MathDB

Problem 3

Part of 1999 Brazil Team Selection Test

Problems(2)

circum- and excircle intersections

Source: Brazil TST 1999 Test 1 P3

4/30/2021
Let BDBD and CECE be the bisectors of the interior angles ∠B\angle B and ∠C\angle C, respectively (D∈ACD\in AC, E∈ABE\in AB). Consider the circumcircle of ABCABC with center OO and the excircle corresponding to the side BCBC with center IaI_a. These two circles intersect at points PP and QQ.
(a) Prove that PQPQ is parallel to DEDE. (b) Prove that IaOI_aO is perpendicular to DEDE.
geometryTriangle
linear recurrence, finite difference divisible by 1999

Source: Brazil TST 1999 Test 2 P3

4/30/2021
A sequence ana_n is defined by a0=0,a1=3;a_0=0,\qquad a_1=3;an=8an−1+9an−2+16 for n≥2.a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.Find the least positive integer hh such that an+h−ana_{n+h}-a_n is divisible by 19991999 for all n≥0n\ge0.
algebraSequencerecurrence relation