MathDB

1

Part of 2018 Romania Team Selection Tests

Problems(3)

maxima of xi^2

Source: Romanian TST 2018 Day 1 Problem 1

5/25/2020
Find the least number c c satisfyng the condition ∑i=1nxi2≤cn\sum_{i=1}^n {x_i}^2\leq cn and all real numbers x1,x2,...,xnx_1,x_2,...,x_n are greater than or equal to −1-1 such that ∑i=1nxi3=0\sum_{i=1}^n {x_i}^3=0
inequalitiesn-variable inequality
Izogonal points

Source: Romanian 2018 TST Day 2 Problem 1

5/25/2020
Let ABCABC be a triangle, and let MM be a point on the side (AC)(AC) .The line through MM and parallel to BCBC crosses ABAB at NN. Segments BMBM and CNCN cross at PP, and the circles BNPBNP and CMPCMP cross again at QQ. Show that angles BAPBAP and CAQCAQ are equal.
geometryMiquel pointSymedian
Incenters in cyclic quadrilaterals

Source: Romanian 2018 TST Problem 1 day 3

5/25/2020
Let ABCDABCD be a cyclic quadrilateral and let its diagonals ACAC and BDBD cross at XX. Let II be the incenter of XBCXBC, and let JJ be the center of the circle tangent to the side BCBC and the extensions of sides ABAB and DCDC beyond BB and CC. Prove that the line IJIJ bisects the arc BCBC of circle ABCDABCD, not containing the vertices AA and DD of the quadrilateral.
cyclic quadrilateralincentergeometry