MathDB

6

Part of 2019 India Regional Mathematical Olympiad

Problems(2)

Collinarity

Source: RMO 2019 P6

10/24/2019
Suppose 9191 distinct positive integers greater than 11 are given such that there are at least 456456 pairs among them which are relatively prime. Show that one can find four integers a,b,c,da, b, c, d among them such that gcd⁡(a,b)=gcd⁡(b,c)=gcd⁡(c,d)=gcd⁡(d,a)=1.\gcd(a,b)=\gcd(b,c)=\gcd(c,d)=\gcd(d,a)=1.
Circles with centers in a set

Source: RMO Maharashtra and Goa 2019 P6

11/10/2019
Let kk be a positive real number. In the X−YX-Y coordinate plane, let SS be the set of all points of the form (x,x2+k)(x,x^2+k) where x∈Rx\in\mathbb{R}. Let CC be the set of all circles whose center lies in SS, and which are tangent to XX-axis. Find the minimum value of kk such that any two circles in CC have at least one point of intersection.
algebrageometrycoordinate geometryanalytic geometryconicsparabola