MathDB

3

Part of 2024 Girls in Mathematics Tournament

Problems(2)

simple geometry

Source: 2024 Girls in Mathematics Tournament, Level A, Problem 3

10/26/2024
In a triangle scalene ABCABC, let II be its incenter and DD the intersection of AIAI and BCBC. Let MM and NN points where the incircle touches ABAB and ACAC, respectively. Let FF be the second intersection of the circumcircle (AMN)(AMN) with the circumcircle (ABC)(ABC). Let TT the intersection of AFAF and BCBC. Let JJ be the intersection of TITI with the line parallel of FIFI that passes through DD. Prove that the line AJAJ is perpendicular to BCBC.
geometrysharky devil point
set of points with integer coordinates

Source: 2024 Girls in Mathematics Tournament- Level B, Problem 3

10/26/2024
Let CC be the set of points (x,y)(x,y) with integer coordinates in the plane where 1≤x≤9001\leq x\leq 900 and 1≤y≤10001\leq y\leq 1000. A polygon PP with vertices in CC is called emerald if PP has exactly zero or two vertices in each row and each column and all the internal angles of PP are 90∘90^\circ or 270∘270^\circ. Find the greatest value of kk such that we can color kk points in CC such that any subset of these kk points is not the set of vertices of an emerald polygon.
https://cdn.discordapp.com/attachments/954427908359876608/1299737432010395678/image.png?ex=671e4a4f&is=671cf8cf&hm=ce008541975226a0e9ea53a93592a7469d8569baca945c1c207d4a722126bb60&
On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.
combinatorics