MathDB

4

Part of 2024 Thailand October Camp

Problems(2)

a_1 is anything but 2

Source: Thailand TSTST 2024 P4

7/18/2024
The sequence (an)n∈N(a_n)_{n\in\mathbb{N}} is defined by a1=3a_1=3 and an=a1a2⋯an−1−1a_n=a_1a_2\cdots a_{n-1}-1 Show that there exist infinitely many prime number that divide at least one number in this sequences
number theoryprime numbers
Lovely geo

Source: Thailand TSTST 2024 P10

7/18/2024
Let ABCABC be an acute triangle with altitudes AD,BEAD,BE and CFCF. Denote ω1,ω2\omega_1,\omega_2 the circumcircles of △AEB,△AFC\triangle AEB, \triangle AFC, respectively. Suppose the line through AA parallel to EFEF intersects ω1\omega_1 and ω2\omega_2 at PP and QQ, respectively. Show that the circumcenter of △PQD\triangle PQD lies on ADAD
geometrycircumcircle