MathDB

P2

Part of 2022 Junior Balkan Team Selection Tests - Romania

Problems(4)

Romania Junior TST 2022 Day 1 P2

Source: Romania JBMO TST 2022

4/20/2022
Let ABCABC be a triangle such that ∠A=30∘\angle A=30^\circ and ∠B=80∘\angle B=80^\circ. Let DD and EE be points on sides ACAC and BCBC respectively so that ∠ABD=∠DBC\angle ABD=\angle DBC and DE∥ABDE\parallel AB. Determine the measure of ∠EAC\angle EAC.
geometryRomanian TST
Largest n with a strange condition

Source: Romania JBMO TST 2022 Day 2 Problem 2

5/15/2022
Find the largest positive integer nn such that the following is true: There exists nn distinct positive integers x1, x2,…,xnx_1,~x_2,\dots,x_n such that whatever the numbers a1, a2,…,an∈{−1,0,1}a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\} are, not all null, the number n3n^3 do not divide ∑k=1nakxk\sum_{k=1}^n a_kx_k.
number theoryRomanian TSTJBMO TST
Romania Junior TST 2022 Day 3 P2

Source: Romania JBMO TST 2022

6/3/2022
Let C1\mathcal{C}_1 and C2\mathcal{C}_2 be two circles, internally tangent at PP (C2\mathcal{C}_2 lies inside of C1\mathcal{C}_1). A chord ABAB of C1\mathcal{C}_1 is tangent to C2\mathcal{C}_2 at C.C. Let DD be the second point of intersection between the line CPCP and C1.\mathcal{C}_1. A tangent from DD to C2\mathcal{C}_2 intersects C1\mathcal{C}_1 for the second time at EE and it intersects C2\mathcal{C}_2 at F.F. Prove that FF is the incenter of triangle ABE.ABE.
geometryromaniaRomanian TST
Romania Junior TST 2022 Day 4 P2

Source: Romania JBMO TST 2022

6/3/2022
Let ABCABC be an acute scalene triangle. Let DD be the foot of the AA-bisectrix and EE be the foot of the AA-altitude. The perpendicular bisector of the segment ADAD intersects the semicircles of diameter ABAB and ACAC which lie on the outside of triangle ABCABC at XX and YY respectively. Prove that the points X,Y,DX,Y,D and EE lie on a circle.
geometryromaniaRomanian TST