MathDB

4

Part of 2022 Kosovo & Albania Mathematical Olympiad

Problems(2)

Diophantine equation over a restricted set of integers

Source: KAMO 2022 Grades 7-8 P4

7/5/2022
Let AA be the set of natural numbers nn such that the distance of the real number n2022−13n\sqrt{2022} - \frac13 from the nearest integer is at most 12022\frac1{2022}. Show that the equation 20x+21y=22z20x + 21y = 22z has no solutions over the set AA.
Diophantine equationalgebra
Counting lines forming a small angle

Source: KAMO 2022 Grade 9 P4

7/5/2022
Consider n>9n>9 lines on the plane such that no two lines are parallel. Show that there exist at least n/9n/9 lines such that the angle between any two of the lines is at most 20∘20^\circ.
combinatoricsgeometry