MathDB

8

Part of 2023 Tuymaada Olympiad

Problems(2)

Locus of circumcentres

Source: Tuymaada 2023 Junior P8

7/7/2023
Circle ω\omega lies inside the circle Ω\Omega and touches it internally at point PP. Point SS is taken on ω\omega and the tangent to ω\omega is drawn through it. This tangent meets Ω\Omega at points AA and BB. Let II be the centre of ω\omega. Find the locus of circumcentres of triangles AIBAIB.
geometry
Fractions in (0,1) with bounded denominators

Source: Tuymaada 2023 Senior P8

7/7/2023
Given is a positive integer nn. Let AA be the set of points x∈(0;1)x \in (0;1) such that ∣x−pq∣>1n3|x-\frac{p} {q}|>\frac{1}{n^3} for each rational fraction pq\frac{p} {q} with denominator q≤n2q \leq n^2. Prove that AA is a union of intervals with total length not exceeding 100n\frac{100}{n}.
Proposed by Fedor Petrov