MathDB

6

Part of 2018 India Regional Mathematical Olympiad

Problems(2)

Right angles on incircle

Source: RMO 2018 P6

10/7/2018
Let ABCABC be an acute-angled triangle with AB<ACAB<AC. Let II be the incentre of triangle ABCABC, and let D,E,FD,E,F be the points where the incircle touches the sides BC,CA,AB,BC,CA,AB, respectively. Let BI,CIBI,CI meet the line EFEF at Y,XY,X respectively. Further assume that both XX and YY are outside the triangle ABCABC. Prove that (i)\text{(i)} B,C,Y,XB,C,Y,X are concyclic. (ii)\text{(ii)} II is also the incentre of triangle DYXDYX.
geometryincenter
Sequences...

Source: RMO 2018 P6

10/28/2018
Define a sequence {an}n≥1\{a_n\}_{n\geq 1} of real numbers by a1=2,an+1=an2+12, for n≥1.a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1. Prove that ∑j=1N1aj+1<1\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1 for every natural number NN.
algebra