MathDB

3

Part of 2018 Danube Mathematical Competition

Problems(2)

a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n , a_i \in Q

Source: Danube 2018 junior p3

7/22/2019
Find all the positive integers nn with the property: there exists an integer k>2k > 2 and the positive rational numbers a1,a2,...,aka_1, a_2, ..., a_k such that a1+a2+..+ak=a1a2...ak=na_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n.
number theorySumProductrationalpositive integers
danube senior parallel wanted 2018 P3

Source: Danube 2018 P3

12/11/2018
Let ABCABC be an acute non isosceles triangle. The angle bisector of angle AA meets again the circumcircle of the triangle ABCABC in DD. Let OO be the circumcenter of the triangle ABCABC. The angle bisectors of ∠AOB\angle AOB, and ∠AOC\angle AOC meet the circle γ\gamma of diameter ADAD in PP and QQ respectively. The line PQPQ meets the perpendicular bisector of ADAD in RR. Prove that AR//BCAR // BC.
geometryparallelangle bisectorcircumcircleperpendicular bisector