MathDB

2

Part of 2021 Azerbaijan EGMO TST

Problems(2)

symmetric lines with respect to angle bisector

Source: Azerbaijan EGMO TST 2021, D1 P2

8/11/2023
Let ω\omega be a circle with center O,O, and let AA be a point with tangents APAP and AQAQ to the circle. Denote by KK the intersection point of AOAO and PQ.PQ. l1l_1 and l2l_2 are two lines passing through AA that intersect ω.\omega. Call BB the intersection point of l1l_1 with ω,\omega, which is located nearer to AA on l1.l_1. Call CC the intersection point of l2l_2 with ω,\omega, which is located further to AA on l2.l_2. Prove that ∠PAB=∠QAC\angle PAB = \angle QAC if and only if the points B,K,CB, K, C are on line.
geometryisogonal linesAZE EGMO TST
constructing unbounded sequence form non-decreasing, unbounded sequence

Source: Azerbaijan EGMO TST 2021, D2 P2

8/11/2023
Given a non-decreasing unbounded sequence an,a_n, construct a new sequence bnb_n as follows bn=a2−a1a2+a3−a2a3+...+an−an−1anb_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n} Prove that bnb_n is also unbounded.
SequenceunboundedalgebraAZE EGMO TST