 2,..., p_n +n\} and {p1−1,p2−2,...,pn−n}\{p_1-1, p_2-2,...,p_n -n\} are complete residue systems mod  n\mod n.
modular arithmeticnumber theory proposednumber theory