3

Part of 2024 239 Open Mathematical Olympiad

Problems(2)

Two products of distances

Source: 239 MO 2024 J6~S3

a) (version for grades 10-11)
Let

P

be a point lying in the interior of a triangle. Show that the product of the distances from

P

to the sides of the triangle is at least

8

times less than the product of the distances from

P

to the tangents to the circumcircle at the vertices of the triangle.
b) (version for grades 8-9)
Is it true that for any triangle there exists a point

P

for which equality in the inequality from a) holds?

Digits on a circle

Source: 239 MO 2024 J3

169

non-zero digits written around a circle. Prove that they can be split into

14

non-empty blocks of consecutive digits so that among the

14

natural numbers formed by the digits in those blocks, at least

13

of them are divisible by

13

(the digits in each block are read in clockwise direction).

number theorycombinatorics