MathDB

4

Part of 2018 Brazil National Olympiad

Problems(2)

Maximum value of a cyclic sum

Source: Brazilian Mathematical Olympiad 2018 - Q4

11/16/2018
Esmeralda writes 2n2n real numbers x1,x2,…,x2nx_1, x_2, \dots , x_{2n}, all belonging to the interval [0,1][0, 1], around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products p1=x1x2p_1 = x_1x_2, p2=x2x3p_2 = x_2x_3, …\dots , p2n=x2nx1p_{2n} = x_{2n}x_1. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
inequalitiesalgebramaximum valueBrazilian Math OlympiadBrazilian Math Olympiad 2018
Incentrics

Source:

12/22/2019
a) In a XYZ XYZ triangle, the incircle tangents the XY XY and XZ XZ sides at the T T and W W points, respectively. Prove that: XT=XW=XY+XZ−YZ2 XT = XW = \frac {XY + XZ-YZ} {2} Let ABC ABC be a triangle and D D is the foot of the relative height next to A. A. Are I I and J J the incentives from triangle ABD ABD and ACD ACD , respectively. The circles of ABD ABD and ACD ACD tangency AD AD at points M M and N N , respectively. Let P P be the tangency point of the BC BC circle with the AB AB side. The center circle A A and radius AP AP intersect the height D D at K. K. b) Show that the triangles IMK IMK and KNJ KNJ are congruent c) Show that the IDJK IDJK quad is inscritibed
geometryeuclidean geometry