MathDB

2

Part of 2021 Argentina National Olympiad

Problems(3)

Argentina MO 2021 National Level 1 P2

Source:

4/23/2022
On each OMA lottery ticket there is a 99-digit number that only uses the digits 1,21, 2 and 33 (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the 99 figures, then they are of different colors. Bill 122222222122222222 is red, 222222222222222222 is green, what color is bill 123123123123123123?
combinatoricscontests
Argentina MO 2021 National Level 2 P2

Source:

4/23/2022
In a semicircle with center OO, let CC be a point on the diameter ABAB different from A,BA, B and O.O. Draw through CC two rays such that the angles that these rays form with the diameter ABAB are equal and that they intersect at the semicircle at DD and at EE. The line perpendicular to CDCD through DD intersects the semicircle at K.K. Prove that if D≠E,D\neq E, then KEKE is parallel to AB.AB.
geometrycontests
Argentina MO 2021 National Level 3 P2

Source:

4/24/2022
Let mm be a positive integer for which there exists a positive integer nn such that the multiplication mnmn is a perfect square and m−nm- n is prime. Find all mm for 1000≤m≤2021.1000\leq m \leq 2021.
number theoryprime numbersPerfect Squarespositive integerscontestsArgentina