MathDB

5

Part of 2022 Saint Petersburg Mathematical Olympiad

Problems(2)

Orthocenter geo with tangencies

Source: St Petersburg 2022 11.5

9/28/2022
Altitudes AA1,BB1,CC1AA_1, BB_1, CC_1 of acute triangle ABCABC intersect at point HH. On the tangent drawn from point CC to the circle (AB1C1)(AB_1C_1), the perpendicular HQHQ is drawn (the point QQ lies inside the triangle ABCABC). Prove that the circle passing through the point B1B_1 and touching the line ABAB at point AA is also tangent to line A1QA_1Q.
geometry
Weird gcd function

Source: St Petersburg 2022 10.5

9/28/2022
Let nn be a positive integer and let a1,a2,⋯aka_1, a_2, \cdots a_k be all numbers less than nn and coprime to nn in increasing order. Find the set of values the function f(n)=gcd(a13−1,a23−1,⋯ ,ak3−1)f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1).
functionnumber theory