MathDB

4

Part of 2024 Brazil National Olympiad

Problems(2)

prove that two circles and a line have a common point

Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 4

10/12/2024
Let ABC ABC be an acute-angled scalene triangle. Let D D be a point on the interior of segment BC BC , different from the foot of the altitude from A A . The tangents from A A and B B to the circumcircle of triangle ABD ABD meet at O1 O_1 , and the tangents from A A and C C to the circumcircle of triangle ACD ACD meet at O2 O_2 . Show that the circle centered at O1 O_1 passing through A A , the circle centered at O2 O_2 passing through A A , and the line BC BC have a common point.
geometrytangent
how many trilegal numbers with 10 digits?

Source: Brazilian Mathematical Olympiad 2024, Level 2, Problem 4

10/12/2024
A number is called trilegal if its digits belong to the set {1,2,3}\{1, 2, 3\} and if it is divisible by 9999. How many trilegal numbers with 1010 digits are there?
number theorycombinatoricscounting